Best pure mathematic books according to redditors
We found 2,758 Reddit comments discussing the best pure mathematic books. We ranked the 819 resulting products by number of redditors who mentioned them. Here are the top 20.
We found 2,758 Reddit comments discussing the best pure mathematic books. We ranked the 819 resulting products by number of redditors who mentioned them. Here are the top 20.
Copying my answer from another post:
I was literally in the bottom 14th percentile in math ability when i was 12.
One day, by pure chance, i stumbled across this (free and open) book written by Carl Stitz and Jeff Zeager, of Lakeland Community College
Precalculus
It covers everything from elementary algebra (think grade 5), all the way up to concepts used in Calculus and Linear Algebra (Partial fractions and matrix algebra, respectively.) The book is extremely well organized. Every sections starts with a dozen or so pages of proofs and derivations that show you the logic of why and how the formulas you'll be using work. This book, more than any other resource (and i've tried a lot of them), helped me build my math intuition from basically nothing.
Math is really, really intimidating when you've spent your whole life sucking at it. This book addresses that very well. The proofs are all really well explained, and are very long. You'll basically never go from one step to the next and be completely confused as to how they got there.
Also, there is a metric shitload of exercises, ranging from trivial, to pretty difficult, to "it will literally take your entire class working together to solve this". Many of the questions follow sort of an "arc" through the chapters, where you revisit a previous problem in a new context, and solve it with different means (Also, Sasquatches. You'll understand when you read it.)
I spent 8 months reading this book an hour a day when i got home from work, and by the end of it i was ready for college. I'm now in my second year of computer science and holding my own (although it's hard as fuck) against Calculus II. I credit Stitz and Zeager entirely. Without this book, i would never have made it to college.
Edit: other resources
Khan Academy is good, and it definitely complements Stitz/Zeager, but Khan also lacks depth. Like, a lot of depth. Khan Academy is best used for the practice problems and the videos do a good job of walking you through application of math, but it doesn't teach you enough to really build off of it. I know this from experience, as i completed all of Khan's precalculus content. Trust me, Rely on the Stitz book, and use Khan to fill in the gaps.
Paul's Online Math Notes
This website is so good it's ridiculous. It has a ton of depth, and amazing reference sheets. Use this for when you need that little extra detail to understand a concept. It's still saving my ass even today (Damned integral trig substitutions...)
Stuff that's more important than you think (if you're interested in higher math after your GED)
Trigonometric functions: very basic in Algebra, but you gotta know the common values of all 6 trig functions, their domains and ranges, and all of their identities for calculus. This one bit me in the ass.
Matrix algebra: Linear algebra is p. cool. It's used extensively in computer science, particularly in graphics programming. It's relatively "easy", but there's more conceptual stuff to understand.
Edit 2: Electric Boogaloo
Other good, cheap math textbooks
/u/ismann has pointed out to me that Dover Publications has a metric shitload of good, cheap texts (~$25CAD on Amazon, as low as a few bucks USD from what i hear).
Search up Dover Mathematics on Amazon for a deluge of good, cheap math textbooks. Many are quite old, but i'm sure most will agree that math is a fairly mature discipline, so it's not like it makes a huge difference at the intro level. Here is a Math
OverflowExchange list of the creme de la creme of Dover math texts, all of which can be had for under $30, often much less. I just bought ~1,000 pages of Linear Algebra, Graph Theory, and Discrete Math text for $50. If you prefer paper to .pdf, this is probably a good route to go.Also, How to Prove it is a very highly rated (and easy to read!) introduction to mathematical proofs. It introduces the basic logical constructs that mathematicians use to write rigorous proofs. It's very approachable, fairly short, and ~$30 new.
Here's my list of the classics:
General Computing
Computer Science
Software Development
Case Studies
Employment
Language-Specific
C
Python
C#
C++
Java
Linux Shell Scripts
Web Development
Ruby and Rails
Assembly
The answer is "virtually all of mathematics." :D
Although lots of math degrees are fairly linear, calculus is really the first big branch point for your learning. Broadly speaking, the three main pillars of contemporary mathematics are:
You might also think of these as the three main "mathematical mindsets" — mathematicians often talk about "thinking like an algebraist" and so on.
Calculus is the first tiny sliver of analysis and Spivak's Calculus is IMO the best introduction to calculus-as-analysis out there. If you thought Spivak's textbook was amazing, well, that's bread-n-butter analysis. I always thought of Spivak as "one-dimensional analysis" rather than calculus.
Spivak also introduces a bit of algebra, BTW. The first few chapters are really about abstract algebra and you might notice they feel very different from the latter chapters, especially after he introduces the least-upper-bound property. Spivak's "properties of numbers" (P1-P9) are actually the 9 axioms which define an algebraic object called a field. So if you thought those first few chapters were a lot of fun, well, that's algebra!
There isn't that much topology in Spivak, although I'm sure he hides some topology exercises throughout the book. Topology is sometimes called the study of "shape" and is where our most general notions of "continuous function" and "open set" live.
Here are my recommendations.
Analysis If you want to keep learning analysis, check out Introductory Real Analysis by Kolmogorov & Fomin, Principles of Mathematical Analysis by Rudin, and/or Advanced Calculus of Several Variables by Edwards.
Algebra If you want to check out abstract algebra, check out Dummit & Foote's Abstract Algebra and/or Pinter's A Book of Abstract Algebra.
Topology There's really only one thing to recommend here and that's Topology by Munkres.
If you're a high-school student who has read through Spivak in your own, you should be fine with any of these books. These are exactly the books you'd get in a more advanced undergraduate mathematics degree.
I might also check out the Chicago undergraduate mathematics bibliography, which contains all my recommendations above and more. I disagree with their elementary/intermediate/advanced categorization in many cases, e.g., Rudin's Principles of Mathematical Analysis is categorized as "elementary" but it's only "elementary" if your idea of doing math is pursuing a PhD. Baby Rudin (as it's called) is to first-year graduate analysis as Spivak is to first-year undergraduate calculus — Rudin says as much right in the introduction.
A Book of Abstract Algebra by Charles C. Pinter
Here's my rough list of textbook recommendations. There are a ton of Dover paperbacks that I didn't put on here, since they're not as widely used, but they are really great and really cheap.
Amazon search for Dover Books on mathematics
There's also this great list of undergraduate books in math that has become sort of famous: https://www.ocf.berkeley.edu/~abhishek/chicmath.htm
Pre-Calculus / Problem-Solving
Calculus
Linear Algebra
Differential Equations
Number Theory
Proof-Writing
Analysis
Complex Analysis
Functional Analysis
Partial Differential Equations
Higher-dimensional Calculus and Differential Geometry
Abstract Algebra
Geometry
Topology
Set Theory and Logic
Combinatorics / Discrete Math
Graph Theory
P. S., if you Google search any of the topics above, you are likely to find many resources. You can find a lot of lecture notes by searching, say, "real analysis lecture notes filetype:pdf site:.edu"
I started from scratch on the formal CS side, with an emphasis on program analysis, and taught myself the following starting from 2007. If you're in the United States, I recommend BookFinder to save money buying these things used.
On the CS side:
On the math side, I was advantaged in that I did my undergraduate degree in the subject. Here's what I can recommend, given five years' worth of hindsight studying program analysis:
Final bit of advice: you'll notice that I heavily stuck to textbooks and Ph.D. theses in the above list. I find that jumping straight into the research literature without a foundational grounding is perhaps the most ill-advised mistake one can make intellectually. To whatever extent that what you're interested in is systematized -- that is, covered in a textbook or thesis already, you should read it before digging into the research literature. Otherwise, you'll be the proverbial blind man with the elephant, groping around in the dark, getting bits and pieces of the picture without understanding how it all forms a cohesive whole. I made that mistake and it cost me a lot of time; don't do the same.
For real analysis I really enjoyed Understanding Analysis for how clear the material was presented for a first course. For abstract algebra I found A book of abstract algebra to be very concise and easy to read for a first course. Those two textbooks were a lifesaver for me since I had a hard time with those two courses using the notes and textbook for the class. We were taught out of rudin and dummit and foote as mainly a reference book and had to rely on notes primarily but those two texts were incredibly helpful to understand the material.
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If any undergrads are struggling with those two courses I would highly recommend you check out those two textbooks. They are by far the easiest introduction to those two fields I have found. I also like that you can find solutions to all the exercises so it makes them very valuable for self study also. Both books also have a reasonable amount of excises so that you can in theory do nearly every problem in the book which is also nice compared to standard texts with way too many exercises to realistically go through.
I'm 2 years into a part time physics degree, I'm in my 40s, dropped out of schooling earlier in life.
As I'm doing this for fun whilst I also have a full time job, I thought I would list what I'm did to supplement my study preparation.
I started working through these videos - Essence of Calculus as a start over the summer study whilst I had some down time. https://www.youtube.com/playlist?list=PLZHQObOWTQDMsr9K-rj53DwVRMYO3t5Yr
Ive bought the following books in preparation for my journey and to start working through some of these during the summer prior to start
Elements of Style - A nice small cheap reference to improve my writing skills
https://www.amazon.co.uk/gp/product/020530902X/ref=oh_aui_detailpage_o02_s00?ie=UTF8&psc=1
The Humongous Book of Trigonometry Problems https://www.amazon.co.uk/gp/product/1615641823/ref=oh_aui_detailpage_o08_s00?ie=UTF8&psc=1
Calculus: An Intuitive and Physical Approach
https://www.amazon.co.uk/gp/product/0486404536/ref=oh_aui_detailpage_o09_s00?ie=UTF8&psc=1
Trigonometry Essentials Practice Workbook
https://www.amazon.co.uk/gp/product/1477497781/ref=oh_aui_detailpage_o05_s00?ie=UTF8&psc=1
Systems of Equations: Substitution, Simultaneous, Cramer's Rule
https://www.amazon.co.uk/gp/product/1941691048/ref=oh_aui_detailpage_o05_s00?ie=UTF8&psc=1
Feynman's Tips on Physics
https://www.amazon.co.uk/gp/product/0465027970/ref=oh_aui_detailpage_o07_s00?ie=UTF8&psc=1
Exercises for the Feynman Lectures on Physics
https://www.amazon.co.uk/gp/product/0465060714/ref=oh_aui_detailpage_o08_s00?ie=UTF8&psc=1
Calculus for the Practical Man
https://www.amazon.co.uk/gp/product/1406756725/ref=oh_aui_detailpage_o09_s00?ie=UTF8&psc=1
The Feynman Lectures on Physics (all volumes)
https://www.amazon.co.uk/gp/product/0465024939/ref=oh_aui_detailpage_o09_s00?ie=UTF8&psc=1
I found PatrickJMT helpful, more so than Khan academy, not saying is better, just that you have to find the person and resource that best suits the way your brain works.
Now I'm deep in calculus and quantum mechanics, I would say the important things are:
Algebra - practice practice practice, get good, make it smooth.
Trig - again, practice practice practice.
Try not to learn by rote, try understand the why, play with things, draw triangles and get to know the unit circle well.
Good luck, it's going to cause frustrating moments, times of doubt, long nights and early mornings, confusion, sweat and tears, but power through, keep on trucking, and you will start to see that calculus and trig are some of the most beautiful things in the world.
It just comes from the way we define sums of infinite sums, aka series. .999... is just shorthand for (.9+.09+.09+.009...), which is an infinite sum. We define the sum of a series to be equal to the limit of the partial sums. The limit is rigorously defined, and you can read the definition on wikipedia if you google "epsilon delta". The limit of an infinite sum, if it exists, is unique. For this infinite sum, that limit is exactly 1. By the way we define infinite sums, .999... is therefore exactly equal to 1.
It's not so bad when you remember that all real numbers have a representation as a non-terminating decimal. 0.5 can be written as 0.4999... and 1/3 can be written as 0.333... and pi can be written as 3.14159... for example.
And lastly, if .999... and 1 are different real numbers, then there must exist a number between them. This is because of an axiom we have called trichotomy: for any two real numbers a and b, exactly one of the following is true: a<b, a=b, a>b. If a=/=b, then there exists a real number between them, because the real numbers have a property called "dense". It is easy to prove that here is no such number between .999... and 1, real or otherwise. Therefore .999... is exactly equal to 1.
e: The sum (.9+.09+.009...) is bigger than every real number less than 1. You can check if you want. The smallest number that is greater than every real number less than 1 is 1 itself. We get this from an axiom called the "least upper bound property". Therefore .999... is at least 1. Using our rigorous definition of a limit, we find that it is exactly 1.
e2: Apostol's Calculus vol 1 is a fantastic place to start learning about rigorous math shit. Chapter one starts you out with axioms for real numbers, and about half way through chapter 1 you prove the whole thing about repeating decimals corresponding to rational numbers. It is slow and easy to follow. Other people recommend Spivak but I haven't seen it so idk.
The rate of your learning is defined by your determination. If you don't give up then you will learn the material.
Look at the book that is required and only learn what you need in the class. Don't learn everything in the book either. Just learn what you need to do well and refer to the books when you get confused.
Note don't try to learn everything that's below. Only use it to learn what you actually need. This can be overwhelming at first but just set aside a set time to study this.
EDIT I added more books and courses.
OCW
http://ocw.mit.edu/courses/mathematics/18-01sc-single-variable-calculus-fall-2010/
http://ocw.mit.edu/courses/mathematics/18-02-multivariable-calculus-fall-2007/index.htm
http://ocw.mit.edu/courses/mathematics/18-03-differential-equations-spring-2010/
http://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/
Helpful books
http://www.amazon.com/Mathematical-Proofs-Transition-Advanced-Mathematics/dp/0321390539/ref=sr_1_3?s=books&amp;ie=UTF8&amp;qid=1312542911&amp;sr=1-3
http://www.amazon.com/Understanding-Probability-Chance-Rules-Everyday/dp/0521540364
http://www.amazon.com/gp/product/048663518X/ref=pd_lpo_k2_dp_sr_1?pf_rd_p=486539851&amp;pf_rd_s=lpo-top-stripe-1&amp;pf_rd_t=201&amp;pf_rd_i=0155510053&amp;pf_rd_m=ATVPDKIKX0DER&amp;pf_rd_r=0YXJR9EVHCH9PCBDN372
Khan Academy
http://khan-academy.appspot.com/#calculus
http://www.youtube.com/user/keithpeterb#p/u/19/dS2p_APpcnI
http://khan-academy.appspot.com/video/probability--part-1?playlist=Old%20Algebra
http://www.youtube.com/user/keithpeterb#p/u/19/dS2p_APpcnI
http://khan-academy.appspot.com/video/linear-algebra--introduction-to-vectors?playlist=Linear%20Algebra
EDIT: I knew nothing about topological quantum computation about 1.5 years ago but then I took a independent study in college and I was assigned 1-3 papers a week to read. Eventually I got it a few months ago. What got me through it was not giving up...
OCW
Single Variable Calculus
Multivariable Calculus
Differential Equasions
Linear Algebra
Helpful books
Mathematical Proofs: A Transition to Advanced Mathematics
Understanding Probability: Chance Rules in Everyday Life
Linear Algebra
Other Resources
Kahn Academy
Probability (keithpeterb on youtube)
Probability (Kahn)
Linear Algebra (Kahn)
There, Fixed up your links.
It should also be noted that simpler subjects and introductory texts tend to be common knowledge to the point that citation is often not needed. You don't need to cite that water is wet, not even on Wikipedia.
Journals and modern texts about modern subjects tend to be very well cited, because they're building heavily on other sources of information.
When you don't do this, you have to have an enormous amount of backgrounding. For instance, check out this algebra text. It's 944 pages because it doesn't tend to cite much of exposition and instead states it all directly. It includes an enormous amount of information -- it's meant to be used as fundamental education material. It's not just high level conclusions that could fit in 20-50 pages.
So the amount of citation depends a great deal on the purpose of the text and how close it is to common knowledge. However, Anita's criticism is clearly not common knowledge because nobody but her sees it the way she does. Therefore, she should be explaining how she comes to her conclusions, and citing information. She should also be citing the direct quotes she uses, because it's plagiarism otherwise (and we have huge volumes of evidence that she outright plagiarizes a great deal). Plagiarism in academia is something that ends your career.
I doubt that you're going to find everything you're looking for in a single book.
I suggest that you start with Axler's Linear Algegra Done Right. Despite the pretentious name it does a good job of introducing linear algebra in a general form.
But Axler doesn't do any applications and almost completely ignores determinants (which I like, but it sounds like you want more of that) so I would supplement with Strang's MIT Lectures and any one of his books.
My main hobby is reading textbooks, so I decided to go beyond the scope of the question posed. I took a look at what I have on my shelves in order to recommend particularly good or standard books that I think could characterize large portions of an undergraduate degree and perhaps the beginnings of a graduate degree in the main fields that interest me, plus some personal favorites.
Neuroscience: Theoretical Neuroscience is a good book for the field of that name, though it does require background knowledge in neuroscience (for which, as others mentioned, Kandel's text is excellent, not to mention that it alone can cover the majority of an undergraduate degree in neuroscience if corequisite classes such as biology and chemistry are momentarily ignored) and in differential equations. Neurobiology of Learning and Memory and Cognitive Neuroscience and Neuropsychology were used in my classes on cognition and learning/memory and I enjoyed both; though they tend to choose breadth over depth, all references are research papers and thus one can easily choose to go more in depth in any relevant topics by consulting these books' bibliographies.
General chemistry, organic chemistry/synthesis: I liked Linus Pauling's General Chemistry more than whatever my school gave us for general chemistry. I liked this undergraduate organic chemistry book, though I should say that I have little exposure to other organic chemistry books, and I found Protective Groups in Organic Synthesis to be very informative and useful. Unfortunately, I didn't have time to take instrumental/analytical/inorganic/physical chemistry and so have no idea what to recommend there.
Biochemistry: Lehninger is the standard text, though it's rather expensive. I have limited exposure here.
Mathematics: When I was younger (i.e. before having learned calculus), I found the four-volume The World of Mathematics great for introducing me to a lot of new concepts and branches of mathematics and for inspiring interest; I would strongly recommend this collection to anyone interested in mathematics and especially to people considering choosing to major in math as an undergrad. I found the trio of Spivak's Calculus (which Amazon says is now unfortunately out of print), Stewart's Calculus (standard text), and Kline's Calculus: An Intuitive and Physical Approach to be a good combination of rigor, practical application, and physical intuition, respectively, for calculus. My school used Marsden and Hoffman's Elementary Classical Analysis for introductory analysis (which is the field that develops and proves the calculus taught in high school), but I liked Rudin's Principles of Mathematical Analysis (nicknamed "Baby Rudin") better. I haven't worked my way though Munkres' Topology yet, but it's great so far and is often recommended as a standard beginning toplogy text. I haven't found books on differential equations or on linear algebra that I've really liked. I randomly came across Quine's Set Theory and its Logic, which I thought was an excellent introduction to set theory. Russell and Whitehead's Principia Mathematica is a very famous text, but I haven't gotten hold of a copy yet. Lang's Algebra is an excellent abstract algebra textbook, though it's rather sophisticated and I've gotten through only a small portion of it as I don't plan on getting a PhD in that subject.
Computer Science: For artificial intelligence and related areas, Russell and Norvig's Artificial Intelligence: A Modern Approach's text is a standard and good text, and I also liked Introduction to Information Retrieval (which is available online by chapter and entirely). For processor design, I found Computer Organization and Design to be a good introduction. I don't have any recommendations for specific programming languages as I find self-teaching to be most important there, nor do I know of any data structures books that I found to be memorable (not that I've really looked, given the wealth of information online). Knuth's The Art of Computer Programming is considered to be a gold standard text for algorithms, but I haven't secured a copy yet.
Physics: For basic undergraduate physics (mechanics, e&m, and a smattering of other subjects), I liked Fundamentals of Physics. I liked Rindler's Essential Relativity and Messiah's Quantum Mechanics much better than whatever books my school used. I appreciated the exposition and style of Rindler's text. I understand that some of the later chapters of Messiah's text are now obsolete, but the rest of the book is good enough for you to not need to reference many other books. I have little exposure to books on other areas of physics and am sure that there are many others in this subreddit that can give excellent recommendations.
Other: I liked Early Theories of the Universe to be good light historical reading. I also think that everyone should read Kuhn's The Structure of Scientific Revolutions.
"Without a bunch of jargon" is your interpretation of what Dr. Feynman said. If you read Six Easy Pieces and Six Not-So-Easy Pieces, you'll find that Dr. Feynman's actual teaching is loaded with physics jargon, and a big part of his genius lay in offering supporting intuitions for the very precise terminology physics uses.
Actual computer science—as opposed to the dressed-up vocational training that calls itself "computer science," not that there's anything wrong with vocational training—is the same way. It's mathematical, painstakingly precise, and the terminology shows it. With that said, there are Feynmans out there to help lead us through it. For example, Conceptual Mathematics: A First Introduction to Categories is a text suitable for a bright high-schooler that nevertheless will have you understanding the terms "monad," "monoid," "category," and "endofunctor" by the time you're through it, should you choose to work through it.
> Are the deep mathematical answers to things usually very complex or insanely elegant and simple when you get down to it?
I would say that the deep mathematical answers to questions tend to be very complex and insanely elegant at the same time. The best questions that mathematicians ask tend to be the ones that are very hard but still within reach (in terms of solving them). The solutions to these types of questions often have beautiful answers, but they will generally require lots of theory, technical detail, and/or very clever solutions all of which can be very complex. If they didn't require something tricky, technical, or the development of new theory, they wouldn't be difficult to solve and would be uninteresting.
For any experts that happen to stumble by, my favorite example of this is the classification of semi-stable vector bundles on the complex projective plane by LePotier and Drezet. At the top of page 7 of this paper you'll see a picture representing the fractal structure that arises in this classification. Of course, this required a lot of hard math and complex technical detail to come up with this, but the answer is beautiful and elegant.
> How hard would it be for a non mathematician to go to a pro? Is there just some brain bending that cannot be handled by some? How hard are the concepts to grasp?
I would say that it's difficult to become a professional mathematician. I don't think it has anything to do with an individual's ability to think about it. The concepts are difficult, certainly, but given time and resources (someone to talk to, good books, etc) you can certainly overcome that issue. The majority of the difficulty is that there is so much math! If you're an average person, you've probably taken at most Calculus. The average mathematics PhD (i.e., someone who is just getting their mathematical career going) has probably taken two years of undergraduate mathematics courses, another two years of graduate mathematics courses, and two to three years of research level study beyond calculus to begin to be able tackle the current theory and solve the problems we are interested in today. That's a lot of knowledge to acquire, and it takes a very long time. That doesn't mean you can't start solving problems earlier, however. If you're interested in this type of thing, you might want to consider picking up this book and see if you like it.
And here's the free version:
https://www.amazon.com/dp/0321797094/ref=cm_sw_r_cp_apa_i_V1UDDb4JBGWFX
> Mathematical Logic
It's not exactly Math Logic, just a bunch of techniques mathematicians use. Math Logic is an actual area of study. Similarly, actual Set Theory and Proof Theory are different from the small set of techniques that most mathematicians use.
Also, looks like you have chosen mostly old, but very popular books. While studying out of these books, keep looking for other books. Just because the book was once popular at a school, doesn't mean it is appropriate for your situation. Every year there are new (and quite frankly) pedagogically better books published. Look through them.
Here's how you find newer books. Go to Amazon. In the search field, choose "Books" and enter whatever term that interests you. Say, "mathematical proofs". Amazon will come up with a bunch of books. First, sort by relevance. That will give you an idea of what's currently popular. Check every single one of them. You'll find hidden jewels no one talks about. Then sort by publication date. That way you'll find newer books - some that haven't even been published yet. If you change the search term even slightly Amazon will come up with completely different batch of books. Also, search for books on Springer, Cambridge Press, MIT Press, MAA and the like. They usually house really cool new titles. Here are a couple of upcoming titles that might be of interest to you: An Illustrative Introduction to Modern Analysis by Katzourakis/Varvarouka, Understanding Topology by Shaun Ault. I bet these books will be far more pedagogically sound as compared to the dry-ass, boring compendium of facts like the books by Rudin.
If you want to learn how to do routine proofs, there are about one million titles out there. Also, note books titled Discrete Math are the best for learning how to do proofs. You get to learn techniques that are not covered in, say, How to Prove It by Velleman. My favorites are the books by Susanna Epp, Edward Scheinerman and Ralph Grimaldi. Also, note a lot of intro to proofs books cover much more than the bare minimum of How to Prove It by Velleman. For example, Math Proofs by Chartrand et al has sections about doing Analysis, Group Theory, Topology, Number Theory proofs. A lot of proof books do not cover proofs from Analysis, so lately a glut of new books that cover that area hit the market. For example, Intro to Proof Through Real Analysis by Madden/Aubrey, Analysis Lifesaver by Grinberg(Some of the reviewers are complaining that this book doesn't have enough material which is ridiculous because this book tackles some ugly topological stuff like compactness in the most general way head-on as opposed to most into Real Analysis books that simply shy away from it), Writing Proofs in Analysis by Kane, How to Think About Analysis by Alcock etc.
Here is a list of extremely gentle titles: Discovering Group Theory by Barnard/Neil, A Friendly Introduction to Group Theory by Nash, Abstract Algebra: A Student-Friendly Approach by the Dos Reis, Elementary Number Theory by Koshy, Undergraduate Topology: A Working Textbook by McClusckey/McMaster, Linear Algebra: Step by Step by Singh (This one is every bit as good as Axler, just a bit less pretentious, contains more examples and much more accessible), Analysis: With an Introduction to Proof by Lay, Vector Calculus, Linear Algebra, and Differential Forms by Hubbard & Hubbard, etc
This only scratches the surface of what's out there. For example, there are books dedicated to doing proofs in Computer Science(for example, Fundamental Proof Methods in Computer Science by Arkoudas/Musser, Practical Analysis of Algorithms by Vrajitorou/Knight, Probability and Computing by Mizenmacher/Upfal), Category Theory etc. The point is to keep looking. There's always something better just around the corner. You don't have to confine yourself to books someone(some people) declared the "it" book at some point in time.
Last, but not least, if you are poor, peruse Libgen.
I highly recommend reading "Mathematical Proofs: A Transition to Advanced Mathematics" by Gary Chartrand et. al. It helped me get a better understanding of how to write a proof as well as organize my own thoughts.
Here's the Amazon link: Mathematical Proofs: https://www.amazon.com/dp/0321797094/ref=cm_sw_r_cp_apa_i_V1UDDb4JBGWFX
Besides the Napkin Project I mentioned, which is a genuinely good resource? I got a coordinate-free treatment of linear algebra in my school's prelim. abstract algebra course. We used Dummit and Foote, which must be prescribed by law somewhere because I haven't yet seen a single department not use it. However, in reviewing abstract algebra I instead used Hungerford, which I definitely prefer for its brevity. But really, you can pick any graduate intro algebra text and it should teach this stuff.
Without knowing much about you, I can't tell how much you know about actual math, so apologies if it sounds like I'm talking down to you:
When you get further into mathematics, you'll find it's less and less about doing calculations and more about proving things, and you'll find that the two are actually quite different. One may enjoy both, neither, or one, but not the other. I'd say if you want to find out what higher level math is like, try finding a very basic book that involves a lot of writing proofs.
This one is aimed at high schoolers and I've heard good things about it, but never used it myself.
This one I have read (well, an earlier edition anyway) and think is a phenomenal way to get acquainted with higher math. You may protest that this is a computer science book, but I assure you, it has much more to do with higher math than any calculus text. Pure computer science essentially is mathematics.
Of course, you are free to dive into whatever subject interests you most. I picked these two because they're intended as introductions to higher math. Keep in mind though, most of us struggle at first with proofwriting, even with so-called "gentle" introductions.
One last thing: Don't think of your ability in terms of your age, it's great to learn young, but there's nothing wrong with people learning later on. Thinking of it as a race could lead to arrogance or, on the other side of the spectrum, unwarranted disappointment in yourself when life gets in the way. We want to enjoy the journey, not worry about if we're going fast enough.
Best of luck!
I'll be that guy. There are two types of Calculus: the Micky Mouse calculus and Real Analysis. If you go to Khan Academy you're gonna study the first version. It's by far the most popular one and has nothing to do with higher math.
The foundations of higher math are Linear Algebra(again, different from what's on Khan Academy), Abstract Algebra, Real Analysis etc.
You could, probably, skip all the micky mouse classes and start immediately with rigorous(proof-based) Linear Algebra.
But it's probably best to get a good foundation before embarking on Real Analysis and the like:
Discrete Mathematics with Applications by Susanna Epp
How to Prove It: A Structured Approach Daniel Velleman
Learning to Reason: An Introduction to Logic, Sets, and Relations by Nancy Rodgers
Book of Proof by Richard Hammock
That way you get to skip all the plug-and-chug courses and start from the very beginning in a rigorous way.
You could read Timothy Gowers' welcome to the math students at Oxford, which is filled with great advice and helpful links at the bottom.
You could read this collection of links on efficient study habits.
You could read this thread about what it takes to succeed at MIT (which really should apply everywhere). Tons of great discussion in the lower comments.
You could read How to Solve It and/or How to Prove It.
If you can work your way through these two books over the summer, you'll be better prepared than 90% of the incoming math majors (conservatively). They'll make your foundation rock solid.
If you want to learn how calculus actually works (rather than just how to do computations), I highly recommend working through Spivak's Calculus. Spivak builds up calculus from the foundations with mathematical rigor and actual proofs, explaining (and proving) what's really going on. (That includes properly developing sequences and limits.) The exercises are also excellent; many of them require real thought and insight, instead of the usual "repeat the steps you were just told fifty times" exercises that fill up mainstream calculus textbooks.
Also, from a more sophisticated perspective, dx is a differential form.
Traditionally, a mathematical proof has one and only one job: convince other people that your proof is correct. (In this day and age, there is such a thing as a computer proof, but if you don't understand traditional proofs, you can't handle computer proofs either.)
Notice what I just said: "convince other people that your proof is correct." A proof is, in some sense, always an interactive undertaking, even if the interaction takes place across gulfs of space and time.
Because interaction is so central to the notion of a proof, it is rare for students to successfully self-study how to write proofs. That seems like what you're asking. Don't get me wrong. Self-study helps. But it is not the only thing you need. You need, at some point, to go through the process of presenting your proofs to others, answering questions about your proof, adjusting your proof to take into account new feedback, and using this experience to anticipate likely issues in future proofs.
What you're proposing to do, in most cases, is the wrong strategy. You need more interactive experience, not less. You should be beating down the doors of your professor or TA in your class during their office hours, asking for feedback on your proofs. (This implies that you should be preparing your proofs in advance for them to read before going to their office hours.) If your school has a tutorial center, that's a wonderful resource as well. A math tutor who knows math proofs is a viable source of help, but if you don't know how to do proofs, it's hard for you to judge whether or not your tutor knows how to do proofs.
If you do self-study anything, you should not be self-studying calculus, linear algebra, real analysis, or abstract algebra. You should be self-studying how to do proofs. Some people here say that How to Prove It is a useful resource. My own position is that while self-studying can be helpful, it needs to be balanced with some amount of external interactive feedback in order to really stick.
I don't think that is a very compelling argument, unless we believe mathematicians can do no notational wrong :-) The imprecise, ambiguous, sometimes obfuscatory notation that arises in multivariable calculus and the calculus of variations is a well known and frequently discussed issue. I think we underestimate the difficulty it causes to students, especially to students coming from other disciplines who aren't steeped in the mathematical vernacular.
It's been problematic enough that there are some high profile and semi-accepted attempts to refine the notation, such as the functional notation used in Spivak's Calculus on Manifolds, which is based in an earlier attempt from the 50s if I remember correctly. Another presentation of physics motivated in large part by fixing the notation is Sussman & Wisdom's Structure and Interpretation of Classical Mechanics which adopts Spivak's notation, and also uses computer programs to describe algorithms more precisely.
if you want determinants, Shilov's is supposed to be "Determinants done right" I wouldn't recommend the other Dover LA book by Stoll
http://www.amazon.com/Linear-Algebra-Dover-Books-Mathematics/product-reviews/048663518X/
-----------
Anyway: Free!
http://www.math.ucdavis.edu/~anne/linear_algebra/
http://www.math.ucdavis.edu/~linear/linear.pdf
http://www.cs.cornell.edu/courses/cs485/2006sp/LinAlg_Complete.pdf (Dawkins notes that were recently pulled off lamar.edu site, gentle intro like Anton's)
http://joshua.smcvt.edu/linearalgebra/
http://www.ee.ucla.edu/~vandenbe/103/reader.pdf
http://www.math.brown.edu/%7Etreil/papers/LADW/LADW.pdf
https://math.byu.edu/~klkuttle/Linearalgebra.pdf
---------
Or, google "positive definite matrix" or "hermitian" or "hessian" or some term like that and it will show you lecture notes from dozens of universities after the inevitable wikipedia and Wolfram hits
You are in a very special position right now where many interesing fields of mathematics are suddenly accessible to you. There are many directions you could head. If your experience is limited to calculus, some of these may look very strange indeed, and perhaps that is enticing. That was certainly the case for me.
