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.

1. Mathematics: A very Short Introduction : A very good book, but also very short book about mathematics by Timothy Gowers, a Field medalist and overall awesome guy, gives you a feelling for what math is all about.
   
   
2. Concepts of Modern Mathematics: A really interesting book by Ian Stewart, it has more topics than the last book, it is also bigger though less formal than Gower's book. A gem.
   
   
3. What is Mathematics?: A classic that has aged well, it's more textbook like compared to the others, which is good because the best way to learn mathematics is by doing it. Read it.
   
   
4. An Infinitely Large Napkin: This is the most modern book in this list, it delves into a huge number of areas in mathematics and I don't think it should be read as a standalone, rather it should guide you through your studies.
   
   
5. The Princeton Companion to Mathematics: A humongous book detailing many areas of mathematics, its history and some interesting essays. Another book that should be read through your life.
   
   
6. Mathematical Discussions: Gowers taking a look at many interesting points along some mathematical fields.
   
   
7. Technion Linear Algebra Course - The first 14 lectures: Gets you wet in a few branches of maths.
   
   Linear Algebra: An extremelly versatile branch of Mathematics that can be applied to almost anything, also the first "real math" class in most universities.
   
   
8. Linear Algebra Done Right: A pretty nice book to learn from, not as computational heavy as other Linear Algebra texts.
   
   
9. Linear Algebra: A book with a rather different approach compared to LADR, if you have time it would be interesting to use both. Also it delves into more topics than LADR.
   
   
10. Calculus Vol II : Apostols' beautiful book, deals with a lot of lin algebra and complements the other 2 books by having many exercises. Also it doubles as a advanced calculus book.
    
    
11. Khan Academy: Has a nice beginning LinAlg course.
    
    
12. Technion Linear Algebra Course: A really good linear algebra course, teaches it in a marvelous mathy way, instead of the engineering-driven things you find online.
    
    
13. 3Blue1Brown's Essence of Linear Algebra: Extra material, useful to get more intuition, beautifully done.
    
    Calculus: The first mathematics course in most Colleges, deals with how functions change and has many applications, besides it's a doorway to Analysis.
    
    
14. Calculus: Tom Apostol's Calculus is a rigor-heavy book with an unorthodox order of topics and many exercises, so it is a baptism by fire. Really worth it if you have the time and energy to finish. It covers single variable and some multi-variable.
    
    
15. Calculus: Spivak's Calculus is also rigor-heavy by Calculus books standards, also worth it.
    
    
16. Calculus Vol II : Apostols' beautiful book, deals with many topics, finishing up the multivariable part, teaching a bunch of linalg and adding probability to the mix in the end.
    
    
17. MIT OCW: Many good lectures, including one course on single variable and another in multivariable calculus.
    
    Real Analysis: More formalized calculus and math in general, one of the building blocks of modern mathematics.
    
    
18. Principle of Mathematical Analysis: Rudin's classic, still used by many. Has pretty much everything you will need to dive in.
    
    
19. Analysis I and Analysis II: Two marvelous books by Terence Tao, more problem-solving oriented.
    
    
20. Harvey Mudd's Analysis lectures: Some of the few lectures on Real Analysis you can find online.
    
    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.
    
    
21. Abstract Algebra: Dummit and Foote's book, recommended by many and used in lots of courses, is pretty much an encyclopedia, containing many facts and theorems about structures.
    
    
22. Harvard's Abstract Algebra Course: A great course on Abstract Algebra that uses D&F as its textbook, really worth your time.
    
    
23. Algebra: Chapter 0: I haven't used this book yet, though from what I gathered it is both a category theory book and an Algebra book, or rather it is a very different way of teaching Algebra. Many say it's worth it, others (half-jokingly I guess?) accuse it of being abstract nonsense. Probably better used after learning from the D&F and Harvard's course.
    
    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.

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.

• The Nature and Power of Mathematics, Donald M. Davis. This book seems to be a survey of some history of mathematics and various modern topics. Check out the table of contents to get an idea. You'll notice a few of the subjects in the list below. It seems like this would be a good buy if you want to taste a few different subjects to see what pleases your palate.
  
  
• Introduction to Graph Theory, Richard J. Trudeau. Check out the Wikipedia entry on graph theory and the one defining graphs to get an idea what the field is about and some history. The reviews on Amazon for this book lead me to believe it would be a perfect match for an interested high school student.
  
  
• Game Theory: A Nontechnical Introduction, Morton D. Davis. Game theory is a very interesting field with broad applications--check out the wiki. This book seems to be written at a level where you would find it very accessible. The actual field uses some heavy math but this seems to give a good introduction.
  
  
• An Introduction to Information Theory, John R. Pierce. This is a light-on-the-maths introduction to a relatively young field of mathematics/computer science which concerns itself with the problems of storing and communicating data. Check out the wiki for some background.
  
  
• Lady Luck: The Theory of Probability, Warren Weaver. This book seems to be a good introduction to probability and covers a lot of important ideas, especially in the later chapters. Seems to be a good match to a high school level.
  
  
• Elementary Number Theory, Underwood Dudley. Number theory is a rich field concerned with properties of numbers. Check out its Wikipedia entry. I own this book and am reading through it like a novel--I love it! The exposition is so clear and thorough you'd think you were sitting in a lecture with a great professor, and the exercises are incredible. The author asks questions in such a way that, after answering them, you can't help but generalize your answers to larger problems. This book really teaches you to think mathematically.
  
  
• A Book of Abstract Algebra, Charles C. Pinter. Abstract algebra formalizes and generalizes the basic rules you know about algebra: commutativity, associativity, inverses of numbers, the distributive law, etc. It turns out that considering these concepts from an abstract standpoint leads to complex structures with very interesting properties. The field is HUGE and seems to bleed into every other field of mathematics in one way or another, revealing its power. I also own this book and it is similarly awesome. The exposition sets you up to expect the definitions before they are given, so the material really does proceed naturally.
  
  
• Introduction to Analysis, Maxwell Rosenlicht. Analysis is essentially the foundations and expansion of calculus. It is an amazing subject which no math student should ignore. Its study generally requires a great deal of time and effort; some students would benefit more from a guided class than from self-study.
  
  
• Principles of Statistics, M. G. Bulmer. In a few words, statistics is the marriage between probability and analysis (calculus). The wiki article explains the context and interpretation of the subject but doesn't seem to give much information on what the math involved is like. This book seems like it would be best read after you are familiar with probability, say from Weaver's book linked above.
  
  
• I have to second sellphone's recommendation of Naive Set Theory by Paul Halmos. It's one of my favorite math books and gives an amazing introduction to the field. It's short and to the point--almost a haiku on the subject.
  
  
• Continued Fractions, A. Ya. Khinchin. Take a look at the wiki for continued fractions. The book is definitely terse at times but it is rewarding; Khinchin is a master of the subject. One review states that, "although the book is rich with insight and information, Khinchin stays one nautical mile ahead of the reader at all times." Another review recommends Carl D. Olds' book on the subject as a better introduction.
  
