Best algebra books according to redditors
We found 1,130 Reddit comments discussing the best algebra books. We ranked the 347 resulting products by number of redditors who mentioned them. Here are the top 20.
We found 1,130 Reddit comments discussing the best algebra books. We ranked the 347 resulting products by number of redditors who mentioned them. Here are the top 20.
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.
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&ie=UTF8&qid=1312542911&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&pf_rd_s=lpo-top-stripe-1&pf_rd_t=201&pf_rd_i=0155510053&pf_rd_m=ATVPDKIKX0DER&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.
No, it's not. Math-major algebra was typically taught from something like Herstein. These days, Dummit and Foote seems more popular.
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.
I was in the same position as you in high school (and am finishing my math major this semester). Calculus is not "math" in the sense you're referring to it, which is pure mathematics, without application, just theory and logic. Calculus, as it is taught in high school, is taught as a tool, not as a theory. It is boring, tedious, and has no aesthetic appeal because it is largely taught as rote memorization.
Don't let this bad experience kill your enthusiasm. I'm not sure what specifically to recommend to you to perk your enthusiasm, but what I did in high school was just click around Wikipedia entries. A lot of them are written in layman enough terms to give you a glimpse and you inspire your interest. For example, I remember being intrigued by the Fibonacci series and how, regardless of the starting terms, the ratio between the (n-1)th and nth terms approaches the golden ratio; maybe look at the proof of that to get an idea of what math is beyond high school calculus. I remember the Riemann hypothesis was something that intrigued me, as well as Fermat's Last Theorem, which was finally proved in the 90s by Andrew Wiles (~350 years after Fermat suggested the theorem). (Note: you won't be able to understand the math behind either, but, again, you can get a glimpse of what math is and find a direction you'd like to work in).
Another thing that I wish someone had told me when I was in your position is that there is a lot of legwork to do before you start reaching the level of mathematics that is truly aesthetically appealing. Mathematics, being purely based on logic, requires very stringent fundamental definitions and techniques to be developed first, and early. Take a look at axiomatic set theory as an example of this. Axiomatic set theory may bore you, or it may become one of your interests. The concept and definition of a set is the foundation for mathematics, but even something that seems as simple as this (at first glance) is difficult to do. Take a look at Russell's paradox. Incidentally, that is another subject that captured my interest before college. (Another is Godel's incompleteness theorem, again, beyond your or my understanding at the moment, but so interesting!)
In brief, accept that math is taught terribly in high school, grunt through the semester, and try to read farther ahead, on your own time, to kindle further interest.
As an undergrad, I don't believe I yet have the hindsight to recommend good books for an aspiring math major (there are plenty of more knowledgeable and experienced Redditors who could do that for you), but here is a list of topics that are required for my undergrad math degree, with links to the books that my school uses:
And a couple electives:
And a couple books I invested in that are more advanced than the undergrad level, which I am working through and enjoy:
Lastly, if you don't want to spend hundreds of dollars on books that you might not end up using in college, take a look at Dover publications (just search "Dover" on Amazon). They tend to publish good books in paperback for very cheap ($5-$20, sometimes up to $40 but not often) that I read on my own time while trying to bear high school calculus. They are still on my shelf and still get use.
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.
This is from Paolo Aluffi's excellent Algebra: Chapter 0, which uses categories as a unifying theme.
A groupoid is a small category in which every morphism is an isomorphism. An automorphism of an object A of a category C is an isomorphism from A to itself. The set of automorphisms of A is denoted Aut_C(A).
Edit: added that groupoids are small categories (thank you cromonolith)
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!
ProfRobBob Youtube - This sir has great videos. His playlists are in order and very useful for Calculus. Loved his pre calculus playlist.
Patrick JMT - I could not have passed Calculus 2 without this guy. For the most part, his Calculus section is in order on his website.
KhanAcademy - Nice courses with problems available for you. Really easy to use and navigate. I worked through Algebra and only watched his videos on Trigonometry and Calculus.
Hope you get back on track buddy. Don't give up.
I self taught myself Algebra through Precalculus, here are books I used:
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/
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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
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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 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.
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.
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
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&pf_rd_s=lpo-top-stripe-1&pf_rd_t=201&pf_rd_i=0134843460&pf_rd_m=ATVPDKIKX0DER&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.
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.
You can try something like Artin or Dummit & Foote.
Harvard’s Benedict Gross gave a course using Artin’s textbook with lectures available on youtube.
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.
/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
Algebra
Trigonometry
Functions and Graphs
These are three books that I would recommend to somebody trying to prepare for calculus. They're all written by the mathematician Gelfand and his colleages, and they're some of the best-written math books I've ever read. You come away from reading them really understanding the subject matter. I'd read them in that order, too.
I found Axler's Linear Algebra Done Right to be a very easy to digest introduction to abstract linear algebra.
E-books are poorly discounted (if at all,) cannot be bought used, and cannot be sold to recover some of the purchase cost. Is it any wonder students prefer paper?
Picking a Calculus book at random. Which would be more appealing to a cash starved student. Spending $208 or spending $167 with a strongly likely hood of reselling for ~$60 after fees. In this case paper is potentially half the cost of an e-book.
Damn, son. That's way bigger than my guesstimate.
The amazon prices I checked out pinned the collection closer to $400, which granted is still really, really impressive.
In case you're curious this was my textbook. It's come down by a lot in price over a couple years. Brand new it was $365 in the shrink wrap from my school's store!
Eh, either way I'm wrong, just by a different amount.
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.
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.
Practical Algebra: A Self-Teaching Guide
really helped me a couple of years ago when I had to get up to speed on algebra quickly.
Beyond that, you can hardly do better in the best-bang-for-the-buck department than the Humongous Books series. 1000 problems in each book, annotated and explained, and he has an entertaining style.
The Humongous Book of Algebra Problems: Translated for People Who Don't Speak Math
The Humongous Book of Geometry Problems: Translated for People Who Don't Speak Math
The Humongous Book of Calculus Problems: For People Who Don't Speak Math
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!
I enjoyed the class. The professor was awesome, so that helped. I thought it was pretty easy, but I think that was because I had already been introduced to proofs. We did some Number Theory, Set Theory, Counting, Relations, Modular Arithmetic, Functions, Limits, Axiom of Choice and the Cantor-Schroder-Bernstein Theorem. We spent roughly two weeks on each subject, so we didn't go too in depth. At the end, we did some combinatorics because the professor likes combinatorics.
The book we used was A Transition to Advanced Mathematics by Douglas Smith. I didn't really use it at all. Our notes were sufficient.
I definitely think introduction to proof classes are helpful (and fun), but I would rather the school recommend a book to read over the summer so there is more room for another math elective. Naturally, this depends on the motivation of the school's students. My school has a bunch of lazy blobs. I doubt more than 5 would read a book over the summer.
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.
