(Part 2) Best pure mathematic books according to redditors
We found 2,758 Reddit comments discussing the best pure mathematic books. We ranked the 819 resulting products by number of redditors who mentioned them. Here are the products ranked 21-40. You can also go back to the previous section.
33. Not Even Wrong: The Failure of String Theory and the Search for Unity in Physical Law
11 mentions
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.
I was once a teacher's aide for an autistic teen. He seemed very bored with the 3rd grade arithmetic the teacher thought was his limit. One day, we had some extra time. I asked him if he wanted to read my set theory book. It's difficult to assess consent and comprehension, but we have our ways. I figured out that not only did he like this book, but he could follow along. It took about 3 months, but he was able to learn basics of sets. What makes me sad is the hard truth that people who know about this kind of math, generally don't find themselves being an educator for special needs students. His higher math education ended when I left. That's not right.
No, it's not. Math-major algebra was typically taught from something like Herstein. These days, Dummit and Foote seems more popular.
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)
I'll be that guy. There are two types of Calculus: the Micky Mouse calculus and Real Analysis. If you go to Khan Academy you're gonna study the first version. It's by far the most popular one and has nothing to do with higher math.
The foundations of higher math are Linear Algebra(again, different from what's on Khan Academy), Abstract Algebra, Real Analysis etc.
You could, probably, skip all the micky mouse classes and start immediately with rigorous(proof-based) Linear Algebra.
But it's probably best to get a good foundation before embarking on Real Analysis and the like:
Discrete Mathematics with Applications by Susanna Epp
How to Prove It: A Structured Approach Daniel Velleman
Learning to Reason: An Introduction to Logic, Sets, and Relations by Nancy Rodgers
Book of Proof by Richard Hammock
That way you get to skip all the plug-and-chug courses and start from the very beginning in a rigorous way.
If you're planning on learning haskell (you should :D), why not do a book that teaches you both discrete maths and haskell at the same time?
There are atleast two books that do this:
Both books use Haskell in an effective way to teach you discrete mathematics. (They don't assume prior knowledge of Haskell).
I've been working through the 2nd book and posting my notes and solutions here . You could compare solutions/notes and discuss stuff with me.
Depends on what kind of math you are looking for. For example, there is a middle school outreach program called Bootstrap World which is about teaching algebra using functional programming. You could take a look at their materials.
If you're looking for university-level math, there are some books like The Haskell Road to Logic, Maths, and Programming. I haven't read it, but I think it covers discrete math sort of topics.
The Haskell Road to Logic, Maths and Programming:
http://www.amazon.com/Haskell-Logic-Maths-Programming-Computing/dp/0954300696
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:
I just started the PhD program this semester at North Carolina State. The program in general isn't ranked well but I'm interested in Environmental and Resource Econ and NCSU is top 10 (arguably top 5) in that field. I thought I'd give you a brief overview of the math that I had to prepare (undergrad rather than a math camp).
One big thing that I didn't work on was programming skills. If you are intending to do applied work rather than theory, you'll want to be a solid programmer. Matlab and/or Maple are valuable and Stata, SAS, and ArcMap don't hurt.
That said, I've met a lot of people in decent PhD programs who do not have much more than Calc, diff EQ, and linear algebra. I don't know if they passed comps or not but they got in. There are a number of good programs ranked 50+ that will teach you the math needed for applied work. However, if you want to go to a top 20 program you should definitely look into a math undergrad.
Good luck to anyone thinking about applying.
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.
In response to the same question, my Logic professor suggested:
A few of my favorites:
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.
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&qid=1315534435&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
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
http://www.amazon.com/How-Prove-It-Structured-Approach/dp/0521446635
That's what I started with and it was very helpful. The next semester when I took abstract vector spaces (proof-based linear algebra) I found writing the proofs to generally be straightforward because I'd already learned how to write a proof.
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.
There are a few options. Firstly, if you are more familiar using infinity in the context of Calculus, then you might want to look into Real Analysis. These subjects view infinity in the context of limits on the real line and this is probably the treatment you are probably most familiar with. For an introductory book on the subject, check out Baby Rudin (Warning: Proofs! But who doesn't like proofs, that's what math is!)
Secondly, you might want to look at Projective Geometry. This is essentially the type of geometry you get when you add a single point "at infinity". Many things benefit from a projective treatment, the most obvious being Complex Analysis, one of its main objects of study is the Riemann Sphere, which is just the Projective Complex Plane. This treatment is related to the treatment given in Real Analysis, but with a different flavor. I don't have any particular introductory book to recommend, but searching "Introductory Projective Geometry" in Amazon will give you some books, but I have no idea if they're good. Also, look in your university library. Again: Many Proofs!
The previous two treatments of infinity give a geometric treatment of the thing, it's nothing but a point that seems far away when we are looking at things locally, but globally it changes the geometry of an object (it turns the real line into a circle, or a closed line depending on what you're doing, and the complex plane into a sphere, it gets more complicated after that). But you could also look at infinity as a quantitative thing, look at how many things it takes to get an infinite number of things. This is the treatment of it in Set Theory. Here things get really wild, so wild Set Theory is mostly just the study of infinite sets. For example, there is more than one type of infinity. Intuitively we have countable infinity (like the integers) and we have uncountable infinity (like the reals), but there are even more than that. In fact, there are more types of infinities than any of the infinities can count! The collection of all infinities is "too big" to even be a set! For an introduction into this treatment I recommend Suppes and Halmos. Set Theory, when you actually study it, is a very abstract subject, so there will be more proofs here than in the previous ones and it may be over your head if you haven't taken any proof-based courses (I don't know your background, so I'm just assuming you've taken Calc 1-3, Diff Eq and maybe some kind of Matrix Algebra course), so patience will be a major virtue if you wish to tackle Set Theory. Maybe ask some professors for help!
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.
Daniel J Velleman's How to Prove It : A Structured Approach
This book is a pretty dang good intro to proofs, I highly reccommend it. This is the first edition, so you'll be able to find a used copy for super cheap.
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.
I haven't read it myself, but I have heard Naive Set Theory recommended here several times before.
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.
You need a good foundation: a little logic, intro to proofs, a taste of sets, a bit on relations and functions, some counting(combinatorics/graph theory) etc. The best way to get started with all this is an introductory discrete math course. Check these books out:
Mathematics: A Discrete Introduction by Edward A. Scheinerman
Discrete Mathematics with Applications by Susanna S. Epp
How to Prove It: A Structured Approach Daniel J. Velleman
Learning to Reason: An Introduction to Logic, Sets, and Relations by Nancy Rodgers
Combinatorics: A Guided Tour by David R. Mazur
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&qid=1453926037&sr=8-1&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&qid=1453926180&sr=8-1&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&qid=1453926270&sr=8-2&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.
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
You may also want to check out The Haskell Road to Logic, Maths and Programming.