  Basically, don't limit yourself to the track you see before you. Explore and enjoy.

It sounds like you might perhaps want a background in Number Theory and/or Basic Logic and/or Set Theory. The thing about math is that there is a lot...

My advice for a text that might serve you well is N.L. Biggs' Discrete Mathematics (http://www.amazon.com/Discrete-Mathematics-Norman-L-Biggs/dp/0198507178). If you are at all interested in computer science, this is also a great book for that because it introduces some of the mathematical rigor behind it. Some people have a smidgen of difficulty with this text because it doesn't give some names to proofs/algorithms that maybe you've heard whispered (e.g. Dijkstra's shortest path and Prim's minimal spanning tree). A text that I tend to think is on par with Biggs', but many think is vastly superior (I love both, but for different reasons) that covers some (most) of the same topics is Eccles' An Introduction to Mathematical Reasoning (http://www.amazon.com/Introduction-Mathematical-Reasoning-Peter-Eccles/dp/0521597188/ref=pd_sim_b_4?ie=UTF8&refRID=1BB6VKRP59S2420M132F). This book has a wonderful focus on building from the ground up and emphasizes clearly worded and mathematically rigorous proofs.

You seem genuinely interested in mathematics, but I do want to warn you about some more ahem esoteric (read: improperly worded, perhaps?) problems that ask such things as why 1 is greater than 0. The mathematics here is largely armchair - lacking any fundamental logic. There would be no issue with redefining a set of bases such that "0" is greater than "1". However, if you want to have rationale of the concept of things being greater than another, that's more like number theory. You can learn the 10 axioms of natural numbers and then build from there.

Both of the books I mentioned will cover stuff like this. For example, they both (unless I'm not remembering correctly) delve into Euclid's proof of infinite primes, something which may interest you.

Briefly (and not so rigorously), assume that the number of primes, p1, p2, p3, ..., pN, is finite. Then there exists a number P which is the product of these primes. Based on the axioms of natural numbers, since all primes p1,p2,...,pN are natural numbers P is a natural number and so is P+1. Consider S = P+1. If S is prime than our list is incomplete, assume S isn't prime. Then some number in our list, say pI, divides S because any natural number can be written as the product of primes. pI must also divide P because P equals the sum of all primes. Therefore if pI divides S and pI divides P, then pI divides S-P = 1. That's a contradiction because no prime evenly divides 1.

Stuff like this is super cool, super simple, and super beautiful and you absolutely can learn it. These two books would be a great place to start.

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.

Good question OP! I drafted a blog article on this topic a while back but haven't published it yet. An excerpt is below.
--------

With equations, I sometimes just visualize what I'd usually do on paper. For arithmetic, there are actually a lot of computational methods that are better suited to mental computation than the standard pencil-and-paper algorithms.

In fact, mathematician Arthur Benjamin has written a book about this called Secrets of Mental Math.

There are tons of different options, often for the same problem. The main thing is to understand some general principles, such as breaking a problem down into easier sub-problems, and exploiting special features of a particular problem.

Below are some basic methods to give you an idea. (These may not all be entirely different from the pencil-and-paper methods, but at the very least, the format is modified to make them easier to do mentally.)

ADDITION
(1) Separate into place values: 27+39= (20+30)+(7+9)=50+16=66

We've reduced the problem into two easier sub-problems, and combining the sub-problems in the last step is easy, because there is no need to carry as in the standard written algorithm.

(2) Exploit special features: 298+327 = 300 + 327 -2 = 625

We could have used the place value method, but since 298 is close to 300, which is easy to work with, we can take advantage of that by thinking of 298 as 300 - 2.

SUBTRACTION

(1) Number-line method: To find 71-24, you move forward 6 units on the number line to get to 30, then 41 more units to get to 71, for a total of 47 units along the number line.

(2) There are other methods, but I'll omit these, since the number-line method is a good starting point.

MULTIPLICATION

(1) Separate into place values: 18*22 = 18*(20+2)=360+36=396.

(2) Special features: 18*22=(20-2)*(20+2)=400-4=396

Here, instead of using place values, we use the feature that 18*22 can be written in the form (a-b)*(a+b) to obtain a difference of squares.

(3) Factoring method: 14*28=14*7*4=98*4=(100-2)*4=400-8=392

Here, we've turned a product of two 2-digit numbers into simpler sub-problems, each involving multiplication by a single-digit number (first we multiply by 7, then by 4).

(4) Multiplying by 11: 11*52= 572 (add the two digits of 52 to get 5+2=7, then stick 7 in between 5 and 2 to get 572).

This can be done almost instantaneously; try using the place-value method to see why this method works. Also, it can be modified slightly to work when the sum of the digits is a two digit number.

DIVISION
(1) Educated guess plus error correction: 129/7 = ? Note that 7*20=140, and we're over by 11. We need to take away two sevens to get back under, which takes us to 126, so the answer is 18 with a remainder of 3.

(2) Reduce first, using divisibility rules. Some neat rules include the rules for 3, 9, and 11.

The rules for 3 and 9 are probably more well known: a number is divisible by 3 if and only if the sum of its digits is divisible by 3 (replace 3 with 9 and the same rule holds).

For example, 5654 is not divisible by 9, since 5+6+5+4=20, which is not divisible by 9.

The rule for 11 is the same, but it's the alternating sum of the digits that we care about.

Using the same number as before, we get that 5654 is divisible by 11, since 5-6+5-4=0, and 0 is divisible by 11.

PRACTICE
I think it's kind of fun to get good at finding novel methods that are more efficient than the usual methods, and even if it's not that fun, it's at least useful to learn the basics.

If you want to practice these skills outside of the computations that you normally do, there's a nice online arithmetic game I found that's simple and flexible enough for you to practice any of the four operations above, and you can set the parameters to work on numbers of varying sizes.

Happy calculating!
Greg at Higher Math Help

Edit: formatting

hey nerdinthearena,

i too find this area to be fascinating and wish i knew more on the upper end myself. i'm just going to list off a few resources. in my opinion, graduate school will concentrate a lot on progressing your technical knowledge, but will likely not give you a lot of time to hone your intuition (at least in the first few years). so, the more time you spend in undergraduate school doing so, the better.

helpful for intuition and basic understanding

• geometrical vectors by gabriel weinreich - why there's more to vectors and how to draw them. this basically shows visual examples of differential forms and the like.
  
• advanced calculus: a differential forms approach by harold edwards - a fantastic approach to differential forms. it's a treasure, so just read it. :) it is actually well built for skimming and skipping around.
  
• vector calculus, linear algebra, and differential forms: a unified approach - this is a big book, but you could just jump ahead to the later sections covering differential forms for an interesting approach and technical skill building.
  