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).
the "vanilla" books are IMO quite boring to read - especially when you don't know more than Set/Functions.
but I really enjoy P. Aluffi; Algebra: Chapter 0 that builds up algebra using CT from the go instead of after all the work
----
remark I don't know if this will really help you understanding Haskell (I doubt it a bit) but it's a worthy intellectual endeavor all in itself and you can put on a knowing smile whenever you hear those horrible words after
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
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.
Category theory is not easy to get into, and you have to learn quite a bit and use it for stuff in order to retain a decent understanding.
The best book for an introduction I have read is:
Algebra (http://www.amazon.com/Algebra-Chelsea-Publishing-Saunders-Lane/dp/0821816462/ref=sr_1_1?ie=UTF8&amp;qid=1453926037&amp;sr=8-1&amp;keywords=algebra+maclane)
For more advanced stuff, and to secure the understanding better I recommend this book:
Topoi - The Categorical Analysis of Logic (http://www.amazon.com/Topoi-Categorial-Analysis-Logic-Mathematics/dp/0486450260/ref=sr_1_1?ie=UTF8&amp;qid=1453926180&amp;sr=8-1&amp;keywords=topoi)
Both of these books build up from the basics, but a basic understanding of set theory, category theory, and logic is recommended for the second book.
For type theory and lambda calculus I have found the following book to be the best:
Type Theory and Formal Proof - An Introduction (http://www.amazon.com/Type-Theory-Formal-Proof-Introduction/dp/110703650X/ref=sr_1_2?ie=UTF8&amp;qid=1453926270&amp;sr=8-2&amp;keywords=type+theory)
The first half of the book goes over lambda calculus, the fundamentals of type theory and the lambda cube. This is a great introduction because it doesn't go deep into proofs or implementation details.
Linear Algebra Done Right is a good introduction, but if you want to go beyond an undergraduate level, try Linear Algebra by Hoffman and Kunze.
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.
Hrrumph. Determinants are a capstone, not a cornerstone, of Linear Algebra.
https://www.amazon.com/Linear-Algebra-Right-Undergraduate-Mathematics/dp/0387982582
>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.
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.
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.
I actually bought http://www.amazon.com/Book-Abstract-Algebra-Second-Mathematics/dp/0486474178
On chapter three now :D
Linear algebra is an essential tool in many areas of mathematics. Computations with matrices aren't always that important; far more important are the concepts of vector space and linear transformation. Pretty much any time you work with coordinates, dimension, changes of coordinates, vectors, linear relations, or anything like that, you're going to need some linear algebra.
If you're interested, I recommend taking a look at Axler's Linear Algebra Done Right. Axler has very clear exposition and proofs, and if you've only seen the computational aspect of linear algebra, it'll provide a different, more abstract and conceptual perspective.
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.
For Calculus:
Calculus Early Transcendentals by James Stewart
^ Link to Amazon
Khan Academy Calculus Youtube Playlist
For Physics:
Introductory Physics by Giancoli
^ Link to Amazon
Crash Course Physics Youtube Playlist
Here are additional reading materials when you're a bit farther along:
Mathematical Methods in the Physical Sciences by Mary Boas
Modern Physics by Randy Harris
Classical Mechanics by John Taylor
Introduction to Electrodynamics by Griffiths
Introduction to Quantum Mechanics by Griffiths
Introduction to Particle Physics by Griffiths
The Feynman Lectures
With most of these you will be able to find PDFs of the book and the solutions. Otherwise if you prefer hardcopies you can get them on Amazon. I used to be adigital guy but have switched to physical copies because they are easier to reference in my opinion. Let me know if this helps and if you need more.
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.
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'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.
This is the book I’m using right now in my first proofs class it’s pretty good at explaining the thought processes as well as it can be paired with How to Prove It by Daniel J. Velleman for a more through brake down of them problem types.
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.
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.
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.
Just as an add-on, Stewart's is definitely the best way to go for learning applied calculus as a beginner. It's EXHAUSTIVE, though I'd actually recommend the full-on "Calculus" textbook instead of "Early Transcendentals" or "Single (Multi) Variable" texts for this reason:
At the end of every chapter, there are "problems plus" that will really challenge the way you think about what you've just learned. You don't get these in the other books. They'll make you think like a mathematician or a scientist instead of a "plug-and-chugger."
And once again, I'm gonna plug Smith's "Transition to Advanced Mathematics" for an introduction to proof-writing and set theory and the most basic of analysis. http://www.amazon.com/Transition-Advanced-Mathematics-Douglas-Smith/dp/0495562025/ref=sr_1_1?ie=UTF8&amp;qid=1371247275&amp;sr=8-1&amp;keywords=douglas+smith+transition
Though definitely get an older edition to save more money. And I realize you can't get books delivered--you can find pdf versions of older editions.......
As for the lower, pre-calculus stuff, just look to the right on this reddit for Khan Academy and just browse through the topics there. If you're as good a student as you say you are, you just need the few holes filled in and a quick refresher, and Khan is perfect for it.
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)
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.
I would advise you not to start with category theory, but abstract algebra. Mac Lane and Birkhoff's book Algebra is excellent and well worth the money in hardback. It covers things like monoids, groups, rings, modules and vector spaces, all of which are -- not coincidentally -- typical examples of structures that form categories. Saunders Mac Lane invented category theory along with Samuel Eilenberg, and Birkhoff basically founded universal algebra, so you cannot find a more authoritative text.
Edit: The other thing that will really help you is a basic understanding of preorders and posets. I don't have a book that deals exclusively with this topic, but any introduction to lattice theory, logical semantics or denotational semantics of programming languages will treat it. I would recommend Paul Taylor's Practical Foundations of Mathematics, though the price on Amazon is very steep. You can look through it here: http://paultaylor.eu/~pt/prafm/
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.
Linear Algebra and [Linear Algebra and Its Applications] (http://www.amazon.com/dp/0321385179).
Axler's Linear Algebra Done Right is something you might enjoy looking at; since his basic point of view is that linear algebra is generally done wrong.
http://www.amazon.com/Linear-Algebra-Right-Undergraduate-Mathematics/dp/0387982582
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!
I had a friend who went through the program. I don't think there was a pre-assessment as Academic Math itself is a prerequisite to other stuff, but don't take my word as law on that. The course resource appears [to be here] (https://www.nscc.ca/learning_programs/programs/PlanDescr.aspx?prg=ACC&amp;pln=ACCONNECT) and doesn't mention pre-assessments. [This PDF] (http://gonssal.ca/documents/AcadMathIVCurr2010.pdf) should cover a fair bit of what the course is about.
As an aside, [this book] (https://www.amazon.ca/Practical-Algebra-Self-Teaching-Peter-Selby/dp/0471530123) is a fantastic way to get yourself up to speed on algebra. I can't recommend it highly enough.
How long ago did you do it? I work with calculus and statistics a lot and I often go back to earlier concepts to make sure my foundations are still strong.