This book focusses on logic and how to use it, so you get to learn proofs. It even hits corecursion and combinatorics. If you think math is pretty but you want to use it interactively as source code, this could be the book for you.
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 really believe that Michael Kelly's "Humongous Book of" series are the best resources for getting through all math classes up to Calculus II. These books contain every single type of problem you will ever encounter in Algebra I & II, Geometry, Trig, and Calc I & II, all solved in great detail. They are like Schaums Outlines but much more reliable.
https://www.amazon.com/Humongous-Basic-Pre-Algebra-Problems-Books/dp/1615640835
https://www.amazon.com/Humongous-Book-Algebra-Problems-Books/dp/1592577229
https://www.amazon.com/Humongous-Book-Geometry-Problems-Books/dp/1592578640
https://www.amazon.com/Humongous-Book-Trigonometry-Problems-Comprehensive/dp/1615641823
https://www.amazon.com/Humongous-Book-Calculus-Problems-Books/dp/1592575129
Not exactly. For an incredibly long time, string theory has dominated the field of physics over a small minority of objections that it cannot be tested - that it wasn't even a theory, it was "not even wrong" as Peter Woit has written; Lee Smolin wrote a similar book around the same time. Smolin and Woit were mocked by hordes of theorists who just knew the evidence for string theory was going to show up any day now. But every time it didn't show up at the LHC, all these same theorists had to do was tweak their work a bit and move the goal post to a new energy level - this gimmick has been repeated, ad nauseam, for years. Only recently have some people finally started to come around to the possibility that string theory might not be the solution to figuring out the last pieces of the Standard Model.
So the analogy goes something like this:
Woit and Smolin:Scholze and Stix :: string theorists:Mochizuki and his inner circle.
The many-worlds-interpretation and string "theory" are completely un-related (and note where I put the quotes)
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.
Computability and Logic by Boolos, Burgess and Jeffrey is good but seems to cover much of the stuff in Hunter. You may want to dig deeper into set theory, model theory, proof theory or recursion theory and look at some references specific to those topics.
I dunno about “undergraduate”, but you could try Birkhoff & Mac Lane or Greub. Those are both kind of old, so someone else may have a better idea.
Linear Algebra and [Linear Algebra and Its Applications] (http://www.amazon.com/dp/0321385179).
http://www.amazon.com/Book-Abstract-Algebra-Edition-Mathematics/dp/0486474178/ref=sr_1_1?ie=UTF8&qid=1394386195&sr=8-1&keywords=pinter+abstract+algebra)
Right now I am studying Proofs from "Learning to Reason: An Introduction to Logic, Sets, and Relations" by Nancy Rodgers. Prior to getting started I looked at tons of "Intro to Proofs/Transition" books and the vast majority of them (including the popular darlings) are, frankly, just mostly doorstops - there's no way you could come out being able to do proofs by studying them.
Rodgers starts out with prop. logic and builds everything on top of that. Everytime she introduces a new topic, she gives logical justification (chapter 1 explores the logic extensively) that makes the proof structure work (very satisfying and makes the concepts stick around longer e. i. you are not just monkeying around with mish-mash of various tools, but actually know what you are doing)- never seen that in Real Analysis/Linear Algebra books that are, supposedly, designed to teach you proofs.
For example, in an intro to Real Anal, they just throw you the structure of Induction Proof and expect you to prove away - unrealistic. They dont show you why the proof works (logic and intuition behind the proof), wont let you explore the syntax of the proof before you get more comfortable with it and since one doesnt have a firm foundation made out of prop. logic, one's on a very shaky ground ready to break down whenever something serious comes on. With Rodgers, whenever something big and scary shows up, you just take everything apart into its logical building blocks like she teaches you in chapter 1 and it will make perfect sense.
But the worst part of RA books is they assume you are intimately familiar with Deduction and wont spend a half a page on it and that's 99% of math Induction Proof structure. Rodgers spends half the book exploring the intricacies of Deduction arguments. Basically, Rodgers' book explores math grammar in all its gory detail, is sort of a very revealing math porn.
If you ever studied a foreign language, you know there are 2 types of books. The ones that spell out all the grammar and give all the necessary vocabulary with an intention that you'll read some real literature in your target language in the future and those that skip the grammar or are very skimpy on it and give you pre-determined phrases and various random knowledge bites instead. The first category of books take the tougher road, but it pays off the at the end. Rodgers' book is one such book.
All in all, I just cant imagine learning proofs from Linear Algebra/Real Analysis books. Because, they are mostly about concepts inherent in these subjects and not proofs. Proofs are there to prove the said concepts, so there wont be enough time/space to explore proofs in-depth which will make your life tougher.
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&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!
Several good books have already been mentioned in this thread, but some good books are hard to get into as a beginner.
I recommend Elementary Number Theory by Underwood Dudley as a good starting point for a beginner, as well as something like Apostol or Ireland-Rosen if you want more details.
I think it makes sense to start with something like Dudley to get an overall framework, and then rely on more detailed books to flesh out the details of whatever topics you're interested in more.
In particular, I think Dudley's book has an approach to Chebyshev's theorem (i.e. there is always a prime between n and 2n) that's great for beginners, even if someone with a bit more experience can streamline that proof a little.
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&qid=1371247275&sr=8-1&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.
If you're interested in non classical logics I'd recommend "An introduction to non classical logic" by Graham Priest (it has modal logic and other very interesting non-classical logics). It's a good overview of the field.
For denser subjects in classical logic like computability, Turing machines, Gödel theorems, proofs for compactness, correctness, completeness, etc.; I'd go for a classical work by now: "Computability and logic" by Boolos, Burgess & Jeffrey. It's not an easy reading though.
That depends on why you're studying Logic.
Do you plan to use Logic as a tool for doing Philosophy? If so, I recommend studying Logic for Philosophy by Theodore Sider. You will get a more rigorous, formal treatment of propositional and predicate logic than what your introductory textbook likely contained. You will be exposed to basic proof theory and model theory. You will also learn, in depth, about several useful extensions to predicate logic, including various modal logics.
Do you want to become a logician, in some capacity? If so, the classic text would be Computability and Logic by Boolos and Jeffrey. This is an extremely rigorous and intensive introduction to metalogical proof. If you want to learn to reason about logics, and gain a basis upon which to go on to study the foundations of mathematics, proof theory, model theory, or computability, then this is probably for you.
Also, perhaps you could tell us what textbook you've just finished? That would give us a better idea of what you've already learned.
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/
Mac Lane and Birkhoff is my favorite math text I've ever read, but the things you have listed as "further topics" in your second semester are mostly absent unfortunately.