Here are a few subject areas in which you may be interested. I'll link you to Dover books on the topics, which are always cheap and generally good.
Basically, don't limit yourself to the track you see before you. Explore and enjoy.
I take it you want something small enough to fit inside a hollowed-out bible or romance novel, so you can hide your secrets from nosy neighbors?
More seriously, this book is great https://amzn.com/0201558025 but might not be quite what you’re looking for at an introductory level.
I’ve seen recommendations of https://amzn.com/0495391328 (I haven’t looked at this book myself.)
The heart of conceptual mathematics (i.e., mathematics that isn't just computation and carrying out algorithms) is mathematical proof. I suggest you work through the book How to Prove It. This will give you the tools to self work through other textbooks (not that it will suddenly be easy).
I think the advice given in the rest of the thread is pretty good, though some of it a little naive. The suggestion that differential equations or applied math somehow should not be of interest is silly. A lot of it builds the motivation for some of the abstract stuff which is pretty cool, and a lot of it has very pure problems associated with it. In addition I think after (or rather alongside) your initial calculus education is a good time to look at some other things before moving onto more difficult topics like abstract algebra, topology, analysis etc.
The first course I took in undergrad was a course that introduced logic, writing proofs, as well as basic number theory. The latter was surprisingly useful as it built modular arithmetic which gave us a lot of groups and rings to play with in subsequent algebra courses. Unfortunately the textbook was god awful. I've heard good things about the following two sources and together they seem to cover the content:
How to prove it
Number theory
After this I would take a look at linear algebra. This a field with a large amount of uses in both pure and applied math. It is useful as it will get you used to doing algebraic proofs, it takes a look at some common themes in algebra, matrices (one of the objects studied) are also used thoroughly in physics and applied mathematics and the knowledge is useful for numerical approximations of ordinary and partial differential equations. The book I used Linear Algebra by Friedberg, Insel and Spence, but I've heard there are better.
At this point I think it would be good to move onto Abstract Algebra, Analysis and Topology. I think Farmerje gave a good list.
There's many more topics that you could possibly cover, ODEs and PDEs are very applicable and have a rich theory associated with them, Complex Analysis is a beautiful subject, but I think there's plenty to keep you busy for the time being.
Learn math at a more "fundamental" level, and that will test if you love it. For me, I didn't love math until I took a class on proofs and real analysis. One of the books we used was "How to Prove it", and to this day it's my favorite textbook ever. How do we know anything in mathematics? Which rules do we follow and how do we know they are true? This starts from basic logic and truth tables, and works its way up to some really complicated stuff. It's not as fancy as complex integrals and PDE's, but I would say it's a more fundamental form of mathematics and the basis for all other subjects in the field.
One of the most fun things I did when I was first learning about proofs was proving the basic facts about algebra from axioms. Where I first read about these ideas was the first chapter of Spivak's Calculus. This would be a very high level book for an 18 year old, but if you decide to look at it, don't be afraid to take your time a little.
Another option is just picking up an introduction to proof, like Velleman's How to Prove It. This wil lteach you the basics for proving anything, really, and is a great start if you want to do more math.
If you want a free alternative to that last one, you can look at The Book of Proof by Richard Hammack. It's well-written although I think it's shorter than How to Prove It.
I double majored in math and CS as an undergrad and I enjoyed math more than CS. I'm a graduate student right now planning on doing research in a mathy area of CS. Everything I write below comes from that perspective.
But if you're interested in really digging in and understanding some math at an advanced undergraduate level (analysis, abstract algebra, topology, etc.) then I don't think there is any substitute for books.
I'm not so sure this is a fundamental difference, so much as a distinction in who is looking at each field. For the most part, category theory is studied by those who are looking to make advances in knowledge. Sure, the things researchers are looking at can be complex. But if you look at current research in abstract algebra, it's equally difficult to get up to speed and comprehend. The reason abstract algebra can be seen as simpler is that there is also introductory material, aimed at undergraduates, and even the general population.
Is it fundamentally impossible to produce such introductory material in category theory? Of course not! Several people have made serious and credible attempts. For example, here and here
I don't say this to be discouraging: Most people don't really have any idea what doing Physics at a high level looks like. I decided in High School that I wanted to be a physicist, and as luck would have it I'm a graduate student and I still enjoy it, but truth be told, the exposure you have in High School doesn't really prepare you for the reality. All that to say: There's no reason to decide at thirteen years old that you need a PhD in Physics! Maybe once you learn math beyond trig you'll decide it isn't for you, or maybe you'll love math and want to switch to a math degree.
All right, now that that's out of the way... You said you're learning trig, that's good, you need it. You also need some basic algebra skills. Then try to teach yourself basic calculus (limits, derivatives, integrals). Then you want to learn Linear Algebra and at least Ordinary Differential Equations.
You can also do some basic physics reading before you've learned the essentials. I really like George Gamow's books for this - he was a very well know and important physicist who also happened to write very accessible books that are very much for lay people but that also don't shy away completely from the math. I really enjoyed this one in particular.
For mathematics, I love Dover books - they're cheap AND good. Shilov, I've found, is clear and readable. This might not be introductory level, but it's inexpensive and let's you see what you're getting yourself into.
Last bit of advice for Physics is what one of my old high school teachers used to say - draw, label, and you can't go wrong. It's still mostly true.
There would have been a time that I would have suggested getting a curriculum
text book and going through that, but if you're doing this for independent work
I wouldn't really suggest that as the odds are you're not going to be using a
very good source.
Going on the typical
Arithmetic > Algebra > Calculus
****
Arithmetic
Arithmetic refresher. Lots of stuff in here - not easy.
I think you'd be set after this really. It's a pretty terse text in general.
*****
Algebra
Algebra by Chrystal Part I
Algebra by Chrystal Part II
You can get both of these algebra texts online easily and freely from the search
chrystal algebra part I filetype:pdfchrystal algebra part II filetype:pdfI think that you could get the first (arithmetic) text as well, personally I
prefer having actual books for working. They're also valuable for future
reference. This
filetype:pdfsearch should be remembered and used liberallyfor finding things such as worksheets etc (eg
trigonometry worksheet<br /> filetype:pdffor a search...).Algebra by Gelfland
No where near as comprehensive as chrystals algebra, but interesting and well
written questions (search for 'correspondence series' by Gelfand).
Calculus
Calculus made easy - Thompson
This text is really good imo, there's little rigor in it but for getting a
handle on things and bashing through a few practical problems it's pretty
decent. It's all single variable. If you've done the algebra and stuff before
this then this book would be easy.
Pauls Online Notes (Calculus)
These are just a solid set of Calculus notes, there're lots of examples to work
through which is good. These go through calc I, II, III... So a bit further than
you've asked (I'm not sure why you state up to calc II but ok).
Spivak - Calculus
If you've gone through Chrystals algebra then you'll be used to a formal
approach. This text is only single variable calculus (so that might be calc I
and II in most places I think, ? ) but it's extremely well written and often
touted as one of the best Calculus books written. It's very pure, where as
something like Stewart has a more applied emphasis.
**
Geometry
I've got given any geometry sources, I'm not too sure of the best source for
this or (to be honest) if you really need it for the above. If someone has
good geometry then they're certainly better off, many proofs are given
gemetrically as well and having an intuition for these things is only going to
be good. But I think you can get through without a formal course on it.... I'm
not confident suggesting things on it though, so I'll leave it to others. Just
thought I'd mention it.
****
Pick up mathematics. Now if you have never done math past the high school and are an "average person" you probably cringed.
Math (an "actual kind") is nothing like the kind of shit you've seen back in grade school. To break into this incredible world all you need is to know math at the level of, say, 6th grade.
Intro to Math:
These books only serve as samplers because they don't even begin to scratch the surface of math. After you familiarized yourself with the basics of writing proofs you can get started with intro to the largest subsets of math like:
Intro to Abstract Algebra:
There are tons more books on abstract/modern algebra. Just search them on Amazon. Some of the famous, but less accessible ones are
Intro to Real Analysis:
Again, there are tons of more famous and less accessible books on this subject. There are books by Rudin, Royden, Kolmogorov etc.
Ideally, after this you would follow it up with a nice course on rigorous multivariable calculus. Easiest and most approachable and totally doable one at this point is
At this point it's clear there are tons of more famous and less accessible books on this subject :) I won't list them because if you are at this point of math development you can definitely find them yourself :)
From here you can graduate to studying category theory, differential geometry, algebraic geometry, more advanced texts on combinatorics, graph theory, number theory, complex analysis, probability, topology, algorithms, functional analysis etc
Most listed books and more can be found on libgen if you can't afford to buy them. If you are stuck on homework, you'll find help on [MathStackexchange] (https://math.stackexchange.com/questions).
Good luck.
Or How to Prove It by Velleman.
Conceptual Mathematics by Lawvere and Schanuel is a good low level introduction to category theory (and a bit of set theory) if you are feeling shaky on those grounds. From there lots of books open up to you.
The best books I know on how to "think" like a functional programmer are all written by Richard Bird. http://www.amazon.com/gp/product/1107452643/ref=pd_lpo_sbs_dp_ss_1?pf_rd_p=1944579842&amp;pf_rd_s=lpo-top-stripe-1&amp;pf_rd_t=201&amp;pf_rd_i=0134843460&amp;pf_rd_m=ATVPDKIKX0DER&amp;pf_rd_r=090NKMWKY6078Z0WPCTW http://www.amazon.com/Pearls-Functional-Algorithm-Design-Richard/dp/0521513383
Not much is available in book form, especially that I can recommend on the FRP front.
Dependent types is a broad area, you're going to find yourself reading a lot of research papers. You might be able to get by with something more practical like Chlipala's Certified Programming with Dependent Types, but if you want a more theoretical treatment then perhaps Zhaohui Luo's Computation and Reasoning might be a better starting point.
/u/another_user_name posted this list a while back. Actual aerospace textbooks are towards the bottom but you'll need a working knowledge of the prereqs first.
Non-core/Pre-reqs:
Mathematics:
Calculus.
1-4) Calculus, Stewart -- This is a very common book and I felt it was ok, but there's mixed opinions about it. Try to get a cheap, used copy.
1-4) Calculus, A New Horizon, Anton -- This is highly valued by many people, but I haven't read it.
1-4) Essential Calculus With Applications, Silverman -- Dover book.
More discussion in this reddit thread.
Linear Algebra
3) Linear Algebra and Its Applications,Lay -- I had this one in school. I think it was decent.
3) Linear Algebra, Shilov -- Dover book.
Differential Equations
4) An Introduction to Ordinary Differential Equations, Coddington -- Dover book, highly reviewed on Amazon.
G) Partial Differential Equations, Evans
G) Partial Differential Equations For Scientists and Engineers, Farlow
More discussion here.
Numerical Analysis
5) Numerical Analysis, Burden and Faires
Chemistry:
Physics:
2-4) Physics, Cutnel -- This was highly recommended, but I've not read it.
Programming:
Introductory Programming
Programming is becoming unavoidable as an engineering skill. I think Python is a strong introductory language that's got a lot of uses in industry.
Core Curriculum:
Introduction:
Aerodynamics:
Thermodynamics, Heat transfer and Propulsion:
Flight Mechanics, Stability and Control
5+) Flight Stability and Automatic Control, Nelson
5+)[Performance, Stability, Dynamics, and Control of Airplanes, Second Edition](http://www.amazon.com/Performance-Stability-Dynamics-Airplanes-Education/dp/1563475839/ref=sr_1_1?ie=UTF8&amp;qid=1315534435&amp;sr=8-1, Pamadi) -- I gather this is better than Nelson
Engineering Mechanics and Structures:
3-4) Engineering Mechanics: Statics and Dynamics, Hibbeler
6-8) Analysis and Design of Flight Vehicle Structures, Bruhn -- A good reference, never really used it as a text.
G) Introduction to the Mechanics of a Continuous Medium, Malvern
G) Fracture Mechanics, Anderson
G) Mechanics of Composite Materials, Jones
Electrical Engineering
Design and Optimization
Space Systems
http://www.amazon.com/Discrete-Mathematics-Applications-Susanna-Epp/dp/0495391328 is one of my math books. The bookstore wants $350 for it.
There are essentially "two types" of math: that for mathematicians and everyone else. When you see the sequence Calculus(1, 2, 3) -> Linear Algebra -> DiffEq (in that order) thrown around, you can be sure they are talking about non-rigorous, non-proof based kind that's good for nothing, imo of course. Calculus in this sequence is Analysis with all its important bits chopped off, so that everyone not into math can get that outta way quick and concentrate on where their passion lies. The same goes for Linear Algebra. LA in the sequence above is absolutely butchered so that non-math majors can pass and move on. Besides, you don't take LA or Calculus or other math subjects just once as a math major and move on: you take a rigorous/proof-based intro as an undergrad, then more advanced kind as a grad student etc.
To illustrate my point:
Linear Algebra:
Linear Algebra Through Geometry by Banchoff and Wermer
3. Here's more rigorous/abstract Linear Algebra for undergrads:
Linear Algebra Done Right by Axler
4. Here's more advanced grad level Linear Algebra:
Advanced Linear Algebra by Steven Roman
-----------------------------------------------------------
Calculus:
Calulus by Spivak
3. Full-blown undergrad level Analysis(proof-based):
Analysis by Rudin
4. More advanced Calculus for advance undergrads and grad students:
Advanced Calculus by Sternberg and Loomis
The same holds true for just about any subject in math. Btw, I am not saying you should study these books. The point and truth is you can start learning math right now, right this moment instead of reading lame and useless books designed to extract money out of students. Besides, there are so many more math subjects that are so much more interesting than the tired old Calculus: combinatorics, number theory, probability etc. Each of those have intros you can get started with right this moment.
Here's how you start studying real math NOW:
Learning to Reason: An Introduction to Logic, Sets, and Relations by Rodgers. Essentially, this book is about the language that you need to be able to understand mathematicians, read and write proofs. It's not terribly comprehensive, but the amount of info it packs beats the usual first two years of math undergrad 1000x over. Books like this should be taught in high school. For alternatives, look into
Discrete Math by Susanna Epp
How To prove It by Velleman
Intro To Category Theory by Lawvere and Schnauel
There are TONS great, quality books out there, you just need to get yourself a liitle familiar with what real math looks like, so that you can explore further on your own instead of reading garbage and never getting even one step closer to mathematics.
If you want to consolidate your knowledge you get from books like those of Rodgers and Velleman and take it many, many steps further:
Basic Language of Math by Schaffer. It's a much more advanced book than those listed above, but contains all the basic tools of math you'll need.
I'd like to say soooooooooo much more, but I am sue you're bored by now, so I'll stop here.
Good Luck, buddyroo.
Is it really such a big step from du Sautoy's explanation to the formal proof? I don't think so, but maybe I'm biased. I bet there are books on elementary number theory that don't assume much of any background that you could understand. If you're interested in proofs in general, you might enjoy Velleman's How to Prove It.
I picked up a book a couple years ago called How to Prove It.
It has helped me develop a greater appreciation for logic and proofs. I wish I took this stuff more seriously when I started programming. A little bit of knowledge of boolean algebra can help tremendously.
Calculus on Manifolds is an elementary treatment that only assumes basic mathematical maturity along with say a year of calculus. A classic from Michael Spivak, who these days is mostly known for his rigorous calculus book.
math 271 is easy if you can think logically
just pirate this book: https://www.amazon.ca/Discrete-Mathematics-Applications-Susanna-Epp/dp/0495391328
and browse the first 10 chapters - thats whats taught at math 271. i had thi dinh or something, some asian dude. he's a hardass but one of my fav profs of all time
ill be honest though, none of this math is plug and chug like calc 1 or even calc 2. i thought 271 required more thinking than linear or calc 1 or calc 2. once you get into counting and probability and set theory and graphs/relations and shits, it gets pretty intense. the first half of the course is easy but a lot of people fail the final exam and then fail the course lmao.
eng319 is hard but u gotta take it bro.
TAKING BOTH??? are you ready to stay indoors/at school all day for 60 days? then you can do it. if u slack off ur gonna fail 271 or 319.
gljuck bro
If you are getting your degree in math or computer science, you will probably have to take a course on "Discrete math" (or maybe an "introduction to proofs") in your first year or two (it should be by your 3rd semester). Unfortunately, this will probably be the first time you will take a course that is more about the why than the how. (On the bright side, almost everything after this will focus on why instead of how.) Depending on how linear algebra is taught at your university, and the order you take classes in, linear algebra may be also be such a class.
If your degree is anything else, you may have no formal requirement to learn the why.
For the math you are learning right now, analysis is the "why". I'm not sure of a good analysis book, but there are two calculus books which treat the subject more like a gentle introduction to analysis-- Apostol's and Spivak's. Your library might have a copy you can check out. If not, you can probably find pdfs (which are probably[?] legal) online.
I found Axler's Linear Algebra Done Right to be a very easy to digest introduction to abstract linear algebra.
Not sure what sort of thing you're trying to prove, but there are a few good books on techniques for proof that you'll end up using if you go into higher math. I like How to Prove It by Velleman. It's geared towards students finishing high school math who are planning to do math at the university level, so it might be the sort of thing you're looking for.
https://www.amazon.com/How-Prove-Structured-Approach-2nd/dp/0521675995/
well my favorite subjects in graduate school were differential geometry, particularly the theory of smooth manifolds, and functional analysis, in particular distributions. once i got a job writing software and dealing with hardware systems, i tried to keep up with my math (a losing battle somewhat) to see what connections i could make, and i eventually found the book conceptual mathematics: a first introduction to categories. i was also at the same time trying to pick up haskell, so between haskell and the category theory book and my job and my mathematics background, i started to realize that there are some connections between what people do in software and systems and the math. then i came across the book/paper category theory for scientists.
i'm now convinced that category can serve as a fantastic foundation for applied mathematics. when people think of applied mathematics, they immediately think discrete, combinatorial mathematics or throw differential equations at whatever problem is at hand. but i think there's a lot of the more abstract mathematics that can be applied, and i think (or at least agree with the authors of the materials i linked to) that category theory can help with this. you should also take a look at the work of robert ghrist as well, who applies algebraic topology to many engineering problems.
This:
http://www.amazon.com/Conceptual-Mathematics-First-Introduction-Categories/dp/052171916X
is a good book as an introduction for a math student!
None of these I've finished, but they're on the backburner whenever I have free time.
A Singular Mathematical Promenade (Etienne Ghys)
Music: A Mathematical Offering (Dave Benson)
Nonlinear Dynamics and Chaos (Strogatz)
You're not really doing higher math right now as much as you're learning tricks to solve problems. Once you start proving stuff that'll be a big jump. Usually people start doing that around Real Analysis like your father said. Higher math classes almost entirely consist of proofs. It's a lot of fun once you get the hang of it, but if you've never done it much before it can be jarring to learn how. The goal is to develop mathematical maturity.
Start learning some geometry proofs or pick up a book called "Calculus" by Spivak if you want to start proving stuff now. The Spivak book will give you a massive head start if you read it before college. Differential equations will feel like a joke after this book. It's called calculus but it's really more like real analysis for beginners with a lot of the harder stuff cut out. If you can get through the first 8 chapters or so, which are the hardest ones, you'll understand a lot of mathematics much more deeply than you do now. I'd also look into a book called Linear Algebra done right. This one might be harder to jump into at first but it's overall easier than the other book.
No, his single variable book.
I do plan on reading Calculus on Manifolds eventually, though.
If you are serious about learning, Linear Algebra by Friedberg Insel and Spence, or Linear Algebra by Greub are your best bets. I love both books, but the first one is a bit easier to read.
I speak of this famous calculus book: https://www.amazon.com/Calculus-Vol-One-Variable-Introduction-Algebra/dp/0471000051
Which is a "theoretical" approach to Calculus rather than a mechanical approach.
Here is the ooh page on Statisticians:
http://www.bls.gov/oco/ocos045.htm
A job straight out of college might see you as a research assistant. I could see you getting a job at Mathematica perhaps. Try to get a SAS certificate before you graduate, a working knowledge of R, and if you feel like tackling it a programming language good for numerical analysis.
Have you taken a course on Regression? I'd consider that, and perhaps even trying to take a Mathematical Statistics Course, if it is offered. You can try to see if you university would allow you to take a class online, or try a Semester Abroad at a university that has that class.
My background: I am an Economist that uses Statistics heavily, and works with Statistical methods often (ie: econometrics). I love it.
Your plans on studying Calc 2 and Linear Algebra are great. That is perfect.
My pay after 10 years is likely to be 100k-150k.
Before you start your first semester at the graduate level know the following things really well: Set theory, integration, matrix algebra, and proofs.
Get this book: http://www.amazon.com/How-Prove-Structured-Daniel-Velleman/dp/0521675995 -- read it before you study linear algebra, and maybe even some Calculus. It doesn't require a heavy Math background and will save you a lot of frustration later on.
I'd echo what /u/Odnahc has said.
Struggling in Intro the Proofs isn't he end of the world. I struggled in proofs and still ended up with a BS and MS in Math, however, I bought this book and self studied proofs over the Summer and made sure I had a stronger foundation.
The courses normally taken after proofs (Advanced Calculus and Modern Algebra) usually spend the first class reviewing proofs to make sure students have a handle of the material. After that though, you're expected to know the stuff. And honestly, you'll be doing lot of work trying to understand the new material and you're really going to struggle if you're fighting proof writing instead of the new ideas.
Proceed with caution. Definitely speak to your advisor.
Alternately, any introductory book on mathematical analysis will have a section on sentential logic. 'How to Prove It' by Velleman is a great intro, and comes with a link to a web tool to practice!
I think the most important part of being able to see beauty in mathematics is transitioning to texts which are based on proofs rather than application. A side effect of gaining the ability to read and write proofs is that you're forced to deeply understand the theory of the math you're learning, as well as actively using your intuition to solve problems, rather than dry route calculations found in most application based textbooks. Based on what you've written, I feel you may enjoy taking this path.
Along these lines, you could start of with Book of Proof (free) or How to Prove It. From there, I would recommend starting off with a lighter proof based text, like Calculus by Spivak, Linear Algebra Done Right by Axler, or Pinter's book as you mentioned. Doing any intro proofs book plus another book at the level I mentioned here would have you well prepared to read any standard book at the undergraduate level (Analysis, Algebra, Topology, etc).
I know the answer to this.
First, though: arithmetic and all that, through calculus, is not math.
True math is the discovery of properties of ideas. One interesting example is the fact that there is a hypothetical machine that is proven to be able to do everything a (real) computer can do, but that there are many things that it can never do. Therefore, there are questions that can never be answered by a computer, no matter how powerful.
If you actually want to know about the beauty, you need to see it for yourself. As I recall, How to Prove it is pretty decent.
Your professors really aren't expecting you to reinvent groundbreaking proofs from scratch, given some basic axioms. It's much more likely that you're missing "hints" - exercises often build off previous proofs done in class, for example.
I appreciated Laura Alcock's writings on this, in helping me overcome my fear of studying math in general:
https://www.amazon.com/How-Study-as-Mathematics-Major/dp/0199661316/
https://www.amazon.com/dp/0198723539/ <-- even though you aren't in analysis, the way she writes about approaching math classes in general is helpful
If you really do struggle with the mechanics of proof, you should take some time to harden that skill on its own. I found this to be filled with helpful and gentle exercises, with answers: https://www.amazon.com/dp/0989472108/ref=rdr_ext_sb_ti_sims_2
And one more idea is that it can't hurt for you to supplement what you're learning in class with a more intuitive, chatty text. This book is filled with colorful examples that may help your leap into more abstract territory: https://www.amazon.com/Visual-Group-Theory-Problem-Book/dp/088385757X
If you're looking for other texts, I would suggest Spivak's Calculus and Calculus on Manifolds. At first the text may seem terse, and the exercises difficult, but it will give you a huge advantage for later (intermediate-advanced) undergraduate college math.
It may be a bit obtuse to recommend you start with these texts, so maybe your regular calculus texts, supplemented with linear algebra and differential equations, should be approached first. When you start taking analysis and beyond, though, these books are probably the best way to return to basics.
I'd like to give you my two cents as well on how to proceed here. If nothing else, this will be a second opinion. If I could redo my physics education, this is how I'd want it done.
If you are truly wanting to learn these fields in depth I cannot stress how important it is to actually work problems out of these books, not just read them. There is a certain understanding that comes from struggling with problems that you just can't get by reading the material. On that note, I would recommend getting the Schaum's outline to whatever subject you are studying if you can find one. They are great books with hundreds of solved problems and sample problems for you to try with the answers in the back. When you get to the point you can't find Schaums anymore, I would recommend getting as many solutions manuals as possible. The problems will get very tough, and it's nice to verify that you did the problem correctly or are on the right track, or even just look over solutions to problems you decide not to try.
Basics
I second Stewart's Calculus cover to cover (except the final chapter on differential equations) and Halliday, Resnick and Walker's Fundamentals of Physics. Not all sections from HRW are necessary, but be sure you have the fundamentals of mechanics, electromagnetism, optics, and thermal physics down at the level of HRW.
Once you're done with this move on to studying differential equations. Many physics theorems are stated in terms of differential equations so really getting the hang of these is key to moving on. Differential equations are often taught as two separate classes, one covering ordinary differential equations and one covering partial differential equations. In my opinion, a good introductory textbook to ODEs is one by Morris Tenenbaum and Harry Pollard. That said, there is another book by V. I. Arnold that I would recommend you get as well. The Arnold book may be a bit more mathematical than you are looking for, but it was written as an introductory text to ODEs and you will have a deeper understanding of ODEs after reading it than your typical introductory textbook. This deeper understanding will be useful if you delve into the nitty-gritty parts of classical mechanics. For partial differential equations I recommend the book by Haberman. It will give you a good understanding of different methods you can use to solve PDEs, and is very much geared towards problem-solving.
From there, I would get a decent book on Linear Algebra. I used the one by Leon. I can't guarantee that it's the best book out there, but I think it will get the job done.
This should cover most of the mathematical training you need to move onto the intermediate level physics textbooks. There will be some things that are missing, but those are usually covered explicitly in the intermediate texts that use them (i.e. the Delta function). Still, if you're looking for a good mathematical reference, my recommendation is Lua. It may be a good idea to go over some basic complex analysis from this book, though it is not necessary to move on.
Intermediate
At this stage you need to do intermediate level classical mechanics, electromagnetism, quantum mechanics, and thermal physics at the very least. For electromagnetism, Griffiths hands down. In my opinion, the best pedagogical book for intermediate classical mechanics is Fowles and Cassidy. Once you've read these two books you will have a much deeper understanding of the stuff you learned in HRW. When you're going through the mechanics book pay particular attention to generalized coordinates and Lagrangians. Those become pretty central later on. There is also a very old book by Robert Becker that I think is great. It's problems are tough, and it goes into concepts that aren't typically covered much in depth in other intermediate mechanics books such as statics. I don't think you'll find a torrent for this, but it is 5 bucks on Amazon. That said, I don't think Becker is necessary. For quantum, I cannot recommend Zettili highly enough. Get this book. Tons of worked out examples. In my opinion, Zettili is the best quantum book out there at this level. Finally for thermal physics I would use Mandl. This book is merely sufficient, but I don't know of a book that I liked better.
This is the bare minimum. However, if you find a particular subject interesting, delve into it at this point. If you want to learn Solid State physics there's Kittel. Want to do more Optics? How about Hecht. General relativity? Even that should be accessible with Schutz. Play around here before moving on. A lot of very fascinating things should be accessible to you, at least to a degree, at this point.
Advanced
Before moving on to physics, it is once again time to take up the mathematics. Pick up Arfken and Weber. It covers a great many topics. However, at times it is not the best pedagogical book so you may need some supplemental material on whatever it is you are studying. I would at least read the sections on coordinate transformations, vector analysis, tensors, complex analysis, Green's functions, and the various special functions. Some of this may be a bit of a review, but there are some things Arfken and Weber go into that I didn't see during my undergraduate education even with the topics that I was reviewing. Hell, it may be a good idea to go through the differential equations material in there as well. Again, you may need some supplemental material while doing this. For special functions, a great little book to go along with this is Lebedev.
Beyond this, I think every physicist at the bare minimum needs to take graduate level quantum mechanics, classical mechanics, electromagnetism, and statistical mechanics. For quantum, I recommend Cohen-Tannoudji. This is a great book. It's easy to understand, has many supplemental sections to help further your understanding, is pretty comprehensive, and has more worked examples than a vast majority of graduate text-books. That said, the problems in this book are LONG. Not horrendously hard, mind you, but they do take a long time.
Unfortunately, Cohen-Tannoudji is the only great graduate-level text I can think of. The textbooks in other subjects just don't measure up in my opinion. When you take Classical mechanics I would get Goldstein as a reference but a better book in my opinion is Jose/Saletan as it takes a geometrical approach to the subject from the very beginning. At some point I also think it's worth going through Arnold's treatise on Classical. It's very mathematical and very difficult, but I think once you make it through you will have as deep an understanding as you could hope for in the subject.
Attention Publishers
This is why readers hate you. Note the version number. Seventh Edition? Really, how much has calculus changed in the past 20 years? The past 50? Or 100? I only graduated 4 years ago, and this is the second time they've cranked out a new version of the book since my freshman year.
Of course they quit printing the older editions, because they can cripple the market for used textbooks and force everyone to buy new versions. So they go and re-hash and reword a chapter here and there and pretend it's a "new" book somehow.
I seriously doubt it takes until the 4th, 5th, or 6th printing of a book for the publisher to recoup their investment; if it does, I think the only reason is that they're writing themselves such large checks.
How to Ace Calculus: The Streetwise Guide is charming. It does an excellent job scaffolding intuitive understanding without unnecessarily sacrificing rigor. It took me at least three attempts to properly spell the word "unnecessarily" in the previous sentence.
Extremely delayed edit: It also has the marked advantage of being quite cheap.
Not sure what level you're approaching it from, but Steve Strogatz's Nonlinear Dynamics and Chaos is a pretty good upper-level undergraduate introduction to the topic.
If she's bright and interested enough you might want to consider getting her an entry level college calculus book such as Spivak's.
It won't pose a replacement to the technical approach of high school, but it will illuminate a lot.
I think this is a better approach than trying to tie connections between calculus and other areas of math, because calculus has an inherent beauty of its own which could be very compelling when taught with the right philosophical approach.
If you're currently at the pre-calc level, you could probably get away with learning from khan academy for a little while. After that (and building some familiarity with proof writing), you'd be ready for some of the pure math classes like abstract algebra and real analysis. For those courses, you'll probably want to check out some Open Courseware. You'd want to treat it like a real class; watch the lectures online and read from the textbooks, while working on problem sets.
While you're working your way through the khan academy stuff, you might want to check out Stewart's calculus book; it's pretty solid for making your way through the calculus sequence.
I'd ask around for a good linear algebra book, since I haven't encountered a decent one that's at that level.
Everyone has to take MATH 150--MATH 152's prerequisite isn't Calculus 12. So after 150, you're at the same level as everyone else.
A tip: make sure when studying, you understand every part of what's being taught. You won't be able to just memorize this stuff. If you don't get something, spend a bit of time trying to figure it out, move forward if the following information doesn't rely on what you're passing, but come back to it later and try again and again till you understand what that thing is, how it works, and why. YouTube the name of what you're having trouble with, cause there are going to be several tutorials from people on there per topic.
You'll have to put in the hours, though, and study smart. Remember: being a student is your job, and 3 courses is full time (equivalent to 9-5 Mon-Fri). SFU uses the "flipped classroom" where you're supposed to read the sections of the textbook before class, the lecture reinforces and clarifies the most important stuff, then you self-study till you understand it 100%.
The rule of thumb for all classes is 2-3 hours of study for every hour in lecture. That means for MATH 150 you should expect to spend 8-12 hours studying on your own outside of class.