I would recommend looking at this book and just quickly running through the exercises. That will give you a good idea about what you need to focus on. If you feel comfortable with those, something like this might be good to check out since it's made for self-teaching as opposed to being used in conjunction with a class.
Edited to add: math is like any language, in that the more you practice and manipulate numbers the better you'll be at it!
/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.
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.
Reading through Algebra: Chapter 0
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.
8 to 12 hours is really not that much, but it should be enough to learn something interesting! I would start with category theory if you can. I liked Emily Riehl's categories in context for an intro, but it will go a little slow for how little time you have to learn the basics. Maybe the first chapter of Algebra: Chapter 0 by Aleffi? [EDIT: you might want to find a "reasonably priced" pdf version of this book if you do decide to use it -- it's pretty expensive] If you can get through that, and understand a little about how types fit into the picture, you should be able to present the basic idea behind curry-howard-lambek. IIRC you do not need functors or natural transformations ("higher level" categorical concepts), as important as they usually are, to get through this topic; Aleffi doesn't go over them in his very first intro to categories which is why I'm recommending him. /u/VFB1210 has some very good recommendations above as well.
I am trying to think of a better introduction to type theory than HoTT -- if you can learn about types without getting infinity categories and homotopy equivalence mixed up in them, I would. Type theory is actually pretty cool and sleek.
Here's a selection of intro-to-type theory resources I found:
Programming in Martin-Löf's Type Theory is
pretty long, but you can probably put together a mini-course as follows: read chapters 1 & 2 quickly, skim 3, and then read 19 and 20.
The lecture notes from Paul Levy's mini-course on the typed lambda calculus form a pretty compact resource, but I'm not sure this will be super useful to you right now -- keep it in mind but don't start off with it. Since it is in lecture-note style it is also pretty hard to keep up with if you don't already kind of know what he's talking about.
Constable's Naïve Computational Type Theory seems to be different from the usual intro to types -- it's done in the style of the old Naive Set Theory text, which means you're supposed to be sort of guided intuitively into knowing how types work. It looks like the intuition all comes from programming, and if you know something functional and hopefully strongly typed (OCaml, SML, Haskell, or Lisp come to mind) you will probably get the most out of it. I think that's true about type theory in general, actually.
PFPL by Bob Harper is probably a stretch -- you won't find it useful right at the moment, but if you want to spend 2 semesters really getting to know how type theory encapsulates pretty much any modern programming paradigm (typed languages, "untyped" languages, parallel execution, concurrency, etc.) this book is top-tier. The preview edition doesn't have everything from the whole book but is a pretty big portion of it.
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.
You'll usually find the following recommended:
I've personally used Friedberg's text, and I found it to be pretty well written.
I posted this in another comment but I'm guessing this bastard?
https://www.amazon.com/Calculus-Early-Transcendentals-James-Stewart/dp/1285741552/
I learned using this book by Larson. It goes out of its way to be intelligible, and I appreciated that. It's hard to recommend things sometimes, because I think everybody has a different path to understanding these topics. A lot of the time it seems you need to just keep throwing resources at it until something sticks. Good luck.
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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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.
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
Fundamentals of Heat and Mass Transfer by Incropera is pretty much the standard text on the subject by my understanding.
I used Hibbeler for Mechanics of Materials, but Beer is also a popular choice.
Hibbeler for dynamics as well.
Larson has a pretty good calculus book, will take you from derivatives up through multivariable.
A good resource if you feel like digging deeper is the physics forums - science and math textbook forum.
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!
Some possibilities:
Calc I & II: Spivak's Calculus
Calc III and a bit of linear algebra: Hubbard & Hubbard's Vector Calculus
LA: Axler or Shilov or both
ODE: Morris Tanenbaum
Discrete/Combinatorics/etc.: Knuth's Concrete Mathematics
For book suggestions beyond concerning Analysis, Algebra, and Topology, the search box will turn up a ton of previous conversations.
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.
Shilov gives a rigorous, determinant-heavy treatment of LA in his $10 book. All the nice properties of determinants are verified in the first chapter
The most popular calculus book for college classes in the United States is Stewart, Calculus: Early Transcendentals. A typical Calculus II course starts somewhere in chapter 5 or 6 (picking up wherever Calculus I left off) and ends with chapter 11.
This book has answers to all of the odd-numbered exercises in the back, so it works reasonably well to read the book and then try the exercises. Typically the first 3/4 of the exercises in each section are straightforward, and the remaining 1/4 are more difficult and would only be assigned in an honors class.
Just buy a Calculus textbook and watch all of the videos on PatrickJMT/KhanAcademy.
I took the calculus sequence at a University, but 90% of what I learned was from the book and online resources.
>It gives me that mental stimulation I desire and that I feel I am genuinely am good at and don't need to have talent for because no matter what, so long as I put in the effort, then I got it down.
That's exactly the right attitude to have. :)
If I can make a recommendation, pick yourself up a copy of "A Transition to Abstract Mathematics" or a similar text and start working your way through it. You start with logic tables and learn about set theory. You'll enjoy it if you are interested in the "whys" of math, and if you end up picking math as a major, it will be helpful stuff to review ahead of time.
> Is there any good book with problems/examples that I could work through in order to thoroughly prepare myself to be able to write proofs for a Real Analysis I course?
Besides Velleman's "How to Prove it," try Mathematical Proofs: A Transition to Advanced Mathematics or maybe How to Read and Do Proofs: An Introduction to Mathematical Thought Processes.
The book I used in my "Intro to Proofs" course was A Transition to Advanced Mathematics. It was pretty good, but the edition that I used had several mistakes in it. Also, it's waaaay too expensive.
Now for the unpleasantries —
Suggestions aside, the main problem here is your "thoroughly prepare myself to be able to write proofs for Real Analysis" goal. Working through a proofs book on your own will be seriously challenging, but the thought of taking Real Analysis without at least two other proofs courses under your belt is terrifying to me. I had to take "An intro to mathematical proofs" followed almost immediately by a proof-based Linear Algebra course before I was even allowed to contemplate a Real Analysis course.
Come to think of it, how in the hell are you even allowed to do this if you haven't taken a proofs course before? Are you sure this is even possible? Are prerequisites not enforced at your school? No one, and I mean no one was permitted to take Abstract Algebra or Real Analysis without the required prerequisites at my university. The only way you could get around it was by being the next Andrew Wiles.
Just to drive all this home, I was a straight-A Physics/Math major with the exception of two courses: Thermodynamics and my first proofs course. I've never worked so damn hard for a B in my life. Come to think of it, I actually recall quantum mechanics being easier than my proofs course.
I'm being sincere when I ask you to reconsider this plan. You are asking for a world of pain followed by the very real possibility of failure if you do this.
TL;DR: Unless you are remarkably sharp and have loads of time on your hands, this is probably a mistake. You should take a more elementary proofs course before tackling Real Analysis. Good luck, whatever you choose to do.