Look at worked-out problems. I highly recommend books in this series: http://www.amazon.com/Humongous-Book-Calculus-Problems-People/dp/1592575129/ref=pd_sim_b_2
Beyond that, slog through practice problems. Math is a language. You can know a mind-blowing concept, but you won't develop an intuition for it without repeated exposure. This includes the stuff you might look at and think there's no reason for you to know that cuz software will handle it - if you're looking at a proof that makes zero sense without knowing what happens when you divide logarithms, you're going to be lost.
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.
Reading through Algebra: Chapter 0
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.
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.
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.
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
Hey mathit.
I'm 32, and just finished a 3 year full-time adult education school here in Germany to get the Abitur (SAT-level education) which allows me to study. I'm collecting my graduation certificate tomorrow, woooo!
Now, I'm going to study math in october and wanted to know what kind of extra prep you might recommend.
I'm currently reading How to Prove It and The Haskell Road to Logic, Maths and Programming.
Both overlap quite a bit, I think, only that the latter is more focused on executing proofs on a computer.
Now, I've just been looking into books that might ease the switch to uni-level math besides the 2 already mentioned and the most promising I found are these two:
How to Study for a Mathematics Degree and Bridging the Gap to University Mathematics.
Do you agree with my choices? What else do you recommend?
I found online courses to be ineffective, I prefer books.
What's your opinion, mathit?
Cheers and many thanks in advance!
A terrific browsing book in number theory is Introduction to the Theory of Numbers by Hardy and Wright. An oldie but a very goodie.
http://www.amazon.com/An-Introduction-Theory-Numbers-Hardy/dp/0199219869
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.
One thing I like to remind people, is that Linear Algebra is really cool and though it tends to come "after" calculus for some reason, it really has no explicit calc prerequisite.
I highly recommend Dr. Gilbert Strang's lectures on it, available on youtube and ocw.mit.edu (which has problems, solutions, etc, also)
I think it's a great topic for right around late HS, early college. And he stresses intuition and imo has the right balance of application and theory.
I'd also say that contrary to most peoples' perceptions, a student's understanding of a math topic will vary greatly depending on the teacher. And some teachers will be better for some students, others for others. That's just my opinion, but I firmly believe it. So if you find yourself struggling with a topic, find another teacher/resource and perhaps it will be more clear. Of course this shouldn't diminish the effort needed on your part, learning math isn't a passive activity, one really has to do problems and work with the material.
And finally, proofs are of course the backbone of mathematics. Here is an intro text I like on that.
Oh okay, one more thing, physics is a great companion to math. I highly recommend "Classical Mechanics" by Taylor, in that regard. It will be challenging right now, but it will provide some great accompaniment to what you'll learn in upcoming years.
> 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.
Mathematical Proofs: A Transition to Advanced Mathematics was the book for my college proofs class. I found it to be a good resource and easy to follow. It covers some introductory set theory as well. Just be prepared to work through the proof exercises if you really want a good intuition on the topic.
I think "Mathematical Proofs: A Transition to Advanced Mathematics (2nd Edition)" is a solid book.
It starts off with what I would expect in a discrete math course (which is generally a first proofs course) and ends with a few chapters that would begin a second step writing intensive proofs course: number theory, calculus (real analysis), and group theory (algebra).
There are also many resources online that will help you once you've gotten through the basic notions in the book.
During my sophomore year I took an "intro to proofs" course (known formally at the institution as Foundations of Advanced Mathematics) and I found it to be extremely beneficial in my development as a mathematician. We used Chartrand's "Mathematical Proofs" textbook (here's the link for those who are interested).
The text covered set theory, logic, the various proof methods, and then dug into stuff like elementary number theory, equivalence relations, functions, cardinality (culminating in Cantor's two main results), abstract algebra, and analysis. Obviously the book only scratched the surface on a lot of these topics, but I felt it accomplished its goal.
Part of my satisfaction with the course is likely due to the fact that we had a brilliant professor who taught the course in the spirit of what u/Rtalbert235 spoke of. He was able to clearly articulate the distinction between computation and theory. The way I like to say it is that he taught us the difference between pounding a bunch of nails into a 2X4 (computation) and building a house (proving theorems).
I don't mean to universally praise "intro to proofs" courses, however. I can definitely see how they can be horrible wastes of time if not done properly, and I can also appreciate the idea of "throwing" students into proof-based courses (analysis, algebra, and so on). For me though, I think it's worth the effort to try and optimize these sorts of classes, which will ultimately serve a LOT of math students who need to understand proofs, but don't necessarily have a desire to pursue the subject beyond the undergraduate level.
tl;dr - Given the right combination of textbook and professor, an "intro to proofs" course can be just what the doctor ordered for developing mathematicians.
What is the "Highest Level" of mathematics you have taken?
Math is substantially more like a foreign language than popular culture would lead you to believe. It takes practice and what I like to call 'settle time.'
If you feel like you have a strong grasp on the concepts of algebra I highly recommend starting from 'scratch' (first principals) and getting a book like http://www.amazon.com/gp/offer-listing/0321390539/ref=sr_1_2_twi_har_1_olp?ie=UTF8&qid=1450533541&sr=8-2&keywords=mathematical+proofs
It was the first textbook that made me really start to understand what is needed to think like a mathematician. Start at the beginning work though problems, set theory is so much more important than most people realize. It will be cloudy and frustrating but really try to work some problems, put it down for a week let it stew and come back to the problems you had trouble with. Do that over and over.
While you are doing that pick up any elementary Stats/Prob and/or Linear Algebra book and start flipping through from the beginning you will see all the tools you are learning in Mathematical Proofs in those books as well. Try to take what you are learning and see it applied in those books to add some extra hooks to attach things to in your brain.
For Numerical Analysis you are going to want to build a strong base in proofs, linear algebra, set theory, and calculus as you go forward. Don't let this stop you from starting to read up it is a great way to stay excited when you are learning things to know fun ways that they are applied but don't get discouraged. My Numerical Analysis class was a Sr level college course that started the semester with 24 Math and CS majors about half gave up before the mid term/
Two great introductions are:
both are short and very well written. Some books particularly close to my heart are:
I think you meant:
not even wrong
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
Sure, there are lots of cool websites that don't ask for crazy prerequisites. One which I share with all of my friends who are starting out in math is the Fun Facts site, hosted by Harvey Mudd College.
As far as learning specific materials, you can try Khan Academy for what are perhaps some of the more elementary topics (it goes up to differential equations and linear algebra). If you want to learn more about number systems and algebra I think that either picking up a good, cheap book on number theory, or even checking out the University of Reddit's Group Theory course (presented by Math Doctor Bob) are both very strong options. Otherwise, you can check out YouTube for other lecture series that people are more and more frequently putting up.
This book is awesome and cheap as hell.