Engineering requires 12 credits/semester, so you'd have at least 13 in the semester you take 150--That means 26-39 hours of studying on your own outside class i.e. 6 hours a day 7 days a week, 6.5 hours every day but Sat/Sun, or 8 hours a day Mon-Fri.
Here are a couple useful resources:
Also, FYI for YouTube, if you hit shift+> on YouTube, it speeds up the video 1.25x 1.5x 2.0x -- which has really saved my ass, cause I don't have patience for slow talkers. (Conversely, shift+< slows the video back down)
Yeah, definitely the best book I've read on differential forms was Spivaks Calculus on Manifolds. Its very readable once you have a solid foundational calculus background and is pretty small given what it covers (160pp). If you need to know this stuff then this is definitely the right place to learn it.
I recommend you start studying proofs first. How to Prove It by Velleman is a great book for new math students. I went through the first three chapters myself before my first analysis course, and it made all the difference.
As you are taking a class than combines analysis and calculus, you might benefit from Spivak's book Calculus, which despite it's title, is precisely a combination of calculus and real analysis.
You need some grounding in foundational topics like Propositional Logic, Proofs, Sets and Functions for higher math. If you've seen some of that in your Discrete Math class, you can jump straight into Abstract Algebra, Rigorous Linear Algebra (if you know some LA) and even Real Analysis. If thats not the case, the most expository and clearly written book on the above topics I have ever seen is Learning to Reason: An Introduction to Logic, Sets, and Relations by Nancy Rodgers.
Some user friendly books on Real Analysis:
Some user friendly books on Linear/Abstract Algebra:
Topology(even high school students can manage the first two titles):
Some transitional books:
Plus many more- just scour your local library and the internet.
Good Luck, Dude/Dudette.
Intro Calculus, in American sense, could as well be renamed "Physics 101" or some such since it's not a very mathematical course. Since Intro Calculus won't teach you how to think you're gonna need a book like How to Solve Word Problems in Calculus by Eugene Don and Benay Don pretty soon.
Aside from that, try these:
Excursions In Calculus by Robert Young.
Calculus:A Liberal Art by William McGowen Priestley.
Calculus for the Ambitious by T. W. KORNER.
Calculus: Concepts and Methods by Ken Binmore and Joan Davies
You can also start with "Calculus proper" = Analysis. The Bible of not-quite-analysis is:
[Calculus by Michael Spivak] (http://www.amazon.com/Calculus-4th-Michael-Spivak/dp/0914098918/ref=sr_1_1?s=books&amp;ie=UTF8&amp;qid=1413311074&amp;sr=1-1&amp;keywords=spivak+calculus).
Also, Analysis is all about inequalities as opposed to Algebra(identities), so you want to be familiar with them:
Introduction to Inequalities by Edwin F. Beckenbach, R. Bellman.
Analytic Inequalities by Nicholas D. Kazarinoff.
As for Linear Algebra, this subject is all over the place. There is about a million books of all levels written every year on this subject, many of which is trash.
My plan would go like this:
1. Learn the geometry of LA and how to prove things in LA:
Linear Algebra Through Geometry by Thomas Banchoff and John Wermer.
Linear Algebra, Third Edition: Algorithms, Applications, and Techniques
by Richard Bronson and Gabriel B. Costa.
2. Getting a bit more sophisticated:
Linear Algebra Done Right by Sheldon Axler.
Linear Algebra: An Introduction to Abstract Mathematics by Robert J. Valenza.
Linear Algebra Done Wrong by Sergei Treil.
3. Turn into the LinAl's 1% :)
Advanced Linear Algebra by Steven Roman.
Good Luck.
>When university starts, what can I do to ensure that I can compete and am just as good as the best mathematics students?
Read textbooks for mathematics students.
For example for Linear Algebra I heard that Axler's book is very good (I studied Linear Algebra in another language, so I can't really suggest anything from personal experience). For Calculus I personally suggest Spivak's book.
There are many books that I could suggest, but one of the greatest books I've ever read is The Art and Craft of Problem Solving.
Read this book: How To Prove it
You need a good foundation: a little logic, intro to proofs, a taste of sets, a bit on relations and functions, some counting(combinatorics/graph theory) etc. The best way to get started with all this is an introductory discrete math course. Check these books out:
Mathematics: A Discrete Introduction by Edward A. Scheinerman
Discrete Mathematics with Applications by Susanna S. Epp
How to Prove It: A Structured Approach Daniel J. Velleman
Learning to Reason: An Introduction to Logic, Sets, and Relations by Nancy Rodgers
Combinatorics: A Guided Tour by David R. Mazur
Since nobody else has recommended it, I always recommend the book How to Prove it by Daniel J. Velleman for learning proofs. I always found proofs to be kind of black magic until I read that, which totally demystified them for me by revealing the structure of proofs and techniques for proving different kinds of statements. One of the best things about it is that it starts from square one with basic logic and builds from there in way that no prior knowledge is required beyond basic algebra skills.
You cannot go wrong with How To Prove It: A Structured Approach by Velleman https://www.amazon.com/How-Prove-Structured-Approach-2nd/dp/0521675995/ref=sr_1_3?keywords=how+to+prove+it&amp;qid=1558195901&amp;s=gateway&amp;sr=8-3
I saw that book highly recommended, and after going through it myself a while ago I highly recommend it as well. When I do proofs I still maintain the mental model and use some of the mechanics that I learned from this book. You don't even have to read the whole thing in my opinion. Pick it up, work through a few pages per day, and stop when you feel like moving onto another subject-specific book, like Understanding Analysis.
Oh, and you might already know this, but do as many practice problems as you can! Learning proofs is all about practice.
https://www.amazon.com/dp/0521675995/ref=cm_sw_r_other_apa_Sn9NBbDH6MYPX not sure it this is exactly what you're asking for(might be more than you're asking for?) but this helped me a lot.
tl;dr: you need to learn proofs to read most math books, but if nothing else there's a book at the bottom of this post that you can probably dive into with nothing beyond basic calculus skills.
Are you proficient in reading and writing proofs?
If you aren't, this is the single biggest skill that you need to learn (and, strangely, a skill that gets almost no attention in school unless you seek it out as an undergraduate). There are books devoted to developing this skill—How to Prove It is one.
After you've learned about proof (or while you're still learning about it), you can cut your teeth on some basic real analysis. Basic Elements of Real Analysis by Protter is a book that I'm familiar with, but there are tons of others. Ask around.
You don't have to start with analysis; you could start with algebra (Algebra and Geometry by Beardon is a nice little book I stumbled upon) or discrete (sorry, don't know any books to recommend), or something else. Topology probably requires at least a little familiarity with analysis, though.
The other thing to realize is that math books at upper-level undergraduate and beyond are usually terse and leave a lot to the reader (Rudin is famous for this). You should expect to have to sit down with pencil and paper and fill in gaps in explanations and proofs in order to keep up. This is in contrast to high-school/freshman/sophomore-style books like Stewart's Calculus where everything is spelled out on glossy pages with color pictures (and where proofs are mostly absent).
And just because: Visual Complex Analysis is a really great book. Complex numbers, functions and calculus with complex numbers, connections to geometry, non-Euclidean geometry, and more. Lots of explanation, and you don't really need to know how to do proofs.
I actually bought http://www.amazon.com/Book-Abstract-Algebra-Second-Mathematics/dp/0486474178
On chapter three now :D
Have you read A Book of Abstract Algebra by Charles Pinter? https://www.amazon.co.uk/d/cka/Abstract-Algebra-Dover-Books-Mathematics-Charles-Pinter/0486474178
Hrrumph. Determinants are a capstone, not a cornerstone, of Linear Algebra.
https://www.amazon.com/Linear-Algebra-Right-Undergraduate-Mathematics/dp/0387982582
I studied with this book on abstract. It's authoritative and brutal.
It's hard to give an objective answer, because any sufficiently advanced book will be bound to not appeal to everyone.
You probably want Daddy Rudin for real analysis and Dummit & Foote for general abstract algebra.
Mac Lane for category theory, of course.
I think people would agree on Hartshorne as the algebraic geometry reference.
Spanier used to be the definitive algebraic topology reference. It's hard to actually use it as a reference because of the density and generality with which it's written.
Spivak for differential geometry.
Rotman is the group theory book for people who like group theory.
As a physics person, I must have a copy of Fulton & Harris.
It's common to have some difficulty adjusting from lower-level courses with a computational emphasis to upper-level courses with an emphasis on proof. Fortunately, this phenomenon is well known, and there are a number of books aimed at bridging the gap between the two types of courses. A few such books are listed below.
I hope that helps!
It's probably not possible to review everything you need, but getting more experience with proofs is a good start. This course might be helpful:
https://www.coursera.org/course/matrix
and these texts are great examples of mathematical thinking in prose:
Grinstead and Snell's Introduction to Probability:
https://math.dartmouth.edu/~prob/prob/prob.pdf
Apostol's Calculus I and II:
http://www.amazon.com/Calculus-Vol-One-Variable-Introduction-Algebra/dp/0471000051
If you're doing both applied and pure abstract algebra rather than elementary algebra, then you'll probably need to learn to write proofs for the pure side. I recommend Numbers, Groups, and Codes by J. F. Humphreys for an introduction to the basics and to some applied abstract algebra. If you need more work on proofs, the free Book of Proofs can help, and Fraleigh's A First Course in Abstract Algebra is a good book for pure abstract algebra. If you want something more advanced, I recommend the massive Abstract Algebra by Dummit and Foote.
Learn Algebra, arguably the most important math subject (of course, I may be biased). Dummit and Foote is a fantastic intro if you have proof experience etc.
Textbooks (calculus): Fundamentals of Physics: http://www.amazon.com/Fundamentals-Physics-Extended-David-Halliday/dp/0470469080/ref=sr_1_4?ie=UTF8&amp;qid=1398087387&amp;sr=8-4&amp;keywords=fundamentals+of+physics ,
Textbooks (calculus): University Physics with Modern Physics; http://www.amazon.com/University-Physics-Modern-12th-Edition/dp/0321501217/ref=sr_1_2?ie=UTF8&amp;qid=1398087411&amp;sr=8-2&amp;keywords=university+physics+with+modern+physics
Textbook (algebra): [This is a great one if you don't know anything and want a book to self study from, after you finish this you can begin a calculus physics book like those listed above]: http://www.amazon.com/Physics-Principles-Applications-7th-Edition/dp/0321625927/ref=sr_1_1?ie=UTF8&amp;qid=1398087498&amp;sr=8-1&amp;keywords=physics+giancoli
If you want to be competitive at the international level, you definitely need calculus, at least the basics of it.
Here is a good book: http://www.amazon.com/Calculus-Intuitive-Physical-Approach-Mathematics/dp/0486404536/ref=sr_1_1?ie=UTF8&amp;qid=1398087834&amp;sr=8-1&amp;keywords=calculus+kline
It is quite cheap and easy to understand if you want to self teach yourself calculus.
Another option would be this book:http://www.amazon.com/Calculus-4th-Michael-Spivak/dp/0914098918/ref=sr_1_1?ie=UTF8&amp;qid=1398087878&amp;sr=8-1&amp;keywords=spivak
If you can finish self teaching that to yourself, you will be ready for anything that could face you in mathematics in university or the IPhO. (However it is a difficult book)
Problem books: Irodov; http://www.amazon.com/Problems-General-Physics-I-E-Irodov/dp/8183552153/ref=sr_1_1?ie=UTF8&amp;qid=1398087565&amp;sr=8-1&amp;keywords=irodov ,
Problem Books: Krotov; http://www.amazon.com/Science-Everyone-Aptitude-Problems-Physics/dp/8123904886/ref=sr_1_1?ie=UTF8&amp;qid=1398087579&amp;sr=8-1&amp;keywords=krotov
You should look for problem sets online after you have finished your textbook, those are the best recourses. You can get past contests from the physics olympiad websites.
There are a lot of good classics on /u/thebenson's list. I want to highlight the books that are good for what you'll be learning, and give you a sense of how the sequence works. And I'll add a few.
Calculus books: Thomas' Calculus, Calculus by James Stewart (not multivariable), and this cheap easy read by Morris Kline.
Have you learned calculus in the past? It sounds like you'll need it for at least one of those courses, but either way, it will definitely help you conceptually for the others. You should really try to get solid on this before you need to use it.
Intro physics books: Fundamentals of Physics (Halliday & Resnick), Physics for Scientists and Engineers (Serway & Jewett), Physics for Scientists and Engineers (Tipler & Mosca), University Physics (Young), and Physics for Scientists and Engineers (Knight) are all good. Gee, they get really unoriginal with the names, huh?
Each of these books assumes no background in physics, but you do need to use calculus. If you're going to take a class in basic mechanics that doesn't involve any calculus, you may find it more useful to get a book at that level. The only such book that I'm familiar with is Physics: Principles with Applications by Giancoli. I know there are many others, but I can't speak for them.
Mathematical methods: Greenberg is way more than you need here. I think you would find Engineering Mathematics by Stroud & Booth more useful as a reference, since it covers a lot of the less advanced stuff that you may need a refresher on.
Sequence: it's typical to start learning physics by learning about Newtonian mechanics, with or without calculus. After that, one often goes on to thermodynamics or to electricity and magnetism. It sounds like this is roughly how your program is going to work.
If you are learning mechanics with calculus, you can expect E&M to be even heavier on the calculus and thermodynamics to be less so. More calculus is not a bad thing. People often get scared of it, but it actually makes things easier to understand.
It is very typical that you will use only one book (from the intro books above) for all of these topics. You shouldn't need to get any books on specific topics.
**
The other books on /u/thebenson's list are all great textbooks, but I think you should avoid them for now. They generally assume a healthy background in basic physics, and they may not be very relevant to the physics you'll be studying.
But I do want to give some mention to Spacetime Physics* by Taylor and Wheeler, since I don't want to imply that this is a background-heavy book. On the contrary, this is one of the most beginner-friendly physics books ever written, and it is my favorite introduction to special relativity. Special relativity is probably not something you need to learn about right now, but if you have any interest, I seriously recommend finding an old used copy of this book—it's a fun read aside from any other uses!
I'm going to shamelessly plug this book which I consider to be one of my favorite books ever. For the price it is definitely worth keeping a copy and reading it on the side if you're learning abstract algebra for the first time and it reads like a novel. It's definitely a small treasure I feel I discovered.
A book of abstract algebra by Charles Pinter is the best math book I've ever read in terms of readability, I think. The first chapter is an essay on the history of algebra and the book is worth it just for this chapter.
Try Pinter. If you think it is too simple for you go for Aluffi.
These were the most enlightening for me on their subjects:
This should keep you busy, but I can suggest books in other areas if you want.
Math books:
Algebra: http://www.amazon.com/Algebra-I-M-Gelfand/dp/0817636773/ref=sr_1_1?ie=UTF8&amp;s=books&amp;qid=1251516690&amp;sr=8
Calc: http://www.amazon.com/Calculus-4th-Michael-Spivak/dp/0914098918/ref=sr_1_1?s=books&amp;ie=UTF8&amp;qid=1356152827&amp;sr=1-1&amp;keywords=spivak+calculus
Calc: http://www.amazon.com/Linear-Algebra-Dover-Books-Mathematics/dp/048663518X
Linear algebra: http://www.amazon.com/Linear-Algebra-Modern-Introduction-CD-ROM/dp/0534998453/ref=sr_1_4?ie=UTF8&amp;s=books&amp;qid=1255703167&amp;sr=8-4
Linear algebra: http://www.amazon.com/Linear-Algebra-Dover-Mathematics-ebook/dp/B00A73IXRC/ref=zg_bs_158739011_2
Beginning physics:
http://www.amazon.com/Feynman-Lectures-Physics-boxed-set/dp/0465023827
Advanced stuff, if you make it through the beginning books:
E&M: http://www.amazon.com/Introduction-Electrodynamics-Edition-David-Griffiths/dp/0321856562/ref=sr_1_1?ie=UTF8&amp;qid=1375653392&amp;sr=8-1&amp;keywords=griffiths+electrodynamics
Mechanics: http://www.amazon.com/Classical-Dynamics-Particles-Systems-Thornton/dp/0534408966/ref=sr_1_1?ie=UTF8&amp;qid=1375653415&amp;sr=8-1&amp;keywords=marion+thornton
Quantum: http://www.amazon.com/Principles-Quantum-Mechanics-2nd-Edition/dp/0306447908/ref=sr_1_1?ie=UTF8&amp;qid=1375653438&amp;sr=8-1&amp;keywords=shankar
Cosmology -- these are both low level and low math, and you can probably handle them now:
http://www.amazon.com/Spacetime-Physics-Edwin-F-Taylor/dp/0716723271
http://www.amazon.com/The-First-Three-Minutes-Universe/dp/0465024378/ref=sr_1_1?ie=UTF8&amp;qid=1356155850&amp;sr=8-1&amp;keywords=the+first+three+minutes
Learn math first. Physics is essentially applied math with experiments. Start with Calculus then Linear Algebra then Real Analysis then Complex Analysis then Ordinary Differential Equations then Partial Differential Equations then Functional Analysis. Also, if you want to pursue high energy physics and/or cosmology, Differential Geometry is also essential. Make sure you do (almost) all the exercises in every chapter. Don't just skim and memorize.
This is a lot of math to learn, but if you are determined enough you can probably master Calculus to Real Analysis, and that will give you a big head start and a deeper understanding of university-level physics.
If you are serious about this, then the best way to self learn Math is to learn to read Math books. This is a valuable skill. It stops you from having to rely on websites/tutorials and frees you to really read the stuff you're interested in.
Generally you probably want a more "back to basics" approach that will cover basic stuff and act as an introduction (again) to the topic (without handling you as if you're a child). I recommend Discrete Mathematics with Applications. Epp does a good job of starting at the beginning (with logic) and building a decent foundation through connectives, conditionals, existentials, universals, etc. eventually leading into proofs.
Her writing style is very readable IMO but still dense enough to help you learn how to read Math books.
If you're self motivated enough then start there. Read a chapter. Do the problems. Be confused. Do more problems. Still confused? Read the chapter again. Do more problems. Repeat. Eventually finish the book.
The next one will be faster and easier because of the work you put in. Eventually you'll be 3-5 books down, and you'll feel you know quite a bit. Then read more.. realize the field is huge and you know nothing. Read more to solve this.
Repeat
????
Success(?)
How to prove it is a great start. I think after that, you should focus on learning to think mathematically through practice instead of reading (at least, that's how I and most people learn best). Take classes or read and work through the textbooks of subjects that interest you. Discrete math would be a good place to start since it teaches proof techniques and basic probability and combinatorics; my class used this book which I thought was nice.
If you don't actually do the work, your thinking process isn't going to change.
Check out /r/compsci, /r/algorithms, and the subreddits in their sidebars.
Sure! There is a lot of math involved in the WHY component of Computer Science, for the basics, its Discrete Mathematics, so any introduction to that will help as well.
http://www.amazon.com/Discrete-Mathematics-Applications-Susanna-Epp/dp/0495391328/ref=sr_sp-atf_title_1_1?s=books&amp;ie=UTF8&amp;qid=1368125024&amp;sr=1-1&amp;keywords=discrete+mathematics
This next book is a great theoretical overview of CS as well.
http://mitpress.mit.edu/sicp/full-text/book/book.html
That's a great book on computer programming, complexity, data types etc... If you want to get into more detail, check out: http://www.amazon.com/Introduction-Theory-Computation-Michael-Sipser/dp/0534950973
I would also look at Coursera.org's Algorithm lectures by Robert Sedgewick, thats essential learning for any computer science student.
His textbook: http://www.amazon.com/Algorithms-4th-Robert-Sedgewick/dp/032157351X/ref=sr_sp-atf_title_1_1?s=books&amp;ie=UTF8&amp;qid=1368124871&amp;sr=1-1&amp;keywords=Algorithms
another Algorithms textbook bible: http://www.amazon.com/Introduction-Algorithms-Thomas-H-Cormen/dp/0262033844/ref=sr_sp-atf_title_1_2?s=books&amp;ie=UTF8&amp;qid=1368124871&amp;sr=1-2&amp;keywords=Algorithms
I'm just like you as well, I'm pivoting, I graduated law school specializing in technology law and patents in 2012, but I love comp sci too much, so i went back into school for Comp Sci + jumped into the tech field and got a job at a tech company.
These books are theoretical, and they help you understand why you should use x versus y, those kind of things are essential, especially on larger applications (like Google's PageRank algorithm). Once you know the theoretical info, applying it is just a matter of picking the right tool, like Ruby on Rails, or .NET, Java etc...
Discrete Mathematics with Applications by Susanna Epp is pretty good, with a lot of exposition. In the introduction there is a guide on how to use the book, and the different sections to focus on if using it for a mainly mathematics-based class or for a computer science-based class.
http://www.amazon.com/gp/product/0495391328/ref=pd_lpo_sbs_dp_ss_2?pf_rd_p=1944687702&amp;pf_rd_s=lpo-top-stripe-1&amp;pf_rd_t=201&amp;pf_rd_i=0132122715&amp;pf_rd_m=ATVPDKIKX0DER&amp;pf_rd_r=0P155N9Y802PPMETYGVC
How To Prove It
once you get into partial differential equations, you'll be able to understand them. the basic ideas are pretty simple. there's just a bunch of computational overhead
this is a great book: https://www.amazon.com/Nonlinear-Dynamics-Chaos-Applications-Nonlinearity/dp/0813349109/ref=dp_ob_title_bk
it's informal and pretty easy to read. I don't remember it being so expensive though. i could've sworn i paid $20 for it
My suggestion: https://www.amazon.com/Nonlinear-Dynamics-Chaos-Applications-Nonlinearity/dp/0813349109/ref=sr_1_1?ie=UTF8&amp;qid=1494109595&amp;sr=8-1&amp;keywords=strogatz+nonlinear+dynamics+and+chaos
Should be approachable for someone who has only completed Calc I.
I'm not sure about PDE's, but ODE's are more than just existence and uniqueness theorems. You could argue that the modern study of ODE's is now dynamical systems.
Strogatz's Nonlinear Dynamics and Chaos is a classic if you want to know what applied dynamical systems is like. A more formal text that still captures some interesting ideas is Hale and Kocak's Dynamics and Bifurcations.
Reading textbooks is, of course, a huge time commitment. So perhaps go talk to the dynamical systems people in your department and ask them what is interesting about ODE's. Hell, even go talk to the numerical analysis and do the same for PDE's. Assuming you haven't taken a numerical analysis class, you might be surprised how "pure" numerical analysis feels.
As with Michael Spivak's Calculus, Apostol's two-volume Calculus is much, much more proof-centric than your introduction to calculus has been until now. That will make the material challenging but really rewarding, too.
Since you've had three semesters of calculus, I'm confident you likely have the relevant calculus background. I'd be more interested, though, in your background in reading, understanding, and writing proofs. Have you taken any such proof-based courses? In many American university curricula, for example, this is often introduced in a class like discrete mathematics. You'll likely have to be comfortable with the following, for example:
This is just off the top of my head, but don't worry if the above list seems intimidating. What you don't already know, but will need, should be included in the text itself.
Self-study is great, and I applaud your ambition in choosing this text. From my experience, you'll likely be even more successful if you can find someone to join you in self-study, especially if you don't have a teacher or professor to guide you through what will inevitably include very new material. If you can't find someone local with whom you can study together in person, then the internet may be a good way to find a fellow study partner. If you know (or are learning) LaTeX, then Overleaf or similar tools can be really useful for sharing math that's actually legible.
I'd add one other remark: if memory serves, Apostol introduces integration before differentiation, something that I believe is uncommon. Since you're already familiar with both integration and differentiation, that will likely matter less for you.
Good luck with your project!
Have I got the book for you, op
https://www.amazon.com/dp/0521675995/ref=cm_sw_r_cp_apa_i_2EmtDbDDEMPG9
Read and work through this book: http://www.amazon.com/How-Prove-Structured-Daniel-Velleman/dp/0521675995/ref=sr_1_1?ie=UTF8&amp;qid=1301319337&amp;sr=8-1
How to Prove It: A Structured Approach by Velleman is good for developing general proof writing skills.
How to Think About Analysis by Lara Alcock beautifully deconstructs all the major points of Analysis(proofs included).
I think everyone is on point for the most part, but I'd like to be the devil's advocate and suggest a different route.
Learn logic, proof techniques and set theory as early as possible. It will aid you in further study of all 'types' of math and broaden your mind in a general sense. This book is a perfect place to start.
http://www.amazon.com/How-Prove-It-Structured-Approach/dp/0521675995
The best part is, when you start doing proofs you realize you've been thinking about math all wrong (at least I did). It's an exercise in creativity, not calculation.
In my mind, set theory & calculus are necessary pre-requisites to probability anyway, and linear algebra means much more once you have been introduced to inductive proofs, as well.
Perhaps rather than concentrating on these particular proofs you should look at something like How To Prove It.
If your intent is to take a class like analysis, you really should look into something like logic.
Daniel Velleman wrote an excellent little book called How to Prove It: A Structured Approach. It's actually designed for High School level students, but it works through the subject incredibly well.
Here's an Amazon link to the book:
http://www.amazon.com/How-Prove-It-Structured-Approach/dp/0521675995/ref=sr_1_1?s=books&amp;ie=UTF8&amp;qid=1333383091&amp;sr=1-1
Congrats! And sorry about the DNF.
My opinion (for whatever its worth i guess), if your right on the edge of cut off times then you have to look at 3 things: age, weight, time spent training.
Unfortunately not much we can do about age, at a certain point no one is finishing a half ironman. I assume that you are not at that age yet.
Weight is probably the hardest thing to adjust. You can't out run a bad diet. So knowing nothing about your weight, are you satisfied with your weight or do you think that there is room for improvement?
Time spent training is the easy stuff! Woooo! More specifically, effective training and an effective training plan is probably your biggest gap. I (and others) suggest a book called The Triathlete's Training Bible by Joel Friel. This gets into how to spend your time to be more effectively training with self guided training plans etc etc. If you give more information about what you did to train for this specific event then maybe we could have more in-depth conversation about what you should be doing.
https://www.amazon.com/Triathletes-Training-Bible-Joe-Friel/dp/1934030198/ref=sr_1_2?ie=UTF8&amp;qid=1491248736&amp;sr=8-2&amp;keywords=triathletes+training+bible
I used Joe Friel's Going Long: Training for Triathlon's Ultimate Challenge and Joe Friel's The Triathlete's training Bible Very in-depth books on how to set up a training plan and schedule your time.
I recommend reading the Triathlete's Training Bible (http://www.amazon.com/The-Triathletes-Training-Bible-Friel/dp/1934030198) which quite extensively covers the base training period.
If I recall correctly, he speaks about doing lots of leg and core strength training, swimming drills concentrating heavily on technique, hill repeats on the treadmill, etc... Things that would serve as a good base for other training later on.
Yes!
If you don't know any calculus Stewart Calculus is the typical primer in colleges. Combine this with Khan Academy for easy mode cruise control.
After that, you want to look at The Big Orange Book, which is essentially the bible for undergrad astrophysics and 100% useful beyond that. This book could, alone, tell you everything you need to know.
As for other topics like differential equations and linear algebra you can shop around. I liked Linear Algebra Done Right for linear personally. No recommendations from me on differential equations though, never found a book that I loved.
Here's a couple of books I'd recommend.
amazon
amazon
You might check out the Minneapolis area for a tri club. I'm certain there is a good one up there. Some clubs have New Triathlete programs that can be really good.
There's really no easy way to do it without getting yourself "in the shit", in my opinion. Take a course on multivariate calculus/analysis, or else teach yourself. Work through the proofs in the exercises.
For a somewhat grounded and practical introduction I recommend Multivariable Mathematics: Linear Algebra, Calculus and Manifolds by Theo Shifrin. It's a great reference as well. If you want to dig in to the theoretical beauty, James Munkres' Analysis on Manifolds is a bit of an easier read than the classic Spivak text. Munkres also wrote a book on topology which is full of elegant stuff; topology is one of my favourite subjects in mathematics,
By the way, I also came to mathematics through the study of things like neural networks and probabilistic models. I finally took an advanced calculus course in my last two semesters of undergrad and realized what I'd been missing; I doubt I'd have been intellectually mature enough to tackle it much earlier, though.
This might be of interest, Spivak's Calculus on Manifolds.
Strogatz Nonlinear Dynamics and Chaos covers phase space, phase portraits, and linear stability analysis in great detail with examples from many disciplines including physics. It's probably a good place to start, but I don't think it has very much that's specifically on turbulent fluids. For that, you'll probably want a more focused textbook. Hopefully, someone more knowledgeable can recommend one.
This is a compilation of what I gathered from reading on the internet about self-learning higher maths, I haven't come close to reading all this books or watching all this lectures, still I hope it helps you.
General Stuff:
The books here deal with large parts of mathematics and are good to guide you through it all, but I recommend supplementing them with other books.
Linear Algebra: An extremelly versatile branch of Mathematics that can be applied to almost anything, also the first "real math" class in most universities.
Calculus: The first mathematics course in most Colleges, deals with how functions change and has many applications, besides it's a doorway to Analysis.
Real Analysis: More formalized calculus and math in general, one of the building blocks of modern mathematics.
Abstract Algebra: One of the most important, and in my opinion fun, subjects in mathematics. Deals with algebraic structures, which are roughly sets with operations and properties of this operations.
There are many other beautiful fields in math full of online resources, like Number Theory and Combinatorics, that I would like to put recommendations here, but it is quite late where I live and I learned those in weirder ways (through olympiad classes and problems), so I don't think I can help you with them, still you should do some research on this sub to get good recommendations on this topics and use the General books as guides.
http://www.amazon.com/Calculus-4th-Michael-Spivak/dp/0914098918/ref=sr_1_1?ie=UTF8&amp;qid=1344481564&amp;sr=8-1&amp;keywords=spivaks+calculus
I haven't read all of it, but even the bit I did read was very challenging and it is generally recommended around here for a rigorous introduction to calculus. Be warned, it is pretty challenging, especially if you aren't comfortable with proofs.
A First Course in Graph Theory by Chartrand and Zhang
Combinatorics: A Guided Tour by Mazur
Discrete Math by Epp
For Linear Algebra I like these below:
Lecture Notes by Tao
Linear Algebra: An Introduction to Abstract Mathematics by Robert Valenza
Linear Algebra Done Right by Axler
Linear Algebra by Friedberg, Insel and Spence
A graph theory project! I just started today (it was assigned on Friday and this is when I selected my topic). I’m on spring break but next month I have to present a 15-20 minute lecture on graph automorphisms. I don’t necessarily have to, but I want to try and tie it in with some group theory since there is a mix of undergrads who the majority of them have seen some algebra before and probably bored PhD students/algebraists in my class, but I’m not sure where to start. Like, what would the binary operation be, composition of functions? What about the identity and inverse elements, what would those look like? In general, what would the elements of this group look like? What would the group isomorphism be? That means it’s a homomorphism with a bijective function. What would the homomorphism and bijective function look like? These are the questions I’m trying to get answers to.
Last semester I took a first course in Abstract Algebra and I’m currently taking a follow up course in Linear Algebra (I have the same professor for both algebra classes and my graph theory class). I’m curious if I can somehow also bring up some matrix representation theory stuff as that’s what we’re going over in my linear algebra class right now.
This is the textbook I’m using for my graph theory class: Graph Theory (Graduate Texts in Mathematics) https://www.amazon.com/dp/1846289696?ref=yo_pop_ma_swf
Here are the other graph theory books I got from my library and am using as references: Graph Theory (Graduate Texts in Mathematics) https://www.amazon.com/dp/3662536218?ref=yo_pop_ma_swf
Modern Graph Theory (Graduate Texts in Mathematics) https://www.amazon.com/dp/0387984887?ref=yo_pop_ma_swf
And for funsies, here is my linear algebra text: Linear Algebra, 4th Edition https://www.amazon.com/dp/0130084514?ref=yo_pop_ma_swf
But that’s what I’m working on! :)
And I certainly wouldn’t mind some pointers or ideas or things to investigate for this project! Like I said, I just started today (about 45 minutes ago) and am just trying to get some basic questions answered. From my preliminary investigating in my textbook, it seems a good example to work with in regards to a graph automorphism would be the Peterson Graph.
The course I took as an undergraduate used Friedberg, Insel and Spence. I remember liking it fine, but it's insultingly expensive. Find it in a library or get a used copy if you can. If you're looking for a bargain, it can't hurt to try Shilov. He's Russian, so the book is very terse, but covers a lot of ground.