• div, grad, curl are dead by william burke - pure intuition and similar to geometrical vectors. don't look for anything else here, but definitely read it.
  
• applied differential geometry by william burke - a more complete work than the above and showcases a lot of the connections between geometry and physics.
  
  

  more advanced but still intuitive
  

• an introduction to manifolds - this is at the same level and covers essentially the same material as lee, but in my opinion, it is much better. it is much more concise and to the point with more focused exercises.
  
• differential forms in algebraic topology by raoul bott and loring tu - the sequel to the above, but i haven't read beyond the first chapter. it is well known as a must read classic, but having some topology background will help with this book.
  
• the laplacian on a riemannian manifold: an introduction to analysis on manifolds - an interesting introduction to geometric analysis.
  
• differentiable manifolds: forms, currents, harmonic forms by georges de rham - a classic by one of the greats and also one of the only sources i've found that covers de rham currents (besides research papers that just use them).
  
• riemannian manifolds: an introduction to curvature by john lee - the sequel to lee's smooth manifolds book.
  
  hopefully this helps. if i were to revisit geometric analysis, i would basically use the above books to help bone up my understanding, intuition, and technical skill before moving on. these are also mainly geometry books, so learning analysis (like functional analysis) would be good as well. i mainly have three suggestions there.
  
  

  three general analysis favorites
  

• introductory functional analysis with applications by erwin kreyszig - used everywhere.
  
• calculus of variations by gelfand and fomin
  
• linear differential operators by cornelius lanczos - a geometrical look at the subject. also take a look at the variational principles of mechanics by him as well.

> Mathematical Logic

It's not exactly Math Logic, just a bunch of techniques mathematicians use. Math Logic is an actual area of study. Similarly, actual Set Theory and Proof Theory are different from the small set of techniques that most mathematicians use.

Also, looks like you have chosen mostly old, but very popular books. While studying out of these books, keep looking for other books. Just because the book was once popular at a school, doesn't mean it is appropriate for your situation. Every year there are new (and quite frankly) pedagogically better books published. Look through them.

Here's how you find newer books. Go to Amazon. In the search field, choose "Books" and enter whatever term that interests you. Say, "mathematical proofs". Amazon will come up with a bunch of books. First, sort by relevance. That will give you an idea of what's currently popular. Check every single one of them. You'll find hidden jewels no one talks about. Then sort by publication date. That way you'll find newer books - some that haven't even been published yet. If you change the search term even slightly Amazon will come up with completely different batch of books. Also, search for books on Springer, Cambridge Press, MIT Press, MAA and the like. They usually house really cool new titles. Here are a couple of upcoming titles that might be of interest to you: An Illustrative Introduction to Modern Analysis by Katzourakis/Varvarouka, Understanding Topology by Shaun Ault. I bet these books will be far more pedagogically sound as compared to the dry-ass, boring compendium of facts like the books by Rudin.

If you want to learn how to do routine proofs, there are about one million titles out there. Also, note books titled Discrete Math are the best for learning how to do proofs. You get to learn techniques that are not covered in, say, How to Prove It by Velleman. My favorites are the books by Susanna Epp, Edward Scheinerman and Ralph Grimaldi. Also, note a lot of intro to proofs books cover much more than the bare minimum of How to Prove It by Velleman. For example, Math Proofs by Chartrand et al has sections about doing Analysis, Group Theory, Topology, Number Theory proofs. A lot of proof books do not cover proofs from Analysis, so lately a glut of new books that cover that area hit the market. For example, Intro to Proof Through Real Analysis by Madden/Aubrey, Analysis Lifesaver by Grinberg(Some of the reviewers are complaining that this book doesn't have enough material which is ridiculous because this book tackles some ugly topological stuff like compactness in the most general way head-on as opposed to most into Real Analysis books that simply shy away from it), Writing Proofs in Analysis by Kane, How to Think About Analysis by Alcock etc.

Here is a list of extremely gentle titles: Discovering Group Theory by Barnard/Neil, A Friendly Introduction to Group Theory by Nash, Abstract Algebra: A Student-Friendly Approach by the Dos Reis, Elementary Number Theory by Koshy, Undergraduate Topology: A Working Textbook by McClusckey/McMaster, Linear Algebra: Step by Step by Singh (This one is every bit as good as Axler, just a bit less pretentious, contains more examples and much more accessible), Analysis: With an Introduction to Proof by Lay, Vector Calculus, Linear Algebra, and Differential Forms by Hubbard & Hubbard, etc

This only scratches the surface of what's out there. For example, there are books dedicated to doing proofs in Computer Science(for example, Fundamental Proof Methods in Computer Science by Arkoudas/Musser, Practical Analysis of Algorithms by Vrajitorou/Knight, Probability and Computing by Mizenmacher/Upfal), Category Theory etc. The point is to keep looking. There's always something better just around the corner. You don't have to confine yourself to books someone(some people) declared the "it" book at some point in time.

Last, but not least, if you are poor, peruse Libgen.

I have a B.S. and M.S. in math, and am currently working on my PhD...here's my shot at your questions:

>1) At what point in your studies did you come to know about your limitations and abilities?

I didn't really have any struggles through my bachelor's, but as I got further into graduate studies I definitely had some hard classes and had to work much longer and harder to understand things than I ever had.

>I read about "Maryam Mirzakhani" two days ago. Do you think that you have a chance of producing worthy work in the future?

I don't think I'll ever win a Field's medal or be anywhere near the level of intelligence of someone like Maryam Mirzakhani, but I don't let that keep me from enjoying the journey. I know that I'll do something worthwhile, even if it's not groundbreaking.

>2) How did you choose your specific graduate program? I'm confused about what I should start with.

I was confused about what area to work in also, until I began studying for my comprehensive examinations (we have to take 3, each in different areas). I found that I really enjoyed studying the logic material, and I wasn't even too worried about the exam because enjoying the preparation made me well prepared. I just wanted to keep learning more. Just pick something that you find really interesting. It doesn't have to be "your area" for the rest of your life...you can always try something else later.

>3) How did you develop your critical thinking skills that are needed in following through with proofs and ideas?

The only way to get better at proofs as to do a ton of them. I had to get reamed pretty bad on some proofs at the beginning of grad school before I really got it...and I still have a long way to go. There's is always something to be improved upon.

There's a great excerpt from The Number Devil that sums up my feelings about proofs exactly:

"Have you ever tried to cross a raging stream?" the number devil asked.

"Have I?" Robert cried. "I'll say I have!"

"You can't swim across: the current would sweep you into the rapids. But there are a few rocks in the middle. So what do you do?"

"I see which ones are close enough together so I can leap from one to the next. If I'm lucky, I make it; if I'm not, I don't."