Check out A Transition to Advanced Mathetmatics. I took an enjoyable course with this book before starting to get deeper into my career and it was a nice primer.
Edit: .pdf version.
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.
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 :-)
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. :)
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.
For basic Algebra(Linear, Multilinear bla, bla, bla) there exists an amazing book called "Algebra" by Saunders Maclane and Garett Birkhoff
I don't know what second/third semester Calculus means. Is it proof-based or non-proof based? Is it a regular Calculus sequence or is it Analysis?
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).
If you've already read and done all the exercises in Hungerford, why would you be reading Dummit and Foote?
EDIT: Ah, never mind, I thought you meant Algebra by Hungerford. To wit, if you want a supplementary text at the next level up from D&F, you could try Hungerford, Lang (the big daddy of all algebra books), or my personal favorite, Isaacs.
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 a lot from getting a copy of Rudin (however, this book is very challenging and probably not the best to self study from. I was able to get to about continuity before taking my analysis course and it was challenging, but worth while). You can probably find it online somewhere for free.
A teacher lent Introduction to Analysis to me and suggested I use it instead of the book by Rudin. It was a well written book and had exercises which were much more approachable (although it included very difficult ones as well). The layout of this book (and I'd bet many others) is quite similar to that of Rudin. It was nice to be able to read them together.
For linear algebra, I can't speak to the quality of many books, but there are plenty which can fairly easily be found online. You will likely be recommended Linear Algebra Done Right however I found it a bit challenging as a first introduction to linear algebra and never got quite far.
My university course used Larson, Falvo Linear Algebra and it was enjoyable and helps you learn the computations very well and gives a decent understanding of proofs.
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.
I'm not sure if you mean abstract algebra or linear algebra, but if it's the former, I liked Herstein's Topics in Algebra. There's also Abstract Algebra by Herstein as well, which I think is a cheaper slimmed down version. I used these books for self study and found Herstein's exposition, particularly at the beginning of chapters, very helpful. He isn't as verbose as your typical 7th edition mass market textbook author though.
For linear algebra, I hear Axler's Linear Algebra Done Right is good. I haven't read it, but I read his paper "Down with Determinants" which is, I think, written in the same style and enjoyed the alternative perspective a lot.
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.
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.
Idk what kind of dead line you're on, but several years ago I was trying to do the same thing you are (without the stress of an actual test, just for myself).
I ended up purchasing the following two books:
All The Math You'll Ever Need - Steve Slavin
Practical Algebra - Peter H. Selby
I agree with all the suggestions to start with How to Prove It by Velleman. It's a great start for going deeper into mathematics, for which rigor is a sine qua non.
As you seem to enjoy calculus, might I also suggest doing some introductory real analysis? For the level you seem to be at, I recommend Understanding Analysis by Abbott. It helped me bridge the gap between my calculus courses and my first analysis course, together with Velleman. (Abbott here has the advantage of being more advanced and concise than Spivak, but more gentle and detailed than baby Rudin -- two eminent texts.)
Alternatively, you can start exploring some other fascinating areas of mathematics. The suggestion to study Topology by Munkres is sound. You can also get a friendly introduction to abstract algebra by way of A Book of Abstract Algebra by Pinter.
If you're more interested in going into a field of science or engineering than math, another popular approach for advanced high schoolers to start multivariable calculus (as you are), linear algebra, and ordinary differential equations.
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.
A few Suggestions:
All this books can easily be found. Good luck.
I can't help you with professors, but back when I took linear algebra in 2010 I found Linear Algebra by Hoffman and Kunze to be very helpful.
Link to the US version.
Friendly info:
"College Algebra" = Elementary Algebra.
College Level Algebra = Abstract Algebra.
Example: Undergrad Algebra book.
Example: Graduate Algebra book.
There's no single book that's right for everyone: a suitable book will depend upon (1) your current background, (2) the material you want to study, (3) the level at which you want to study it (e.g., undergraduate- versus graduate-level), and (4) the "flavor" of book you prefer, so to speak. (E.g., do you want lots of worked-out examples? Plenty of exercises? Something which will be useful as a reference book later on?)
That said, here's a preliminary list of titles, many of which inevitably get recommended for requests like yours:
Good luck finding something useful!
Algebra by Serge Lang.
It has a good introduction to category theory, as well as being one of the better all around resources for algebra. He can be quite terse sometimes, but it is well-referenced.
I hear that you can also obtain digital copies of this book.
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.
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.
Linear algebra is about is about linear functions and is typically taken in the first or second year of college. College algebra normally refers to a remedial class that covers what most people do in high school. I highly recommend watching this series of videos for getting an intuitive idea of linear algebra no matter what book you go with. The book you should use depends on how comfortable you are with proofs and what your goal is. If you just want to know how to calculate and apply it, I've heard Strang's book with the accompanying MIT opencourseware course is good. This book also looks good if you're mostly interested in programming applications. A more abstract book like Linear Algebra Done Right or Linear Algebra Done Wrong would probably be more useful if you were familiar with mathematical proofs beforehand. How to Prove it is a good choice for learning this.
I haven't seen boolean algebra used to refer to an entire course, but if you want to learn logic and some proof techniques you could look at How to Prove it.
Most calculus books cover both differential and integral calculus. Differential calculus refers to taking derivatives. A derivative essentially tells you how rapidly a function changes at a certain point. Integral calculus covers finding areas under curves(aka definite integrals) and their relationship with derivatives. This series gives some excellent explanations for most of the ideas in calculus.
Analysis is more advanced, and is typically only done by math majors. You can think of it as calculus with complete proofs for everything and more abstraction. I would not recommend trying to learn this without having a strong understanding of calculus first. Spivak's Calculus is a good compromise between full on analysis and a standard calculus class. It's possible to use this as a first exposure to calculus, but it would be difficult.
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.
Since your current knowledge is limited to calculus only, your goal seems kind of out of reach, at least in my opinion (but it depends on your progress/motivation). Writing good proofs is not something that you learn in a day by reading notes, it's something that comes with lots of experience reading and writing mathematics.
That being said, if you put a lot of focus on your studies it is certainly possible to learn the basics of algebra pretty fast. Linear algebra is an excellent tool, but it isn't required for learning abstract algebra. You can take both linear algebra and group theory classes at once and see where you want to go from there. It is a beautiful field of study for sure!
I'd strongly recommend Herstein's Topics in Algebra for a very solid introduction to most everything algebra-related. It covers Group Theory, Ring Theory, Vector Spaces and Modules, Fields, Linear Transformations, and some selected special topics.
Fraleigh is a little bit easier to wrap your head around. Get an old edition (or find it at the library), obviously.
Also, I highly recommend Herstein's Topics in Algebra. Again, try to get it from a university library.