Hey! I am a math major at Harvey Mudd College (who went to high school in the Pacific NW!). I'll answer from what I've seen.
End: Also, if you wanna learn something cool, I'd check out Discrete math. It's usually required for both a math or CS major, and it's some of the coolest undergraduate math out there. Oh, and, unlike some other math, it's not terrible to self-teach. :)
Good luck! Math is awesome!
Apostol's classic calculus textbook, used at Caltech and MIT. The Art of Problem Solving textbook for calculus. The Stanford and Harvard-MIT Math Tournaments have calculus subject tests. The college-level Putnam competition has calculus problems, in addition to linear algebra, abstract algebra, etc.
Start With "Foundations Of Analysis" By Edmund Landau
http://www.amazon.com/Foundations-Analysis-AMS-Chelsea-Publishing/dp/082182693X
It's a tiny book, but is very good at explaining basic abstract algebra.
Here is the description from Amazon:
"Why does $2 \times 2 = 4$? What are fractions? Imaginary numbers? Why do the laws of algebra hold? And how do we prove these laws? What are the properties of the numbers on which the Differential and Integral Calculus is based? In other words, What are numbers? And why do they have the properties we attribute to them? Thanks to the genius of Dedekind, Cantor, Peano, Frege and Russell, such questions can now be given a satisfactory answer. This English translation of Landau's famous Grundlagen der Analysis-also available from the AMS-answers these important questions."
With the above book you should then have enough knowledge to move on to calculus.
I recommend the two volume series called "Calculus" by Tom M. Apostol.
The first volume is single variable calculus and the second is multivariate calculus
http://www.amazon.com/Calculus-Vol-One-Variable-Introduction-Algebra/dp/0471000051/ref=sr_1_4?ie=UTF8&amp;s=books&amp;qid=1239384587&amp;sr=1-4
http://www.amazon.com/Calculus-Vol-Multi-Variable-Algebra-Applications/dp/0471000078/ref=sr_1_3?ie=UTF8&amp;s=books&amp;qid=1239384587&amp;sr=1-3
Calculus: An Intuitive and Physical Approach (Second Edition) (Dover Books on Mathematics) https://www.amazon.com/dp/0486404536/ref=cm_sw_r_cp_apip_qmMduBiBKxeqD
This book currently. I learned precalculus using Kahn academy over the year along with trig.
Definitely agree with the people recommending Calculus Made Easy by Silvanus P. Thompson. Often imitated, never equalled.
Another book similar to that is The Calculus for the Practical Man by J.E. Thompson. Besides its fame for being the book that Richard Feynman used to teach himself calculus, it has a completely nonstandard proof that the derivative of sin(x) is cos(x), using an argument based on arc length, which I haven't seen in any other book.
For more modern books I'd recommend Kline's book, which is underrated in my opinion. I'd avoid Spivak's book, which I feel is vastly overrated; it makes calculus even drier than the standard books do.
/u/LengthContracted this is a good book, as is Daphne Kollers book on PGMs as well as the associated course http://pgm.stanford.edu
A sample of what is on my reference shelf includes:
Real and Complex Analysis by Rudin
Functional Analysis by Rudin
A Book of Abstract Algebra by Pinter
General Topology by Willard
Machine Learning: A Probabilistic Perspective by Murphy
Bayesian Data Analysis Gelman
Probabilistic Graphical Models by Koller
Convex Optimization by Boyd
Combinatorial Optimization by Papadimitriou
An Introduction to Statistical Learning by James, Hastie, et al.
The Elements of Statistical Learning by Hastie, et al.
Statistical Decision Theory by Liese, et al.
Statistical Decision Theory and Bayesian Analysis by Berger
I will avoid listing off the entirety of my shelf, much of it is applications and algorithms for fast computation rather than theory anyway. Most of those books, though, are fairly well known and should provide a good background and reference for a good deal of the mathematics you should come across. Having a solid understanding of the measure theoretic underpinnings of probability and statistics will do you a great deal--as will a solid facility with linear algebra and matrix / tensor calculus. Oh, right, a book on that isn't a bad idea either... This one is short and extends from your vector classes
Tensor Calculus by Synge
Anyway, hope that helps.
Yet another lonely data scientist,
Tim.
http://www.amazon.com/Book-Abstract-Algebra-Edition-Mathematics/dp/0486474178/ref=sr_1_1?ie=UTF8&amp;qid=1394386195&amp;sr=8-1&amp;keywords=pinter+abstract+algebra)
We used the Dover textbook by Pinter. It's my favorite math textbook ever, the writing was just so clear, and even entertaining and funny. We had a good professor too.
Data Structures & Algorithms is usually the second course after Programming 101. Here is a progression (with the books I'd use) I would recommend to get started:
Edit: If you're feeling adventurous then after those you should look at
Sorry, the solution is to do lots of proofs.
There's more to it, but honestly it's more of a thing that you have to read a book about rather than a message on reddit. How are you learning about this right now? Is it part of a course or self-study? I personally found How to Prove It to be a very useful textbook. Doesn't require any particular knowledge, and it builds out a nice foundation in logic and set theory.
How To Prove It. Read through the reviews. It's the best book for learning propositional and predicate logic for the first time.
You will first want to learn fundamental logic and set theory before diving into topics like analysis, algebra, and discrete topics. You will need an understanding of a rigorous proof -- not the hand-wavey kind of proof we've seen in our introductory calculus courses. This book is very readable and will prepare you for advanced mathematics. I've seen it work for many students.
After you're finished with it, you may want to study analysis which will build up the Calculus for you. If you don't care for calculus anymore, consider reading an abstract algebra text. Algebra is pretty fun. You can also pick a discrete topic like graph theory or combinatorics whose applications are very easy to see.
There are many ways to go, but in all of them you will absolutely need a a basic understanding of the use of logic in a mathematical proof.
What this expressions says
First of all let's specify that the domain over which these statements operate is the set of all people say.
Let us give the two place predicate P(x,y) a concrete meaning. Let us say that P(x,y) signifies the relation x loves y.
This allows us to translate the statement:
∀x∀yP(x,y) -> ∀xP(x,x)
What does ∀x∀yP(x,y) mean?
This is saying that For all x, it is the case that For all y, x loves y.
So you can interpret it as saying something like everyone loves everyone.
What does ∀xP(x,x) mean?
This is saying that For all x it is the case that x loves x. So you can interpret this as saying something like everyone loves themselves.
So the statement is basically saying:
Given that it is the case that Everyone loves Everyone, this implies that everyone loves themselves.
This translation gives us the impression that the statement is true. But how to prove it?
Proof by contradiction
We can prove this statement with a technique called proof by contradiction. That is, let us assume that the conclusion is false, and show that this leads to a contradiction, which implies that the conclusion must be true.
So let's assume:
∀x∀yP(x,y) -> not ∀xP(x,x)
not ∀xP(x,x) is equivalent to ∃x not P(x,x).
In words this means It is not the case that For all x P(x,x) is true, is equivalent to saying there exists x such P(x,x) is false.
So let's instantiate this expression with something from the domain, let's call it a. Basically let's pick a person for whom we are saying a loves a is false.
not P(a,a)
Using the fact that ∀x∀yP(x,y) we can show a contradiction exists.
Let's instantiate the expression with the object a we have used previously (as a For all statement applies to all objects by definition) ∀x∀yP(x,y)
This happens in two stages:
First we instantiate y
∀xP(x,a)
Then we instantiate x
P(a,a)
The statements P(a,a) and not P(a,a) are contradictory, therefore we have shown that the statement:
∀x∀yP(x,y) -> not ∀xP(x,x) leads to a contradiction, which implies that
∀x∀yP(x,y) -> ∀xP(x,x) is true.
Hopefully that makes sense.
Recommended Resources
Wilfred Hodges - Logic
Peter Smith - An Introduction to Formal Logic
Chiswell and Hodges - Mathematical Logic
Velleman - How to Prove It
Solow - How to Read and Do Proofs
Chartand, Polimeni and Zhang - Mathematical Proofs: A Transition to Advanced Mathematics
For proof writing techniques I highly recommend Velleman's "How to Prove It" link
I'm going to recommend the book How to Prove It. Its all about learning the logic for proofs and strategies for writing proofs. Its one of those books that you work through slowly and complete all the exercises. Its recommended around here a-lot. I'd also suggest using the search feature if you ever want to look for other recommended books because those threads come up often around here.
Best wishes.
There are some really good books that you can use to give yourself a solid foundation for further self-study in mathematics. I've used them myself. The great thing about this type of book is that you can just do the exercises from one side of the book to the other and then be confident in the knowledge that you understand the material. It's nice! Here are my recommendations:
First off, three books on the basics of algebra, trigonometry, and functions and graphs. They're all by a guy called Israel Gelfand, and they're good: Algebra, Trigonometry, and Functions and Graphs.
Next, one of two books (they occupy the same niche, material-wise) on general proof and problem-solving methods. These get you in the headspace of constructing proofs, which is really good. As someone with a bachelors in math, it's disheartening to see that proofs are misunderstood and often disliked by students. The whole point of learning and understanding proofs (and reproducing them yourself) is so that you gain an understanding of the why of the problem under consideration, not just the how... Anyways, I'm rambling! Here they are: How To Prove It: A Structured Approach and How To Solve It.
And finally a book which is a little bit more terse than the others, but which serves to reinforce the key concepts: Basic Mathematics.
After that you have the basics needed to take on any math textbook you like really - beginning from the foundational subjects and working your way upwards, of course. For example, if you wanted to improve your linear algebra skills (e.g. suppose you wanted to learn a bit of machine learning) you could just study a textbook like Linear Algebra Done Right.
The hard part about this method is that it takes a lot of practice to get used to learning from a book. But that's also the upside of it because whenever you're studying it, you're really studying it. It's a pretty straightforward process (bar the moments of frustration, of course).
If you have any other questions about learning math, shoot me a PM. :)
So here are some options I recommend:
You can find all the textbooks I mentioned online, if you know what I mean. All of these assume you haven't seen math in a while, and they all start from the very basics. Take your time with the material, play around with it a bit, and enjoy your summer :D
EditL this article describes one way you can go about your studies
How to Prove It is only 20 bucks.
Hey mathit.
I'm 32, and just finished a 3 year full-time adult education school here in Germany to get the Abitur (SAT-level education) which allows me to study. I'm collecting my graduation certificate tomorrow, woooo!
Now, I'm going to study math in october and wanted to know what kind of extra prep you might recommend.
I'm currently reading How to Prove It and The Haskell Road to Logic, Maths and Programming.
Both overlap quite a bit, I think, only that the latter is more focused on executing proofs on a computer.
Now, I've just been looking into books that might ease the switch to uni-level math besides the 2 already mentioned and the most promising I found are these two:
How to Study for a Mathematics Degree and Bridging the Gap to University Mathematics.
Do you agree with my choices? What else do you recommend?
I found online courses to be ineffective, I prefer books.
What's your opinion, mathit?
Cheers and many thanks in advance!
Спасибо за ссылку. Я обязательно это проверю. Думаю, надо было быть более конкретным. Я читал книгу, которая учит своих читателей, как строить математические доказательства. В книге дается очень общий обзор этих тем, которые я перечислил выше. Я проверю ссылку, но если вы знаете книгу на русском языке, которая учит строить математические доказательства студентам, которые начинают изучать продвинутую математику, напишите Мне пожалуйсте.
Вот книга Для справки. (в случае, если вы знаете английского языка).
How to Prove it - A structured approach
I would recommend the book "How To Prove It".
https://www.amazon.com/How-Prove-Structured-Approach-2nd/dp/0521675995
It helped me in my transition into proof based mathematics. It will teach common techniques used in proofs and provides a bunch of practice problems as well.
You received A's in your math classes at a major public university, so I think you're in pretty good shape. That being said, have you done proof-based math? That may help tremendously in giving intuition because with proofs, you are giving rigor to all the logic/theorems/ formulas, etc that you've seen in your previous math classes.
Statistics will become very important in machine learning. So, a proof-based statistics book, that has been frequently recommended by /r/math and /r/statistics is Statistical Inference by Casella & Berger: https://www.amazon.com/Statistical-Inference-George-Casella/dp/0534243126
I've never read it myself, but skimming through some of the beginning chapters, it seems pretty solid. That being said, you should have an intro to proof-course if you haven't had that. A good book for starting proofs is How to Prove It: https://www.amazon.com/How-Prove-Structured-Approach-2nd/dp/0521675995
How to Prove It
It's cheap, highly rated, starts with the basics, and as the title says, shows you how to prove it!
“How to Prove it”. D. Velleman: Amazon US Link
Probably the best resource on the topic!
There are a couple of easy-ish sources on category theory that are good to have under your belt.
Category Theory for Programmers is available for free: https://github.com/hmemcpy/milewski-ctfp-pdf
It's not amazing, but it's good for programmers who want to start having basic intuitions about category theory.
Lawvere's Conceptual Mathematics is enjoyable and accessible
https://www.amazon.com/Conceptual-Mathematics-First-Introduction-Categories/dp/052171916X/ref=mp_s_a_1_1?keywords=conceptual+mathematics&amp;qid=1568389352&amp;s=gateway&amp;sr=8-1
To answer your general question: in my experience, your question is less about math and maybe more about chasing something you think has the answers. You'll meander as long as you feel like something is lacking.
I've seen this a lot with people who have massive textbook collections. A massive collection of textbooks is debt, and it provokes anxiety. You may have to figure out some squishy human stuff in addition to the technical math stuff.
People will give me flack for this but I think Stewart is a great text for an intro to calc, and moreover, one that a person with little math experience can feasibly use for self study. Obviously buying it new is expensive but I've heard rumors of PDF's flying around on torrent sites and stuff, and there's always a few used copies of it in like a 1 mile radius of wherever you are. Working through the first 8 chapters and maybe chapter 11 (infinite sequences and series) will give you a pretty thorough understanding of all of a first year calculus course, and the sections on multivariable calculus aren't bad either. Once you actually know some basics you'll want to find a more advanced text, but I find myself turning back to this text constantly when I need to remember how to do something basic that I've forgotten from first year.
Do the problems. You'll get stuck on lots of them. /r/learnmath is great for that—if you post a problem from this book up there you'll have a detailed answer in about 45 seconds. http://math.stackexchange.com is also great for that.
As for statistics, there's only so far you can go in traditional statistics without knowing any calculus. You can learn the extreme basics like descriptive statistics and basic probability, but at some point, probability theory requires that you know how to take a derivative or an integral, so you'll need to have those skills under your belt. So I'd start on Stewart's book and just try to work through it.
I am not sure that a pure math textbook is what you want. A lot of the problems that mathematicians think about may not be what you need. Let's take functional analysis for example. Most textbooks focus on bounded/ compact operators, and they only have one chapter at the end dedicated to unbounded operators. Unfortunately, the derivative (momentum) is an unbounded operator, so the part that has the least detail is what you need.
I would recommend a "math for physics students" book. A nice book that tries to paint the intuitive idea of most branches of math relevant to physics (and then some) and show you how to calculate is Goldbart and Stone's book, which they have made freely available online. This book assumes familiarity with linear algebra. If you are weak on this subject, I would highly recommend the book by Friedberg, Insel, and Spence. This is a more traditional math textbook, but it gets you very comfortable with the details of linear algebra (except for tensor products, but you should understand their construction with this background).
What's up, man. I failed geometry twice (sophomore and junior year) in high school. I barely graduated high school with a 2.0 gpa. I am now a senior studying math and computer science (going to be getting masters in math). I am at the top of my class, and I will be graduating with a ~3.91 GPA.
Math, just like anything else, is about practice and perseverance. I thought I sucked at math (and basically everyone told me I was more of an "english" kind of guy). But when I got to college, I found that I really enjoyed the challenge, and I found the material interesting as hell. So I worked my ass off at it.
If you work hard (some may need to work harder than others!) and persevere, then you will be fine. There will definitely be challenges, but that's what makes math so fun.
edit: Also, unless you are a math major, I can't imagine you will be getting into too much rigorous theory. You will likely continue mostly just be doing calculations (Calc 1, Calc 2 and Calc 3). That is how it is at my university, at least. However, if you are a math major, it can't hurt to get a head start on writing simple proofs. For that, I recommend the following book: https://www.amazon.com/Mathematical-Proofs-Transition-Advanced-Mathematics/dp/0321797094/ref=sr_1_4?s=books&amp;ie=UTF8&amp;qid=1474297774&amp;sr=1-4&amp;keywords=a+transition+to+advanced+mathematics
Seriously, that book is such a fucking good introductory text. It helped me so much.
How do you learn proofs? Do you just memorize them straight up? Can you prove simple things in Set Theory and Point Set Topology on your own? There are only so many techniques for proving things. You absolutely need to master them. After that all you have to remember is a few definitions/theorems/lemmas and an odd trick here and there.
It could also be notation/symbols constantly throwing you off.
Anyway, I like these 2 books below:
Mathematical Proofs by Gary Chartrand et al.
Mathematical Writing by Franco Vivaldi.
If you're dislike of linear algebra comes from using the determinant and matrix calculations, you would love Axler's Linear Algebra Done Right.
I learned lin. alg. from Axler's Linear Algebra Done Right. I found it extremely readable, with exercises that were not too hard to get through quickly.
I'd suggest Probability, Linear Algebra, Convex Optimization and ML in that order.
As for study materials, I'd suggest
That should keep you busy for a while.
Apostol's Calculus
http://www.amazon.com/Calculus-Vol-One-Variable-Introduction-Algebra/dp/0471000051
Calculus Made Easy -- Can't get much better as far as bang for the buck. Follow it up something more rigorous. Maybe, Calculus, Vol 1 by Apostol. The problem with Apostol, as most calculus texts, is price.
I'm soon starting my trek through every problem in the algebra text that Harvard's PhD prelim recommends for study:
Abstract Algebra by Dummit and Foote
I've started the first section of the first chapter, but that was only in a few hours of spare time. I'll be posting solutions by chapter soon and post my stories/insights on Hacker News. Here's section 1.1 (except the last problem, 36):
http://therobert.org/alg/1.1.pdf
Comments are appreciated. Better now than when I start the real journey. :)
Well, Hardy & Wright is the classic book for elementary stuff. It has almost everything there is to know. There is also a nice book by Melvyn Nathanson called Elementary Methods in Number Theory which I really like and would probably be my first recommendation. Beyond that, you need to decide which flavour you like. Algebraic and analytic are the big branches.
For algebraic number theory you'll need a solid grounding in commutative algebra and Galois theory - say at the level of Dummit and Foote. Lang's book is pretty classic, but maybe a tough first read. I might try Number Fields by Marcus.
For analytic number theory, I think Davenport is the best option, although Montgomery and Vaughan is also popular.
Finally, Serre (who is often deemed the best math author ever) has the classic Course in Arithmetic which contains a bit of everything.
If you are asking for classics, in Algebra, for example, there are(different levels of difficulty):
Basic Algebra by Jacobson
Algebra by Lang
Algebra by MacLane/Birkhoff
Algebra by Herstein
Algebra by Artin
etc
But there are other books that are "essential" to modern readers:
Chapter 0 by Aluffi
Basic Algebra by Knapp
Algebra by Dummit/Foot
I learned it out of Dummit and Foote originally, and I thought that was a pretty good book.
I know that in the long run competition math won't be relevant to graduate school, but I don't think it would hurt to acquire a broader background.
That said, are there any particular texts you would recommend? For Algebra, I've heard that Dummit and Foote and Artin are standard texts. For analysis, I've heard that Baby Rudin and also apparently the Stein-Shakarchi Princeton Lectures in Analysis series are standard texts.
If you are still an undergrad and your school offers a "how to prove stuff and how to think about abstract maths" course take it anyway. No matter how far along you have come.
An example text for such a course is this one:
https://www.amazon.com/Introduction-Mathematical-Reasoning-Numbers-Functions/dp/0521597188
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As for Linear Algebra (the most useful part of all higher mathematics for sure (R/math: if you disagree, fight me on this one...i'll win) ) I will tell you i learned a LOT from these two texts:
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https://www.amazon.com/Linear-Algebra-Introduction-Mathematics-Undergraduate/dp/0387940995
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https://www.amazon.com/Linear-Algebra-Right-Undergraduate-Mathematics/dp/3319110799/ref=pd_lpo_sbs_14_img_0?_encoding=UTF8&psc=1&refRID=APH3PQE76V9YXKWWGCR9
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This Book Is a great read. Explains every part of training and competing at your best.
You should also pick up a copy of The Triathlete's Training Bible. It's a great read with lots of good training & nutrition advice.
If it's too simple, stop wasting your time and start reading something more your speed. Say, Linear Algebra Done Right by Axler. If you're still unimpressed, try Advanced Linear Algebra by Roman. If you can solve most problems in this book cold, just drop everything you're doing now and walk straight into the nearest best grad school.
These are listed in the order I'd recommend reading them. Also, I've purposely recommended older editions since they're much cheaper and still as good as newer ones. If you want the latest edition of some book, you can search for that and get it.
The Humongous Book of Basic Math and Pre-Algebra Problems https://www.amazon.com/dp/1615640835/ref=cm_sw_r_cp_api_pHZdzbHARBT0A
Intermediate Algebra https://www.amazon.com/dp/0072934735/ref=cm_sw_r_cp_api_UIZdzbVD73KC9
College Algebra https://www.amazon.com/dp/0618643109/ref=cm_sw_r_cp_api_hKZdzb3TPRPH9
Trigonometry (2nd Edition) https://www.amazon.com/dp/032135690X/ref=cm_sw_r_cp_api_eLZdzbXGVGY6P
Reading this whole book from beginning to end will cover calculus 1, 2, and 3.
Calculus: Early Transcendental Functions https://www.amazon.com/dp/0073229733/ref=cm_sw_r_cp_api_PLZdzbW28XVBW
You can do LinAlg concurrently with calculus.
Linear Algebra: A Modern Introduction (Available 2011 Titles Enhanced Web Assign) https://www.amazon.com/dp/0538735457/ref=cm_sw_r_cp_api_dNZdzb7TPVBJJ
You can do this after calculus. Or you can also get a book that's specific to statistics (be sure to get the one requiring calc, as some are made for non-science/eng students and are pretty basic) and then another book specific to probability. This one combines the two.
Probability and Statistics for Engineering and the Sciences https://www.amazon.com/dp/1305251806/ref=cm_sw_r_cp_api_QXZdzb1J095Y1
Differential Equations with Boundary-Value Problems, 8th Edition https://www.amazon.com/dp/1111827060/ref=cm_sw_r_cp_api_sSZdzbDKD0TQ9
After doing all of the above, you'd have the equivalent most engineering majors have to take. You can go further by exploring partial diff EQs, real analysis (which is usually required by math majors for more advanced topics), and an intro to higher math which usually includes logic, set theory, and abstract algebra.
If you want to get into higher math topics you can use this fantastic book on the topic:
This book is also available for free online, but since you won't have internet here's the hard copy.
Book of Proof https://www.amazon.com/dp/0989472108/ref=cm_sw_r_cp_api_MUZdzbP64AWEW
From there you can go on to number theory, combinatorics, graph theory, numerical analysis, higher geometries, algorithms, more in depth in modern algebra, topology and so on. Good luck!
You should spend some lovely evenings with my friend, Stewart. If you find my friend Stewart too hard on you, take some exercises from my little friend Thomas! And if you want even more fun, my friend Piskunov has some lovely exercises for you!
And ask your questions here :-)
I like Algebra and Trigonometry by I.M. Gelfand. They are cheap books too.
I also have scans of them, PM me if you want to check them out.
Edit:
Also, Khan Academy is great resource for explanations. But I would recommend aiding Khan Academy with a text just for the problem set and solutions.
If it helps, here are some free books to go through:
Linear Algebra Done Wrong
Paul's Online Math Notes (fantastic for Calc 1, 2, and 3)
Basic Analysis
Basic Analysis is pretty basic, so I'd recommend going through Rudin's book afterwards, as it's generally considered to be among the best analysis books ever written. If the price tag is too high, you can get the same book much cheaper, although with crappier paper and softcover via methods of questionable legality. Also because Rudin is so popular, you can find solutions online.
If you want something better than online notes for univariate Calculus, get Spivak's Calculus, as it'll walk you through single-variable Calculus using more theory than a standard math class. If you're able to get through that and Rudin, you should be good to go once you get good at linear algebra.
https://www.amazon.com/Calculus-4th-Michael-Spivak/dp/0914098918
From my experience, Calculus in America is taught in 2 different ways: rigorous/mathematical in nature like Calculus by Spivak and applied/simplified like the one by Larson.
Looking at the link, I dont think you need to know sets and math induction unless you are about to start learning Rigorous Calculus or Real Analysis. Also, real numbers are usually introduced in Real Analysis that comes after one's exposure to Applied/Non-Rigorous Calculus. Complex numbers are, I assume, needed in Complex Analysis that follows Real Analysis, so I wouldn't worry about sets, real/complex numbers beyond the very basics. Math induction is not needed in non-proof based/regular/non-rigorous Calculus.
From the link:
For Calc 1(applied)- again, in my experience, this is the bulk of what's usually tested in Calculus placement exams:
Solving inequalities and equations
Properties of functions
Composite functions
Polynomial functions
Rational functions
Trigonometry
Trigonometric functions and their inverses
Trigonometric identities
Conic sections
Exponential functions
Logarithmic functions
For Calc 2(applied) - I think some Calc placement exams dont even contain problems related to the concepts below, but to be sure, you, probably, should know something about them:
Sequences and series
Binomial theorem
In Calc 2(leading up to multivariate Calculus (Calc 3)). You can pick these topics up while studying pre-calc, but they are typically re-introduced in Calc 2 again:
Vectors
Parametric equations
Polar coordinates
Matrices and determinants
As for limits, I dont think they are terribly important in pre-calc. I think, some pre-calc books include them just for good measure.
If you're on a budget, check out Calculus: An Intuitive and Physical Approach by Morris Kline.
:D
http://betterexplained.com/
http://www.amazon.com/Calculus-Intuitive-Physical-Approach-Mathematics/dp/0486404536/ref=sr_1_1?ie=UTF8&amp;qid=1422649729&amp;sr=8-1&amp;keywords=calculus+an+intuitive&amp;pebp=1422649747330&amp;peasin=486404536
The first site is fun, because it teaches you how to intuitively understand math. I love it. Second is a book that makes you think. Read the reviews for it. I really hope it helps because it's helped me, and I didn't even like math that much in the beginning, now I'm all excited for it :D
A Book of Abstract Algebra by Charles C. Pinter
I really enjoyed reading the book, almost reads like a novel. There is a great first chapter laying out the history of the subject and it just builds from there.
Math is essential the art pf careful reasoning and abstraction.
Do yes, definitely.
But it may be difficult at first, like training anything that’s not been worked.
Note: there are many varieties of math. I definitely recommend trying different ones.
A couple good books:
An Illustrated Theory of Numbers
Foolproof (first chapter is math history, but you can skip it to get to math)
A Book of Abstract Algebra
Also, formal logic is really fun, imk. And excellent st teaching solid thinking. I don’t know a good intro book, but I’m sure others do.
> I'm not sure what to read into before the Galois class begins.
"A Book of Abstract Algebra" by Charles Pinter
It depends on your interests. I thought the machine learning course on coursera was great. Antirez sometimes blogs about the internals of Redis on his blog, and he is a great writer. If you like math, this is the best math book I've read. Finally, you can always start contributing code to an open source project -- learn by doing!
Perhaps you might find Shilov's Linear Algebra or Roman's Advanced Linear Algebra to be useful. Both of them treat bilinear and quadratic forms.
I think Shilov does actually discuss Gram-Schmidt orthonormalization, but he doesn't call it that, and it seems to be spread over several sections in chapters 7 and 8. Roman might be better for that. Anyway, you can peruse both of these at libgen.
I don't think it contains any group theory, but everything else is there:
Discrete Mathematics with Applications by Susanna S. Epp (
This one below contains some algebra(groups):
Mathematics: A Discrete Introduction by Edward A. Scheinerman
Both are pretty elementary.
I have not taken the class yet (I'm taking 161 and 225 in January), but I looked at the syllabi already and here's the textbook for the class:
http://www.amazon.com/Discrete-Mathematics-Applications-Kenneth-Rosen/dp/0073383090/ref=sr_1_1?ie=UTF8&amp;qid=1417826968&amp;sr=8-1&amp;keywords=Rosen+Discrete+Math
You may want to go ahead and pick this up and start looking through it prior to January. I already grabbed a copy; I finish Calculus II tomorrow at my community college and I am going to be starting Rosen very soon.
This book is also commonly recommended:
http://www.amazon.com/Discrete-Mathematics-Applications-Susanna-Epp/dp/0495391328/ref=sr_1_1?ie=UTF8&amp;qid=1417827137&amp;sr=8-1&amp;keywords=Epp
I'm not sure what your math background is, but one of the most important success factors (in my experience) in math classes is a lot of practice. If you start working through either of those books now, you'll probably be in a good place once class starts in January.
We could also probably get a study group going on in here; I'm pretty comfortable with math, so I am happy to help out anyone else who needs help.
I used Susanna Epp's Discrete Mathematics text and rather enjoyed it. Velleman's How To Prove It is also quite good.
http://www.abebooks.com/servlet/SearchResults?bi=0&amp;bx=off&amp;ds=20&amp;kn=Epp+discrete+applications+3&amp;recentlyadded=all&amp;sortby=17&amp;sts=t
How to Prove It: A Structured Approach by Daniel J. Velleman http://www.amazon.com/dp/0521675995/ref=cm_sw_r_udp_awd_ff3Vtb08QR4FZ
What text are you using?
Edit: Most calc II or multivariable textbooks that I've encountered (e.g.: this one, this one, this one, or this one) are full of examples, problems, and sections dealing with physical applications, if that's what you mean by outside the classroom.
From what I recollect, Calc II was mostly about developing facility with integration techniques, with some extensions of the concept of integration to boot. Although some of the material may seem to be of little relevance, think of it as an important stepping stone. It is preparing you for some super interesting subjects (like line integrals on vector fields!) that are used to model physical systems.
I dont know much about boot camp, but it sounds like having a physical book will be your best bet.
Personally, my favorite text book to use is Calculus: an Intutitive Approach by Morris Kline, but you might want something more advanced than that.
I'm working through the exercises in Pinter's Abstract Algebra.
If you're not afraid of math there are some cheap introductory textbooks on topics that might be accessible:
For abstract algebra: http://www.amazon.com/Book-Abstract-Algebra-Second-Mathematics/dp/0486474178/ref=sr_1_1?ie=UTF8&amp;qid=1459224709&amp;sr=8-1&amp;keywords=book+of+abstract+algebra+edition+2nd
For Number Theory: http://www.amazon.com/Number-Theory-Dover-Books-Mathematics/dp/0486682528/ref=sr_1_1?ie=UTF8&amp;qid=1459224741&amp;sr=8-1&amp;keywords=number+theory
These books have complimentary material and are accessible introductions to abstract proof based mathematics. The algebra book has all the material you need to understand why quintic equations can't be solved in general with a "quintic" formula the way quadratic equations are all solved with the quadratic formula.
The number theory book proves many classic results without hard algebra, like which numbers are the sum of two squares, etc, and has some of the identities ramanujan discovered.
For an introduction to analytic number theory, a hybrid pop/historical/textbook is : http://www.amazon.com/Gamma-Exploring-Constant-Princeton-Science/dp/0691141339/ref=sr_1_1?ie=UTF8&amp;qid=1459225065&amp;sr=8-1&amp;keywords=havil+gamma
This book guides you through some deep territory in number theory and has many proofs accessible to people who remember calculus 2.
I like this abstract algebra book: A Book of Abstract Algebra
This is one of the best books of abstract algebra I've seen, very well explained, favoring clear explanations over rigor, highly recommended (take your time to read the reviews, the awesomeness of this book is real :P): http://www.amazon.com/Book-Abstract-Algebra-Edition-Mathematics/dp/0486474178/ref=sr_1_6?ie=UTF8&amp;qid=1345229432&amp;sr=8-6&amp;keywords=introduction+to+abstract+algebra
On a side note, trust me, Dummit or Fraileigh are not what you want.
If you're interested in this sort of thing, I'd recommend you look at abstract algebra later. There's lots of proofs that certain constructions are impossible with it. This book is a pretty accessible introduction.