They may not be the best books for complete self-learning, but I have a whole bookshelf of the small introductory topic books published by Dover- books like An Introduction to Graph Theory, Number Theory, An Introduction to Information Theory, etc. The book are very cheap, usually $4-$14. The books are written in various ways, for instance the Number Theory book is highly proof and problem based if I remember correctly... whereas the Information Theory book is more of a straightforward natural-language summary of work by Claude Shannon et al. I still find them all great value and great to blast through in a weekend to brush up to a new topic. I'd pair each one with a real learning text with problem sets etc, and read the Dover book first quickly which introduces the reader to any unfamiliar terminology that may be needed before jumping into other step by step learning texts.
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.
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.
It depends what you're trying to get out of it.
There are literally hundreds of introductory texts for first-order logic. Other posters can cover them. There's so much variety here that I would feel a bit silly recommending one.
For formal tools for philosophy, I would say David Papineau's Philosophical Devices. There's also Ted Sider's Logic for Philosophy but something about his style when it comes to formalism rubs me the wrong way, personally.
For a more mathematical approach to first-order logic, Peter Hinman's Fundamentals of Mathematical Logic springs to mind.
For a semi-mathematical text that is intermediate rather than introductory, Boolos, Burgess, and Jeffrey's Computability and Logic is the gold standard.
Finally, if you want to see some different ways of doing things, check out Graham Priest's An Introduction to Non-Classical Logic.
I'm glad prof. faux decided to randomly cite an elementary introduction to logic such as hunter (without citing any particular page).
Don't get me wrong, it is a pretty decent introduction afaict, especially for undergrad students, but I'd be really delighted if he could mention what he meant in something a bit more used, if not slightly more rigorous, like boolos: https://www.amazon.com/Computability-Logic-Fifth-George-Boolos/dp/0521701465
And, of course, I'd feel real joy if he could cite a particular page to back up what he meant.
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?
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.
As I see it there are four kinds of books that fall into the sub $30 zone:
You can get a lot of great books if you are willing to spend a bit more however. For example, Hardy and Wright is an excellent book (and if you think about it: is a 600 page book for $60 really more expensive than a 300 page one for 30?). Richard Stanley's books on combinatorics: Enumerative Combinatorics Vol. I and Algebraic Combinatorics are also excellent choices. For algebra, Commutative Algebra by Eisenbud is great. If computer science interests you you can find commutative algebra books with an emphasis on Gröbner bases or on algorithmic number theory.
So that's a lot of suggestions, but two of them are free so you can't go wrong with those.
I would say that it would depend on the problem. If you cannot solve the first ten, I would be worried, as they can all be solved by simple brute force methods. I have a degree in Astrophysics, and some of the 300 and 400 problems are giving me pause, so if you are stuck there you are in good company.
There are elegant solutions to each problem, if you want to delve into them, but the first handful, the first ten especially, can be simply solved.
Once you get beyond the first ten or so, the mathematical difficulty ratchets up. There are exceptions to that rule of course, but by and large, it holds.
If you are interested in Number Theory, the best place to start is a number theory course at a local university. Mathematics, especially number theory, is difficult to learn by yourself, and a good instructor can expound, not only on the math, but also on the history of this fascinating subject.
Gauss, quite arguably the finest mathematician to ever live loved number theory; of it, he once said:
> Mathematics is the queen of sciences and number theory is the queen of mathematics. She often condescends to render service to astronomy and other natural sciences, but in all relations she is entitled to the first rank.
Although my personal favorite quote of his on the subject is:
> The enchanting charms of this sublime science reveal themselves in all their beauty only to those who have the courage to go deeply into it.
If you are interested in purchasing some books about number theory, here are a handful of recommendations:
Number Theory (Dover Books on Mathematics) by George E. Andrews
Number Theory: A Lively Introduction with Proofs, Applications, and Stories by James Pommersheim, Tim Marks, Erica Flapan
An Introduction to the Theory of Numbers by G. H. Hardy, Edward M. Wright, Andrew Wiles, Roger Heath-Brown, Joseph Silverman
Elementary Number Theory (Springer Undergraduate Mathematics Series) by Gareth A. Jones , Josephine M. Jones
and it's companion
A Classical Introduction to Modern Number Theory (Graduate Texts in Mathematics) (v. 84) by Kenneth Ireland, Michael Rosen
and a fun historical book:
Number Theory and Its History (Dover Books on Mathematics) Paperback by Oystein Ore
I would also recommend some books on
Markov Chains
Algebra
Prime number theory
The history of mathematics
and of course, Wikipedia has a good portal to number theory.
There's a couple options. You could pick up a basic elementary number theory book, which will have basically no prerequisites, so you'll be totally fine going into it. For instance Silverman has an elementary number theory book that I've heard great things about. I haven't read most of it myself, but I've read other things Silverman has written and they were really good.
There's a couple other books you might consider. Hardy and Wright wrote the classic text on it, which I've heard still holds up. I learned my first number theory from a book by Underwood Dudley which is by far the easiest introduction to number theory I've seen.
Another route you might take is that, since you have some background in calculus, you could learn a little basic analytic number theory. Much of this will still be out of your reach because you haven't taken a formal analysis class yet, but there's a book by Apostol whose first few chapters really only require knowledge of calculus.
If you decide you want to learn more number theory at that point, you're going to want to make sure you learn some basic algebra and analysis, but these are good places to start.
Proofs: Hammack's Book of Proof. Free and contains solutions to odd-numbered problems. Covers basic logic, set theory, combinatorics, and proof techniques. I think the third edition is perfect for someone who is familiar with calculus because it covers proofs in calculus (and analysis).
Calculus: Spivak's Calculus. A difficult but rewarding book on calculus that also introduces analysis. Good problems, and a solution manual is available. Another option is Apostol's Calculus which also covers linear algebra. Knowledge of proofs is recommended.
Number Theory: Hardy and Wright's An Introduction to the Theory of Numbers. As he explains in a foreword to the sixth edition, Andrew Wiles received this book from his teacher in high school and was a starting point for him. It also covers the zeta function. However, it may be too difficult for absolute beginners as it doesn't contain any problems. Another book is Stark's An Introduction to Number Theory which has a great section on continued fractions. You should have familiarity with proof before learning number theory.
Don't think of your abilities as fixed. The number of proofs you encounter grows from where you are now. You did not know algebra when you started. You will be increasingly exposed to proofs as you go along. Spend time on them. I recommend you get a tutor, or at least read some extra material. https://www.amazon.com/Mathematical-Proofs-Transition-Advanced-Mathematics/dp/0321390539/ref=sr_1_69?ie=UTF8&qid=1494805054&sr=8-69&keywords=proofs+math
Halmos's Naive Set Theory is a classic.
http://www.amazon.com/Naive-Theory-Undergraduate-Texts-Mathematics/dp/0387900926/ref=sr_1_1?ie=UTF8&qid=1320969231&sr=8-1
Naive Set Theory if you want a more textbook approach, the book mentioned in my other response if you're looking for something more like a story with proofs.