If you are a newcomer to abstract algebra, you might consider using a text other than Dummit and Foote. I used baby Herstein (as opposed to big Herstein) in an undergraduate class and found it to be a good introduction.
For an introductory text, I recommend Herstein's Topics in Algebra. It slowly walks you through groups, rings, vector spaces, modules, fields, linear transformations and other selected topics.
Has plenty of exercises and doesn't skip over any details.
Herstein's Topics in Algebra is the book I learned both group and field theory from. It's a very easy read with lots of good examples and problems that help you develop and learn about the topics.
Also, the field of quaternions with integer coefficients is pretty cool. You can use it to prove that every natural number can be written as the sum of four squares, almost for free just by examining the field.
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.
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.
Did this exact thing a few years ago, refreshed my skills with this book Practical Algebra .
Took pre calc over the summer on my own dime and then started full time in the fall with calc 1 and others.
A lot of people recommend Khan Academy, but you cannot really learn from the Khan Academy; there is just too much material to cover. I recommend either going into an algebra class at your local community college, and/or get some good algebra/maths books. This one gets a lot of praise on Amazon.com:
http://www.amazon.com/Practical-Algebra-Self-Teaching-Guide-Second/dp/0471530123/ref=sr_1_fkmr0_1?ie=UTF8&amp;qid=1288684060&amp;sr=8-1-fkmr0
and, this one is the one nobel laureate Richard Feynman taught himself with:
http://www.amazon.com/Algebra-practical-Mathematics-self-study/dp/B0007DZPT6
I'm really enjoying this book:
Practical Algebra
It starts from scratch and doesn't even assume too much about your knowledge of arithmetic. I was surprised how many gaps in my basic knowledge I had, but it helps explain why teaching myself via Khan or tutors didn't work well.
I'm working through the exercises in Pinter's Abstract Algebra.
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
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?
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
This webpage has a solid list of recommended textbooks: https://mathblog.com/mathematics-books/
For Linear Algebra, Linear Algebra Done Right (3rd Ed.).
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.
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.
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.
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.
I love Aluffi! It's a fun read, and more "modern" than texts like Dummit and Foote (in that it uses basic category theory freely). I like category theory, so I really enjoy Aluffi's approach.
> The 14th is brand new this year, so I'd take that single one-star review with a huge grain of salt.
Yes you raise a good point. One review is usually not a good metric.
>So it's possible that they botched it up pretty badly and Thomas is rolling in his grave.
Looks like this edition was released in January. I guess we will find out at the end of this semester as more reviews start to roll in.
I was always a fan of the Stewart book to be honest. It was lovely to go through.
A lesser known book Calculus by Larson and Edwards is also a personal favorite. Have you used the Stewart or Larson books?
>I recommend a riot, burning the Pearson HQ to the ground
That is just yet another reason. There are already many reasons to riot Pearson already. :)
Thanks! Link for anyone interested
Honestly, Calculus by Ron Larson. You can get a previous edition(I used the 9th when I was learning) for cheaper. This is the clearest Calculus book I have ever read at Stewart's level (I textbook I detest btw). Larson also has a website http://www.calcchat.com/ where he has step by step solutions to all odd problems, so very very good for self learning.
I took Calc I & II last semesters and we used [this textbook] (https://www.amazon.com/Calculus-Ron-Larson/dp/1285057090). Check again in a month on coursebook, but you'll most likely use the same book.
> btw ce crezi de masterul asta de la unibuc http://fmi.unibuc.ro/ro/pdf/2008/curs_master/informatica/4InteligentaArtificialaEnachescuSite.pdf , e din 2008,nu am gasit o varianta mai buna.Daca voi avea posibilitatea sa fiu acceptat l;a o facultate mai moderna care face cercetare din afara o voi face,dar mai intai trebuie sa capat o diploma din Romania).
Acum, trebuie sa intelegi ca ML si AI sunt 2 lucruri diferite. AI includes ML, si ce ai tu aici e un master general de AI. Nu pot sa iti spun cat de bun e masterul, dar vad ca faci 1 curs de ML doar in anul 2, ceea ce pentru mine ar fi un motiv sa nu il fac. Information retrieval si NLP sunt interesante, dar eu as incerca sa invat ML la nivel teoretic first, si apoi sa abordez probleme specifice domeniilor.
> Eu ma gandeam ca Unibuc e mai potrivit pt ca la Poli voi face multa electronica si programare low-level si nu cred ca le voi folosi
Ar putea fi utile daca te gandesti la un moment dat ca te intereseaza mai degraba sa fii Research engineer si sa nu lucrezi atat de mult pe teorie, cat pe implementare. Toate librariile de scientific programming sunt implementate in C/C++. Dar pe langa asta, in general programarea low-level ar fi interesant sa o inveti pentru ca te ajuta sa intelegi cum functioneaza lucrurile at a more basic level, fara x abstractii construite pentru a fi totul beginner-friendly. Daca nu vrei sa continui cu asta dupa 1-2 cursuri e ok, tot cred ca iti va folosi mai incolo. Sa inveti python si c++ in paralel e un challenge interesant :).
> Va veni vacanta de vara si voi avea mult timp liber si vreau sa ma apuc de machine learning de-acum.Ce crezi de planul asta de invatare?
Iti va lua mai mult decat 1 vara sa termini ce ai listat aici. Sfatul meu ar fi sa imbini programare aplicata cu matematica. Cursurile sunt ok, dar eu pentru matematica as incepe cu single variable calculus -> multiple variable calculus inainte de altceva (daca ai cunostintele necesare sa abordezi cursul). Uite o carte pe care ti-o recomand: https://www.amazon.com/Calculus-Early-Transcendentals-James-Stewart/dp/1285741552
Are in jur de 8 sectiuni care reprezinta pre-requisites (lucruri pe care ar trebui sa le stii inainte sa abordezi cartea), algebra, geometrie de baza, etc. Fiecare invata diferit, eu prefer cartile.
Legat de programare, incearca sa faci probleme de aici: https://projecteuler.net/, te va ajuta mai incolo :). Si daca te plictisesti incearca construiesti lucruri care ti-ar fi utile. Vei invata destule din proiecte de genul.
Calculus or Pre-calc?
For calculus, I recommend: http://www.amazon.com/Calculus-Ron-Larson/dp/1285057090/ref=sr_1_2?s=books&amp;ie=UTF8&amp;qid=1398268486&amp;sr=1-2&amp;keywords=larson+edwards+calculus
It's written by the same guy who does the Calculus 1 and 2 lectures for The Teaching Company.
He doesn't address the problem I mentioned in my previous post, but I still think this is a much more concise book.
I don't know the first but I didn't really like the second book. Right now I do seem to make some good progress understanding stuff (not all but most) in Algebra Chapter 0 which is a lot bigger but introduces a lot of the Algebra I was missing (or forgot since school) along with the Category Theory terms.