You might want to pick up a copy of Pinter and skim the first few chapters.
I would recommend getting familiar with how the cyclic groups work (they are basically clock arithmetic), how dihedral groups work (flipping and rotating polygons), and how the symmetric group works (ways you can shuffle things).
Work out the multiplication table for a handful, including the cyclic group of order 12, the symmetric group on 3 symbols, and the dihedral groups for the triangle and square.
WARNING: Don't think you need to get quick at mental multiplication in these groups. It's better you get a "feel" for how they work. Just like matrix multiplication, multiplying group elements is (in general) very tedious for people to do.
Try to think about groups in other areas of math or in everyday life. They appear anywhere you think of symmetry. Rotations and other rigid motions in space are a common example. But even something as simple as tic-tac-toe.... if you rotate (or invert) the board in a game of tic-tac-toe, a player with the advantage still has the advantage. If you know some physics, you should immediately look up Noether's theorem.
Having a head start on those, you can spare yourself some mental strain early on and focus on the harder first-year ideas: subgroups, homomorphisms, Lagrange's theorem, and quotient groups.
"A Book of Abstract Algebra" by Charles C. Pinter is nice, from what I've seen of it--which is about the first third. I'm going through it in an attempt to relearn the abstract algebra I've forgotten.
I was using Herstein (which was what I learned from the first time), and was doing fine, but saw the Pinter book at Barnes & Noble. I've found it is often helpful when relearning a subject to use a different book from the original, just to get a different approach, so gave it a try (it's a Dover, so was only ten bucks).
What is nice about the Pinter book is that it goes at a pretty relaxed pace, with a good variety of examples. A lot of the exercises apply abstract algebra to interesting things, like error correcting codes, and some of these things are developed over the exercises in several chapters.
You don't have to be a prodigy to be able to understand some real mathematics in middle school or early high school. By 9th grade, after a summer of reading calculus books from the local public library, I was able to follow things like Niven's proof that pi is irrational, for instance, and I was nowhere near a prodigy.
Dummit and Foote's Abstract Algebra is an excellent book for the algebra side of things. It can be a little dense, but it's chock full of examples and is very thorough.
To help get through the first ten or so chapters, Charles Pinter's A Book of Abstract Algebra is an incredible resource. It does wonders for building up an intuition behind algebra.
From the ground up, I dunno. But I looked through my amazon order history for the past 10 years and I can say that I personally enjoyed reading the following math books:
An Introduction to Graph Theory
Introduction to Topology
Coding the Matrix: Linear Algebra through Applications to Computer Science
A Book of Abstract Algebra
An Introduction to Information Theory
Also, this book is a tough piece of work, for sure, but it's very helpful. It probably goes deeper than your class will, and may present ideas/methods in a different way, but if you grapple w/ this one, it'll really help you figure out L.A.
There are some recommendations on Amazon :
>I find it ironic that my two favourite Linear Algebra texts are this book and the Axler, for they are exact opposites: Axler shuns determinants, and Shilov starts with them and builds much of his theory off them. However, there is no book I have found that has such a deep and clear exposition of determinants. The first chapter alone makes this book worth buying.
http://www.amazon.com/Linear-Algebra-Dover-Books-Mathematics/dp/048663518X/ref=sr_1_1?s=books&amp;ie=UTF8&amp;qid=1346872221&amp;sr=1-1&amp;keywords=linear+algebra
I would suggest this book for more advanced reading : http://www.amazon.com/gp/product/0415267994/ref=cm_cr_mts_prod_img
^ That book is really good. It starts with linear algebra topics and moves into functional analysis.
For discrete math I like Discrete Mathematics with Applications by Suzanna Epp.
It's my opinion, but Learning to Reason: An Introduction to Logic, Sets, and Relations by Nancy Rodgers is much better structured and more in depth than How To Prove It by Velleman. If you follow everything she says, proofs will jump out at you. It's all around great intro to proofs, sets, relations.
Also, knowing some Linear Algebra is great for Multivariate Calculus.
This is how I learned logic, for computer science.
First chapter of this Discrete mathematics book in my discrete math class
https://www.amazon.ca/Discrete-Mathematics-Applications-Susanna-Epp/dp/0495391328
Then, using The Logic Book for a formal philosophy logic 1 course.
https://www.amazon.ca/Logic-Book-Merrie-Bergmann/product-reviews/0078038413/ref=dpx_acr_txt?showViewpoints=1
The second book was horrid on itself, luckily my professor's academic lineage goes back to Tarski. He's an amazing Professor and knows how to teach...that was a god send. Ironically, he dropped the text and I see that someone has posted his openbook project.
The first book (first chapter), is too applied I imagine for your needs. It would also only be economically feasible if well, you disregarded copyright law and got a "free" PDF of it.
Try these books(the authors will hold your hand tight while walking you through interesting math landscapes):
Discrete Mathematics with Applications by Susanna Epp
Learning to Reason: An Introduction to Logic, Sets, and Relations by Nancy Rodgers
A Friendly Introduction to Number Theory Joseph Silverman
A First Course in Mathematical Analysis by David Brannan
The Foundations of Analysis: A Straightforward Introduction: Book 1 Logic, Sets and Numbers by K. G. Binmore
The Foundations of Topological Analysis: A Straightforward Introduction: Book 2 Topological Ideas by K. G. Binmore
Introductory Modern Algebra: A Historical Approach by Saul Stahl
An Introduction to Abstract Algebra VOLUME 1(very elementary)
by F. M. Hall
There is a wealth of phenomenally well-written books and as many books written by people who have no business writing math books. Also, Dover books are, as cheap as they are, usually hit or miss.
One more thing:
Suppose your chosen author sets the goal of learning a, b, c, d. Expect to be told about a and possibly c explicitly. You're expected to figure out b and d on your own. The books listed above are an exception, but still be prepared to work your ass off.
>My first goal is to understand the beauty that is calculus.
There are two "types" of Calculus. The one for engineers - the plug-and-chug type and the theory of Calculus called Real Analysis. If you want to see the actual beauty of the subject you might want to settle for the latter. It's rigorous and proof-based.
There are some great intros for RA:
Numbers and Functions: Steps to Analysis by Burn
A First Course in Mathematical Analysis by Brannan
Inside Calculus by Exner
Mathematical Analysis and Proof by Stirling
Yet Another Introduction to Analysis by Bryant
Mathematical Analysis: A Straightforward Approach by Binmore
Introduction to Calculus and Classical Analysis by Hijab
Analysis I by Tao
Real Analysis: A Constructive Approach by Bridger
Understanding Analysis by Abbot.
Seriously, there are just too many more of these great intros
But you need a good foundation. You need to learn the basics of math like logic, sets, relations, proofs etc.:
Learning to Reason: An Introduction to Logic, Sets, and Relations by Rodgers
Discrete Mathematics with Applications by Epp
Mathematics: A Discrete Introduction by Scheinerman
You should be able to see the textbook that is required by looking on your MyOSU page. This was the book used when I took it last year: https://www.amazon.com/Discrete-Mathematics-Applications-Susanna-Epp/dp/0495391328/ref=sr_1_2?crid=3TYI2C38O2DKB&keywords=susanna+epp+discrete+mathematics+with+applications+4th+edition&qid=1568344545&sprefix=susannah+epp%2Caps%2C175&sr=8-2
I personally didn't find discrete maths difficult, but I know it's the big hurdle for most people. Also since my degrees were primarily in mathematics, discrete maths wasn't my first proof-based subject so my experience is very different from most people.
I certainly don't have a comprehensive background in computer science, I've only taken 4 computer science subjects in my 5 years of university education and find my training in mathematics to be highly useful for programming at work and self-study of computer science.
If you can understand and absorb the first four chapters of Discrete Mathematics with Applications (only ~200 pages and should be available in any Uni library) then you'll be well set up for most of the maths that seems to trip people up.
This is the textbook we used and it was incredibly dry (and expensive). I highly recommend this book to ease you into it.
This is the book we are using for our class.
Is the book for 225 necesary? The book store said that there were no required materials for the class but I came across this one several times Discrete Mathematics with Applications 4th Edition
You can read the "Highlights of the Fourth Edition" on Page xvii through Amazon's preview.
Edit: This Amazon review is also relevant to you:
> I've taught discrete math from the 3rd Edition of this book at least 6 times, and struggled with several issues. (The textbook for our Discrete Math course is chosen by a committee in our department.) Much of a discrete math course involves looking closely at some very simple mathematics. Most of the mathematics is already known to a typical university freshman; what a set is, what a prime is, what an ordered pair is, etc. Of course they have had little rigor in these elementary topics, but still, they have the notions and vocabulary. The 3rd Edition pretended that sets, e.g., did not exist until one finally arrived at the chapter on sets. It's unnatural to lecture one's way through two chapters on logic and a chapter on techniques of proof, without being able to draw on simple examples from set theory. One gets tired quickly of examples of dogs and cats in highly artificial situations, and would like to say something about primes or the set of even integers.
>The 4th Edition corrects this problem by the addition of an introductory chapter which fixes the vocabulary and notation. This was a needed change. The 3rd Edition required considerable acrobatics in avoiding words like "is an element of" until Chapter 5 (Set Theory.) Really? I'm supposed to cover the proof technique of "division into cases" and I can't say "the set of integers of the form 4k+1?" So good change.
>Every semester, I get e-mails from my students asking if the previous edition of the text will suffice for my course. Usually, I say yes. In the case of my discrete math course, I'll have to say no. The modifications of this text are substantial. Besides the above, the old Chapter 8 (Recursion) is now incorporated into the new (much expanded) Chapter 5 (Sequences and Induction.) That is also a sensible change.
>My remaining complain about the text is that it's a bit condescending. I think it's bad form to always present mathematics apologetically. "There, there now, I know it's difficult, but we'll go extremely slowly and take tiny, tiny bites covered in catsup so you can scarcely taste them." There's no need for us at the university level to re-enforce the bad attitudes the students learned in high school. It's math. It's hard. You can do it, not because math is made easy, but because you are, in fact, clever enough.
>I would not have recommended the 3rd Edition to anyone, but I would recommend the 4th. I'm very happy with the changes.
Yes, the arrow is obvious when you stop to think about it, but everything becomes obvious after given enough thought. If they had used "=" it would have required slightly less thought and made it easier on the reader. What is the downside? I had no trouble reading the psuedo code, I just take issue with the useless proof ridden overly dense textbooks. You're saying that because it's possible to understand something, we should make no effort to make it easier to understand? This is where a lot of textbooks fail, in my opinion.
And just because a text is easier to understand doesn't make it dumbed down. I had a discrete math class that used this textbook:
http://www.amazon.com/Discrete-Mathematics-Computer-Scientists-Cliff/dp/0132122715/ref=sr_1_1?s=books&amp;ie=UTF8&amp;qid=1345561997&amp;sr=1-1&amp;keywords=discrete+math+for+computer+scientists
It was horribly dense and consisted of proof after proof with no real world examples. I could have learned from it if I so chose to, but I'd rather not spend 30 minutes parsing each page for what they are actually trying to teach me. I instead downloaded:
http://www.amazon.com/Discrete-Mathematics-Applications-Susanna-Epp/dp/0495391328/ref=sr_1_2?ie=UTF8&amp;qid=1345561973&amp;sr=8-2&amp;keywords=susanna+epp
And got a 95 in the class. Did I learn less because I used an easier textbook? I would say that I actually learned more, because it took significantly less time to learn the concepts presented, leaving me with more time to learn overall.
Nowhere is this disparity more clear than in data structures and algorithms books.
The book that really helped me prepare for CS 6505 this fall was Discrete Mathematics with Applications by Susanna Epp. I found it easy to digest and it seemed to line up well with the needed knowledge to do well in the course.
Richard Hammack's Book of Proof also proved invaluable. Because so much of your success in the class relies on your ability to do proofs, strengthening those skills in advance will help.
Course Name: Discrete Structures
Course Link: http://uncw.edu/csc/courses-133.html
Text Book: http://www.amazon.com/Discrete-Mathematics-Applications-Susanna-Epp/dp/0495391328/ref=sr_1_1?s=books&amp;ie=UTF8&amp;qid=1289513076&amp;sr=1-1
Comments: It wasn't that book when I was there back in the 90's though. We were using a book written by a UNC-W professor and published in-house. But I took another course later, using the 3rd edition of the Susanna Epp book, and I have to say, I prefer Epp's book. But holy fuck would you look at the price for the current edition? WTF?!??
Also, Dr. Berman is nice enough, but is one of the most boring professors ever born... talk about the stereotypical dry, boring, dull, monotone delivery... this guy has it perfected.
The deck corresponding to the intellectual property book has ~325 cards.
The deck corresponding to IEA has ~400 cards.
The deck corresponding to linear algebra has ~1000 cards. That seems weird to me, since I feel that I make fewer cards for math books; most of the extra time comes from doing a lot of scratch work. Weird. In addition to timing, I've more recently started keeping track of how many Ankis I've added each section, so maybe I'll have more insight there later. We'll see.
And please message me when you start doing math! If you're looking towards advanced mathematics (beyond calculus/linear algebra-for-engineers), I recommend starting with either Mathematics for Computer Science (review) or, if you really have no interest in doing that, How to Prove It.
I'm not sure what kind of shape you're in, but I'm guessing that the ironman requires a lot more planning just to finish it. I'd suggest getting a copy of this book which will help you plan out and train for all three sports.
Depending on the area you're in, I'd suggest joining a club that does group worksouts (runs, rides, swims, etc). Very useful for all sorts of things, but especially for organized pool workouts. If you're in the DC area, I'll suggest (Team Z)[http://www.triteamz.com/], but I'm sure there are other teams out there.
If your priority is training for the Tri, a muscle building program like SL will not be very helpful.
You would be much better off following an endurance program that peaks on your event date. You still have a couple months to establish base and then another couple months added anaerobic and intervals.
Read this entire book- it will help you plan a good peak - http://www.amazon.com/Triathletes-Training-Bible-Joe-Friel/dp/1934030198/ref=dp_ob_title_bk
Where I teach they use Linear Algebra by Lay for the introductory class. I'm not sure what level you need but Linear Algebra Done Right is also commonly recommended; could be more abstract than what you need?
Can you run on the deck of the ship?
If you are already pretty fit (which I assume you are since you are in the Navy), you shouldn't have too much of an issue finishing an Oly. If you are shooting for a specific time goal you will be a bit more constrained however.
You have quite a bit of time until early summer so I would build up a strong aerobic base and maybe incorporate a bit of weights in for lower body and upper body. I would be careful with maximal weights at this point. Try to go for low weight and a lot of reps. Try to avoid putting on a ton of mass -- keep it lean.
Joe Friel writes some amazing books that you would find very interesting and helpful in structuring your plan. See the Triathlete's Training Bible.
Thanks! Link for anyone interested
I would say it is absolutely doable. Joe Friel says tris are a swim warm up, a bike race, and a jog to the finish. So you being a cyclist, yes. Yes you can do it.
I keep referencing this site and keep referring back to it. I'm making my own plan, but I started with this as my template: http://www.beginnertriathlete.com/Scott%20Herrick/halfim/preparing_for_your_first_half_ir.htm
I bought these books this past weekend and I'm learning a lot from them:
http://www.amazon.com/Triathlete-Magazines-Complete-Triathlon-Book/dp/0446679283/ref=cm_lmf_img_7
http://www.amazon.com/The-Triathletes-Training-Bible-Friel/dp/1934030198/ref=cm_lmf_img_2
The Joe Friel Books are great. The Triathletes Training Bible by Joe Friel is fantastic (https://www.amazon.com/Triathletes-Training-Bible-Joe-Friel/dp/1934030198) in addition I found a subscription to training peaks with a training plan to be great for accountability.
This webpage has a solid list of recommended textbooks: https://mathblog.com/mathematics-books/
For Linear Algebra, Linear Algebra Done Right (3rd Ed.).
Hey y'all! I'm 16, and am about to finish Spivak's Calculus. Assuming that I know everything up to Algebra II, AP Statistics, Trigonometry, a bit of linear algebra (please specify if the subject requires extensive knowledge here), and have thoroughly gone through Spivak's Calculus, what should be the next thing I study? And what textbook(s) would you recommend for learning that subject?
Right now I'm leaning towards Real Analysis, or Multi Variable Calculus, or maybe Topology, or... case in point, I am very undecided and am in need of recommendations.
You do realize that there is guesswork but the extremes of the confidence interval are strictly positive right? In other words, no one is certain but what we are certain about is that optimum homework amount is positive. Maybe it's 4 hours, maybe it's 50 hours. But it's definitely not 0.
I don't like homework either when I was young. I dreaded it, and I skipped so many assignments, and I regularly skipped school. I hated school. In my senior year I had such severe senioritis that after I got accepted my grades basically crashed to D-ish levels. (By the way this isn't a good thing. It makes you lazy and trying to jumpstart again in your undergrad freshman year will feel like a huge, huge chore)
Now that I'm older I clearly see the benefits of homework. My advice to you is not to agree with me that homework is useful. My advice is to pursue your dreams, but when doing so be keenly aware of the pragmatical considerations. Theoretical physics demands a high level of understanding of theoretical mathematics: Lie groups, manifolds and differential algebraic topology, grad-level analysis, and so on. So get your arse and start studying math; you don't have to like your math homework, but you'd better be reading Spivak if you're truly serious about becoming a theoretical physicist. It's not easy. Life isn't easy. You want to be a theoretical physicist? Guess what, top PhD graduate programs often have acceptance rates lower than Harvard, Yale, Stanford etc. You want to stand out? Well everyone wants to stand out. But for every 100 wannabe 15-year-old theoretical physicists out there, only 1 has actually started on that route, started studying first year theoretical mathematics (analysis, vector space), started reading research papers, started really knowing what it takes. Do you want to be that 1? If you don't want to do homework, fine; but you need to be doing work that allows you to reach your dreams.
If you're looking at it from a mathematical "I want to prove things" standpoint, I'd recommend Apostol. I've also heard good things about Spivak, although I've never read that book.
If you're looking at it from an engineering "Just tell me how to do the damn problem" perspective, I'm no help to you.
I have yet to read it myself, but the classic text for Calculus is Spivak's Calculus. It is very highly recommended.
Here's my radical idea that might feel over-the-top and some here might disagree but I feel strongly about it:
In order to be a grad student in any 'mathematical science', it's highly recommended (by me) that you have the mathematical maturity of a graduated math major. That also means you have to think of yourself as two people, a mathematician, and a mathematical-scientist (machine-learner in your case).
AFAICT, your weekends, winter break and next summer are jam-packed if you prefer self-study. Or if you prefer classes then you get things done in fall, and spring.
Step 0 (prereqs): You should be comfortable with high-school math, plus calculus. Keep a calculus text handy (Stewart, old edition okay, or Thomas-Finney 9th edition) and read it, and solve some problem sets, if you need to review.
Step 0b: when you're doing this, forget about machine learning, and don't rush through this stuff. If you get stuck, seek help/discussion instead of moving on (I mean move on, attempt other problems, but don't forget to get unstuck). As a reminder, math is learnt by doing, not just reading. Resources:
math on irc.freenode.net
Here are two possible routes, one minimal, one less-minimal:
Minimal
Less-minimal:
NOTE: this is pure math. I'm not aware of what additional material you'd need for machine-learning/statistical math. Therefore I'd suggest to skip the less-minimal route.
ummarycoc has a good point. Snoop around his room and see if he already has How To Prove it: A Structured Approach. Someone bought this book for me and I return to it frequently.
Your mileage with certifications may vary depending on your geographical area and type of IT work you want to get into. No idea about Phoenix specifically.
For programming work, generally certifications aren't looked at highly, and so you should think about how much actual programming you want to do vs. something else, before investing in training that employers may not give a shit about at all.
The more your goals align with programming, the more you'll want to acquire practical skills and be able to demonstrate them.
I'd suggest reading the FAQ first, and then doing some digging to figure out what's out there that interests you. Then, consider trying to get in touch with professionals in the specific domain you're interested in, and/or ask more specific questions on here or elsewhere that pertain to what you're interested in. Then figure out a plan of attack and get to it.
A lot of programming work boils down to:
As a basic primer, you might want to look at Code for a big picture view of what's going with computers.
For basic logic skills, the first two chapters of How to Prove It are great. Being able to think about conditional expressions symbolically (and not get confused by your own code) is a useful skill. Sometimes business requirements change and require you to modify conditional statements. With an understanding of Boolean Algebra, you will make fewer mistakes and get past this common hurdle sooner. Lots of beginners struggle with logic early on while also learning a language, framework, and whatever else. Luckily, Boolean Algebra is a tiny topic. Those first two chapters pretty much cover the core concepts of logic that I saw over and over again in various courses in college (programming courses, algorithms, digital circuits, etc.)
Once you figure out a domain/industry you're interested in, I highly recommend focusing on one general purpose programming language that is popular in that domain. Learn about data structures and learn how to use the language to solve problems using data structures. Try not to spread yourself too thin with learning languages. It's more important to focus on learning how to get the computer to do your bidding via one set of tools - later on, once you have that context, you can experiment with other things. It's not a bad idea to learn multiple languages, since in some cases they push drastically different philosophies and practices, but give it time and stay focused early on.
As you gain confidence there, identify a simple project you can take on that uses that general purpose language, and perhaps a development framework that is popular in your target industry. Read up on best practices, and stick to a small set of features that helps you complete your mini project.
When learning, try to avoid haplessly jumping from tutorial to tutorial if it means that it's an opportunity to better understand something you really should understand from the ground up. Don't try to understand everything under the sun from the ground up, but don't shy away from 1st party sources of information when you need them. E.g. for iOS development, Apple has a lot of development guides that aren't too terrible. Sometimes these guides will clue you into patterns, best practices, pitfalls.
Imperfect solutions are fine while learning via small projects. Focus on completing tiny projects that are just barely outside your skill level. It can be hard to gauge this yourself, but if you ever went to college then you probably have an idea of what this means.
The feedback cycle in software development is long, so you want to be unafraid to make mistakes, and prioritize finishing stuff so that you can reflect on what to improve.
How to Prove It by Vellemen is a superb introduction to what proofs are, and how to make them.
Keep in mind certain proof based courses can be frustrating to some students (discrete math and real analysis) as these classes often make formal concepts students may understand intuitively. Abstract Algebra or Topology may give you a more accurate idea of your feelings towards math.
Honestly, I think you should be more realistic: doing everything in that imgur link would be insane.
You should try to get a survey of the first 3 semesters of calculus, learn a bit of linear algebra perhaps from this book, and learn about reading and writing proofs with a book like this. If you still have time, Munkres' Topology, Dummit and Foote's Abstract Algebra, and/or Rudin's Principles of Mathematical Analysis would be good places to go.
Roughly speaking, you can theoretically do intro to proofs and linear algebra independently of calculus, and you only need intro to proofs to go into topology (though calculus and analysis would be desirable), and you only need linear algebra and intro to proofs to go into abstract algebra. For analysis, you need both calculus and intro to proofs.
Depends on what you are looking for. You might not be aware that the concepts in that book are literally the foundations of math. All math is (or can be) essentially expressed in set theory, which is based on logic.
You want to improve math reasoning, you should study reasoning, which is logic. It's really not that hard. I mean, ok its hard sometimes but its not rocket science, its doable if you dedicate real time to it and go slowly.
Two other books you may be interested in instead, that teach the same kinds of things:
Introduction to Mathematical Thinking which he wrote to use in his Coursera course.
How to Prove It which is often given as the gold standard for exactly your question. I have it, it is fantastic, though I only got partway through it before starting my current class. Quite easy to follow.
Both books are very conversational -- I know the second one is and I'm pretty sure the first is as well.
What books like this do is teach you the fundamental logical reasoning and math structures used to do things like construct the real number system, define operations on the numbers, and then build up to algebra step by step. You literally start at the 1+1=2 type level and build up from there by following a few rules.
Also, I just googled "basic logic" and stumbled across this, it looks like a fantastic resource that teaches the basics without any freaky looking symbols, it uses nothing but plain-English sentences. But scanning over it, it teaches everything you get in the first chapter or two of books like those above. http://courses.umass.edu/phil110-gmh/text/c01_3-99.pdf
Honestly if I were starting out I would love that last link, it looks fantastic actually.
Read this book: http://www.amazon.com/How-Study-as-Mathematics-Major/dp/0199661316/ref=asap_bc?ie=UTF8
And work through this book in its entirety: http://www.amazon.com/How-Prove-Structured-Approach-2nd/dp/0521675995/ref=sr_1_1?s=books&amp;ie=UTF8&amp;qid=1463201632&amp;sr=1-1
Like one of the other's suggested, learning more about proofs is probably what you're interested in since that's where these rules and equations come from. I've seen this book recommended a few times. It should give you a better understanding of how math is formed.
Probably something on coursera; however, I really recommend this gem of a book, http://www.amazon.com/How-Prove-Structured-Approach-2nd/dp/0521675995/ref=sr_1_1?s=books&amp;ie=UTF8&amp;qid=1458411780&amp;sr=1-1&amp;keywords=how+to+prove+it .
Get the book [How to Prove It: A Structured Approach by Daniel J. Velleman] (http://www.amazon.com/How-Prove-Structured-Approach-2nd/dp/0521675995) it will teach you how to write, and I think more importantly, read proofs.
How comfortable are you with proofs? If you are not yet comfortable, then read this: How to Prove It: A Structured Approach
I may be in the minority here, but I think that high school students should be exposed to statistics and probability. I don't think that it would be possible to exposed them to full mathematical statistics (like the CLT, regression, multivariate etc) but they should have a basic understanding of descriptive statistics. I would emphasize things like the normal distribution, random variables, chance, averages and standard deviations. This could improve numerical literacy, and help people evaluate news reports and polls critically. It could also cut down on some issues like the gambler's fallacy, or causation vs correlation.
It would be nice if we could teach everyone mathematical statistics, the CLT, and programming in R. But for the majority of the population a basic understanding of the key concepts would be an improvement, and would be useful.
Edit At the other end of the spectrum, I would like to see more access to an elective class that covers the basics of mathematical thinking. I would target this at upperclassmen who are sincerely interested in mathematics, and feel that the standard trig-precalculus-calculus is not enough. It would be based off of a freshman math course at my university, that strives to teach the basics of proofs and mathematical thinking using examples from different fields of math, but mostly set theory and discrete math. Maybe use Velleman's book or something similar as a text.
How to Prove it by Velleman seems to be right up your alley.
Thanks for the answer!
Glad to hear about Spivak! I've heard good things about that textbook and am looking forward to going through it soon :). Are the course notes for advanced algebra available online? If so, could you link them?
Is SICP used only in the advanced CS course or the general stream one, too? (last year I actually worked my way through the first two chapters before getting distracted by something else - loved it though!) Also, am I correct in thinking that the two first year CS courses cover functional programming/abstraction/recursion in the first term and then data structures/algorithms in the second?
That's awesome to know about 3rd year math courses! I was under the impression that prerequisites were enforced very strongly at Waterloo, guess I was wrong :).
As for graduate studies in pure math, that's the plan, but I in no way have my heart set on anything. I've had a little exposure to graph theory and I loved it, I'm sure that with even more exposure I'd find it even more interesting. Right now I think the reason I'm leaning towards pure math is 'cause the book I'm going through deals with mathematical logic / set theory and I think it's really fascinating, but I realize that I've got 4/5 years before I will even start grad school so I'm not worrying about it too much!
Anyways, thanks a lot for your answer! I feel like I'm leaning a lot towards Waterloo now :)
How to Prove It is a nice introduction to writing proofs.
You should absolutely not give up.
None of this is groundbreaking, and a lot of it is pretty cliché, but it's true. Everyone struggles with math at some point. Einstein said something like "whatever your struggles with math are, I assure you that mine are greater."
As for specific recommendations, make the most of this summer. The most important factor in learning math in my experience is "time spent actively doing math." My favorite math quote is "you don't learn math, you get used to it." I might recommend a book like How to Prove It. I read it the summer before I entered college, and it helped immensely with proofs in real analysis and abstract algebra. Give that a read, and I bet you will be able to prove most lemmas in undergraduate algebra and topology books, and solve many of their problems. Just keep at it!
http://www.amazon.com/How-Prove-It-Structured-Approach/dp/0521675995
helps with the first part of the class. the stuff after that I would suggest just having good google-fu.
Math isn't going to be like the math classes you've already taken. It's a lot of writing and logic and very little calculating. If you go for mathematical sciences, you'll probably take more classes that involve calculations, but you won't make it that far if you can't handle proofs.
Check out this book: http://www.amazon.com/How-Prove-It-Structured-Approach/dp/0521675995
The full book is first-page googleable. If you find that material interesting, you'll probably enjoy being a math major.
To learn basic proof writing I highly recommend How to Prove It by Velleman.
I study topology and I can give you some tips based on what I've done. If you want extra info please PM me. I'd love to help someone discover the beautiful field of topology. TLDR at bottom.
If you want to study topology or knot theory in the long term (actually knot theory is a pretty complicated application of topology), it would be a great idea to start reading higher math ASAP. Higher math generally refers to anything proof-based, which is pretty much everything you study in college. It's not that much harder than high school math and it's indescribably beneficial to try and get into it as soon as you possibly can. Essentially, your math education really begins when you start getting into higher math.
If you don't know how to do proofs yet, read How to Prove It. This is the best intro to higher math, and is not hard. Absolutely essential going forward. Ask for it for the holidays.
Once you know how to prove things, read 1 or 2 "intro to topology" books (there are hundreds). I read this one and it was pretty good, but most are pretty much the same. They'll go over definitions and basic theorems that give you a rough idea of how topological spaces (what topologists study) work.
After reading an intro book, move on to this book by Sutherland. It is relatively simple and doesn't require a whole lot of knowledge, but it is definitely rigorous and is definitely necessary before moving on.
After that, there are kind of two camps you could subscribe to. Currently there are two "main" topology books, referred to by their author's names: Hatcher and Munkres. Both are available online for free, but the Munkres pdf isn't legally authorized to be. Reading either of these will make you a topology god. Hatcher is all what's called algebraic topology (relating topology and abstract algebra), which is super necessary for further studies. However, Hatcher is hella hard and you can't read it unless you've really paid attention up to this point. Munkres isn't necessarily "easier" but it moves a lot slower. The first half of it is essentially a recap of Sutherland but much more in-depth. The second half is like Hatcher but less in-depth. Both books are outstanding and it all depends on your skill in specific areas of topology.
Once you've read Hatcher or Munkres, you shouldn't have much trouble going forward into any more specified subfield of topology (be it knot theory or whatever).
If you actually do end up studying topology, please save my username as a resource for when you feel stuck. It really helps to have someone advanced in the subject to talk about tough topics. Good luck going forward. My biggest advice whatsoever, regardless of what you study, is read How to Prove It ASAP!!!
TLDR: How to Prove It (!!!) -> Mendelson -> Sutherland -> Hatcher or Munkres
http://www.amazon.com/How-Prove-It-Structured-Approach/dp/0521675995
This book was very helpful to me.
I would recommend the following two books:
The first book introduces most of the topics in the book that you linked, and was what was used in my Foundations of Mathematics class (essentially the same thing as your class).
Understanding Analysis, on the other hand, is probably the perfect book to follow up with, since it is such a well-motivated, yet rigorous book on the analysis of one real variable, that you may, in fact, become too accustomed to such lucid and entertaining prose for your own good.
Here are some great books that I believe you may find helpful :)
and last but definitely not least:
Later on:
Yeah, I've just never been shown a problem where this stuff gives deep insight, and until I see one and understand it these are just gonna be arbitrary definitions that slide right out of my brain when I'm done reading them. I'll definitely give the book a look - is it motivated with examples?
The only book I have on category theory is Conceptual Mathematics: A First Introduction to Categories, and I must say, I'm not a fan of it - too intuitive, not detailed enough, not well organized, not formal enough - should have gone for MacLane instead.
Category theory is an overkill. If you think you're gonna have an easier time with it, you're mistaken. Category Theory is an extreme generalization of abstract math. Although, there's a very nice intro that you can get started with: Conceptual Mathematics: A First Introduction to Categories by Schanuel and Lawvere. It's accessible to most high school students.
What you are trying to understand is trivial. Most any intro to proofs/higher math book has an explanation of the subject.
In general, you need to learn how to think logically because the way you're going right now won't get you anywhere.
Again, read a book on the very basics of logic and sets. It would contain everything you need to know. For example,
Learning to Reason: An Introduction to Logic, Sets, and Relations by Nancy Rodgers.