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.
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!
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.
In a quite literal sense:
Learning to Reason: An Introduction to Logic, Sets, and Relations by Nancy Rodgers
Try these books(the authors will hold your hand tight while walking you through interesting math landscapes):
Discrete Mathematics with Applications by Susanna Epp
Learning to Reason: An Introduction to Logic, Sets, and Relations by Nancy Rodgers
A Friendly Introduction to Number Theory Joseph Silverman
A First Course in Mathematical Analysis by David Brannan
The Foundations of Analysis: A Straightforward Introduction: Book 1 Logic, Sets and Numbers by K. G. Binmore
The Foundations of Topological Analysis: A Straightforward Introduction: Book 2 Topological Ideas by K. G. Binmore
Introductory Modern Algebra: A Historical Approach by Saul Stahl
An Introduction to Abstract Algebra VOLUME 1(very elementary)
by F. M. Hall
There is a wealth of phenomenally well-written books and as many books written by people who have no business writing math books. Also, Dover books are, as cheap as they are, usually hit or miss.
One more thing:
Suppose your chosen author sets the goal of learning a, b, c, d. Expect to be told about a and possibly c explicitly. You're expected to figure out b and d on your own. The books listed above are an exception, but still be prepared to work your ass off.
>My first goal is to understand the beauty that is calculus.
There are two "types" of Calculus. The one for engineers - the plug-and-chug type and the theory of Calculus called Real Analysis. If you want to see the actual beauty of the subject you might want to settle for the latter. It's rigorous and proof-based.
There are some great intros for RA:
Numbers and Functions: Steps to Analysis by Burn
A First Course in Mathematical Analysis by Brannan
Inside Calculus by Exner
Mathematical Analysis and Proof by Stirling
Yet Another Introduction to Analysis by Bryant
Mathematical Analysis: A Straightforward Approach by Binmore
Introduction to Calculus and Classical Analysis by Hijab
Analysis I by Tao
Real Analysis: A Constructive Approach by Bridger
Understanding Analysis by Abbot.
Seriously, there are just too many more of these great intros
But you need a good foundation. You need to learn the basics of math like logic, sets, relations, proofs etc.:
Learning to Reason: An Introduction to Logic, Sets, and Relations by Rodgers
Discrete Mathematics with Applications by Epp
Mathematics: A Discrete Introduction by Scheinerman
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.
I really like Nancy Rodgers' "Learning to Reason".
Keith Devlin has a course called Introduction to Mathematical Thinking that covers a subset of the material in Nancy's book.
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&qid=1288684060&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.
Set Theory:
Naive Set Theory
Number Theory:
Elementary Number Theory
Introduction to Analytic Number Theory
A Classical Introduction to Modern Number Theory
Topology:
Topology
Introduction to Topological Manifolds
A Pathway into Number Theory by Burns might appeal to you. You might want to put extra effort into digging up a book that approaches elementary number theory from a combinatorial point of view, which is more in line with the stuff you're doing now.
EDIT: This seems perfect for you: https://www.amazon.com/Number-Theory-Dover-Books-Mathematics/dp/0486682528/
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&qid=1459224709&sr=8-1&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&qid=1459224741&sr=8-1&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&qid=1459225065&sr=8-1&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 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.
Set theory/proof-writing is much more difficult than high school algebra. I'm teaching myself Calc III and Proof Writing right now in preparation for Abstract Algebra - I can say that compared to Calculus, the more advanced set theory is much more difficult. For me, anyway. For the Proof Writing I am using this book - How To Prove It: A Structured Approach. I'd say looking through that before moving on to anything more advanced than Calculus is a good idea. Which.. is why I'm doing it, myself.
Get a logic book. For math majors at my University Sets and Logic is required before Linear Algebra which is the first proof intensive class.
http://www.amazon.com/How-Prove-Structured-Daniel-Velleman/dp/0521446635
This is the textbook. Very helpful.
This turned into kind of a treatise, but you are in the same position I was once, so here goes...
First of all, this is about the best introduction to proofs you can get. It's $17. You should buy this now and read it. Do the problems, too - they're fun and not particularly hard.
As for other advice, if I were you, I'd just graduate so you have a bachelor's and then go back for pure math. That way if you don't end up liking it, at least you'll have something.
You could also just switch majors now if you're sure you want to do it, but take it from me, you're not going to do it in 2 years. The important thing is, even if you could, you wouldn't want to. If you're getting into pure math to go to graduate school, you need to keep in mind that your intense 2 years of studying is exactly what the rest of us do for 4 years. The minimum requirements for a math degree are exactly that - the bare minimum. In fact, I myself switched during the 4th year of an art degree, planning to graduate after 2 years, and am now at the tail end of my 3rd year and no longer have any intention to graduate "early." I'm just doing what I would have done if I had started in math normally, because I realized I want to be my best for graduate schools.
Point is, don't cheat yourself out of this by trying to get some fuckin BA in math. If you decide to do it, do it for real.
(Note: This is assuming you're looking for grad school. If your plan is to stop at bachelor's and then work, consider stats or applied math or double majoring math with something else, cause you ain't doin' shit with only a bachelor's in pure math. That's just a fact.)
This being said, the decision to become a mathematician is the best one I ever made. I was in your position and I am so much happier - even now, when all my old friends have graduated and I'm in "major switch purgatory" - than I would have been if I would have kept trying to be something I'm not. So, I'm not trying to be discouraging. It really is worth a thought.
Here is how you make the decision... next semester, find out if your university has a proofs class. It will probably be for sophomore mathematics majors and use a book similar to the one I linked. Take this class alongside whatever humanities requirements you'd be taking anyway. If it has prerequisites other than 1st year calc (it shouldn't), talk to the math advisor and get them waved. The class probably won't be very hard, but it will give you an idea of what the process of "doing a math problem" evolves into when you get to higher level math. After this, find an introductory abstract algebra class (not a linear algebra class - one that includes group theory), and an introductory analysis class. This way you'll get a taste of two very different flavors of upper level math, and you'll be able to see how doing proofs actually works out. If you find yourself wanting more, then switch (or graduate and go back). If you don't, then don't be a math major. All in all taking three classes is a pretty inexpensive way to find out whether you want to do something, and since you can fit them into your fourth year, it won't fuck up the option of graduating with cinema studies if you decide math isn't your thing.
If you want a strong mathematical approach, check out Peter Smith's Teach Yourself Logic Guide. If you don't want to take as heavy of an approach, you can use the suggestions as a roadmap and pick-and-choose from the suggestions. Even the introductory logic book suggestions in that guide might be too math heavy, but you might at least read their reviews on Amazon. A lot of reviewers tend to link to books on either side: easier and harder approaches.