"That's how it is with mathematical proofs," the number devil told Robert. "But since mathematicians have spent a few thousand years finding ways to cross the stream, you don't need to start from scratch. You've got all kinds of rocks to rely on. They've been tested millions of times and are guaranteed slip-resistant. When you have a new idea, a conjecture, you look for the nearest safe rock, and from there you keep leaping--with the greatest of caution, of course--until you reach the other side, the shore."
...
"Sometimes the rocks are so far apart that you can't make it from one to the next, and if you try jumping, you fall in. Then you have to take tricky detours, and even they may not help in the end. You may come up with an idea, but then you find that it doesn't lead anywhere. Or you may find that your brilliant idea wasn't so brilliant at all."

This is just my perspective, but . . .

I think there are two separate concerns here: 1) the "process" of mathematics, or mathematical thinking; and 2) specific mathematical systems which are fundamental and help frame much of the world of mathematics.

​

Abstract algebra is one of those specific mathematical systems, and is very important to understand in order to really understand things like analysis (e.g. the real numbers are a field), linear algebra (e.g. vector spaces), topology (e.g. the fundamental group), etc.

​

I'd recommend these books, which are for the most part short and easy to read, on mathematical thinking:

​

How to Solve It, Polya ( https://www.amazon.com/How-Solve-Mathematical-Princeton-Science/dp/069111966X ) covers basic strategies for problem solving in mathematics

Mathematics and Plausible Reasoning Vol 1 & 2, Polya ( https://www.amazon.com/Mathematics-Plausible-Reasoning-Induction-Analogy/dp/0691025096 ) does a great job of teaching you how to find/frame good mathematical conjectures that you can then attempt to prove or disprove.

Mathematical Proof, Chartrand ( https://www.amazon.com/Mathematical-Proofs-Transition-Advanced-Mathematics/dp/0321797094 ) does a good job of teaching how to prove mathematical conjectures.

​

As for really understanding the foundations of modern mathematics, I would start with Concepts of Modern Mathematics by Ian Steward ( https://www.amazon.com/Concepts-Modern-Mathematics-Dover-Books/dp/0486284247 ) . It will help conceptually relate the major branches of modern mathematics and build the motivation and intuition of the ideas behind these branches.

​

Abstract algebra and analysis are very fundamental to mathematics. There are books on each that I found gave a good conceptual introduction as well as still provided rigor (sometimes at the expense of full coverage of the topics). They are:

​

A Book of Abstract Algebra, Pinter ( https://www.amazon.com/Book-Abstract-Algebra-Second-Mathematics/dp/0486474178 )

​

Understanding Analysis, Abbott ( https://www.amazon.com/Understanding-Analysis-Undergraduate-Texts-Mathematics/dp/1493927116 ).

​

If you read through these books in the order listed here, it might provide you with that level of understanding of mathematics you talked about.

If you enjoy analysis, maybe you'd like to learn some more?

I really enjoyed learning introductory functional analysis, which is presented incredibly well in Kreyszig's book Introductory Functional Analysis with Applications. It's very easy to read, and covers a lot and assumes very little on the part of the reader (basic concepts from analysis and linear algebra). This will teach you about doing analysis on finite and infinite dimensional spaces and about operators between such spaces. It's incredibly interesting, and I highly recommend it if you enjoy analysis and linear algebra.

Another great analysis topic is Fourier Analysis and wavelets. I enjoyed the books by Folland Fourier Analysis and Its Applications. I don't believe that book has any wavelets in it, so if you're interested in learning Fourier analysis plus wavelet theory, then I highly recommend the very approachable and fun book by Boggess and Narcowich A First Course in Wavelets with Fourier Analysis. If you have any interest at all in applications (like signals processing), this subject is fundamental.

There are a fair number of popular level books about mathematics that are definitely interesting and generally not too challenging mathematically. William Dunham is fantastic. His Journey through Genius goes over some of the most important and interesting theorems in the history of mathematics and does a great job of providing context, so you get a feel for the mathematicians involved as well as how the field advanced. His book on Euler is also interesting - though largely because the man is astounding.

The Man who Loved only Numbers is about Erdos, another character from recent history.

Recently I was looking for something that would give me a better perspective on what mathematics was all about and its various parts, and I stumbled on Mathematics by Jan Gullberg. Just got it in the mail today. Looks to be good so far.

Yes. In fact, certain theorems get re-proven using different methods as a kind of "math golf" or fun puzzle for mathematicians.

Although their contents will likely be over your head, look at some of the various proofs of the fundamental theorem of algebra, for example.

Similarly, if an important but hard-to-prove theorem "falls out" of some newly-developed mathematical abstraction, that's considered a sign that the abstraction is the "right one" (doubly so if the abstraction wasn't developed with a proof of the original theorem in mind).

For example, Brouwer's original proof of what is now called Brouwer's fixed-point theorem was somewhat cumbersome, requiring lots of calculation, consideration of special cases, and other necessary-but-unenlightening bookkeeping. Using the more modern language of homology, however, the proof becomes very straightforward.

One could say that a "simple" or "elegant" proof manages to isolate exactly those things which convey the essence of "what's going on" with the theorem and related concepts. At a purely formal level a proof is a proof is a proof, but in practice an elegant proof offers a more visceral resolution to the question of "Why should this be true?"

Most mathematicians will collect their favorite proofs of various theorems. You'll often here them say things like "Oh, have you ever seen so-and-so's proof of XYZ theorem?" It's a lot like music fans being excited about sharing covers or remixes ("Oh, did you hear DJ so-and-so's remix of XYZ song?"). There's a sociology paper in there somewhere.

You might be interested in Proofs from THE BOOK.

This will give you some solid theory on ODEs (less so on PDEs), and a bunch of great methods of solving both ODEs and PDEs. I work a lot with differential equations and this is one of my principal reference books.

This is an amazing book, but it mostly covers ODEs sadly. Both the style and the material covered are great. It might not be exactly what you're looking for, but it's a great read nonetheless.

This covers PDEs from a very basic level. It assumes no previous knowledge of PDEs, explains some of the theory, and then goes into a bunch of elementary methods of solving the equations. It's a small book and a fairly easy read. It also has a lot of examples and exercises.

This is THE book on PDEs. It assumes quite a bit of knowledge about them though, so if you're not feeling too confident, I suggest you start with the previous link. It's something great to have around either way, just for reference.

Hope this helped, and good luck with your postgrad!

I think the advice given in the rest of the thread is pretty good, though some of it a little naive. The suggestion that differential equations or applied math somehow should not be of interest is silly. A lot of it builds the motivation for some of the abstract stuff which is pretty cool, and a lot of it has very pure problems associated with it. In addition I think after (or rather alongside) your initial calculus education is a good time to look at some other things before moving onto more difficult topics like abstract algebra, topology, analysis etc.