> ... relation between finite and infinite.
...relation between finite sets and infinite sets. Just about everything in math is a set. There are many different types of relations. Some are functions, some are equivalence relations, some are isomorphisms.
> Just because something is an adjective or property does not mean it can't be negated.
Ok. Opposite of infinite is finite. In fact, we can say that a set is finite if it is not infinite. But limit is a number and infinity is not. You can't compare apples to oranges.
> In fact almost everything has an inverse.
Relations and special kind of relations called functions have an inverse. Also, operations can be inverse.
For me, it was Conceptual Mathematics: A First Introduction to Categories. Today, I might suggest How to Bake Pi: An Edible Exploration of the Mathematics of Mathematics or Category Theory in Context to start.
There is this online Category Theory book (PDF). Also, the book Conceptual Mathematics has been well recommended as an introduction to CT starting from the basics.
Linear Algebra (Fourth Edition) by Stephen H. Friedberg
EDIT: I just realized that you already mentioned this book in your comments. I used this book in my upper level course too and it was a real treat.
Almost forgot to reply. Linear Algebra by Friedberg is one of the more mathematically rigorous texts I've seen for undergraduates. My school used it in the honors linear algebra course. I think you'll find that it covers most of what you need. Hope it helps (if you can find it at the library or something).
Personally, I would take the time to read them both. A strong linear algebra background will be very helpful in ML. Its especially useful if you want to expand out a little bit more into other areas of signal processing. Make sure you also spend some time getting a good background in probability and statistics.
EDIT: I haven't actually read Axler's book but me and some of my friends are partial to this book.
Both 215 and 220 need plenty of practice. So as long as you set time aside for that you should be well on your way.
For 220 I would review some basic proof techniques (contradiction, contra-positive, induction) but not worry too much about knowing the details. In general we were never ask to prove anything where we couldn't apply the basics from a proof we had already learned.
We used Mathematical Proofs: A Transition to Advanced Mathematics (https://www.amazon.com/dp/0321797094), which was a very clear text with plenty of practice problems. If you have time I would recommend reading chapter 2 and 3.
Hello,
You can take a step back and learn about the philosophy and history of math. Once you learn some famous mathematicians and what their discoveries mean you will have more interest. One book is "What is Mathematics"
Then you can pickup some difficult texts. Doing a million problems mechanically is useless, except perhaps to pass tests. Get some idea of how to construct and read and appreciate a mathematical proof. Learn how to write proofs and prove common theorems and what those theorems mean. I recommend this https://www.amazon.ca/gp/aw/d/0321797094 to give you an overview.
Finally nothing beats taking advanced classes in university.
If this all seems a bit too much then maybe you can pick up something specifically for your purpose like 3D Math Primer for Graphics and Game Development, 2nd Edition 3D Math Primer for Graphics and Game Development, 2nd Edition https://www.amazon.ca/dp/1568817231/
Get yourself to high school math level first (understand the unit circle, exponents, algebra, trigonometry) and you can move up from there.
No problem. FWIW my intro to proofs class used this book and I thought it served its purpose well
https://www.amazon.com/Mathematical-Proofs-Transition-Advanced-Mathematics/dp/0321797094
Linear Algebra can be of different levels of difficulty:
I've never taught the course, but a couple of my colleagues are very fond of Linear Algebra Done Wrong and would willingly teach from it if (1) the title wouldn't immediately turn students off of it and (2) the school would be okay with sacrificing some income from students having to purchase a book.
If you're curious, the book title is a play on the title of another well-known linear algebra book.
It's aight. Just read linear algebra, and mv calculus. Maybe some statistical mechanics, read some thermo and kinetics. Atkins for kinetics and thermo, McQuarrie for stat mech. For linear algebra read get this. You'll still have to take classes on it, so it's cool. The worst you may have to do is take some UG classes to get up to speed.
I'm currently on this journey as well! I'm a programmer teaching my self rigorous maths, so I can definitely help you out.
I find it's best to simultaneously look at several resources on topics such as proofs, so you can get a few perspectives on the same essential topics and have an easier time of finding something.
As a preliminary to proofing, I would suggest a survey of basic logic and Set Theory. I picked up my Set Theory from google searches and the introduction in Apostol's Calculus, and wiki articles on logic and set operations.. It's really easy to learn enough set theory and logic to begin understanding rigorous proofs.
To learn the proofing skills needing for Real Analysis I recommend
a) "Foundations of Analysis" by Edmund Landau.
b) Math 378: Number Systems: An Axiomatic Approach
For an actual book on real analysis, there can be no greater book than Apostol's Calculus.
The popular opinion by some mathematical elite is that Stewart dumbs down calculus, focuses too much on applications, and not enough on theory, which is important for those moving beyond to real analysis and other upper division courses. You should read the reviews of Spivak's or Apostol's calculus text books to see what I mean.
http://www.amazon.com/Calculus-Vol-One-Variable-Introduction-Algebra/dp/0471000051
Here you go. Apostol wrote this classic a while back, and it's currently used at MIT. It treats integration before differentiation. It is mathematically more mature than anything most engineers will ever encounter.
Friendly info:
"College Algebra" = Elementary Algebra.
College Level Algebra = Abstract Algebra.
Example: Undergrad Algebra book.
Example: Graduate Algebra book.
If you can read through gallian's book, I consider dummit and foote's book (http://www.amazon.com/Abstract-Algebra-3rd-David-Dummit/dp/0471433349) as the best math textbook i've ever read. tons of examples, thorough treatment of material, and tons of exercises.
I am a master's student with interests in algebraic geometry and number theory. And I have a good collection of textbooks on various topics in these two fields. Also, as part of my undergraduate curriculum, I learnt abstract algebra from the books by Dummit-Foote, Hoffman-Kunze, Atiyah-MacDonald and James-Liebeck; analysis from the books by Bartle-Sherbert, Simmons, Conway, Bollobás and Stein-Shakarchi; topology from the books by Munkres and Hatcher; and discrete mathematics from the books by Brualdi and Clark-Holton. I also had basic courses in differential geometry and multivariable calculus but no particular textbook was followed. (Please note that none of the above-mentioned textbooks was read from cover to cover).
As you can see, I didn't learn much geometry during my past 4 years of undergraduate mathematics. In high school, I learnt a good amount of Euclidean geometry but after coming to university geometry appears very mystical to me. I keep hearing terms like hyperbolic/spherical geometry, projective geometry, differential geometry, Riemannian manifold etc. and have read general maths books on them, like the books by Hartshorne, Ueno-Shiga-Morita-Sunada and Thorpe.
I will be grateful if you could suggest a series of books on geometry (like Stein-Shakarchi's Princeton Lectures in Analysis) or a book discussing various flavours of geometry (like Dummit-Foote for algbera). I am aware that Coxeter has written a series of textbooks in geometry, and I have read Geometry Revisited in high school (which I enjoyed). If these are the ideal textbooks, then where to start? Also, what about the geometry books by Hilbert?
If you are enjoying your Calc 3 book, I highly recommend reading Topology, which provides the foundations of analysis and calculus. Two other books I would highly recommend to you would be Abstract Algebra and Introduction to Algorithms, though I suspect you're well aware of the latter.
A good intro book on calculus I found helpful was Calculus: A Physical and Intuitive Approach by Morris Kline. Jumping right into Spivak, while doable, is not for the faint of heart. (But one should definitely approach it eventually!)
Edit: spelling
https://www.amazon.com/Calculus-Intuitive-Physical-Approach-Mathematics/dp/0486404536
An invaluable book when I took calculus the second time: Precalculus Mathematics in a Nutshell
I took calc a second time, because I had taken it previously over ten years before. My instructor at the time was quite the hardass and didn't allow calculators on his tests or homework. I remember doing integration by parts where problems would take two whole sheets of handwritten work.
Consequently, I have a bit of a "been there, done that" attitude towards calculus...
EDIT - My instructor was a big fan of Kline
It is an integral of the variable x, as you point out. You can refer to, this book
> I can solve though but the thought why i am doing this is always alarming inside, go and ask any teacher or students as why they do these maths? They will say it's for Grades!
Eek, you have a terrible history of teachers.
>Don't know how many students give up maths just because of wrong Teacher.
For sure.
Starting with calculus/analysis, the book most undergraduate students in America start with is this one. Not every concept starts with real-life examples, but every chapter and section includes actual real-life examples.
For math you're going to need to know calculus, differential equations (partial and ordinary), and linear algebra.
For calculus, you're going to start with learning about differentiating and limits and whatnot. Then you're going to learn about integrating and series. Series is going to seem a little useless at first, but make sure you don't just skim it, because it becomes very important for physics. Once you learn integration, and integration techniques, you're going to want to go learn multi-variable calculus and vector calculus. Personally, this was the hardest thing for me to learn and I still have problems with it.
While you're learning calculus you can do some lower level physics. I personally liked Halliday, Resnik, and Walker, but I've also heard Giancoli is good. These will give you the basic, idealized world physics understandings, and not too much calculus is involved. You will go through mechanics, electromagnetism, thermodynamics, and "modern physics". You're going to go through these subjects again, but don't skip this part of the process, as you will need the grounding for later.
So, now you have the first two years of a physics degree done, it's time for the big boy stuff (that is the thing that separates the physicists from the engineers). You could get a differential equations and linear algebra books, and I highly suggest you do, but you could skip that and learn it from a physics reference book. Boaz will teach you the linear and the diffe q's you will need to know, along with almost every other post-calculus class math concept you will need for physics. I've also heard that Arfken, Weber, and Harris is a good reference book, but I have personally never used it, and I dont' know if it teaches linear and diffe q's. These are pretty much must-haves though, as they go through things like fourier series and calculus of variations (and a lot of other techniques), which are extremely important to know for what is about to come to you in the next paragraph.
Now that you have a solid mathematical basis, you can get deeper into what you learned in Halliday, Resnik, and Walker, or Giancoli, or whatever you used to get you basis down. You're going to do mechanics, E&M, Thermodynamis/Statistical Analysis, and quantum mechanics again! (yippee). These books will go way deeper into theses subjects, and need a lot more rigorous math. They take that you already know the lower-division stuff for granted, so they don't really teach those all that much. They're tough, very tough. Obvioulsy there are other texts you can go to, but these are the one I am most familiar with.
A few notes. These are just the core classes, anybody going through a physics program will also do labs, research, programming, astro, chemistry, biology, engineering, advanced math, and/or a variety of different things to supplement their degree. There a very few physicists that I know who took the exact same route/class.
These books all have practice problems. Do them. You don't learn physics by reading, you learn by doing. You don't have to do every problem, but you should do a fair amount. This means the theory questions and the math heavy questions. Your theory means nothing without the math to back it up.
Lastly, physics is very demanding. In my experience, most physics students have to pretty much dedicate almost all their time to the craft. This is with instructors, ta's, and tutors helping us along the way. When I say all their time, I mean up until at least midnight (often later) studying/doing work. I commend you on wanting to self-teach yourself, but if you want to learn physics, get into a classroom at your local junior college and start there (I think you'll need a half year of calculus though before you can start doing physics). Some of the concepts are hard (very hard) to understand properly, and the internet stops being very useful very quickly. Having an expert to guide you helps a lot.
Good luck on your journey!
Calculus by James Stewart.
http://www.amazon.com/Calculus-James-Stewart/dp/0538497815
Sucks man. I don't know what level of Calculus you're doing, my GF had a really rough time passing Calc 2 which was the last class she needed to finish her degree (took it 3 times).
The last time she ended up getting a pair of books and those more than anything seemed to get her over the hump of failing with 50% and into the "C" range.
https://www.amazon.com/How-Ace-Calculus-Streetwise-Guide/dp/0716731606
https://www.amazon.com/How-Ace-Rest-Calculus-MultiVariable/dp/0716741741
Also just as a general rule, studying all night so that you're sleep deprived for a test is usually counterproductive. Doesn't matter how much you cram if your brain is fried and not working on all cylinders when it's test time.
Hey man, Calculus is a tough class. Depending on what your algebra background is, Calc 1 can be an especially challenging course. It doesn't say anything about how you'll do in your CS courses. That aside, if you're struggling w/ calc check out this book. It takes the mystery out of the major concepts of Calculus and I attribute a large part of my success in Calc 1 to this book. It doesn't read like a textbook, and I guarantee you won't regret dropping $17 on this. That aside, sorry about the shittiness.
I never said it made it a "bad" book in a deep sense. But it can quite easily explain why someone who isn't in the very narrow set of potential beneficiaries of Spivak's style might feel like the book is opaque, frustrating or unclear--adjectives we commonly associate with "bad" math books. And I also want to double down on the narrowness of Spivak's approach. The people coming away frustrated from Spivak were not looking for How To Ace Calculus, they were looking for a relatively rigorous treatment of the subject matter. What they got was the real meaning of the word rigor--that unexpected revelation is enough to cause some frustration. Frustration that I am willing to partially grant people without castigating them for not matching their expectations properly.
If you feel that ZyBooks does not do a good job in explaining the topics, then you should find other sources to help you understand the material. As you have suggested, take note of the topic and exercises and look for other sources to explain them.
Sources I used when I took Calc:
Here's my dropbox link of the last book for a preview: https://www.dropbox.com/sh/spnay16f7cybdzg/AAAOWMFGgjNqEVdG6o2lXSmba?dl=0
Strictly speaking it's "Analysis in Several Variables" and it uses the Spivak "Calculus on Manifolds" book.
http://www.amazon.com/Calculus-Manifolds-Approach-Classical-Theorems/dp/0805390219/ref=sr_1_1?ie=UTF8&amp;qid=1314643509&amp;sr=8-1
For vector calculus, you might enjoy the less formal British text Div, Grad, Curl, and All That by H. M. Schey; for group theory in brief, consider the free textbook Elements of Abstract and Linear Algebra by Edwin H. Connell.
Alternatives to Schey's book include the much more formal Calculus on Manifolds by Michael Spivak, which does have more exercises than Schey but uses most of them to develop the theory, rather than as the mindless drills that fill an ordinary textbook; Michael E. Corral's free textbook Vector Calculus isn't huge but is written closer to an ordinary textbook.
For me, a "good read" in mathematics should be 1) clear, 2) interestingly written, and 3) unique. I dislike recommending books that have, essentially, the same topics in pretty much the same order as 4-5 other books.
I guess I also just disagree with a lot of people about the
"best" way to learn topology. In my opinion, knowing all the point-set stuff isn't really that important when you're just starting out. Having said that, if you want to read one good book on topology, I'd recommend taking a look at Kinsey's excellent text Topology of Surfaces.
If you're interested in a sequence of books...keep reading.
If you are confident with calculus (I'm assuming through multivariable or vector calculus) and linear algebra, then I'd suggest picking up a copy of Edwards' Advanced Calculus: A Differential Forms Approach. Read that at about the same time as Spivak's Calculus on Manifolds. Next up is Milnor Topology from a Differentiable Viewpoint, Kinsey's book, and then Fulton's Algebraic Topology. At this point, you might have to supplement with some point-set topology nonsense, but there are decent Dover books that you can reference for that. You also might be needing some more algebra, maybe pick up a copy of Axler's already-mentioned-and-excellent Linear Algebra Done Right and, maybe, one of those big, dumb algebra books like Dummit and Foote.
Finally, the books I really want to recommend. Spivak's A Comprehensive Introduction to Differential Geometry, Guillemin and Pollack Differential Topology (which is a fucking steal at 30 bucks...the last printing cost at least $80) and Bott & Tu Differential Forms in Algebraic Topology. I like to think of Bott & Tu as "calculus for grown-ups". You will have to supplement these books with others of the cookie-cutter variety in order to really understand them. Oh, and it's going to take years to read and fully understand them, as well :) My advisor once claimed that she learned something new every time she re-read Bott & Tu...and I'm starting to agree with her. It's a deep book. But when you're done reading these three books, you'll have a real education in topology.
Start here: https://www.complexityexplorer.org/
then here: https://www.amazon.com/Nonlinear-Dynamics-Student-Solutions-Manual/dp/0813349109/ref=asc_df_0813349109/?tag=hyprod-20&amp;linkCode=df0&amp;hvadid=312168166316&amp;hvpos=1o1&amp;hvnetw=g&amp;hvrand=11660544257293770322&amp;hvpone=&amp;hvptwo=&amp;hvqmt=&amp;hvdev=c&amp;hvdvcmdl=&amp;hvlocint=&amp;hvlocphy=9006604&amp;hvtargid=pla-455692727025&amp;psc=1
In essence what you are interested in is "attractor reconstruction (Takens Theorem)", "measuring the lypaunov exponents", or "finding the correlation dimension". Search around for these things or look them up in a nonlinear dynamics textbook and it should get you on your way.
Check out this paper for a good overview of each of these terms, what they mean, and what they can tell you about your timeseries.
It gives a nice runthrough of the things that you can do with a simple time series to detect any chaos in the signal. They also provide some software which can run their analysis on your own time series.
I also would recommend the book: Nonlinear dynamics and Chaos by Steven Strogatz. Its a fantastic book that lays out a primer for chaotic systems, and its relatively short and not too maths heavy for a textbook.
Finally, this website has some nice pictures of analysis of a number of different chaotic systems that might give a better idea of where you can get started in this area.
For anyone interested in this topic, I can recommend two sources for newcomers.
Conversational, largely non-technical: Chaos: Making a New Science by James Gleick
Technical (requires knowledge of ordinary differential equations, but highly readable): Nonlinear Dynamics and Chaos by Steven H. Strogatz
You need calculus, linear algebra and some differential equations. Real analysis is extremely helpful but not completely necessary. Here is a good book on an introduction to the subject:
http://www.amazon.com/gp/aw/d/0813349109/ref=dp_ob_neva_mobile
Try to find entry points that interest you personally, and from there the next steps will be natural. Most books that get into the nitty-gritty assume you're in school for it and not directly motivated, at least up to early university level, so this is harder than it should be. But a few suggestions aimed at the self-motivated: Lockhart Measurement, Gelfand Algebra, 3blue1brown's videos, Calculus Made Easy, Courant & Robbins What Is Mathematics?. (I guess the last one's a bit tougher to get into.)
For physics, Thinking Physics seems great, based on the first quarter or so (as far as I've read).
I usually recommend Lang's Basic Mathematics for those wanting to go over or learn the necessary math before calculus. It covers everything you need and more in a nice fashion that is much better than any book in highschool you may have ever used. Another option is to pick up the series of books by I.M. Gelfand, which are split up in to algebra, coordinate graphs, functions, and trigonometry (i think it's only 4). The advantage here is that each book is small so you can digest it in chunks (plus they are Dover books now so they can be had for cheap). Both of these authors will both prepare and place you beyond your class for Math1050. If you've read and done the questions in these books, you will be more than ready. Personally, I like to not move on in material until I finally understand it or at least can decently explain what was covered to someone. So the time it takes to read these books will vary but I say it is feasible to cover a chapter a week more or less.
The following are all really good and all very different. Check out the reviews and decide what fits you best. If I had to pick only one, I would pick one of the first three listed.
http://www.amazon.com/gp/product/0471530123
http://www.amazon.com/gp/product/1592577229
http://www.amazon.com/The-Humongous-Book-Calculus-Problems/dp/1592575129
http://www.amazon.com/gp/product/0817636773
http://www.amazon.com/gp/product/0760706603
http://www.amazon.com/gp/product/B000ZEC19Y
http://www.amazon.com/gp/product/0070293317
I've heard good things about: http://www.amazon.com/Algebra-Israel-M-Gelfand/dp/0817636773/ref=cm_cr_pr_product_top?ie=UTF8
but I admit I haven't read it.
Read this: https://www.amazon.com/Algebra-Israel-M-Gelfand/dp/0817636773 and you're more than set for algebraic manipulation.
And if you're looking to get super fancy, then some of that: https://www.amazon.com/Method-Coordinates-Dover-Books-Mathematics/dp/0486425657/
And some of this for graphing practice: https://www.amazon.com/Functions-Graphs-Dover-Books-Mathematics/dp/0486425649/
And if you're looking to be a sage, these: https://www.amazon.com/Kiselevs-Geometry-Book-I-Planimetry/dp/0977985202/ + https://www.amazon.com/Kiselevs-Geometry-Book-II-Stereometry/dp/0977985210/
If you're uncomfortable with mental manipulation of geometric objects, then, before anything else, have a crack at this: https://www.amazon.com/Introduction-Graph-Theory-Dover-Mathematics/dp/0486678709/
These are, in my opinion, some of the best books for learning high school level math:
These are all 1900's Russian math text books (probably the type that /u/oneorangehat was thinking of) edited by I.M. Galfand, who was something like the head of the Russian School for Correspondence. I basically lived off them during my first years of high school. They are pretty much exactly what you said you wanted; they have no pictures (except for graphs and diagrams), no useless information, and lots of great problems and explanations :) There is also I.M Gelfand Trigonometry {[.pdf] (http://users.auth.gr/~siskakis/GelfandSaul-Trigonometry.pdf) | Amazon} (which may be what you mean when you say precal, I'm not sure), but I do not own this myself and thus cannot say if it is as good as the others :)
I should mention that these books start off with problems and ideas that are pretty easy, but quickly become increasingly complicated as you progress. There are also a lot of problems that require very little actual math knowledge, but a lot of ingenuity.
Sorry for bad Englando, It is my native language but I haven't had time to learn it yet.
I'll be working through Spivak's calculus for fun. Wish me luck!
For getting more intuition on proofs I would suggest the following book
http://www.amazon.com/Nuts-Bolts-Proofs-Third-Introduction/dp/0120885093/ref=sr_1_1?ie=UTF8&amp;qid=1311007015&amp;sr=8-1
I think Rudin might be really tricky at your level, you can keep with it if you want, but I think Calculus by Michael Spivak would be much more approachable for you.
http://www.amazon.com/Calculus-4th-Michael-Spivak/dp/0914098918/ref=sr_1_1?s=books&amp;ie=UTF8&amp;qid=1311007057&amp;sr=1-1
i personally prefer Yurope! Hillary's Invasion. Very insightful reading.
I didn't mean to make it sound so serious :) However, stress, drinking, and insomnia can all have some unexpectedly large effects, so it may be worth dropping into a counseling session if your university has one.
In regards to math education and intuition, something I found very useful was to read some books that start from scratch, like Burn Math Class, or Spivak's calculus for a real challenge. You're at a point in your education where you have the sophistication to understand the foundations of math, so you can start to rebuild intuition about a lot of things that will make university-level math much more sensible.
Start with 3 Blue 1 Brown's Essence of Calculus Series - https://www.youtube.com/watch?v=WUvTyaaNkzM&list=PLZHQObOWTQDMsr9K-rj53DwVRMYO3t5Yr
and follow the following books -
Calculus by Spivak - https://www.amazon.com/Calculus-4th-Michael-Spivak/dp/0914098918
Calculus Made Easy - http://calculusmadeeasy.org/
Follow all the concepts and solve the examples and exercises.
Feel free to ask the questions here or in mathsoverflow.
Last but not the least, PRACTICE, PRACTICE, PRACTICE........!
I am surprised no one has mentioned M. Spivak's very well known text Calculus. I thought this book was a pleasure to read. His writing was very fun and lighthearted and the book certainly teaches the material very well. In my opinion this is the best introductory calculus text there is.
When I first started learning math on my own, I started learning calculus from something like this. Though I enjoyed it, it didn't really show me what 'real math' was like. For learning something closer to higher math, a more rigorous version would be something like this. It's all preference, though.
If you don't know much about calculus at all, start with the first one, and then work your way up to Spivak.
If you want to learn serious mathematics, start with a theoretical approach to calculus, then go into some analysis. Introductory Real Analysis by Kolmogorov is pretty good.
As far as how to think about these things, group theory is a strong start. "The real numbers are the unique linearly-ordered field with least upper bound property." Once you understand that sentence and can explain it in the context of group theory and the order topology, then you are in a good place to think about infinity, limits, etc.
Edit: For calc, Spivak is one of the textbooks I have heard is more common, but I have never used it so I can't comment on it. I've heard good things, though.
A harder analysis book for self-study would be Principles of Mathematical Analysis by Rudin. He is very terse in his proofs, so they can be hard to get through.
If you have a chance, I recommend checking out some textbooks on real analysis, which will guide you through the derivations and proofs of many theorems in calculus that you've thus far been expected to take for granted.
Some would recommend starting with Rudin's Principles of Mathematical Analysis, and it's certainly a text that I plan to read at some point. For your purposes, I might recommend Spivak's Calculus since it expects you to rigorously derive some of the most important results in calculus through proof-writing exercises. This was my first introduction to calculus during high-school. While it was overwhelming at first, it prepared me for some of my more advanced undergraduate courses (including real analysis and topology), and it seems to be best described as an advanced calculus textbook.
I like this calculus book: https://www.amazon.com/Calculus-7th-James-Stewart/dp/0538497815
And this is Halliday and Resnick https://www.amazon.com/Fundamentals-Physics-David-Halliday/dp/111823071X/ref=sr_1_3?crid=9SK20IQF11Z5&amp;keywords=halliday+and+resnick+fundamentals+of+physics&amp;qid=1567697861&amp;s=books&amp;sprefix=hallida%2Cstripbooks%2C124&amp;sr=1-3
If you want a gentler introduction to calculus, with many examples, lots of intuition, diagrams, and nicer explanations, take any edition of James Stewart's Calculus - Early Transcendentals.
If you feel up to a serious challenge and want to study it as a mathematician would, get Michael Spivak's Calculus.
http://www.amazon.com/Calculus-7th-Edition-James-Stewart/dp/0538497815
King of the formulaic problems.
Cool.
Linear Algebra Don't waste your time with anything other than Lay, pretty much. Sounds like you're 100% new to LinAlg (it's not about polynomial equations) so it may be a bit tough to get off the ground working by yourself, but not impossible. It'd be worth finding a MOOC on the subject, there should be plenty. Otherwise, it's a pretty standard freshman maths course and a lot of people struggle with it (not because it's hard, just because it's different to HS maths), so there's a ton of resources on the internet.
Calculus Kinda just gotta slog away with where you're at tbh. I had Stewart as a freshman, didn't think it was overly great though. Still, that's the kind of level you need, so search for "alternatives to Stewart calculus" and anything that comes up should be appropriate. I wouldn't be able to tell you which to pick though.
Stats Basically, completing both of the above is pretty much a prerequisite for being able to understand linear regression properly, so don't expect to gain much by diving straight into stats. You could probably find a "business analytics" style textbook that would let you do more stats without understanding what's really going on under the hood, but if you want to stick with it in the long term you'll benefit more from getting stuff right at the beginning.
For 241, my favorite 'guide' was this book! . It's got everything you need and is a fun/easy read. There's also one for calc 2/3 if you'll be taking those.
I might have some past exams but i'll have to check. Who's your professor?
It's really smart to be playing to your strengths: if you excel at language and writing, then read a book that talks about math in more detail. Textbooks are good for problems and for reference, but I find them very hard to read. They use equations where they should be using words.
Go to your local library, and look in the math section until you find something interesting. I found this book when I was struggling with calculus: How to Ace Calculus: The Streetwise guide. It was smart, funny, and really explained topics in ways I could relate to.
That's the kind of thing I would look for if I were you. Good luck! I hope you see post in all the ~430 comments!
Well the good news is that we have more resources available now than even 5 years ago. :) I'm in calc 1 right now, and was having trouble putting the pieces together into a whole that made sense. A few of my resources are classroom specific but many would be great for anyone not currently in a class.
Free:
www.khanacademy.org
free video lectures and practice problems on all manner of topics, starting with elementary algebra. You can start at the beginning and work your way through, or just start wherever.
http://ocw.mit.edu/index.htm
free online courses and lessons from MIT (!!) where you can watch lectures on a subject, do practice problems, etc. Use just for review or treat it like a course, it's up to you.
Cheap $$
http://www.amazon.com/How-Ace-Calculus-Streetwise-Guide/dp/0716731606/ref=sr_1_1?ie=UTF8&amp;qid=1331675661&amp;sr=8-1
$10ish shipped for a book that translates calculus from math-professor to plain english, and is funny too.
http://www.amazon.com/Calculus-Lifesaver-Tools-Excel-Princeton/dp/0691130884/ref=pd_cp_b_1
$15 for a book that is 2-3x as thick as the previous one, a bit drier, but still very readable. And it covers Calc 1-3.
I also had about a 12 year break between HS and college, and like you got through Trig just fine and then found myself drowning in Calc 1. Here's what helped me:
-attended another section of the class with another professor
-books that translated the mathy language into intuition
(http://www.amazon.com/How-Ace-Calculus-Streetwise-Guide/dp/0716731606 and http://www.amazon.com/Calculus-Easy-Way-Douglas-Downing/dp/0764129201/ref=sr_1_1?s=books&amp;ie=UTF8&amp;qid=1415864089&amp;sr=1-1&amp;keywords=calculus+the+easy+way)
-MIT OCW videos
-Khan Academy
Good luck. If you make it through this.. well, I'm not going to say it's easy going after, but you will know how to be confused and work through that confusion, and that is a priceless skill in the rest of the curriculum.
Calculus: https://www.amazon.com/How-Ace-Calculus-Streetwise-Guide/dp/0716731606 and https://www.amazon.com/How-Ace-Rest-Calculus-MultiVariable/dp/0716741741 will get you right up to speed.
How to Ace Calculus
Got an A in Calculus (regular, not business) with this book, and I was really rusty at math.
Oh, and accounting is NOT math intensive...at all. If you can do + - * / and use a calculator, then you're fine.
While not a replacement text (you need more problems!), this is pretty swell for single variable and they even have a follow up text.
How to Ace Calculus: The Streetwise Guide
This book seems silly, but it's honestly great for learning Calculus, especially the second time: https://www.amazon.com/How-Ace-Calculus-Streetwise-Guide/dp/0716731606
(I read it in 1999 when I went from HS -> College, and the college I went into assumed you had already passed calc, and freshmen all had to start with second year calc. The professors recommended all incoming students refresh before the start of class, and I'm glad they did, because that book retaught some things I don't think I learned correctly the first time, made a huge difference).
Several books helped me with calc II
http://www.amazon.com/Calculus-Lifesaver-Tools-Princeton-Guides/dp/0691131538
Good refresher
http://www.amazon.com/How-Ace-Calculus-Streetwise-Guide/dp/0716731606
Itunes Video Series
http://itunes.apple.com/us/podcast/calculus-lifesaver-lecture/id266853222
When I was (approximately) in 8th grade I read https://www.amazon.com/How-Ace-Calculus-Streetwise-Guide/dp/0716731606 and I loved it. :)
This isn't an online resource, but this book is awesome for learning Calc 1.
Hey, if you or anybody is having trouble with Calc 1, check out this book: How to Ace Calculus: The Streetwise Guide
It's a math book that is actually fun to read and will take you through the key points of Calc 1 with no bullshit. Lots of fun little jokes and illustrations. It's pretty short and cheap. It helped me a lot back when I learned that stuff.
Not a textbook, but check out these:
http://www.amazon.com/How-Ace-Calculus-Streetwise-Guide/dp/0716731606
http://www.amazon.com/gp/product/0716741741/ref=pd_lpo_k2_dp_sr_1?pf_rd_p=304485901&amp;pf_rd_s=lpo-top-stripe-1&amp;pf_rd_t=201&amp;pf_rd_i=0716731606&amp;pf_rd_m=ATVPDKIKX0DER&amp;pf_rd_r=0KX1ZS8ARF7D5EB6AF92
I wouldn't bother with Apostol's Calculus. For analysis, you should really look at the first two volumes of Stein and Shakarchi's Princeton Lectures in Analysis.
Vol I: Fourier Analysis
Vol II: Complex Analysis
Then, you should pick up:
Munkres, Analysis on Manifolds or something similar, you could try Spivak's book but it's a bit terse. (on a personal note, I tried doing Spivak's book when I was a freshman. It was a big mistake).
In truth, most introductory undergrad analysis texts are actually more invested in trying to teach you the rigorous language of modern analysis than in expositing on ideas and theorems of analysis. For example, Rudin's Principles is basically to acquaint you with the language of modern analysis -- it has no substantial mathematical result. This is where the Stein Shakarchi books really shines. The first book really goes into some actual mathematics (fourier analysis even on finite abelian groups and it even builds enough math to prove Dirichlet's famous theorem in Number Theory), assuming only Riemann Integration (the integration theory taught in Spivak).