For what it's worth, while I was in University we used Computability and Logic in the second logic course, which is after the introductory course teaching basic propositional and predicate logic. It's not a book for learning logic, but it's an awesome book for tying together a lot of what you initially learn with computability, model and proof theory. In another course we used An Introduction to Non-Classical Logic. I really enjoyed both of these books, and they're relatively cheap, but as I said they are not introductory logic books.
I'll be happy to reply again if you have any further questions.
http://www.amazon.com/Computability-Logic-George-S-Boolos/dp/0521701465
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.
Birkhoff and Mac Lane's Algebra goes a long way, and gets you used to Mac Lane's style.
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&f_primeEligible=true
On that book, which is a pretty widely used text, If they sell about 5 books, the price rises almost $70.
ja os espertos usam isso
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.
> 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.
(Note: I wrote this elsewhere)
Discrete Mathematics. It teaches the basics of the following 5 key concepts in theoretical computer science:
When you master these concepts, you will see that all hairy, formal definitions boil down to functions, sets, and propositions (with quantifiers). Recursion appears in many interesting structures in computer science, as well as in proofs of theorems (which are themselves propositions).
I don't know of any good online introduction to discrete math, but here are a few book ideas.
> 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. :)
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.
Good luck! It's some fun stuff. I'd also recommend this book if you don't already have it:
http://www.amazon.com/The-Humongous-Book-Calculus-Problems/dp/1592575129
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 Haskell Road to Logic, Maths and Programming
http://amzn.com/0954300696
I read only the first chapter or two a long time ago. I don't remember much, but I do remember I was able to progress through the book and learn new things about both math and Haskell from the text.
I didn't have any trouble getting the outdated examples to work. I had read LYAH previously though, so I wasn't a complete beginner.
I would really enjoy hearing what others have thought about this book.
Calculus or Pre-calc?
For calculus, I recommend: http://www.amazon.com/Calculus-Ron-Larson/dp/1285057090/ref=sr_1_2?s=books&ie=UTF8&qid=1398268486&sr=1-2&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.
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 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?
I need either this or this. I'm taking Calculus II this semester for the second time. I'm aiming to be a math major, but I had difficulty last time. I'm already off to a better start this semester, but I want as much practice as possible. I'm aiming for a Masters in Math. I'm lucky that I have high grades and the F from last semester only dropped me down to a 3.2 GPA. I can't afford to have it drop any lower. I can't afford to spend any more time at this level. I have a Calculus workbook that my mom bought me, but it only covers Calc I and about two chapters of Calc II.
Actually.. Anything from my School Stuff WL is stuff I feel I need in order to do well at school. I really need to get organized with my school work and papers.. ._.
The Haskell Road to Logic, Maths and Programming. I had already fallen in love with programming, and with Haskell, and this book showed me how well math, logic, and computer science play together. Shoutout to my aunt Trisha for giving me this book as a Christmas present in my junior year of high school
Don't feel badly. Calc II favors rote memorization, which a lot of people have to work at. You just have to practice.
Some stuff I did to get it down:
-Write the formulas on your bathroom mirror with dry erase. Every time you go to wash hands or pass it, try to review it a bit.
-Write the new formula 30+ times. It sucks. You are going to hate it, but damn if it doesn't work. As you're writing, try to review which variables mean what.
-Practice problems while waiting in line, commuting, etc. I liked this book (The Humongous Book of Calculus Problems) for some great explanations and practice problems: https://www.google.com/url?sa=t&source=web&rct=j&url=https://www.amazon.com/Humongous-Book-Calculus-Problems-Books/dp/1592575129&ved=2ahUKEwimlKb11f7jAhXUqZ4KHdXBC0QQFjAAegQIARAB&usg=AOvVaw38Qmi3pxSjppZwJW6CBno8
"Humongous Book of Calculus" explains in english without treating you like a dummy or a 5 year old in need of a story.
I recommend this book.
Most of the trig and precal you need will be built in to calculus problems. I would recommend just jumping in and doing lots of problems. The Humongous Book of Calculus Problems starts with trig and precal and moves into calculus, with everything explained. http://www.amazon.com/gp/aw/d/1592575129/ref=mp_s_a_1_1?qid=1452092788&sr=8-1&pi=SY200_QL40&keywords=humongous+book+of+calculus+problems&dpPl=1&dpID=515J89M2yTL&ref=plSrch.
It is also cheap. They also make one for algebra and trig but you probably don't need it. There is also an awesome free calculus book here:
https://www.math.wisc.edu/~keisler/calc.html
Along the way if you get stuck on something specific and a written explanation won't suffice, check khan Academy or YouTube for it.
Also if you plan on studying mathematics or anything closely related, you will likely need an analysis course, in which case Spivak's "Calculus" provides an excellent bridge.
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
I posted this in /learnmath but didn't get any response so I'll give it a try here.
I'm a senior high school student and I'm learning linear algebra using Pavel Grinfeld's videos and programming in Haskell with this book.
What can I do to practice and apply concepts of linear algebra and programming?
Any recommended textbooks to complement the LA course?
Is it a good idea to solve project Euler problems in order to acquire programming/math skills?
Have you seen The Haskell Road to Logic, Maths, and Programming? It's a pretty decent intro to higher math, and each chapter has a Haskell module.
The Haskell Road to Logic, Maths and Programming takes you through a lot of the basic “essential” math for CS, much of what would be covered in a typical discrete math course, but taught along side Haskell which is fun!
You might be interested in The Haskell Road to Logic, Maths and Programming or just google haskell+math. Formal work seems to be navigating towards haskell now. My background is in power engineering, so I'm very familiar with numerical stuff, but lacking in discrete math. That's what I'm trying to patch.
pre-cal: http://www.amazon.com/Precalculus-5th-Edition-Robert-Blitzer/dp/0321837347
calculus: http://www.amazon.com/Calculus-Ron-Larson/dp/1285057090
I also have this one (10th ed) but its not the one you want :/ https://www.amazon.com/Calculus-Ron-Larson/dp/1285057090
The Haskell Road to Logic, Maths, and Programming.
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.
Stewart, Larson, Rogawski
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
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&ie=UTF8&qid=1536864274&sr=1-6
physical copy
https://www.amazon.com/Calculus-Early-Transcendentals-James-Stewart/dp/1285741552/ref=mt_hardcover?_encoding=UTF8&me=&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.
Hoffman and Kunze
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.
This one? How advanced would you say it goes into primes?
This one's well-known and highly regarded as a good source.
I'm also going to start learning number theory because it's a pretty fun subject. So far, Hardy's been pretty good (I've only read excerpts of the 1st chapter though).
As for your background, you would only need to know basic facts about numbers (divisibility/primes etc) when starting Hardy so you should be fine I think.
There is Elementary number theory by William Stein, and A Computational Introduction to Number Theory and Algebra. The latter is better if you are also interested in some of the computation They are both available for free online (legally). I think you would prefer Stein's book but skim through both and see which one you like more.