The first course I took in undergrad was a course that introduced logic, writing proofs, as well as basic number theory. The latter was surprisingly useful as it built modular arithmetic which gave us a lot of groups and rings to play with in subsequent algebra courses. Unfortunately the textbook was god awful. I've heard good things about the following two sources and together they seem to cover the content:

How to prove it

Number theory

After this I would take a look at linear algebra. This a field with a large amount of uses in both pure and applied math. It is useful as it will get you used to doing algebraic proofs, it takes a look at some common themes in algebra, matrices (one of the objects studied) are also used thoroughly in physics and applied mathematics and the knowledge is useful for numerical approximations of ordinary and partial differential equations. The book I used Linear Algebra by Friedberg, Insel and Spence, but I've heard there are better.

At this point I think it would be good to move onto Abstract Algebra, Analysis and Topology. I think Farmerje gave a good list.

There's many more topics that you could possibly cover, ODEs and PDEs are very applicable and have a rich theory associated with them, Complex Analysis is a beautiful subject, but I think there's plenty to keep you busy for the time being.

I agree with all the suggestions to start with How to Prove It by Velleman. It's a great start for going deeper into mathematics, for which rigor is a sine qua non.

As you seem to enjoy calculus, might I also suggest doing some introductory real analysis? For the level you seem to be at, I recommend Understanding Analysis by Abbott. It helped me bridge the gap between my calculus courses and my first analysis course, together with Velleman. (Abbott here has the advantage of being more advanced and concise than Spivak, but more gentle and detailed than baby Rudin -- two eminent texts.)

Alternatively, you can start exploring some other fascinating areas of mathematics. The suggestion to study Topology by Munkres is sound. You can also get a friendly introduction to abstract algebra by way of A Book of Abstract Algebra by Pinter.

If you're more interested in going into a field of science or engineering than math, another popular approach for advanced high schoolers to start multivariable calculus (as you are), linear algebra, and ordinary differential equations.

I have a variety of books to recommend.
Brushing up on your foundations:
http://www.amazon.com/Beginning-Functional-Analysis-Karen-Saxe/dp/0387952241
If you get this from your library or browse inside of it and it seems easy there are then three books to look at:

1. http://www.amazon.com/Functional-Analysis-Introduction-Princeton-Lectures/dp/0691113874/ref=sr_1_4?s=books&ie=UTF8&qid=1368475848&sr=1-4&keywords=functional+analysis challenging exercises for sure.
   
2. http://www.amazon.com/Introductory-Functional-Analysis-Applications-Kreyszig/dp/0471504599/ref=sr_1_2?s=books&ie=UTF8&qid=1368475848&sr=1-2&keywords=functional+analysis (A great expositor)
   
3. Rudin's Functional Analysis (A challenging book for sure)
   
   More advanced level:
   
4. http://www.amazon.com/Functional-Analysis-Introduction-Graduate-Mathematics/dp/0821836463/ref=cm_cr_pr_product_top
   (An awesome book with exercise solutions that will really get you thinking)
   
   Working on this book and Rudin's (which has many exercise solutions available online is very helpful) would be a very strong advanced treatment before you go into the more specialized topics.
   
   The key to learning this sort of subject is to not delude yourself into thinking you understand things that you really don't. Leave your pride at the door and accept that the SUMS book may be the best starting point. Also remember to use the library at your institution, don't just buy all these books.

As others have said, intelligence isn't everything. If you're willing to work hard, you can earn your bachelor's in mathematics.

But do you want to? What do you want to do with that degree?

Moreover, are you sure you really like math? College algebra and pre-calculus have very little in common with most math courses. At some point in a math curriculum, you'll be taking courses about abstract concepts that bear no obvious relation to the real world (unlike say Calculus and Differential Equations, in which real-world examples are abundant).

Furthermore, in those later classes, the question stops becoming: "What is the value of x?"^ Instead, those classes are more like: "Prove P(x) for all real numbers x". Proofs are different in kind from anything you've done so far in your math classes, and it will dominate all of the upper-level math courses you take.

Before you go down the path of majoring in mathematics, I recommend you get some exposure to proofs and try some on your own, to see if that's really something you're interested in. If your library has it, check out Proofs from THE BOOK, a collection of particularly beautiful proofs.

^ If you're good at solving equations and decide against majoring in mathematics, there are several other good fields to consider. Engineering and computer science, for example, offer great careers for the mathematically inclined.

I haven't heard of some of the lesser known books, but I just wanted to point out that Algebra Chapter 0 by Aluffi is a very advanced book (in comparison to other books on the list), and that you may want a more gentle introduction to Abstract Algebra before attempting that book. (Dummit and Foote is very standard, and there's plenty other good ones as well that are better motivated). Baby Rudin is also gonna be a tough one if you have no background in Analysis, even though it is concise and elegant I think it's best appreciated after knowing some analysis (something at the level of maybe Understanding Analysis by Abbott).

I have to second Dummit and Foote as a supplement to Lang's text, they're pretty much complete opposites; where Lang is very to the point (terse, some may say) and from a very abstract viewpoint, Dummit and Foote has a lot of exposition and examples and is done from, what at least what I would call, an appropriate level for a first graduate course in abstract algebra. It also has an appendix that deals with category theory, it's nothing extensive but it may help you become more familiar with the ideas of category theory. I am currently using this book for a graduate course in algebra so I have some familiarity with it; it is a bit too wordy for my tastes but that may be your thing.

A book with which I have limited experience but quite like so far is Mac Lane and Birkhoff's Algebra it's done with the same general perspective as Dummit and Foote but it has a bit more category theory (it is introduced at the end of the third chapter and the entire fifteenth chapter is dedicated to category theory), it isn't terse but it is less wordy than Dummit and Foote.

Another (very) popular choice (but one with which I have no experience) is Aluffi's Algebra: Chapter 0 it develops category theory pretty much from the start and supposedly is much less terse than Lang (I only say supposedly as I have no first hand experience with it).

If you want something that only deals with category theory, the classic text is Mac Lane's Category Theory for the Working Mathematician I have found looking at this book for a long period of time has helped me with understanding/getting used to categorical ideas. I also have experience with this book for which you can find on the internet (legally) for free and I find it rather good.

A graph theory project! I just started today (it was assigned on Friday and this is when I selected my topic). I’m on spring break but next month I have to present a 15-20 minute lecture on graph automorphisms. I don’t necessarily have to, but I want to try and tie it in with some group theory since there is a mix of undergrads who the majority of them have seen some algebra before and probably bored PhD students/algebraists in my class, but I’m not sure where to start. Like, what would the binary operation be, composition of functions? What about the identity and inverse elements, what would those look like? In general, what would the elements of this group look like? What would the group isomorphism be? That means it’s a homomorphism with a bijective function. What would the homomorphism and bijective function look like? These are the questions I’m trying to get answers to.

Last semester I took a first course in Abstract Algebra and I’m currently taking a follow up course in Linear Algebra (I have the same professor for both algebra classes and my graph theory class). I’m curious if I can somehow also bring up some matrix representation theory stuff as that’s what we’re going over in my linear algebra class right now.