For Algebra, I'd suggest you look into Artin's Algebra. This is truly a fantastic textbook by one of the great modern algebraic geometers (Artin was Grothendieck's student and he set up the foundations of etale cohomology).
This should hold you up till you become a sophomore. At that point, talk to someone in the math department.
I just checked Amazon. It
says 1965, too but it is the 27^th printing.
I just noticed there is a DjVu copy here but it comes up as PDF on my browser.
You should check out Spivaks Calculus on Manifolds.
http://www.amazon.com/Calculus-Manifolds-Approach-Classical-Theorems/dp/0805390219
Read the first chapter or 2 and see how you like it, if you feel overwhelmed check some of the other recommendations out.
It is however a good book, and you should read it sooner or later.
Rudins principles of mathematical analysis is also excellent, however it
is not strictly multi-dimensional analysis.
Read at least chapter 2 and 3, they lay a very important groundwork.
For single variable calculus, like everyone else I would recommend Calculus - Spivak. If you have already seen mechanical caluculus, mechanical meaning plug and chug type problems, this is a great book. It will teach you some analysis on the real line and get your proof writing chops up to speed.
For multivariable calculus, I have three books that I like. Despite the bad reviews on amazon, I think Vector Calculus - Marsden & Tromba is a good text. Lots of it is plug and chug, but the problems are nice.
One book which is proofed based, but still full of examples is Advanced Calculus of Several Variables - Edwards Jr.. This is a nice book and is very cheap.
Lastly, I would like to give a bump to Calculus on Manifolds - Spivak. This book is very proofed based, so if you are not comfortable with this, I would sit back and learn from of the others first.
Steven Strogatz is a great one too:
for anyone interested in chaos, Nonlinear Dynamics and Chaos by Steven Strogatz is a great introduction and among many others topics addresses chaos in chemical reactions.
Apparently there's a new edition out too.
http://www.amazon.com/Nonlinear-Dynamics-Chaos-Applications-Nonlinearity/dp/0813349109/ref=dp_ob_title_bk
> This is an amazing book, but it mostly covers ODEs sadly. Both the style and the material covered are great. It might not be exactly what you're looking for, but it's a great read nonetheless.
This book changed my life. I was all set to become an experimental condensed matter physicist. Then I took a course based on Strogatz... and now I've been a mathematical physicist for the last ten years instead.
This will give you some solid theory on ODEs (less so on PDEs), and a bunch of great methods of solving both ODEs and PDEs. I work a lot with differential equations and this is one of my principal reference books.
This is an amazing book, but it mostly covers ODEs sadly. Both the style and the material covered are great. It might not be exactly what you're looking for, but it's a great read nonetheless.
This covers PDEs from a very basic level. It assumes no previous knowledge of PDEs, explains some of the theory, and then goes into a bunch of elementary methods of solving the equations. It's a small book and a fairly easy read. It also has a lot of examples and exercises.
This is THE book on PDEs. It assumes quite a bit of knowledge about them though, so if you're not feeling too confident, I suggest you start with the previous link. It's something great to have around either way, just for reference.
Hope this helped, and good luck with your postgrad!
Here is a good book on trigonometry.
Here is one for algebra.
Here's another
Here's three very good books:
First, please make sure everyone understands they are capable of teaching the entire subject without a textbook. "What am I to teach?" is answered by the Common Core standards. I think it's best to free teachers from the tyranny of textbooks and the entire educational system from the tyranny of textbook publishers. If teachers never address this, it'll likely never change.
Here are a few I think are capable to being used but are not part of a larger series to adopt beyond one course:
Most any book by Serge Lang, books written by mathematicians and without a host of co-writers and editors are more interesting, cover the same topics, more in depth, less bells, whistles, fluff, and unneeded pictures and other distracting things, and most of all, tell a coherent story and argument:
Geometry and solutions
Basic Mathematics is a precalculus book, but might work with some supplementary work for other classes.
A First Course in Calculus
For advanced students, and possibly just a good teacher with all students, the Art of Problem Solving series are very good books:
Middle & high school:
and elementary linked from their main page. I have seen the latter myself.
Some more very good books that should be used more, by Gelfand:
The Method of Coordinates
Functions and Graphs
Algebra
Trigonometry
Lines and Curves: A Practical Geometry Handbook
https://www.amazon.com/dp/0817636773/
https://www.amazon.com/dp/B000HMQ9VU/
I like this book. https://www.amazon.com/Algebra-Israel-M-Gelfand/dp/0817636773
I second the recommendation to find someone more experienced to help you one-on-one. Is there any way you could hire a private tutor? A big benefit of a tutor is that they'll be able to point out the gaps in your knowledge and point you to relevant resources. This can be tough to do on your own or through web discussions. For example, let's say one thing that's holding you back is that you haven't memorized your times table. This would be a major problem and a blind spot for you that would be immediately obvious to me if we were working face to face, but it would be impossible to see from reading your reddit comments.
Let me make a few more concrete suggestions. First, experiment with different study techniques. Take a look at this comment and the linked video. Try the "Feynman Technique" (video) -- this is not easy but it's the only way to really get a solid understanding. Don't expect to be spoon-fed knowledge when you're watching videos: you need to be spending most of your study time with a pen and paper, puzzling out for yourself why things work.
Second, for algebra, I can recommend two textbooks:
Khan Academy is a good supplement, but in my opinion it's too passive to be used as your main resource. It doesn't encourage independent thinking and it has no problems (easy drill exercises don't count as problems.) You need to do lots of problems. In particular, you need to struggle through problems that you're not explicitly told how to solve ahead of time.
Finally, mechanical knowledge is incredibly important, but of course it does need to be built upon a conceptual foundation. For every technique you learn (like solving 2/3 = 3R) you should first be able to explain why the technique works in simple, obvious terms, and then practice it (invent your own problems!) and add it to your collection of techniques. Math is (arguably) simply a grab bag of such techniques together with explanations of why they work. It's often not obvious which technique to apply in a specific case: this can only be learned through experience. Avoid problem sets with ten variants of one specific problem -- they don't teach this skill! Instead look for varied problems which require creativity (Rusczyk's book is a good start.)
You might also want to check out /r/learnmath and #math on freenode if you have more specific questions.
As an introduction to Algebra I can recommend https://www.amazon.com/Algebra-Israel-M-Gelfand/dp/0817636773/ref=sr_1_1?ie=UTF8&amp;qid=1486044711&amp;sr=8-1&amp;keywords=Shen+Algebra You may be also interested in https://www.amazon.com/Prime-Obsession-Bernhard-Greatest-Mathematics/dp/0452285259/ref=sr_1_1?rps=1&amp;ie=UTF8&amp;qid=1486042204&amp;sr=8-1&amp;keywords=prime+obsession+derbyshire
Moreover, it would be nice to watch these 'Youtube' channels: https://www.youtube.com/user/Vihart https://www.youtube.com/user/njwildberger
Gelfand's Algebra is interesting, encourages mathematical thinking, and has the additional advantage of being much more approachable than the books you've listed.
This is probably a much better place to start for someone who's interested in "starting from the basics."
It's just called "algebra" by I.M. Gelfand and another dude.
For highschool level math I reccomend i.m.gelfands books, one of which is Algebra.
They're excellent for self-study, and provide you with many insights not found elsewhere afaik.
Most schools just use 1 textbook for calc 1-3 : http://www.amazon.com/Calculus-James-Stewart/dp/0538497815
Doesn't really matter which edition you get, you're still going to suffer through it.
A popular other book recommended by math majors/professors is
http://www.amazon.com/Calculus-4th-Michael-Spivak/dp/0914098918
You can get the pdf on "certain websites."
Videos will make you lazy and you will likely lose focus and turn to reddit or games or whatever because the professors can be really boring. Just stay focused on the text.
"Just do it."
You could try Principles of mathematical analysis by Rudin. This is too much for me, so be warned.
I find Spivak's Calculus to be a lot more palatable, but I've read less of it than Rudin.
If your Calculus is rusty before Rudin read Spivak Calculus it is great intro to analysis and you will get your calculus in order. Rudin is going to be overkill for you. Also before trying to do proofs read How to prove it It is a great crash course to naive set theory and proof strategies. And i promise i won't bore you with math any more.:D
No, he wrote a book on single-variable calculus, too: https://www.amazon.com/Calculus-4th-Michael-Spivak/dp/0914098918
None of the questions you asked is “silly” or “simple.” There’s a whole lot going on in calculus, most of which is typically explained in a real analysis course. Rigorous proofs of things like the mean value theorem or various forms of integration are challenging, but they will provide the clarity you’re looking for.
I recommend that you check out something like Spivak’s Calculus, which is going to give a more rigorous intro to the subject. Alternately, you can just find a good analysis or intro to proofs class somewhere. It’s a fascinating subject, so good luck!
I would highly advise going with the 31/37 route. As both of the above courses are proof based, they will be play an integral role in upper year courses. Please be warned that they are extremely challenging but worthwhile courses. I would highly recommend you start preparing for the above two courses. For A37, I would suggest starting with Spivak:
https://www.amazon.com/Calculus-4th-Michael-Spivak/dp/0914098918
You can start with Calculus by Spivak. If you're going to buy it then wait until after the Fall semester begins; the price is inflated right now because students need it for school.
This is a PDF of the third edition of the above book.
This is an excellent introduction to logic and proofs. You will want a strong understanding of how mathematicians communicate via proof and that book will really help.
The math subreddit is primarily undergrads talking about various topics. Make a point of just hanging out and reading stuff. If you don't understand something just tell us and they'll do their best to help out.
Hang out on the math stack exchange and ask questions about things you do not understand while trying to help with things you do understand.
Hope that helps!
"I'm also sure that due to my limited educational resources, self-directed study will be a huge part. Any suggestions on which books are must reads to gain competency in CS?"
Here are a few good choices for the more theoretical areas of computing:
http://www.amazon.com/Algorithms-4th-Edition-Robert-Sedgewick/dp/032157351X/ref=sr_1_1?ie=UTF8&amp;qid=1408406629&amp;sr=8-1&amp;keywords=algorithms+4th+edition
http://www.amazon.com/How-Prove-It-Structured-Approach/dp/0521675995/ref=sr_1_1?ie=UTF8&amp;qid=1408406673&amp;sr=8-1&amp;keywords=how+to+prove+it
You'll also want to look at a decent discrete mathematics book. Sadly the book I used as an undergrad was rubbish, so I don't have a good recommendation.
Don't pay too much attention to the other replies - if you really want to take Math 145/146 it's possible, it will just be a lot of work.
My marks were good in high school (but not 95+) and my score on the Euclid was terrible (in order to enrol without an override you need 80+ on the Euclid). The thing to know is these courses have heavy emphasis on proofs, so the summer before coming I worked my way through the first half of a book on proofs and ended up doing relatively well in these courses.
You can certainly do it, but you have to be really dedicated.
If you do decide on it, definitely read this beforehand:
http://www.amazon.ca/How-Prove-It-Structured-Approach/dp/0521675995
This book helped me out a bit: http://www.amazon.com/How-Prove-Structured-Daniel-Velleman/dp/0521675995 -- However, even though I have a background in programming, I felt it moved rather quickly, especially after about halfway through the book.
Oh man 2011 was probably the hardest MATA31 revision. Don't worry, about that midterm though, the course content is really different now, that was when CSC/MATA67 used to be merged with MATA31, so they did a lot more set theory/number theory in MATA31 than they do now. I doubt most people who took MATA31 (and did well) could even pass that midterm just because we don't learn that stuff in MATA31 anymore. If you're trying to get started on studying for MATA31 now, I actually recommend you don't learn MATA31 material. Instead, improve on your critical thinking skills which your high school has definitely not given you. "Find" a book called how to prove it and go through maybe the first two or so chapters which just introduce proofs, and start to build up your proof skills. Becoming comfortable with proofs will come in handy immensely for CSCA67, MATA37, and in a big chunk of MATA31.
You might want to check out Stein and Shakarchi's book Complex Analysis http://press.princeton.edu/titles/7563.html. This book is a bit hard but iirc doesn't require you to have had real analysis before hand. I would highly recommend that you work through a proof based book before hand though. Often times this will be a course book but something like https://www.amazon.com/How-Prove-Structured-Approach-2nd/dp/0521675995?ie=UTF8&amp;*Version*=1&amp;*entries*=0 that should also get the job done.
Or you can go the traditional route like other people mentioned of getting about a semester's worth of real analysis under your belt. The reason why this is usually the suggested path is because it's not expected that you are 100% competent at writing proofs in the beginning of real but you are in complex.
For whichever professor you have for Math 42, I highly recommend you get this book: https://www.amazon.com/How-Prove-Structured-Approach-2nd/dp/0521675995
It definitely saved me a ton. It’s straight to the point, and not as dry as most textbooks can be. Math 32 will be a bit more work, but in my experience just start homework early and don’t be afraid to go to professor office hours and ask questions. Even if they seem distant during class, most professors do appreciate students who make the effort to ask questions. If you need free tutoring in any of your classes, contact Peer Connections. Specifically for math, I believe MacQuarrie Hall room 221 offers drop-in tutoring for free as well! And for physics, Science building room 319 has free drop-in tutoring.
This is the book I used at university. I thought it was pretty good. Velleman's book is also popular. I've heard good things about this book, but I've not read it.
Not to pile on, but as has been previously stated what you wrote is not a proof. I'm not going to focus on whether or not what you said is true or false because the larger problem is that it's not written as a proof structure-wise. By this I mean, proofs are written using logic. If you're really interested in proof writing and basic analysis I suggest this book: http://www.amazon.com/How-Prove-It-Structured-Approach/dp/0521675995/ref=sr_1_1?s=books&amp;ie=UTF8&amp;qid=1331568877&amp;sr=1-1
If you happen to have the UCLA edition of Friedberg's Linear Algebra (the one you'll likely use for 115A) already, there's a section at the end with an intro to proofs. This book is pretty popular at universities with a dedicated intro to proofs class, so it might be worth checking out; I read a bit of it before taking the upper divs. Hope that helps!
Hmm...sorry but a lot of your post shows a lack of mathematical rigor and philosophical understanding of the terms you say. Not trying to offend you, but you really want to practice on proofs.
> Let me see if I understand you OP. You are asserting that by adopting a position where a positive claim (and BTW a claim that something does not exist or does not work is still a positive claim even though the claim involves a negative) must be justified and supported, such as the position of non-belief in the existence of Gods (for or against), or a person is innocent until proven guilty, "harms discourse and is dishonest"? Really?
Except, this is exactly what the burden of proof is? Any claim, positive or negative, must be proven. Yes, even unicorns existing. This has been discussed at length throughout math and philosophy so I don't see how you think (unless you're ignorant) otherwise. Atheist conflict the burden of proof as a legal tenant and one from an epistemological essence. Legal wise, this is more as "innocent until proven guilty" but in no way does that mean x person didn't do it.
Deeper discussion here: https://www.reddit.com/r/philosophy/comments/72o984/the_natural_world_is_all_there_is_as_far_as_we/
>Any claim that purports to be of knowledge has a burden of proof.
>
>Any claim that limits itself merely to belief does not have a burden of proof.
>
>It makes no difference if the claim is theistic (gnostic or agnostic) or naturalistic (strong or weak), nor does it make any difference if it's a claim that a particular thing exists or is true, or that a particular thing does not exist or is not true, or anything else really for that matter. If it's a claim that purports to be of knowledge, it has a burden of proof, and if it's merely a belief, it does not.
Your version of the burden of proof (taken from rational wiki) has no basis in math nor philosophy. Do not get information from rational wiki. Get a copy of many proofs based mathematical books and start from there by actually proving problems.
Again from stack: https://philosophy.stackexchange.com/questions/678/does-a-negative-claimant-have-a-burden-of-proof
>I would say that generally, the burden of proof falls on whomever is making a claim, regardless of the positive or negative nature of that claim. It's fairly easy to imagine how any positive claim could be rephrased so as to be a negative one, and it's difficult to imagine that this should reasonably remove the asserter's burden of proof.
>
>Now, the problem lies in the fact that it's often thought to be extremely difficult, if not actually impossible, to prove a negative. It's easy to imagine (in theory) how one would go about proving a positive statement, but things become much more difficult when your task is to prove the absence of something.
>
>But many philosophers and logicians actually disagree with the catchphrase "you can't prove a negative". Steven Hales argues that this is merely a principle of "folk logic", and that a fundamental law of logic, the law of non-contradiction, makes it relatively straightforward to prove a negative.
Any claim, false or positive requires to be proven. Whether I say for all natural numbers in set N there exists no element such that N\^N <= N\^2. Or I state the inverse "for all natural numbers in set N there exists an element such that N\^N <= N\^2. The burden of proof is on me.
> Or OP, would you just accept that the grobbuggereater exists because I give witness to this existence?
I truly wish my professors were as simple-minded....so many hours could have been saved by proving negative statements in Mathematics and theoretical computer science. However, yes. Philosophically speaking, to claim grobbugereater does not exist requires proof. Grobbugereater is an idea x, where the probability is x / |r| where r is the set of all ideas. as r tends to infinity the probability of grobbugereater existing tends to 0. Thusly, since grobbugereater has no epistemological evidence then we can conclude his probability of existing is infinitely small. This is how you prove grobbugereater does not exist.
One of your claims (presumably) is that induction is better than deduction. That somehow science is far better than math, philosophy, theism, or any other deductive method. Such a claim is metaphysical and cannot be proven via induction thusly a contradiction.
I find it odd, that so many people who use rational claims lack mathematical rigor. Honestly dilutes the topic into a mindless debate and petty insults. Here is a good read to strengthen your skills:
https://www.amazon.com/How-Prove-Structured-Approach-2nd/dp/0521675995
For compsci you need to study tons and tons and tons of discrete math. That means you don't need much of analysis business(too continuous). Instead you want to study combinatorics, graph theory, number theory, abstract algebra and the like.
Intro to math language(several of several million existing books on the topic). You want to study several books because what's overlooked by one author will be covered by another:
Discrete Mathematics with Applications by Susanna Epp
Mathematical Proofs: A Transition to Advanced Mathematics by Gary Chartrand, Albert D. Polimeni, Ping Zhang
Learning to Reason: An Introduction to Logic, Sets, and Relations by Nancy Rodgers
Numbers and Proofs by Allenby
Mathematics: A Discrete Introduction by Edward Scheinerman
How to Prove It: A Structured Approach by Daniel Velleman
Theorems, Corollaries, Lemmas, and Methods of Proof by Richard Rossi
Some special topics(elementary treatment):
Rings, Fields and Groups: An Introduction to Abstract Algebra by R. B. J. T. Allenby
A Friendly Introduction to Number Theory Joseph Silverman
Elements of Number Theory by John Stillwell
A Primer in Combinatorics by Kheyfits
Counting by Khee Meng Koh
Combinatorics: A Guided Tour by David Mazur
Just a nice bunch of related books great to have read:
generatingfunctionology by Herbert Wilf
The Concrete Tetrahedron: Symbolic Sums, Recurrence Equations, Generating Functions, Asymptotic Estimates by by Manuel Kauers, Peter Paule
A = B by Marko Petkovsek, Herbert S Wilf, Doron Zeilberger
If you wanna do graphics stuff, you wanna do some applied Linear Algebra:
Linear Algebra by Allenby
Linear Algebra Through Geometry by Thomas Banchoff, John Wermer
Linear Algebra by Richard Bronson, Gabriel B. Costa, John T. Saccoman
Best of Luck.
https://www.amazon.ca/How-Prove-Structured-Daniel-Velleman/dp/0521675995/ref=sr_1_1?ie=UTF8&amp;qid=1501343615&amp;sr=8-1&amp;keywords=how+to+prove+it
Learn the information of chapter 4 in this book, in particular the stuff from page 90 to the end of the chapter
I also tried to learn calculus through spivak and found it very difficult; I stopped at then 4th chapter and switched to an easier textbook. If it's your first time learning calculus choosing an easier and verbose text like Stewart may suite you better. It's important to remember Spivak's Calculus is more like a textbook on Analysis (the theory of calculus), which is what often comes junior or senior year for math majors/minors.
If you have already learned calculus I'd suggest the bookHow to Prove It which helps think of math in a more concrete way that can help with proofs, even though no calculus is presented. Also, remember that Spivak likely didn't intend for people to find his questions easy, so don't feel like you are unprepared if it takes a while to do a single question.
My two cents
If you want to improve your skills you can do two things in the short term -- read and practice.
I would recommend Basic Mathematics by Lang (it gets mentioned a lot around here). Or if you are interested in higher math look at How to Prove It by Velleman
The great thing is that both include exercises.
I'm not doing this for a class. It's just that I have been drawing a blank whenever there was a question "Show that" or "Prove that". So now I'm working through Velleman's How to prove it. There are answers for some problems but not all. Not for this one. This question is in chapter three and before that there has been covered some logic and set theory. Nothing fancy like rings and abstract stuff.
I like your suggestion that a<b<0 implies 0<-b<-a, and squaring a negative number of course gets a positive number. Wouldn't that be enough for a proof?
Depends what kind of math you're interested in. If you're looking for an introduction to higher (college) math, then How to Prove It is probably your best bet. It generally goes over how proofs work, different ways of proving stuff, and then some.
If you already know about proofs (i.e. you are comfortable with at least direct proofs, induction, and contradiction) then the world is kind of your oyster. Almost anything you pick up is at least accessible. I don't really know what to recommend in this case since it's highly dependent on what you like.
If you don't really know the basics about proofs and don't care enough to yet, then anything by Dover is around your speed. My favorites are Excursions in Number Theory and Excursions in Geometry. Those two books use pretty simple high school math to give a relatively broad look at each of those fields (both are very interesting, but the number theory one is much easier to understand).
If you're looking for high school math, then /u/ben1996123 is probably right that /r/learnmath is best for that.
If you want more specific suggestions, tell me what you have enjoyed learning about the most and I'd be happy to oblige.
> I assume I ought to check it out after my discrete math class? Or does CLRS teach the proofs as if the reader has no background knowledge about proofs?
Sadly it does not teach proofs. You will need to substitute this on your own. You don't need deep proof knowledge, but just the ability to follow a proof, even if it means you have to sit there for 2-3 minutes on one sentence just to understand it (which becomes much easier as you do more of this).
> We didn't do proof by induction, though I have learned a small (very small) amount of it through reading a book called Essentials of Computer Programs by Haynes, Wand, and Friedman. But I don't really count that as "learning it," more so being exposed to the idea of it.
This is better than nothing, however I recommend you get very comfortable with it because it's a cornerstone of proofs. For example, can you prove that there are less than 2 ^ (h+1) nodes in any perfect binary tree of height h? Things like that.
> We did go over Delta Epsilon, but nothing in great detail (unless you count things like finding the delta or epsilon in a certain equation). If it helps give you a better understanding, the curriculum consisted of things like derivatives, integrals, optimization, related rates, rotating a graph around the x/y-axis or a line, linearization, Newton's Method, and a few others I'm forgetting right now. Though we never proved why any of it could work, we were just taught the material. Which I don't disagree with since, given the fact that it's a general Calc 1 course, so some if not most students aren't going to be using the proofs for such topics later in life.
That's okay, you will need to be able to do calculations too. There are people who spend all their time doing proofs and then for some odd reason can't even do basic integration. Being able to do both is important. Plus this knowledge will make dealing with other math concepts easier. It's good.
> I can completely understand that. I myself want to be as prepared as possible, even if it means going out and learning about proofs of Calc 1 topics if it helps me become a better computer scientist. I just hope that's a last resort, and my uni can at least provide foundation for such areas.
In my honest opinion, a lot of people put too much weight on calculus. Computer science is very much in line with discrete math. The areas where it gets more 'real numbery' is when you get into numerical methods, machine learning, graphics, etc. Anything related to theory of computation will probably be discrete math. If your goal is to get good at data structures and algorithms, most of your time will be spent on discrete topics. You don't need to be a discrete math genius to do this stuff, all you need is some discrete math, some calc (which you already have), induction, and the rest you can pick up as you go.
If you want to be the best you can be, I recommend trying that book I linked first to get your feet wet. After that, try CLRS. Then try TAOCP.
Do not however throw away the practical side of CS if you want to get into industry. Reading TAOCP would make you really good but it doesn't mean shit if you can't program. Even the author of TAOCP, Knuth, says being polarized completely one way (all theory, or all programming, and none of the other) is not good.
> From reading ahead in your post, is Skiena's Manual something worth investing to hone my skills in topics like proof skills? I'll probably pick it up eventually since I've heard nothing but good things about it, but still. Does Skiena's Manual teach proofing skills to those without them/are not good at them? Or is there a separate book for that?
You could, at worst you will get a deeper understanding of the data structure and how to implement them if the proof goes over your head... which is okay, no one on this planet starts off good at this stuff. After you do this for a year you will be able to probably sit down and casually read the proofs in these books (or that is how long it took me).
Overall his book is the best because it's the most fun to read (CLRS is sadly dry), and TAOCP may be overkill right now. There are probably other good books too.
> I guess going off of that, does one need a certain background to be able to do proofs correctly/successfully, such as having completed a certain level of math or having a certain mindset?
This is developed over time. You will struggle... trust me. There will be days where you feel like you're useless but it continues growing over a month. Try to do a proof a day and give yourself 20-30 minutes to think about things. Don't try insane stuff cause you'll only demoralize yourself. If you want a good start, this is a book a lot of myself and my classmates started on. If you've never done formal proofs before, you will experience exactly what I said about choking on these problems. Don't give up. I don't know anyone who had never done proofs before and didn't struggle like mad for the first and second chapter.
> I mean, I like the material I'm learning and doing programming, and I think I'd like to do at least be above average (as evident by the fact that I'm going out of my way to study ahead and read in my free time). But I have no clue if I'll like discrete math/proving things, or if TAOCP will be right for me.
Most people end up having to do proofs and are forced to because of their curriculum. They would struggle and quit otherwise, but because they have to know it they go ahead with it anyways. After their hard work they realize how important it is, but this is not something you can experience until you get there.
I would say if you have classes coming up that deal with proofs, let them teach you it and enjoy the vacation. If you really want to get a head start, learning proofs will put you on par with top university courses. For example at mine, you were doing proofs from the very beginning, and pretty much all the core courses are proofs. I realized you can tell the quality of a a university by how much proofs are in their curriculum. Any that is about programming or just doing number crunching is literally missing the whole point of Computer Science.
Because of all the proofs I have done, eventually you learn forever how a data structure works and why, and can use it to solve other problems. This is something that my non-CS programmers do not understand and I will always absolutely crush them on (novel thinking) because its what a proper CS degree teaches you how to do.
There is a lot I could talk about here, but maybe such discussions are better left for PM.
check out this mahfukka http://www.amazon.com/How-Prove-It-Structured-Approach/dp/0521675995/ref=tmm_pap_title_0
Start with a book like this:
http://www.amazon.com/books/dp/0521597188
or this:
http://www.amazon.com/How-Prove-It-Structured-Approach/dp/0521675995
or the one teuthid recommended. When you're doing self-study, it's doubly important to be able to read and follow most of the material.
Advanced math is subjective. Discrete math is a lot of topics mixed together into one class. A little bit of logic, graph theory, set theory, number theory, modular arithmetic, combinatorics, introduction to proofs, algorithm analysis and some other stuff I might be missing. The only prerequisite for it is pre-calculus. The difficulty of the class is subjective some people find it hard and some people find it easy. If you can remember definitions and theorems and string them together to construct a proof you should be fine. How to prove it is recommended a lot as an intro to writing proofs.
I've been studying How To Prove It by Daniel Velleman for a few months now and I don't know if it's the best book, but it's really good and it has opened my mind in so many ways. Plus, it's really cheap for a textbook.
Logic, Number theory, Graph Theory and Algebra are all too much for you to handle on your own without first learning the basics. In fact, most of those books will probably expect you to have some mathematical maturity (that is, reading and writing proofs).
I don't know how theoretical your CS program is going to be, but I would recommend working on your discrete math, basic set theory and logic.
This book will teach you how to write proofs, basic logic and set theory that you will need: http://www.amazon.com/How-Prove-It-Structured-Approach/dp/0521675995
I can't really recommend a good Discrete Math textbook as most of them are "meh", and "How to Prove It" does contain a lot of the material usually taught in a Discrete Math course. The extra topics you will find in discrete maths books is: basic probability, some graph theory, some number theory and combinatorics, and in some books even some basic algebra and algorithm analysis. If I were you I would focus mostly on the combinatorics and probability.
Anyway, here's a list of discrete math books. Pick the one you like the most judging from the reviews:
Don't bother trying to learn too much too soon, as you really do need to let time for the math to sink in.
I've always wanted to recommend this book to someone who knows no math. I find the writing infuriating. It is a dialog but this approach to dialog totally sucks. On the other hand, this is a stunning introduction to categorical logic. It will not help you solve problems etc. but I can guarantee that this book will change your entire outlook on the world.
http://www.amazon.com/Conceptual-Mathematics-First-Introduction-Categories/dp/052171916X/ref=sr_1_1?ie=UTF8&amp;qid=1320560710&amp;sr=8-1
I love this book, personally.
I don't claim to know Category Theory, but I came across it when doing exercises in the beginning part of Chapter 0 by Aluffi. It was very terse, but still understandable. The video seems to be much more relaxed in comparison. It is even more relaxed than Awodey's book which is a much better intro to CT than Aluffi's Chapter 0. In short, it reminds me of Conceptual Mathematics: A First Introduction to Categories by Lawvere/Schnauel a little.
Unfortunately, many books like Spivak or Thomas are going to be very expensive, although you can find scans of them online if you look hard enough.
Dover books are cheap and are often classics, for example Calculus by Kline.
Spivak would be worth it if you plan to go on to study mathematics. It's going to have the rigor (and interesting stuff from a mathematical standpoint) that are omitted or hidden in other texts.
That's great, it reminds me a lot of Calculus by Kline. He takes a similar approach and his introduction perfectly foresaw 60 years ago the problems with math education now.
https://www.amazon.com/Calculus-Intuitive-Physical-Approach-Mathematics/dp/0486404536
A true classic that will make you a beast at calculus:
Calculus: An Intuitive and Physical Approach by Morris Kline
It's old-school but totally awesome. Gives you great explanations for why we use what we use in the mathematical world.
Made me the man I am today.
http://www.amazon.com/Calculus-Intuitive-Physical-Approach-Mathematics/dp/0486404536
Whew, not looking for Stewart or spivak? That's the two ends of the spectrum as far as calculus is concerned.
Maybe check out Morris Kline. Its intuitive and sounds right up your alley (I think)! For vector calc you may need to pick up something more advanced. I hope this helps :)
http://www.amazon.com/gp/aw/review/0486404536/RTE3I14V7OSHN/ref=cm_cr_dp_mb_rvw_1?ie=UTF8&amp;cursor=1
No worries for the timeliness!
For Measure and Integration Theory I recommend Elements of Integration and Measure by Bartle.
For Functional Analysis I recommend Introductory Functional Analysis with Applications by Kreyszig.
And for Topology, I think it depends on what flavor you're looking for. For General Topology, I recommend Munkres. For Algebraic Topology, I suggest Hatcher.
Most of these are free pdf's, but expensive ([;\approx \$200;]) to buy a physical copy. There are some good Dover books that work the same. Some good ones are this, this, and this.
Hello! I'm interested in trying to cultivate a better understanding/interest/mastery of mathematics for myself. For some context:
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To be frank, Math has always been my least favorite subject. I do love learning, and my primary interests are Animation, Literature, History, Philosophy, Politics, Ecology & Biology. (I'm a Digital Media Major with an Evolutionary Biology minor) Throughout highschool I started off in the "honors" section with Algebra I, Geometry, and Algebra II. (Although, it was a small school, most of the really "excelling" students either doubled up with Geometry early on or qualified to skip Algebra I, meaning that most of the students I was around - as per Honors English, Bio, etc - were taking Math courses a grade ahead of me, taking Algebra II while I took Geometry, Pre-Calc while I took Algebra II, and AP/BC Calc/Calc I while I took Pre-Calc)
By my senior year though, I took a level down, and took Pre-Calculus in the "advanced" level. Not the lowest, that would be "College Prep," (man, Honors, Advanced, and College Prep - those are some really condescending names lol - of course in Junior & Senior year the APs open up, so all the kids who were in Honors went on to APs, and Honors became a bit lower in standard from that point on) but since I had never been doing great in Math I decided to take it a bit easier as I focused on other things.