Depends on the market depths, for a lot of books, there may be a couple low priced books where a few purchases will raise the price pretty significantly. I think a lot of booksellers have repricers that don't work very effectively where they'lll lower the price over time until it sells, and there really isnt a market for text books except for at the beginning of semesters.
http://www.amazon.com/gp/offer-listing/1285741552/ref=olp_f_primeEligible?ie=UTF8&amp;f_primeEligible=true
On that book, which is a pretty widely used text, If they sell about 5 books, the price rises almost $70.
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.
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.
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.
i can back this assessment up, as i used this text for the exact same thing. http://www.amazon.com/Calculus-Early-Transcendentals-Stewarts-Series/dp/0495011665 a broad text, well explained, with many helpful practice problems.
Game Engine:
Game Engine Architecture by Jason Gregory, best you can get.
Game Coding Complete by Mike McShaffry. The book goes over the whole of making a game from start to finish, so it's a great way to learn the interaction the engine has with the gameplay code. Though, I admit I also am not a particular fan of his coding style, but have found ways around it. The boost library adds some complexity that makes the code more terse. The 4th edition made a point of not using it after many met with some difficulty with it in the 3rd edition. The book also uses DXUT to abstract the DirectX functionality necessary to render things on screen. Although that is one approach, I found that getting DXUT set up properly can be somewhat of a pain, and the abstraction hides really interesting details about the whole task of 3D rendering. You have a strong background in graphics, so you will probably be better served by more direct access to the DirectX API calls. This leads into my suggestion for Introduction to 3D Game Programming with DirectX10 (or DirectX11).
C++:
C++ Pocket Reference by Kyle Loudon
I remember reading that it takes years if not decades to become a master at C++. You have a lot of C++ experience, so you might be better served by a small reference book than a large textbook. I like having this around to reference the features that I use less often. Example:
namespace
{
//code here
}
is an unnamed namespace, which is a preferred method for declaring functions or variables with file scope. You don't see this too often in sample textbook code, but it will crop up from time to time in samples from other programmers on the web. It's $10 or so, and I find it faster and handier than standard online documentation.
Math:
You have a solid graphics background, but just in case you need good references for math:
3D Math Primer
Mathematics for 3D Game Programming
Also, really advanced lighting techniques stretch into the field of Multivariate Calculus. Calculus: Early Transcendentals Chapters >= 11 fall in that field.
Rendering:
Introduction to 3D Game Programming with DirectX10 by Frank. D. Luna.
You should probably get the DirectX11 version when it is available, not because it's newer, not because DirectX10 is obsolete (it's not yet), but because the new DirectX11 book has a chapter on animation. The directX 10 book sorely lacks it. But your solid graphics background may make this obsolete for you.
3D Game Engine Architecture (with Wild Magic) by David H. Eberly is a good book with a lot of parallels to Game Engine Architecture, but focuses much more on the 3D rendering portion of the engine, so you get a better depth of knowledge for rendering in the context of a game engine. I haven't had a chance to read much of this one, so I can't be sure of how useful it is just yet. I also haven't had the pleasure of obtaining its sister book 3D Game Engine Design.
Given your strong graphics background, you will probably want to go past the basics and get to the really nifty stuff. Real-Time Rendering, Third Edition by Tomas Akenine-Moller, Eric Haines, Naty Hoffman is a good book of the more advanced techniques, so you might look there for material to push your graphics knowledge boundaries.
Software Engineering:
I don't have a good book to suggest for this topic, so hopefully another redditor will follow up on this.
If you haven't already, be sure to read about software engineering. It teaches you how to design a process for development, the stages involved, effective methodologies for making and tracking progress, and all sorts of information on things that make programming and software development easier. Not all of it will be useful if you are a one man team, because software engineering is a discipline created around teams, but much of it still applies and will help you stay on track, know when you've been derailed, and help you make decisions that get you back on. Also, patterns. Patterns are great.
Note: I would not suggest Software Engineering for Game Developers. It's an ok book, but I've seen better, the structure doesn't seem to flow well (for me at least), and it seems to be missing some important topics, like user stories, Rational Unified Process, or Feature-Driven Development (I think Mojang does this, but I don't know for sure). Maybe those topics aren't very important for game development directly, but I've always found user stories to be useful.
Software Engineering in general will prove to be a useful field when you are developing your engine, and even more so if you have a team. Take a look at This article to get small taste of what Software Engineering is about.
Why so many books?
Game Engines are a collection of different systems and subsystems used in making games. Each system has its own background, perspective, concepts, and can be referred to from multiple angles. I like Game Engine Architecture's structure for showing an engine as a whole. Luna's DirectX10 book has a better Timer class. The DirectX book also has better explanations of the low-level rendering processes than Coding Complete or Engine Architecture. Engine Architecture and Game Coding Complete touch on Software Engineering, but not in great depth, which is important for team development. So I find that Game Coding Complete and Game Engine Architecture are your go to books, but in some cases only provide a surface layer understanding of some system, which isn't enough to implement your own engine on. The other books are listed here because I feel they provide a valuable supplement and more in depth explanations that will be useful when developing your engine.
tldr: What Valken and SpooderW said.
On the topic of XNA, anyone know a good XNA book? I have XNA Unleashed 3.0, but it's somewhat out of date to the new XNA 4.0. The best looking up-to-date one seems to be Learning XNA 4.0: Game Development for the PC, Xbox 360, and Windows Phone 7 . I have the 3.0 version of this book, and it's well done.
*****
Source: Doing an Independent Study in Game Engine Development. I asked this same question months ago, did my research, got most of the books listed here, and omitted ones that didn't have much usefulness. Thought I would share my research, hope you find it useful.
I'd recommend hitting up somewhere like half-price books and grabbing a textbook for like $10-$15. I purchased this book for probably $12 when I needed to brush up. I know it's not online, but it will provide good direction, offer a solid foundation, provide sample problems to test your knowledge, and can easily be supplemented by online materials. As someone else mentioned, Khan Academy is also great, but I would highly recommend using them as a supplement, and using a book as your base.
For a very good textbook, I would recommend Calculus Early transcendentals by Stewart. He goes through every concept in single variable calculus (there's also a version with multi variable calculus) and proves almost every concept he teaches. Its one of my favorite textbooks in general.
I recommend purchasing yourself a copy of this book: https://www.amazon.com/Transition-Advanced-Mathematics-Douglas-Smith/dp/0495562025
Chapter 0 is especially great, as it guides you through some of the basic grammar of mathematics. Most of the material is seen in some form or another in 115A(H), but I personally found this book to be a much better introduction to the upper division courses.
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.
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
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.
If you need to brush up on some of the more basic topics, there's a series of books by IM Gelfand:
Algebra
Trigonometry
Functions and Graphs
The Method of Coordinates
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.
For algebra, I'd recommend Mac Lane/Birkhoff. They may not be as comprehensive as some other texts but to me, they are more motivating, and will probably provide a better introduction to categorical thinking.