This is the textbook I’m using for my graph theory class: Graph Theory (Graduate Texts in Mathematics) https://www.amazon.com/dp/1846289696?ref=yo_pop_ma_swf

Here are the other graph theory books I got from my library and am using as references: Graph Theory (Graduate Texts in Mathematics) https://www.amazon.com/dp/3662536218?ref=yo_pop_ma_swf

Modern Graph Theory (Graduate Texts in Mathematics) https://www.amazon.com/dp/0387984887?ref=yo_pop_ma_swf

And for funsies, here is my linear algebra text: Linear Algebra, 4th Edition https://www.amazon.com/dp/0130084514?ref=yo_pop_ma_swf

But that’s what I’m working on! :)

And I certainly wouldn’t mind some pointers or ideas or things to investigate for this project! Like I said, I just started today (about 45 minutes ago) and am just trying to get some basic questions answered. From my preliminary investigating in my textbook, it seems a good example to work with in regards to a graph automorphism would be the Peterson Graph.

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.

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:

1. Understanding Analysis by Steve Abbot
   
   
2. Yet Another Introduction to Analysis by Victor Bryant
   
   
3. Elementary Analysis: The Theory of Calculus by Kenneth Ross
   
   
4. Real Mathematical Analysis by Charles Pugh
   
   
5. A Primer of Real Functions by Ralph Boas
   
   
6. A Radical Approach to Real Analysis by David Bressoud
   
   
7. The Way of Analysis by Robert Strichartz
   
   
8. Foundations of Analysis by Edmund Landau
   
   
9. A Problem Book in Real Analysis by Asuman Aksoy and Mohamed Khamzi
   
   
10. Calculus by Spivak
    
    
11. Real Analysis: A Constructive Approach by Mark Bridger
    
    
12. Differential and Integral Calculus by Richard Courant, Edward McShane, Sam Sloan and Marvin Greenberg
    
    
13. You can find tons more if you search the internet. There are more superstars of advanced Calculus like Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra by Tom Apostol, Advanced Calculus by Shlomo Sternberg and Lynn Loomis... there are also more down to earth titles like Limits, Limits Everywhere:The Tools of Mathematical Analysis by david Appelbaum, Analysis: A Gateway to Understanding Mathematics by Sean Dineen...I just dont have time to list them all.
    
    Some user friendly books on Linear/Abstract Algebra:
    
    
14. A Book of Abstract Algebra by Charles Pinter
    
    
15. Matrix Analysis and Applied Linear Algebra Book and Solutions Manual by Carl Meyer
    
    
16. Groups and Their Graphs by Israel Grossman and Wilhelm Magnus
    
    
17. Linear Algebra Done Wrong by Sergei Treil-FREE
    
    
18. Elements of Algebra: Geometry, Numbers, Equations by John Stilwell
    
    Topology(even high school students can manage the first two titles):
    
    
19. Intuitive Topology by V.V. Prasolov
    
    
20. First Concepts of Topology by William G. Chinn, N. E. Steenrod and George H. Buehler
    
    
21. Topology Without Tears by Sydney Morris- FREE
    
    
22. Elementary Topology by O. Ya. Viro, O. A. Ivanov, N. Yu. Netsvetaev and and V. M. Kharlamov
    
    Some transitional books:
    
    
23. Tools of the Trade by Paul Sally
    
    
24. A Concise Introduction to Pure Mathematics by Martin Liebeck
    
    
25. How to Think Like a Mathematician: A Companion to Undergraduate Mathematics by Kevin Houston
    
    
26. Introductory Mathematics: Algebra and Analysis by Geoffrey Smith
    
    
27. Elements of Logic via Numbers and Sets by D.L Johnson
    
    Plus many more- just scour your local library and the internet.
    
    Good Luck, Dude/Dudette.

I made a comment in a another thread.

I second /u/ProfThrowawary17's recommendation for Strogatz and also suggest the undergrad text Hale and Kocak. Strogatz is a rare text that delivers both interesting math and well-motivated applications in a fairly accessible manner. I have not systematically read Hale and Kocak, but it also seems to provide a gentle yet rigorous introduction to ODE's from the modern dynamical systems point of view.

Like /u/dogdiarrhea, I also recommend the graduate text Hale. If you have a strong analysis background, working through Hale would be quite worthwhile. It's also a Dover publication! So if Hale doesn't work out for you in a first time reading, it would still be a useful reference later on.

I’ve only skimmed parts of it (I don’t have a copy, but might buy one sometime). Seemed like there was some good stuff in there though. The Princeton Companion to Mathematics is also great in pure math type subjects.

Another book I like, less numerical-analysis-y and more computer algorithms-y, is Graham/Knuth/Patashnik’s Concrete Mathematics. It’s aimed at undergraduate computer science students, but you might find it useful.

What was your undergraduate background / what other experience do you have? And what are your interests? Any specific things you want to build? There are also obviously a whole pile of famous/classic computer science books. (Asking questions in programming or CS related subreddits might get more responses on such a theme.)

I have Mathematics:From the Birth of Numbers and it’s excellent.

Highly recommend

> This extraordinary work takes the reader on a long and fascinating journey--from the dual invention of numbers and language, through the major realms of arithmetic, algebra, geometry, trigonometry, and calculus, to the final destination of differential equations, with excursions into mathematical logic, set theory, topology, fractals, probability, and assorted other mathematical byways. The book is unique among popular books on mathematics in combining an engaging, easy-to-read history of the subject with a comprehensive mathematical survey text. Intended, in the author's words, "for the benefit of those who never studied the subject, those who think they have forgotten what they once learned, or those with a sincere desire for more knowledge," it links mathematics to the humanities, linguistics, the natural sciences, and technology.

You could read Timothy Gowers' welcome to the math students at Oxford, which is filled with great advice and helpful links at the bottom.

You could read this collection of links on efficient study habits.

You could read this thread about what it takes to succeed at MIT (which really should apply everywhere). Tons of great discussion in the lower comments.

You could read How to Solve It and/or How to Prove It.

If you can work your way through these two books over the summer, you'll be better prepared than 90% of the incoming math majors (conservatively). They'll make your foundation rock solid.

I too love fun math[s] books! Here are some of my favorites.

The Number Devil: http://www.amazon.com/dp/0805062998

The Mathematical Magpie: http://www.amazon.com/dp/038794950X

I echo the GEB recommendation. http://www.amazon.com/dp/0465026567

The Magic of Math: http://www.amazon.com/dp/0465054722

Great Feuds in Mathematics: http://www.amazon.com/dp/B00DNL19JO

One Equals Zero (Paradoxes, Fallacies, Surprises): http://www.amazon.com/dp/1559533099

Genius at Play - Biography of J.H. Conway: http://www.amazon.com/dp/1620405938

Math Girls (any from this series are fun) http://www.amazon.com/dp/0983951306

Mathematical Amazements and Surprises: http://www.amazon.com/dp/1591027233

A Strange Wilderness: The Lives of the Great Mathematicians: http://www.amazon.com/dp/1402785844

Magnificent Mistakes in Mathematics: http://www.amazon.com/dp/1616147474

Enjoy!