For something more in depth, I looked at some of the books in this list at mathoverflow. Hardy & Wright , and Niven & Zuckerman's books seem best suited to you (from what I looked at, but go through that list yourself). Many of the other books require some background in abstract algebra.
I haven't read either but just looking through their table of contents I would go with Niven and Zuckerman's book. It seems to go into the more useful things more quickly, and it's not as densely packed with information you probably won't be interested in right now.
TLDR: Start here, or here.
For what it's worth, number theory is a fascinating field. I don't think you'll be disappointed going into it. Good luck!
I think An Introduction to the Theory of Numbers was the book I used as an undergrad.
In a math course I recently took that was basically an introduction to math proofs we used Mathematical Proofs: A Transition to Advanced Mathematics which I found to be a great text. It begins by going through the language and syntax used in proofs and slowly progresses through theory, different types of proofs, and eventually proofs from advanced calculus. There are so many examples that are very well laid out and explained. I would highly recommend it for learning proofs from scratch.
Having a proof explained to you isn't even close to the thrill of proving something yourself, IMO. My advice would be to get your hands on an intro to proofs text and work through some of it. If you don't like writing proofs or think it's boring, your time at university is probably going to bore you to tears.
If you want an intro proofs book, you might start here. The text is very clearly written and chapters 9-13 will give you a very basic notion of what ideas will be at the core of some of your upper-level math classes (abstract algebra, real analysis, etc).
Concepts of Modern Mathematics by Ian Stewart is an excellent book about modern math. As is Foundations and Fundamental Concepts of Mathematics by Howard Eves I would recommend these two along with the far more expensive Naive Set Theory by Halmos
I might suggest the references & bibliography from: http://www.amazon.com/Not-Even-Wrong-Failure-Physical/dp/0465092764/ref=sr_1_1?ie=UTF8&qid=1320808955&sr=8-1
I read The Trouble With Physics about when it came out, so quite a while ago. In trying to find that reference I stumbled on the Not Even Wrong book / blog, which seems a slightly more up to date version of the same thing.
My understanding of the point of the criticism - and this isn't at all my field, so take all of this with that in mind - is stronger than we don't currently have a way to test string theory. The argument from the Trouble With Physics was, if I recall it correctly, that string theory was not so much a theory as a class of theories, and a sufficiently broad class of theories that with the right constants inserted, they could be made to model any result and consequently were unfalsifiable, regardless of any improvements that may come in experimental physics. How much truth do you see in that criticism?
>Hossenfelder’s argument, in brief: There’s no reason to think nature cares what we find beautiful
I'm not string theory supporter anyway and I pointed to its conceptual problems in the time, when Dr. Hossenfelder posted on article about extradimensions after another (see bellow) - but a bit more sanity and less ideology would be useful even when judging the string theory fiasco:
Reality check 1: Dr. Hossenfelder pursuits “ugly” bottom-up phenomenological approach to physics rather than up-bottom “pretty math based” stringy/susy theories – but even uglier fact is, that this (her?) phenomenological approach failed as well. There is no beautiful but failed and ugly but successful approach to theoretical physics: only failed theoretical physics of all kinds thinkable during last four decades.
Reality check 2: At least Lee Smolin or Peter Woit wrote their insightful books well before string theory fiasco – but where Dr. Hossenfelder was, when they pointed to its problems? After battle everyone is general, after wit is everyone’s wit… ;-)
Reality check 3: Her hypocrisy and opportunism goes even deeper: When string theory was still hyped, Dr. Hossenfelder also jumped into its bandwagon for example by many studies involving extradimensions – but now she bravely pretends, she was never involved into this hype.
Dr. Hossenfelder popularity solely depends on short memory of laymen public i.e. that people forgot, she was herself a great promoter of extra dimensional stuffs and black holes and that she made money and scientific "credit" with writing about them (Observables from Large Extra_Dimensions, Signatures_of_Large_Extra_Dimensions, Black hole relics in large extra dimensions, Black Hole Production in Large Extra Dimensions at the Tevatron, Observables of Extra Dimensions Approaching the Planck Scale, [Suppression of High-P_T Jets as a Signal for Large Extra Dimensions](https://www.researchgate.net/publication/2001593_Suppression_of_High-P_T_Jets_as_a_Signal_for_Large_Extra_Dimensions and New_Estimates_of_Lifetimes_for_Meta_stable_Micro_BlackHoles-From_the_Early_Universe_to_Future_Colliders), Schwarze Löcher in Extra-Dimensionen, Black hole production in large extra dimensions at the Tevatron) just before ten years.
The opposition to String Theory boils down to it being "not even wrong" on the basis that it cannot be falsified by experiment. Pretty well summed up in the book of the same title.
Scientists have been successful in ruling out some forms of a String theory (there are almost boundless forms), but the most "successful" forms don't really yield any unique predictions that can be tested (in the real world).
edit: I also found the book I linked to be a very approachable way to understand the mathematics of the Standard Model (irrespective of String Theory).
I think Hawking and Green both are string theorists? I just started reading Peter Woit's book about theory of everything/quantum mechanics. He argues that string theory is not able to be proved right or wrong scientifically, and is basically not valid science.
So it doesn't violate the quick-reject sniff tests. Now what?
I'll let someone smarter than me make the arguments. If you're really interested, go check this out: http://www.amazon.com/Not-Even-Wrong-Failure-Physical/dp/0465092764
Herstein!! http://www.amazon.com/Topics-Algebra-I-N-Herstein/dp/0471010901?tag=vglnk-c905-20
For compsci you need to study tons and tons and tons of discrete math. That means you don't need much of analysis business(too continuous). Instead you want to study combinatorics, graph theory, number theory, abstract algebra and the like.
Intro to math language(several of several million existing books on the topic). You want to study several books because what's overlooked by one author will be covered by another:
Discrete Mathematics with Applications by Susanna Epp
Mathematical Proofs: A Transition to Advanced Mathematics by Gary Chartrand, Albert D. Polimeni, Ping Zhang
Learning to Reason: An Introduction to Logic, Sets, and Relations by Nancy Rodgers
Numbers and Proofs by Allenby
Mathematics: A Discrete Introduction by Edward Scheinerman
How to Prove It: A Structured Approach by Daniel Velleman
Theorems, Corollaries, Lemmas, and Methods of Proof by Richard Rossi
Some special topics(elementary treatment):
Rings, Fields and Groups: An Introduction to Abstract Algebra by R. B. J. T. Allenby
A Friendly Introduction to Number Theory Joseph Silverman
Elements of Number Theory by John Stillwell
A Primer in Combinatorics by Kheyfits
Counting by Khee Meng Koh
Combinatorics: A Guided Tour by David Mazur
Just a nice bunch of related books great to have read:
generatingfunctionology by Herbert Wilf
The Concrete Tetrahedron: Symbolic Sums, Recurrence Equations, Generating Functions, Asymptotic Estimates by by Manuel Kauers, Peter Paule
A = B by Marko Petkovsek, Herbert S Wilf, Doron Zeilberger
If you wanna do graphics stuff, you wanna do some applied Linear Algebra:
Linear Algebra by Allenby
Linear Algebra Through Geometry by Thomas Banchoff, John Wermer
Linear Algebra by Richard Bronson, Gabriel B. Costa, John T. Saccoman
Best of Luck.