For linear algebra, I'm going to suggest something slightly unusual: Kreyszig's Introductory Functional analysis with applications. Functional analysis is essentially linear algebra on infinite dimensional spaces, and it generalizes a lot of the results in finite dimensions. Kreyszig does a good job motivating the reader. I can definitely sit down and read it for hours, much longer than I can for other books, and I definitely don't consider myself an analyst. However, it could be difficult if you are not familiar with basic topology and never seen linear algebra before.
Algebra by Saunders MacLane and Garret Birkhoff is the best algebra book I have ever encountered.
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
I hear D&F is too tough for one's first brush with Abstract Algebra. On the other hand, people swear by "A Book of Abstract Algebra" by Charles Pinter to get one started on the path of AA.
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.
Hello! I'm interested in trying to cultivate a better understanding/interest/mastery of mathematics for myself. For some context:
&nbsp;
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.
&nbsp;
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
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
College books are also much more expensive in the USA than in Europe.
For example:
$152.71
VS
£43.62($68.03)
$146.26 VS
£44.34($69.16)
Awesome, I will take a look at that. Here is the book I have to teach myself with (used it for Calculus 2 a year ago). It seems like a solid book.
Barron's for gov and Calc ab. I would say James Stewart for calculus. Amazon should have his Calc book for cheap price https://www.amazon.com/Calculus-Early-Transcendentals-James-Stewart/dp/0495011665/ just read the chapters and do the problems. Khan academy is useful.
All I know is that they're no longer doing Fitzpatrick or Chartrand (according to what a professor told me). Here's the new book. I think it's possible the course will be less analysis-focused. I think they should incorporate some abstract algebra into it. This goes into effect next semester by the way.
Isn't that true of any subject one likes?
Regardless, besides the Linear Algebra textbook, here are some books you should look at as well. These should give you a taste of what your introductory classes might be:
http://www.amazon.in/Transition-Advanced-Mathematics-Survey-Course/dp/0195310764
http://www.amazon.in/Transition-Advanced-Mathematics-Douglas-Smith/dp/0495562025
PM me if you want pdfs.
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.
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."
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.
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."
You might find this book to be a good place to start: Algebra, by Gelfand and Shen.
Another book in a similar vein might be Basic Mathematics by Serge Lang.
I haven't used either of these books myself, but I came across them recently, and it looks like they might be among the few titles that cover high-school math in the way that you describe (they were written by prominent research mathematicians).
You might consider using the materials on Khan Academy (articles, videos, and exercises) to structure your studies, since these may be more closely aligned with current standards in the U.S. Then, as you go along, you can use these books as supplements (e.g. if you feel that a different perspective on a particular topic might be helpful).
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.
pre-cal: http://www.amazon.com/Precalculus-5th-Edition-Robert-Blitzer/dp/0321837347
calculus: http://www.amazon.com/Calculus-Ron-Larson/dp/1285057090
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.
I like this book a lot: http://www.amazon.com/Linear-Algebra-Right-Undergraduate-Mathematics/dp/3319110799/
The professor who assigned it preferred Linear Algebra Done Wrong but he's a robot.
I also have this one (10th ed) but its not the one you want :/ https://www.amazon.com/Calculus-Ron-Larson/dp/1285057090
I keep seeing this book recommended in a lot of places. How is it different from the one by Axler and one by Roman?
I think category theory is best learned when taught with a given context. The first time I saw category theory was in my first abstract algebra course (rings, modules, etc.), where the notion of a category seemed like a necessary formalism. Given you already know some algebra, I'd suggest glancing through Paolo Aluffi's Algebra: Chapter 0. It is NOT a book on category theory, but rather an abstract algebra book that works with categories from the ground level. Perhaps it could be a good exercise to prove some statements about modules and rings that you already know, but using the language of category theory. For example, I'd get familiar with the idea of Hom(X,-) as a "functor"from the category of R-modules to the category of abelian groups, which maps Y \to Hom(X,Y). We can similarly define Hom(-,X). How do these act on morphisms (R-module homomorphisms)? Which one is covariant and which one is contravariant? If one of these functors preserves short exact sequences (i.e. is exact), what does that tell you about X?
Stewart, Larson, Rogawski
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
For calc MATH 1300/1014 and 1310/1014 you need , buy it new from the bookstore cause you will need the online code for assignments also it’s useful for calc 3 if you wanna take that. Man Wong is a good prof I had him for both 1300 and 13010
For EECS 1019 you need it’s not that useful and PDF can be found online for free and no online assignments so no need to buy it new. I had Zhihua Chang he’s a new prof but really nice but his lectures are boring. Trev tutor on YouTube is really helping with the course.
For Math 1025/1021 you need I found the book helpful but unlike calc some profs tend not to use this book so I’d hold out of buying it but most profs use lyryz which is an online assignment program so you will need to buy that. I had Paul Skoufranis, amazing prof but had tests. The book is also useful for linear 2 but again depend if the prof uses it
For EECS 1022 you need
It’s a good book and the guy you wrote it teaches the class.
PM if you have any other questions
Textbooks in the US are priced for what students will pay, not for their actual cost, because the textbook market isn't a free market for students. You either buy the course's reccomended textbook, or find some other way to access the material. You can't shop between different publishers of the same book, unless you start looking at international editions.
>Paying for content btw
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.
you do know that a digital copy of the text book isn't free. And no you can't use the price for a digital copy that you can buy for personal use. There would be a rental charge. The calc book I used for 3 semester of calculus in College is $32/semester to rent. so that means schools are probably paying round $50/year for each digital copy of a text book.
So if you think a school book costs $250 it becomes cheaper than rental after the 5th year (not even including the increased cost of the chromebook and "insurance" required by the student.
Rental
https://www.amazon.com/Calculus-Early-Transcendentals-James-Stewart-ebook/dp/B00T9X7THG/ref=sr_1_6?s=digital-text&amp;ie=UTF8&amp;qid=1536864274&amp;sr=1-6
physical copy
https://www.amazon.com/Calculus-Early-Transcendentals-James-Stewart/dp/1285741552/ref=mt_hardcover?_encoding=UTF8&amp;me=&amp;qid=1536864274
https://www.amazon.ca/Calculus-Early-Transcendentals-James-Stewart/dp/1285741552
Pretty sure it's this one. You should be able to find a pdf online.
I strongly suggest you take your time learning calculus, because anything you don't grasp completely will come back to haunt you.
But the good news is that there are lots of great resources you can use. MIT OCW has a full course with lectures, notes, and exams. Here are three free online books. If you're looking to buy a textbook, some good choices are Thomas, Stewart, and Spivak. (You can find dirt-cheap copies of older editions at abebooks.com.)
If you want more guidance, another great place to find it is at /r/learnmath.