What do you want to do, though? Is your goal to read math textbooks and later, maybe, math papers or is it for science/engineering? If it's the former, I'd simply ditch all that calc business and get started with "actual" math. There are about a million books designed to get you in the game. For one, try Book of Proof by Richard Hammack. It's free and designed to get your feet wet. Mathematical Proofs: A Transition to Advanced Mathematics by Chartrand/Polimeni/Zhang is my favorite when it comes to books of this kind. You'll also pick up a lot of math from Discrete Math by Susanna Epp. These books assume no math background and will give you the coveted "math maturity".

There is also absolutely no shortage of subject books that will nurse you into maturity. For example, check out [The Real Analysis Lifesaver: All the Tools You Need to Understand Proofs by Grinberg](https://www.amazon.com/Real-Analysis-Lifesaver-Understand-Princeton/dp/0691172935/ref=sr_1_1?ie=UTF8&qid=1486754571&sr=8-1&keywords=real+analysis+lifesaver() and Book of Abstract Algebra by Pinter. There's also Linear Algebra by Singh. It's roughly at the level of more famous LADR by Axler, but doesn't require you have done time with lower level LA book first. The reason I recommend this book is because every theorem/lemma/proposition is illustrated with a concrete example. Sort of uncommon in a proof based math book. Its only drawback is its solution manual. Some of its proofs are sloppy, messy. But there's mathstackexchange for that. In short, every subject of math has dozens and dozens of intro books designed to be as gentle as possible. Heck, these days even grad level subjects are ungrad-ized: The Lebesgue Integral for Undergraduates by Johnson. I am sure there are such books even on subjects like differential geometry and algebraic geometry. Basically, you have choice. Good Luck!

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.

Gallian is basic undergrad stuff through Galois, right?

I can only recommend books that will start from scratch, so will cover many things you already know, but go much deeper than an undergrad text would. Mac Lane and Birkhoff is my favorite math text I've ever read. The only significant drawback is some of the terminology is awkward, the most significant example being that the word "homomorphism" is used once in the entire book, to note it as an alternative to their word "morphism". I'm also currently reading Aluffi to review for a qualifier, and while I personally don't like the exposition as much, it's definitely well-written, and is somewhat more modern. Both of them will cover things you know already, but they should have enough new stuff sprinkled in to keep you interested and help solidify your knowledge.

If you want a more direct transition I can't really be too helpful, sorry.

(edit: minor typo)

Disclaimer: I only have a masters in maths, and I've just started working as a programmer.

Here are topics I enjoyed and would recommend

• At least basic programming. I've heard that if you are interested in mathematics,you may be especially interested in the Haskel language.
  
• Mathematical Logic and Meta-mathematics, some (very basic) model-theory. Personally, I also found set theory and the proof of the independence of the Axiom of Choice from the Zermelo-Fraenkel axioms incredibly interesting. The stuff from Gödel, Escher, Bach.
  
• Probability theory and Bayesian statistics, along with some basic information theory and theory of complexity / entropy.
  
• Basic category theory (though it's a hard subject to learn if you don't have some advanced algebra to motivate why things are done the way they are. It also took me about three or four tries before I finally started understanding it, and after that I found it quite beautiful. Your mileage may vary).
  
• EDIT: Also, Knuth's Concrete Mathematics. The book is a very good source on the topic.

>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

Interviews with mathematicians from MIT (haven't read it, but it is leisurely):
http://www.amazon.com/Recountings-Conversations-Mathematicians-Joel-Segel/dp/1568817134

Some good magazines from AMS:
http://www.amazon.com/Whats-Happening-Mathematical-Sciences-Mathermatical/dp/0821849999

If you want to learn some math in a leisurely way (although it does get pretty deep at times):
http://www.amazon.com/Concepts-Modern-Mathematics-Ian-Stewart/dp/0486284247

A good book on the history of mathematics:
http://www.amazon.com/Mathematics-Nonmathematician-Dover-explaining-science/dp/0486248232

I'll definitely check out that Poincare book, it looks good!

Azcel wrote a good book on Fermat's Last Theorem and Wiles' solution. Amazon

Simon Singh's book on the same subject is also good, but Amazon has it at $10.17 whereas Azcel's is $0.71 better at $10.88.

Either way you get an enjoyable read of one man's dedication to solve a notoriously tricky problem and just enough of the mathematical landscape to get a sense of what was involved.

Another fun & light holiday read is Polya's 'How To Solve it' - read the glowing reviews over at Amazon

I really good textbook is probably what you want. Good math textbooks are engaging and have lots of interesting problems. They have an advantage (in pure math) that they don't have to worry about teaching you specific tools (which IMO can make things boring). Lots of people love this one: https://www.amazon.com/Nonlinear-Dynamics-Chaos-Applications-Nonlinearity/dp/0738204536

Also here is a really good lecture series (on a different topic): https://www.youtube.com/watch?v=7G4SqIboeig&list=PLMsYJgjgZE8hh6d6ia2dP1NI0BKNRXbiw

Also if you have a bit of a programming bent or want to learn a little bit of programming, you might like Project Euler:https://projecteuler.net/

To answer your second question, KhanAcademy is always good for algebra/trig/basic calc stuff. Another good resource is Paul's online Math Notes, especially if you prefer reading to watching videos.

To answer your second question, here are some classic texts you could try (keep in mind that parts of them may not make all that much sense without knowing any calculus or abstract algebra):

Men of Mathematics by E.T. Bell

The History of Calculus by Carl Boyer

Some other well-received math history books:

An Intro to the History of Math by Howard Eves, Journey Through Genius by William Dunham, Morris Kline's monumental 3-part series (1, 2, 3) (best left until later), and another brilliant book by Dunham.

And the MacTutor History of Math site is a great resource.

Finally, some really great historical thrillers that deal with some really exciting stuff in number theory:

Fermat's Enigma by Simon Sigh

The Music of the Primes by Marcus DuSautoy

Also (I know this is a lot), this is a widely-renowned and cheap book for learning about modern/university-level math: Concepts of Modern Math by Ian Stewart.

Usual hierarchy of what comes after what is simply artificial. They like to teach Linear Algebra before Abstract Algebra, but it doesn't mean that it is all there's to Linear Algebra especially because Linear Algebra is a part of Abstract Algebra.

Example,

Linear Algebra for freshmen: some books that talk about manipulating matrices at length.

Linear Algebra for 2nd/3rd year undergrads: Linear Algebra Done Right by Axler

Linear Algebra for grad students(aka overkill): Advanced Linear Algebra by Roman

Basically, math is all interconnected and it doesn't matter where exactly you enter it.

Coming in cold might be a bit of a shocker, so studying up on foundational stuff before plunging into modern math is probably great.