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.
Number theory is pretty cool. I enjoyed Dudley's book for a number of reasons.
I would highly recommend spending some time learning number theory first. Much of crypto relies on understanding a fair amount of number theory in order to understand what and why stuff works.
The book antiantiall linked is fantastic (I have a copy), however if you don't have a strong foundation in number theory will likely be a bit over your head.
Here is the textbook that was used in my number theory course. It isn't necessarily the best out there, but is cheap and does a good job covering the basics.
Also pick a book on Number Theory, pretty useful in computer science. I would recommend: https://www.amazon.com/Elementary-Number-Theory-Second-Mathematics/dp/048646931X/ref=sr_1_1?ie=UTF8&qid=1500074415&sr=8-1&keywords=elementary+number+theory+dudley
Hello! I'm interested in trying to cultivate a better understanding/interest/mastery of mathematics for myself. For some context:
 
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.
 
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
My university used George Andrews's book, which is Dover and really cheap. It was a pretty good book.
https://www.amazon.com/Number-Theory-Dover-Books-Mathematics/dp/0486682528/ref=pd_lpo_sbs_14_t_0/130-7956659-8948968?_encoding=UTF8&psc=1&refRID=7N6E11C8GDASPV8X3BAN
This was my first arithmetic book. I found it quite good and covered a wide range of topics.
I used this book when I was in high school:
Number Theory
Costs $8, explains things beautifully
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&qid=1348165257&sr=8-1&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&ie=UTF8&qid=1348165294&sr=1-2&keywords=abstract+algebra
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.
How to Read and Do Proofs by Daniel Solow this book saved my life in Abstract Algebra.
I can't really give a better testimonial other than I read through this book and applied a couple of the concepts and did very well in the course.
One thing to remember, you can always reverse your steps, if you are stuck at some point, then work backwards from the end and you can sometimes meet up to the point you were stuck at.
Also, How to Prove It by Daniel J Velleman is another classic book that can help.
A lot of early math tends to come down to how often you do problems, and computation classes can generally be seen as rote learning. I'd suggest you start doing some proofs, they force you to understand what you are doing, rather then just doing what you've seen. Pick up http://www.amazon.com/How-Prove-It-Structured-Approach/dp/0521446635
or, http://www.amazon.com/Calculus-4th-Michael-Spivak/dp/0914098918/ref=sr_1_1?s=books&ie=UTF8&qid=1345011596&sr=1-1&keywords=spivaks+calculus
Not knowing random operations as you listed is fine, with time you will get quicker, but don't worry if you need to consider it for a moment.
I've been taking it this year, and we've been using Velleman's "How to Prove It." Unfortunately, there aren't answers for all the problems, but I've found it to be a pretty good book. Amazon
I know a few people who highly recommend How to Prove It by Velleman. I've never read it so I can't say for sure. The first book I used to learn mathematical logic was Lay's Analysis with an Intro to Proof. I can't recommend that book enough. The first quarter of the book or so is a pretty gentle introduction to mathematical logic, sets, functions, and proof techniques. I imagine it will get you where you need to be pretty quickly.
I recommend this to all beginners -- I like the Barwise & Etchemendy book because it's aimed at people with no background at all in logic or upper-level math, it's restricted to propositional and first-order logic (which I think logicians of all stripes should know), and it comes with proof-checker software so that you can check your own understanding instead of needing to find someone to give you feedback.
After that, you'll have some familiarity with the topic and can decide where you want to go. For a more mathematical route, I think Enderton (mentioned previously) or Boolos are good follow-ups. For a more philosophical route, I think Sider or Priest are good next steps.
The Logic Book is a good text for FOL and the early theorems of meta-logic (soundness and completeness of propositional and first-order logics). It's somewhat slow going though.
A more mathematically inclined text is Herbert Enderton's Introduction to Mathematical Logic. Enderton goes into more of the meta-logic, including incompleteness, Lowenheim-Skolem, and computability. He also touches on second-order logic toward the end.
Along the lines of meta-logic, Boolos and Jeffrey's Computability and Logic is very good as well. (Er, and Burgess. I can only vouch for the 3rd edition, which is pre-Burgess.)
Given that you're already familiar with FOL, I'd lean toward Enderton or Boolos and Jeffrey with the caveat that The Logic Book has endless practice problems and, iirc, answers to many of them in the back of the book (the others have fewer (but more interesting) problems).
If you want to go beyond FOL, I second stoic9's suggestion of Priest's book.
Goedel's Incompleteness Theorems, by Raymond Smullyan.
From the preface:
> [intended] for the general mathematician, philosopher, computer scientist and any other curious reader who has at least a nodding acquaintance with the symbolism of first-order logic..and who can recognize the logical validity of a few elementary formulas.
I'm guessing most of the people on /r/math meet that description and more. If you want a general introduction to mathematical logic and computation, you should read Computability and Logic by George Boolos. If you can read Boolos, you can probably read Smullyan, and if you read them both you should emerge with some understanding of incompleteness.
Possible path:
Learn to think like mathematicians because you'll need it. For example, Mathematical Proofs: A Transition to Advanced Mathematics by Chartrand et al is a good book for that. When you got the basics of math argumentation down, it's time for abstract algebra with emphasis on vector spaces(you really need good working knowledge of linear algebra). People like Axler's Linear Algebra Done Right. Maybe, study that. Or maybe work through Maclane's Algebra or Chapter 0 by Aluffi.
After that you want to get familiar with more or less rigorous calculus. One possibility is to study Spivak's Calculus, then pick up Munkres Analysis on Manifolds.
Up next: differential geometry which is your main goal. At this point your mathematical sophistication will have matured to the level of a grad student of math.
Good luck.
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.
I'm planning on relearning calculus also. The books that were recommended to me were:
http://www.amazon.com/gp/aw/d/1592575129?pc_redir=1412262976&robot_redir=1
http://www.amazon.com/gp/aw/d/0716731606/ref=pd_aw_sims_3?pi=SL500_SY115&simLd=1
They're not exactly textbooks, but they appear to be good guides. Best of luck.
That was the premise of "Not Even Wrong", that string theory remains outside the scope of science due to its complete lack of testability.
So that leaves the string theorist with "ad hominem" attacks like this post essentially calling everyone who disagrees with them "stupid" i.e. "non-specialist".
> 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.