So my point is, throughout High School I never really grappled with Math outside of necessity for completing courses, I never did all that well (I mean, grade-wise I was fine, Cs, Bs and occasional As) and pretty much forgot much of it after I needed to.
Currently I'm a sophmore in University. For my first year I kinda skirted around taking Math, since I had never done that well & hadn't enjoyed it much, so I wound up taking Statistics second semester of freshman year. I did okay, I got a C+ which is one of my worse grades, but considering my skills in the subject was acceptable. My professor was well-meaning and helpful outside of classes, but she had a very thick accent & I was very distracted for much of that semester.
Now this semester I'm taking Applied Finite Mathematics, and am doing alright. Much of the content so far has been a retread, but that's fine for me since I forgot most of the stuff & the presentation is far better this time, it's sinking in quite a bit easier. So far we've been going over the basics of Set Theory, Probability, Permutations, and some other stuff - kinda slowly tbh.
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Well that was quite a bit of a preamble, tl;dr I was never all that good at or interested in math. However, I want to foster a healthier engagement with mathematics and so far have found entrance points of interest in discussions on the history and philosophy of mathematics. I think I could come to a better understanding and maybe even appreciation for math if I studied it on my own in some fashion.
So I've been looking into it, and I see that Dover publishes quite a range of affordable, slightly old math textbooks. Now, considering my background, (I am probably quite rusty but somewhat secure in Elementary Algebra, and to be honest I would not trust anything I could vaguely remember from 2 years ago in "Advanced" Pre-Calculus) what would be a good book to try and read/practice with/work through to make math 1) more approachable to me, 2) get a better and more rewarding understanding by attacking the stuff on my own, and/or 3) broaden my knowledge and ability in various math subjects?
Here are some interesting ones I've found via cursory search, I've so far just been looking at Dover's selections but feel free to recommend other stuff, just keep in mind I'd have to keep a rather small budget, especially since this is really on the side (considering my course of study, I really won't have to take any more math courses):
Prelude to Mathematics
A Book of Set Theory - More relevant to my current course & have heard good things about it
Linear Algebra
Number Theory
A Book of Abstract Algebra
Basic Algebra I
Calculus: An Intuitive and Physical Approach
Probability Theory: A Concise Course
A Course on Group Theory
Elementary Functional Analysis
I bought a copy of Dover's Linear Algebra (Border's Blowout) which I plan to go through after I finish A Book of Abstract Algebra.
I feel like I have a long way to go to get anywhere. :S
http://www.amazon.com/Book-Abstract-Algebra-Edition-Mathematics/dp/0486474178
You can probably handle this book, and it's all of ten bucks anyways.
What do you want to do, though? Is your goal to read math textbooks and later, maybe, math papers or is it for science/engineering? If it's the former, I'd simply ditch all that calc business and get started with "actual" math. There are about a million books designed to get you in the game. For one, try Book of Proof by Richard Hammack. It's free and designed to get your feet wet. Mathematical Proofs: A Transition to Advanced Mathematics by Chartrand/Polimeni/Zhang is my favorite when it comes to books of this kind. You'll also pick up a lot of math from Discrete Math by Susanna Epp. These books assume no math background and will give you the coveted "math maturity".
There is also absolutely no shortage of subject books that will nurse you into maturity. For example, check out [The Real Analysis Lifesaver: All the Tools You Need to Understand Proofs by Grinberg](https://www.amazon.com/Real-Analysis-Lifesaver-Understand-Princeton/dp/0691172935/ref=sr_1_1?ie=UTF8&amp;qid=1486754571&amp;sr=8-1&amp;keywords=real+analysis+lifesaver() and Book of Abstract Algebra by Pinter. There's also Linear Algebra by Singh. It's roughly at the level of more famous LADR by Axler, but doesn't require you have done time with lower level LA book first. The reason I recommend this book is because every theorem/lemma/proposition is illustrated with a concrete example. Sort of uncommon in a proof based math book. Its only drawback is its solution manual. Some of its proofs are sloppy, messy. But there's mathstackexchange for that. In short, every subject of math has dozens and dozens of intro books designed to be as gentle as possible. Heck, these days even grad level subjects are ungrad-ized: The Lebesgue Integral for Undergraduates by Johnson. I am sure there are such books even on subjects like differential geometry and algebraic geometry. Basically, you have choice. Good Luck!
All the books listed can be found on libgen.io
If interest is theoretical mathematics:
Become adept at writing proofs.
I recommend
https://www.amazon.com/Discrete-Transition-Advanced-Mathematics-Undergraduate/dp/0821847899
Do some exercises in the first chapter, and go around the book doing whatever is of interest. I suggest learning about proofs/truth tables, functions, infinite sets, and number theory. This book will have chapters approaching all of these.
After this, you have some choice. I would take a beginners book in any of the following fields
Abstract algebra: https://www.amazon.com/Book-Abstract-Algebra-Second-Mathematics/dp/0486474178
Linear algebra: Linear Algebra Done Right by axler
Analysis: foundations of mathematical analysis by rudin (this will be hard but don’t be afraid!)
Approach each of these books slowly. Do not rush. Self-studying math is HARD. You might only get through 3 pages in a week, but I guarantee that you will get the ropes, and a few weeks later, look back and wonder how it was difficult at all.
In making the choice of what to study first, go to the subjects Wikipedia page or google “should I study x or y first” and you’ll likely find good resources
I highly recommend Pinter's "A Book of Abstract Algebra" for a quick course and handy refresher book.
Many functions don't take real numbers or integers as their arguments. Consider the multiplication of an MxN matrix and an NxM matrix where M != N. The result of which is an NxN matrix. In this context, matrix addition doesn't even have a relation to matrix multiplication.
If you're interested, these relationships are what group theory tries to explore. My favorite book on the subject is A Book of Abstract Algebra
Sure. https://www.amazon.com/Book-Abstract-Algebra-Second-Mathematics/dp/0486474178
There are these videos and there is also this book. The book is better if you struggled the first time, and it includes a short section on number theory.
Dummit (or just D&F), Artin, [Lang] (https://www.amazon.com/Algebra-Graduate-Texts-Mathematics-Serge/dp/038795385X), [Hungerford] (https://www.amazon.com/Algebra-Graduate-Texts-Mathematics-v/dp/0387905189). The first two are undergraduate texts and the next two are graduate texts, those are the ones I've used and seen recommended, although some people suggest [Pinter] (https://www.amazon.com/Book-Abstract-Algebra-Second-Mathematics/dp/0486474178) and Aluffi. Please don't actually buy these books, you won't be able to feed yourself. There are free versions online and in many university libraries. Some of these books can get quite dry at times though. Feel free to stop by /r/learnmath whenever you have specific questions
This is just my perspective, but . . .
I think there are two separate concerns here: 1) the "process" of mathematics, or mathematical thinking; and 2) specific mathematical systems which are fundamental and help frame much of the world of mathematics.
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Abstract algebra is one of those specific mathematical systems, and is very important to understand in order to really understand things like analysis (e.g. the real numbers are a field), linear algebra (e.g. vector spaces), topology (e.g. the fundamental group), etc.
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I'd recommend these books, which are for the most part short and easy to read, on mathematical thinking:
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How to Solve It, Polya ( https://www.amazon.com/How-Solve-Mathematical-Princeton-Science/dp/069111966X ) covers basic strategies for problem solving in mathematics
Mathematics and Plausible Reasoning Vol 1 & 2, Polya ( https://www.amazon.com/Mathematics-Plausible-Reasoning-Induction-Analogy/dp/0691025096 ) does a great job of teaching you how to find/frame good mathematical conjectures that you can then attempt to prove or disprove.
Mathematical Proof, Chartrand ( https://www.amazon.com/Mathematical-Proofs-Transition-Advanced-Mathematics/dp/0321797094 ) does a good job of teaching how to prove mathematical conjectures.
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As for really understanding the foundations of modern mathematics, I would start with Concepts of Modern Mathematics by Ian Steward ( https://www.amazon.com/Concepts-Modern-Mathematics-Dover-Books/dp/0486284247 ) . It will help conceptually relate the major branches of modern mathematics and build the motivation and intuition of the ideas behind these branches.
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Abstract algebra and analysis are very fundamental to mathematics. There are books on each that I found gave a good conceptual introduction as well as still provided rigor (sometimes at the expense of full coverage of the topics). They are:
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A Book of Abstract Algebra, Pinter ( https://www.amazon.com/Book-Abstract-Algebra-Second-Mathematics/dp/0486474178 )
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Understanding Analysis, Abbott ( https://www.amazon.com/Understanding-Analysis-Undergraduate-Texts-Mathematics/dp/1493927116 ).
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If you read through these books in the order listed here, it might provide you with that level of understanding of mathematics you talked about.
For classes like number theory and abstract algebra, I would suggest just picking up a book and attempting to read it. It will be hard, but the main prerequisite for courses like this is some mathematical maturity. That only comes with practice.
Realistically there is probably no preparation that you could have which would prepare you in such a way that a book on advanced mathematics would be super easy.
I like this number theory book
http://www.amazon.com/Number-Theory-Dover-Books-Mathematics/dp/0486682528/ref=sr_1_1?ie=UTF8&amp;qid=1348165257&amp;sr=8-1&amp;keywords=number+theory
I like this abstract algebra book
http://www.amazon.com/Book-Abstract-Algebra-Second-Mathematics/dp/0486474178/ref=sr_1_2?s=books&amp;ie=UTF8&amp;qid=1348165294&amp;sr=1-2&amp;keywords=abstract+algebra
AH HA, one of the few times I will link a dover book in good heart!
http://www.amazon.com/Book-Abstract-Algebra-Edition-Mathematics/dp/0486474178
Pinter offers a fine introduction to abstract algebra.
I haven't yet started practicing for the GRE, but does it include Linear Algebra or Modern/Abstract Algebra? Also is there Calculus on it? I'm taking (or have taken, or will take by the time of the GRE) all of those classes and they're all very interesting. I just bought this book on Abstract Algebra, if you're interested.
I meant it quite literally, something along the lines Linear Algebra by Georgi E. Shilov, but less rigorous.
> Leslie Ballentine's book
these
https://www.amazon.co.uk/Quantum-Mechanics-Development-Leslie-Ballentine/dp/9810241054
https://www.amazon.co.uk/Linear-Algebra-Dover-Books-Mathematics/dp/048663518X/ref=sr_1_1?s=books&amp;ie=UTF8&amp;qid=1542818674&amp;sr=1-1&amp;keywords=linear+algebra+Halmos
Ebay. You have to be careful you're getting the right edition, though.
Here's an international paperback version of a 4th Edition book (about $50 includes shipping) that has a very different cover from the U.S. 4th Edition of the same book (about $270 on Amazon).
The book is incredibly good, it's really very well written, so I'm glad the author got paid their share for my purchase but at $270 there's no way I was going to buy the book for full price. I'd have gone without the book and just googled the topics every week, at that price.
I've started with this course, then I dropped because it touches on things too superficially and there are very few exercises. I find that I learn best doing exercises so I picked up a book (different from the on on the website).
The book which most seem to agree is great for self-study of Discrete Math is Discrete mathematics with applications by Susanna S. Epps. There is a similar book with a similar name - Dicrete Mathematics and its Applications by Rosen - but a lot of people seem to dislike this book. They're frustrated with the fact that proofs are not well explained and the book doesn't prepare the reader for the exercises.
Discrete math gives you a good foundation for learning higher math.
This one is a good book:
Discrete Mathematics with Applications by Susanna Epp.
It deals with logic, sets, relations, counting, proofs etc...you know the stuff you need in higher math. Seeing as you're a CS student, I am guessing Discrete Math would be much more important to you than Calculus.
I'm finishing up a discrete math course using Susanna Epp's textbook, and while it's pretty good, I just see it as a preparation for Concrete Mathematics.
As a side note, I never know how to go about math in how it relates to my major. It takes a lot of effort on my part to really learn the concepts in depth, and I have a hard time justifying that kind of investment before I really know what I'm eventually going to do for work.
The easiest one that I know of is the one by Epp. She doesn't go into the history as much, but her writing style is extremely easy on the brain.
The only "math" you don't learn in secondary education that I would say is truly important to all students of computer science is discrete mathematics. One of my old instructors told me he was trying to get permission to use Discrete Mathematics With Ducks in his curriculum. I flipped through it, it seemed like a pretty good discrete math book, but maybe only if you already had some basic understanding. The more traditional choice might be a better option, but it's much drier.
Every other field of math that has applications in computer science has a narrower breadth of use than discrete mathematics, in my opinion. Learning discrete math well will probably do you more good than learning lots of subjects poorly.
You need Susanna Epp's Discrete Math. Her stuff can be ever so slightly rough at certain spots, but absolutely phenomenal everywhere else. But there's Book of Proof by Richard Hammack[FREE] to help you overcome these difficulties.
For a slightly different take on essentially the same material you could also try Mathematics: A Discrete Introduction by Scheinerman. Also a very gentle book.
Can you get Discrete Mathematics with Applications https://www.amazon.com/dp/0495391328/ref=cm_sw_r_cp_apa_i_lf2vCbG0MVRXJ
> How do you think I'll struggle?
The reason I think you'll struggle is because you can't reverse engineer grad school exams, assuming your program is a decent stats program. You have to be more open to learning more when you need to learn it, whether it be for school or for your professional life. (Yes, I do learn in my full-time job. I've learned around 4-5 different programming languages in the two years I've been in this job.)
One thing I should emphasize is that although you did well in Statistical Methods, the grad-school version of this (in my experience) is much more theoretical and proof-oriented. I got an A in the grad-school version, but it was by no means an easy A, and I tutored the calculus-based probability and statistics material for about 3 years in my undergrad. The class definitely put me on edge at times.
> Any tips to study technique? Should I really never look up any solution? Even if it takes weeks to solve a problem? I don't think my University has any introduction to proof courses, I think you just have to know proof or learn it on your own.
Get your hands on a Discrete Mathematics book - my favorite one is the text by Epp (see https://www.amazon.com/Discrete-Mathematics-Applications-Susanna-Epp/dp/0495391328/ref=sr_1_1?s=books&amp;ie=UTF8&amp;qid=1503020282&amp;sr=1-1&amp;keywords=discrete+mathematics+with+applications). You can read the index of this to get an idea of what it covers. But basically, Discrete Math goes through the logic of how a proof is constructed to guide you through proofs.
> Any tips on that? Never taken notes before, except for in lectures to not fall asleep, but I never look back on the notes I take.
Notes are essential. Our minds only have a finite amount of memory space, and we forget things. You need to take notes with the mindset that you'll probably be using this stuff later, and if you need to re-learn it again, you can read your notes and brush up on it. In my case, I've kept all of my grad school notes.
Thank you. I really want to go into the 'Curse the entire field of CS and maths' level. This is the book I was suggested via PM. I'll try the first approach you suggest, then I'll scale up since we are a small group of people trying to learn together so I think that a couple of heads tackling into a problem might solve it.
I've heard the book How To Prove it is pretty good. Also I'd recommend the Art of Problem Solving books as well for algebra and the likes. (It seems to go over stuff you'd learn in 7th grade, but written at a level adequate for adults).
I would also recommend sites like www.expii.com and www.brilliant.org
Khan academy also has a problem generator iirc.
Get used to proof based mathematics. How to Prove It: A Structured Approach, by Daniel J. Velleman, would be a great start.
EDIT: Ok math that's useful for a STEM major, maybe forget about the proof based math unless you're considering mathematical physics. It's still a good book though.
http://www.amazon.com/The-Triathletes-Training-Bible-Friel/dp/1934030198/ref=sr_1_sc_1?ie=UTF8&amp;qid=1398039218&amp;sr=8-1-spell&amp;keywords=triathalets+training+bible
Totally worth it.
Don't skip proofs and wrestle through them. That's the only way; to struggle. Learning mathematics is generally a bit of a fight.
It's also true that computation theory is essentially all proofs. (Specifically, constructive proofs by contradiction).
You could try a book like this: https://www.amazon.com/Book-Proof-Richard-Hammack/dp/0989472108/ref=sr_1_1?ie=UTF8&amp;qid=1537570440&amp;sr=8-1&amp;keywords=book+of+proof
But I think these books won't really make you proficient, just more familiar with the basics. To become proficient, you should write proofs in a proper rigorous setting for proper material.
Sheldon Axler's "Linear Algebra Done Right" is really what taught me to properly do a proof. Also, I'm sure you don't really understand Linear Algebra, as will become very apparent if you read his book. I believe it's also targeted towards students who have seen linear algebra in an applied setting, but never rigorous and are new to proof-writing. That is, it's meant just for people like you.
The book will surely benefit you in time. Both in better understanding linear algebra and computer science classics like isomorphisms and in becoming proficient at reading/understanding a mathematical texts and writing proofs to show it.
I strongly recommend the second addition over the third addition. You can also find a solutions PDF for it online. Try Library Genesis. You don't need to read the entire book, just the first half and you should be well-prepared.
The Book of Proof was such a great book, I bought a copy that I regularly refer back to. It's full of worked examples, exercises, and explanations. This should be on the bookshelf of every undergrad.
Well, skinnypenis420,
It's hard to study math on your own--it's not as fun as programming or whatnot, and especially not so when you're doing not-so-interesting high school/freshman-level mathematics. And especially when you're not really sure how to even study mathematics on your own--a skill that takes a long time to develop!
I'd say doing something like https://www.amazon.com/Book-Proof-Richard-Hammack/dp/0989472108/ would be a summer well spent. It's a short book and not very difficult. But it will probably be entirely new to you and it's a very important skill set and, most of all, it's realistic for you to study it yourself.
I keep seeing this book recommended in a lot of places. How is it different from the one by Axler and one by Roman?
You'll remember and forget formulae as you use them. It's the using them that makes things concrete in your head.
Once you're comfortable with algebra, trig. I'm assuming you've had geometry, since you were taking algebra 2; if not, geometry as well.
Once you're comfortable with those topics, you'll have enough of the basics to start branching out. Calculus is one obvious direction; a lot people have recommended Spivak's book for that. Introductory statistics is another (far too few people are even basically statistically literate.) Discrete math is yet another possibility. You can also start playing with "problem math", like the Green Book or Red Book. Algebraic structures is yet another possibility (I found Herstein's abstract algebra book pretty easy to read when we used it in school).
Edit: added Amazon links.
Like 50 on amazon but could also try Abebooks and see if there's a cheaper used or international copy.
I don't know much about AI, though I do know that (there's a theme, here) linear algebra gets a starring role. So, if you're currently enjoying linear algebra, continue with that. Axler is frequently recommended, if you want a textbook to go through.
After that it's really up to you what you want to go for next, since you have many paths available. Sipser is a great intro to theoretical CS, but, again, don't spend $200 on it. Try to find it in a library, or use something like this to find a much cheaper international edition.
Edit: Forgot to mention, CLRS is the standard for algorithms, but I'm not sure how useful it is as a primary source for learning. Maybe try to borrow a copy to see if you like it.
u/emiliaslarke My undergrad used “The Book of Proof” in our transition to upper level math class Book of Proof https://www.amazon.com/dp/0989472108/ref=cm_sw_r_cp_api_i_YBEUDbF7NREFN
Its not too hard, is cheap, and gives a good preview
Question about Spivak's Calculus and Ross' Elementary Classical Analysis:
Are they books treating mathematics on the same level? Do they treat the rigorous theoretical foundation and computational techniques equally well? Can each one be an alternative to the other? Could someone please give brief comparative reviews/comments on them?
This question is also on r/learnmath: HERE.
This is good advice. Source: I flunked a private engineering school at age 17, in spite of of being 99th percentile in the ACT. Reason? Besides socialization issues, poor mathematics and academic preparation at my rural high school, where few went to college, let alone out-of-state.
I'm a strong believer in self-education (and self-employment) and am currently rectifying the above-stated issues.
Came here to plug Spivak's Calculus. It's a bit harder and more detailed than most calculus texts used today, but that's because he actually explains all the tricky bits, rather than just using hand-waving to finish those tricky bits. (It was the hand-waving that always left me confused in classroom teaching.) Spivak's Calculus might not be the place to start, but it's where you want to end up, so I want you to know about it.
Peace out, bro, and keep working. We'll make it. ME/EE is a great combo, btw. ME is the first branch of engineering, though it was called something else, when "engines of war", catapaults and whatnot, was the only game in town. But, all machines need sensors, controls, and power, which is the EE bit. Put it together, and you get mechatronics, which is part of the future.
One piece of added advice: stick to one of the main-line branches of engineering: mechanical, electrical, chemical, maybe civil, instead of one of the new, hybrid branches, like biomedical, etc. The jobs are more plentiful, you'll get a sounder foundation in engineering principles, and specializing is still possible.
Ed: Do you already know about MIT's Open Course Ware site? Most MIT courses are online with videoed lectures, recommended textbooks, homework and tests. It's a great resource. They also have edX, a co-operative venture with a bunch of fancy schools.
Calculus - Michael Spivak
Sorry, I went on vacation and totally blanked about posting these for you!
Anyway, here are some books
Linear Algebra Done Right (Undergraduate Texts in Mathematics) https://www.amazon.com/dp/3319110799/ref=cm_sw_r_cp_api_1L8Byb5M5W9D3
This one is actually for analysis but depending on your appetite, it might help greatly with the proof side of your class. You can buy it here: Counterexamples in Analysis (Dover Books on Mathematics) https://www.amazon.com/dp/0486428753/ref=cm_sw_r_cp_api_GS8BybQWYBFXX
But there's also a PDF hosted here: http://www.kryakin.org/am2/_Olmsted.pdf
A cool Mark Allen article, to get you in the mood
THE ONE TRUE BOOK (Accept no imitations)
Hal Higdon's Site
Joe Friel's book about how too piss on your bike seat, I mean, complete a triathlon
You could try Book of Proof by Richard Hammack. I've never read Velleman so I can't directly compare, but it's free for pdf (link to author's site above) and quite cheap in paperback (~$15). I found the explanations quite clear, the examples well worked and the exercises plentiful and helpful. Amazon reviewers seem to like it as well.
First, to get a sense as to the world of math and what it encompasses, and what different sub-subjects are about, watch this: https://www.youtube.com/watch?v=OmJ-4B-mS-Y
Ok, now that's out of the way -- I'd recommend doing some grunt work, and have a basic working knowledge of algebra + calculus. My wife found this book useful to do just that after having been out of university for a while: https://www.amazon.com/No-bullshit-guide-math-physics/dp/0992001005
At this point, you can tackle most subjects brought up from first video without issue -- just find a good introductory book! One that I recommend that is more on computer science end of things is a discrete math
book.
https://www.amazon.com/Concrete-Mathematics-Foundation-Computer-Science/dp/0201558025
And understanding proofs is important: https://www.amazon.com/Book-Proof-Richard-Hammack/dp/0989472108
There is a lot in the way of resources for new triathletes these days. For your first tri, grab a free training plan online that matches where you are now. Read Beginner Triathlete in your free time; it's a fantastic resource, and I still refer back to its articles all the time. Train your butt off. You don't need to buy a sweet road bike up front, though you sound like you're pretty sure that you want to get into this stuff.
Feel free to skimp on some of the gear for your first race. No one wants to find out that they dislike triathlon after dumping $3k on tri gear. You can race on an old bike with platform pedals. Unless it's really cold, you don't need a wetsuit. The first race is where you truly find out if this is the sport for you. EDIT: Someone mentioned a bike fit. If you're riding an old bike, Competitive Cyclist's Bike Fit Calculator will get you pretty darn close--good enough to get through your first race. Use the road calculator mode if you don't have aerobars off the bat.
After you finish your first race, sit down and think about what you liked, what you did well with, what needs improvement. Get Joe Friel's Triathlete's Training Bible, read it cover to cover. Read it again. Figure out your long-term training plan for the rest of that season. If you start your base training in the winter/early spring and pick an early first race, you can get a full season of sprints and/or Olympics in.
Look for a triathlon club in your area or find a coach or drag a friend into the insanity of triathlon; the camaraderie is priceless in keeping your spirits up during long seasons packed full of hard training and races.
As far as spending money on triathlon "stuff" goes: Remember during your first couple seasons that gadgets and gizmos and aero gear are great, but what really makes the difference is eating well and training hard.
After that, the gear that makes your races more comfortable is the best place to spend your money (tri shorts if you don't them, cycling kit and proper running shorts for training). Then, points of contact with the bike and pool "toys" will improve your efficiency and form (new bike w/ fit if req'd, clipless pedals, shoes, aerobars, pull buoy, kickboard, fins, paddles... a bike computer probably fits in here, as well). Beyond that, you're at a wetsuit and then the "extras" like aero helmet, race wheels, power meters, GPS, HRM, tri bike, speedsuits, etc., etc. That's the approximate map for spending in my book, anyhow. There's practically no limit to the amount of stuff you can buy for triathlon, and as you train more, you'll know what needs to come next.
There's this, The Book of Proof https://smile.amazon.com/dp/0989472108/ref=cm_sw_r_cp_awdb_t1_jW04BbNE8ZA19
Here ya go: https://www.amazon.com/Book-Proof-Richard-Hammack/dp/0989472108/ref=sr_1_1?ie=UTF8&amp;qid=1500351882&amp;sr=8-1&amp;keywords=the+book+of+proof
this is the book: http://www.amazon.com/Linear-Algebra-Edition-Stephen-Friedberg/dp/0130084514
There is no "one fastest" method to solving them.
Systems of equations are systematic, and it really depends on the problem. The only real way to learn about this is to take a course in Linear Algebra. That is all about systems of linear equations.
But these show up all of the time, here is what I usually do:
If I just need one of the 2-3 variables, Cramer's Rule is a good way to test solvability and extract a single value.
On normal 2x2 systems, I usually do a quick determinant/matrix inverse. Checks the rank as well as the det, and it is always going to work.
On 3x3 or higher systems, it depends. This is why Linear Algebra is important.
Supposedly Linear Algebra Done Right is a good book on the subject, so if you're interested there is one way. The book I used was A custom edition of this one. I thought it was very good as well.
I've been reviewing linear algebra recently and found that I like my old textbook much more now than when I took the course.
https://www.amazon.com/Linear-Algebra-4th-Stephen-Friedberg/dp/0130084514
Its not very good on visual intuition but there are a lot of examples. You could supplement it with the 3blue1brown series for that.
It covers a lot of the topics i needed to review for group theory. For example, it covers dual spaces and the transpose in the second chapter (it stresses invariant subspaces, projection operators, bilinear forms- essentials for group theory.). It's clear, concise and seems popular. One of the prof.s featured on Numberphile said he used it for his course. It might not be a good first linear algebra book for some people. But check it out.
I'm a huge fan of linear algebra. My favorite book for a theoretical understanding is this book. A pdf copy of the solutions manual can be found here.
Linear algebra by Friedberg, Insel and Spence
http://www.amazon.ca/Linear-Algebra-Edition-Stephen-Friedberg/dp/0130084514
You can also check Chartrand's et al book:
https://www.amazon.com/Mathematical-Proofs-Transition-Advanced-Mathematics/dp/0321797094
Mathematical Proofs: A Transition to Advanced Mathematics was my undergrad discrete text and I've gone back to it for review over the years, I think it's a very fine text especially for an introduction to proofs.
That's not the standard 310 textbook. This is. Also, in 310 you go over the first 3 chapters of the 410 book. I'm not disagreeing with your comment otherwise though.
A course I took previously used this book; it has a chapter on introductory real analysis, which is what you want to get at. I would not suggest going directly to a book like Rudin, as he (in my opinion) tends to amplify the "general route" problem that you mention.
I graduated w/ degree in Math n' Physics but have been doing programming for startup for last 5+ years so many of my math skills got rusty.
While trying to get back into it went through several books and have found this to be the best if you're interested in more advanced mathematics: https://www.amazon.com/Mathematical-Proofs-Transition-Advanced-Mathematics/dp/0321797094. It's not only been an excellent review but has fleshed out some areas I was weak (in higher level courses like complex analysis, topology, group theory the methodology of proofs was assumed and often not taught).
The explanations are solid, varied, and they go through each proof they present (often w/ exhaustive step-by-step details).
From there pick a domain you're interested in and pickup the relevant undergraduate (and maybe some graduate) level books/textbooks and see if you can pick it up.
As others mentioned, it is very hard to make progress learning programming without using a computer (think of reading about driving without ever driving a car). Instead, get yourself excited about science and computer science:
Science:
Computer Science (actually math, but this will help change the way you think to be more analytical, and will be useful for programming, vector graphics, etc.):
I think linear algebra is a much more interesting topic without getting bogged down in matrix computations, such as what Axler does with Linear Algebra Done Right. That's just my opinion I suppose.
Whoa, great questions, but I think you want a textbook, not a reddit post response. I used Dummit & Foote but it is probably a bit "heavier" than what you want/need at this point.
Griffiths' Quantum Mechanics has a crash course in most of the linear algebra required to do a first course in quantum mechanics. It's not very complicated - you just need basic understanding of vector spaces, linear transformations and functionals, and inner products, with a little bit of practice using dual notation of vectors (not too much, just enough for the Dirac notation which the book explains). Griffiths' also has a good explanation of simple fourier series/transform.
The key thing is being able to do basic linear algebra without matrices since in most of the cases, the vector space is infinite dimensional. But spin is a good example where almost everything can be done with matrices.
Additionally, solving ordinary differential equations and using separation of variables for partial differential equations in 3-d quantum mechanics would help.
Group theory will be of help in more advanced classes. Dummit and Foote or Arton's books on algebra are decent introduction. They are a bit dense though. If you want a real challenge, try Lang's Algebra book. I don't know of any easier books though. My first algebra book was Dummit and Foote which can be done without any real prerequisites beyond matrix algebra, but isn't really well written.
Links to books: Griffiths, Dummit and Foote.
PS: I have ebooks of these two books in particular.
Yes: Dummit and Foote. I used it in my freshman algebra class. It has excellent proofs and exercises. It will teach you the mathematical maturity faster than analysis and will most likely be more useful to you later on.
http://www.amazon.com/Calculus-Intuitive-Physical-Approach-Mathematics/dp/0486404536/ref=sr_1_1?s=books&amp;ie=UTF8&amp;qid=1405668438&amp;sr=1-1&amp;keywords=calculus+an+intuitive+and+physical+approach
Starts out with a brief history of calculus in chapter 1.
Chapter 2 is derivatives.
Chapter 3 is anti-derivatives
Chapter 4 talks about the geometric importance of the derivative...etc..
Chapter 21 talks about multivariable functions and geometric representation then 22 is over partial differentiation, 23 multiple integrals then an introduction to diff eq.
I don't know if that's what you're looking for.. but its been an excellent read so far, and it tends to be written in layman's terms(great for me) rather than math speak.
Not specific to calculus, but Daniel Velleman's How to Prove It is an excellent and thorough introduction to the practice/art of proof-writing, and it requires only a high school mathematics education as a prerequisite.
Math major courses after the first year consist of proof based courses. If you already have interest in proof writing and argumentation then I suggest you go for it. If not then you try it out. Firstly, you should brush up on basic logic. Check out the quick introduction at berkeley. Once that's cleared, go complete a book such as How To Prove It. Now, you should have the most basic tools needed to tackle and solve problems in future courses in analysis, algebra, topology, etc. The time required to learn and write good proofs is steep. It takes constant feedback and solving numerous problem to get a knack of. I would say that you should get used to proof writing in a semester. Overall, learning to write proofs isn't very difficult but learning the material for the courses and developing a solid theoretical understanding is.
In regard to your concerns about time commitment with courses, it depends on how far you are willing to go to obtain a career in mathematics and to do deep research.
I'm planning on relearning calculus also. The books that were recommended to me were:
http://www.amazon.com/gp/aw/d/1592575129?pc_redir=1412262976&amp;robot_redir=1
http://www.amazon.com/gp/aw/d/0716731606/ref=pd_aw_sims_3?pi=SL500_SY115&amp;simLd=1
They're not exactly textbooks, but they appear to be good guides. Best of luck.
Would probably have to say Calculus on Manifolds by Spivak.
I think you were looking for things that weren't necessarily textbooks, but I think this book is still popular...amongst analysis courses.
Intro to Discrete Math will ease you into Abstract Algebra. I find this book by Susanna Epp phenomenal for that.
Buy a training plan off amazon and follow it rigorously.
Something like
this
or this
Spivak's Calculus is a great resource that I used for a real analysis class. The first exercise is something on par with proving that 1+1=2 and it goes on to build all of Calculus from there.