That depends on your own level, your goals and your ambition. For example, OP wants to learn machine learning. Assuming OP's highschool math is solid, it might be possible for OP to simply download pytorch and immediately start programming neural networks without worrying too much about the hardcore math in the background.
On the other hand, if OP is more serious about improving as a mathematician, and assuming nothing but highschool math, I would start with linear algebra and differential and integral calculus. The famous professor Gil Strang has an excellent book on linear algebra, which is strangely available online. For differential and integral calculus, probably the standard reference is Stewart's book. At this point, OP would have all the basic things needed to start with machine learning. I'm not aware of the literature for machine learning so I can't recommend any specific books.
If OP wanted to get sidetracked learning more things before plunging into machine learning then the obvious choice would be Scientific Computing (my friends wrote an excellent book on the subject). Scientific Computing is the science of calculating things using computers and supercomputers. In addition, the area of Mathematical Optimization is good to know because Stochastic Gradient Descent is omnipresent in machine learning, but I don't know enough about optimization to recommend a book. There is Boyd and Vandenberghe but that is only for convex optimization. Some more areas that are related and useful are Probability and Statistics.
Sometimes you can find what textbook your school uses before the semester starts (I'm also the weird kid that emails the professor asking about books if I cant find it >.>). Some of my professors have what material they use for each class on their personal web pages though. For calculus, you'll most likely use this book. My brother used it at his Uni my friend at another and I myself used it at mine. Not sure if you're registered yet though. I had a weird case going into my Uni because I did community college then took summer courses so I was enrolled earlier than students who transfer and probably the freshman. YouTube videos will also be your best friend. People I liked for my math classes are TrevTutor (I don't think he ever finished his Calc 2 series) and PatrickJMT. Hope this helps a bit if you have any other questions or need more clarifications don't hesitate to ask.
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
I recommend this book.
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.
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
Strang's book is a fantastic resource for learning linear algebra but as you stated that your problem in you current text is the fact that it does not offer as much theory as you would like I am going to recommend another one of the MIT books Linear Algebra by Hoffman and Kunze I used this text book in my honors class and it is definitely not short on theory but you might want to keep you other textbook around for clarification on some issues as it can be quite opaque at some parts.
I’d recommend reading a text on linear algebra. Hoffman is pretty thorough: Linear Algebra (2nd Edition) https://www.amazon.com/dp/0135367972/ref=cm_sw_r_cp_api_oZkMAb4NH15B3
This text is the bible of linear algebra.
Grab a copy of this book from a local university or public library (or pay the massive price tag if you can afford it). It's a great text.
I think a rigorous course in linear algebra is the right place to start. Not only does the subject in some sense unify geometry and algebra, it's also necessary to understand it if you want to understand more advanced topics.
Strang is alright. Hoffman and Kunze is where it's really at.
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.
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.
Herstein!! http://www.amazon.com/Topics-Algebra-I-N-Herstein/dp/0471010901?tag=vglnk-c905-20
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.
My math skills sucked when I started. Definitely go though a book on math if you can.
There are two books I recommend. One book I found recently and plan to go through once I am done with the program (I am too busy now), just because I want to solidify my math skills is: Mastering Technical Mathematics
I found the book randomly and after skimming through a few pages knew it was a great book. It starts out with basic discrete mathematics concepts like counting and then goes all the way up to some calculus ideas.
The other book I reccomend is one I went through called Practical Algebra: A Self-Teaching Guide, Second Edition. It focuses more on algebra obviously but Algebra is actually the hardest part of CS 225 and CS325!
I had a nice book called Precalculus Mathematics in a Nutshell but it is not geared to starting from scratch. It's a good book if you remember some of your algebra, geometry, and trigonometry.
I've known some people who had good experiences with Practical Algebra
I took a long, long break between undergrad and grad school (think decades). I found this GRE math prep book very helpful. (The GRE math section tests high school math knowledge), I'd take the sample tests, see where I fell short, and focus on understanding why. I also found Practical Algebra to be a good review-and-practice guide, for the fundamentals. I boned up on discrete math by buying an old copy of Rosen and the matching solutions guide. And, I watched a bunch of videos of this guy explaining various facets of the math you need for computer science.
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.
Yeah either of those are easier. I don't like Fraleigh cause I think it lacks motivation (also the chapters on splitting/separable fields really suck) but I love Herstein. If you're set on cheap, this guy ain't too bad. If I were self studying though I would try to find a cheap older edition of Artin, as he's very example motivated, and it can sometimes be hard to wrap your head around all the abstraction without a class.
EDIT: Also you might want to find a cheap number theory text, since elementary number theory is probably the most accessible way to see groups and rings in action. And for "how do I prove xxx" questions I always recommend starting with this.
If you're teaching algebra at all, I highly reccomend reading the first chapter of Pinter's A Book of Abstract Algebra which gives a very short and entertaining history of the subject.
Hawking is a theoretical physicist. His craft is closer to math than it is to classical physics.
You made a lot of erroneous and hot-headed statements, but that's understandable. Since you seem to be very, very ignorant of math, I don't even know where to even begin to show you the differences - I am at a disadvantage here :) How about we talk about levels, then?
Most math an engineer knows is barely a first year material for a math undergrad. Math is so vast that even the grad students of math are at the very base of a huge mountain.
Here's Basic Algebra for a math major(flip through the first pages and checkout the contents).
Here's Algebra for engineers.
Notice how the algebra for engineers is a very small part of general algebra and non-rigorous at that.
Here's Calculus for engineers.
Here's Calculus for math majors.
This is not to say engineers are mentally inferior to mathematicians, it's just these two professions are concerned with fundamentally different things.
Buy a training plan off amazon and follow it rigorously.
Something like
this
or this
> Turning in assignments should not be locked behind a pay wall. A student should not fail the class just because they didn't buy it.
You're not wrong, but I have a few issues with that.
First, do you really believe that the school does not have a system in place to help those that genuinely cannot afford it? Every class I've had that mentioned Cengage had the teacher explicitly mention that if paying for it was a problem, to get in contact with them.
As I mentioned, I used to go to a different school. $125 per semester, required by every math class I took. It's a good step down for me to pay that much in a year.
Second, we've got 750 students this semester in Calc 3 alone. I've got 3 assignments that were due yesterday, and 2 more due Monday.
If all the assignments only had 6 questions each, that's ~22k questions to be graded this week. Somebody has to do it. UCF is apparently even making their own software/site for this, but regardless of when it gets finished you know we're gonna foot the bill. One way or another we pay for this shit to get done.
> Also, access codes hurt the used textbook market.
You're not wrong but if we can get the textbook + assignments graded for the same price, what's the big deal?
Renting my textbook for Calculus would have cost the same as paying for access, and I covered both Physics classes too (along with whatever else I want to study on)
Beside the point either way. My issue was that the OP was full of shit, not 'Oh poor Cengage.' My bad for expecting people here to read instead of jumping in on another circlejerk.