Books you might like:

Discrete Mathematics with Applications by Susanna Epp

Learning to Reason: An Introduction to Logic, Sets, and Relations by Nancy Rodgers

Building Proofs: A Practical Guide by Oliveira/Stewart

Book Of Proof by Hammack

Mathematical Proofs: A Transition to Advanced Mathematics by Chartrand et al

How to Prove It: A Structured Approach by Velleman

The Nuts and Bolts of Proofs by Antonella Cupillary

How To Think About Analysis by Alcock

Principles and Techniques in Combinatorics by Khee-Meng Koh , Chuan Chong Chen

The Probability Tutoring Book: An Intuitive Course for Engineers and Scientists (and Everyone Else!) by Carol Ash

Problems and Proofs in Numbers and Algebra by Millman et al

Theorems, Corollaries, Lemmas, and Methods of Proof by Rossi

Mathematical Concepts by Jost - can't wait to start reading this

Proof Patterns by Joshi

...and about a billion other books like that I can't remember right now.

Good Luck.

For discrete math I like Discrete Mathematics with Applications by Suzanna Epp.

It's my opinion, but Learning to Reason: An Introduction to Logic, Sets, and Relations by Nancy Rodgers is much better structured and more in depth than How To Prove It by Velleman. If you follow everything she says, proofs will jump out at you. It's all around great intro to proofs, sets, relations.

Also, knowing some Linear Algebra is great for Multivariate Calculus.

Gödel proved several theorems; I'm guessing you're referring to the incompleteness theorems, which are the most well-known. The key point is that Gödel's incompleteness theorems are precise mathematical statements about certain formal systems — not vague philosophical generalities about the nature of truth or anything like that.

In particular, the content of the first incompleteness theorem is essentially:

>In any consistent, effectively generated formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true (in the standard model of arithmetic), but not provable in the theory.

This statement, as with any other statement mathematicians call a "theorem", has been formally proven. Philosophical questions like whether mathematical objects are "real" in whatever sense are irrelevant to the question of whether something is a theorem or not.

By the way, if you want a good introduction to the details of what Gödel's incompleteness theorems say and how they can be proved, I highly recommend Gödel's Proof by Nagel and Newman.

Yes, they're awesome. Brought up pretty frequently on /r/math, too. I'm pretty sure I have at least 10 Dover books. Two excellent titles that come to mind are Pinter's A Book of Abstract Algebra and Rosenlicht's Introduction to Analysis.

I know the symbols are scary! But you will be introduced to them gradually. Right now, everything probably looks like a different language to you.

Your university will either have an entire "Methods of Proof" course that proves basic results in number theory or some course (like real analysis) in which you learn methods of proof whilst immersed in a given course. In a course like this, you will learn what all those symbols you have been seeing mean, as well as some of the terminology.

Try reading an introductory analysis book (this one is a very easy read, as analysis books go). Or something like this. Or this

Anyways, don't be afraid! Everything looks scary right now but you really do get eased into it. Just enjoy the ride! Or you can always change your major to statistics! (I'm a double math/stat major, and I know tons of math majors who found the upper division stuff just wasn't for them and were very happy with stats).

• The Elements of Statistical Learning
  
  It's available free online, but I've def got a hard cover copy on my bookshelf. I can't really deal with digital versions of things, I need physical books.
  
  
• If you're looking for something less technical, try The Lady Tasting Tea
  
  
• You haven't mentioned how old your sister is. If she's on the younger side of the spectrum, she might enjoy Flatland.
  
  
• Also, you mention how much your sister loves proofs. Godel's Proof is a really incredible result (sort of brain melting) and the book I linked does a great job of making it accessible. I think I read this book in high school (probably would have understood more if I read it in college, but I got the gist of it).

Strogatz Nonlinear Dynamics and Chaos covers phase space, phase portraits, and linear stability analysis in great detail with examples from many disciplines including physics. It's probably a good place to start, but I don't think it has very much that's specifically on turbulent fluids. For that, you'll probably want a more focused textbook. Hopefully, someone more knowledgeable can recommend one.

The most important thing you can do is memorize the definitions. I mean seriously have them down cold. The next thing I would recommend is to get another couple of analysis books (go cheap by getting old books, it isn't like the value of epsilon has changed over the past two hundred years) and look at their explanations, work those problems. Having a different set can be enlightening. Be prepared to spend a lot of time on it all.

Good luck!

EDIT: Back home now and able to put in some specific books. I used Rosenlicht and you wouldn't believe how happy I was to buy a textbook that, combined with a slice of pizza and a coke, was still less than $20. One of my books that I looked at for a different view point was Sprecher.

I also got a great deal of value out of Counterexamples in Analysis because after seeing things go wrong (a function that is continuous everywhere but nowhere differentiable? Huh?) I started to get a better feel for what the definitions really meant.

I hope you're also sensing a theme: Dover math books rock!

I used to be just like you, then really became fascinated by physics, which was very difficult given my deficiencies in math. I figured I would start with flash cards and what not, so I started browsing amazon and came across this. This guy is a genius, and teaches you a lot of tricks to do math quickly in your head. The next thing I did was checked out Khan Academy. I can not over-exaggerate how utterly fucking awesome this site is. Not only does he have like 2300+ videos on every topic, but he has something like 125 math modules that allow you to practice. It's completely free and all you need is a facebook or gmail account to log in...

An Introduction to Manifolds by Tu is a very approachable book that will get you up to Stokes. Might as well get the full version of Stokes on manifolds not just in analysis. From here you can go on to books by Ramanan, Michor, or Sharpe.

A Guide to Distribution Theory and Fourier Transforms by Strichartz was my introduction to Fourier analysis in undergrad. Probably helps to have some prior Fourier experience in a complex analysis or PDE course.

Bartle's Elements of Integration and Legesgue Measure is great for measure theory. Pretty short too.

Intro to Functional Analysis by Kreysig is an amazing introduction to functional analysis. Don't know why you'd learn it from any other book. Afterwards you can go on to functional books by Brezis, Lax, or Helemskii.

this book is quite short but perfect for an aspiring mathematician that is going to start hearing about Gödel's proof in casual conversation. This provides a concise easy treatment of it's importance and how the proof works. Also, see it's reviews on goodreads

I'm doing that, I guess, if you call 'advanced maths' anything proof-based (which is, generally, what people mean). I use the internet, my brain, and a lot of books. It was hard for sure. Only way to do it is to enjoy it and not burn yourself out working too hard.

This book is how I got started and probably the easiest way into anything proof based: http://www.amazon.com/Understanding-Analysis-Undergraduate-Texts-Mathematics/dp/0387950605.

Ofcourse you might not want to do analysis especially if you have't done any calc yet. At that level people (I think) do stuff like http://www.artofproblemsolving.com/. Also khan academy, MiT OCW, and competition-oriented books like https://www.google.com/webhp?sourceid=chrome-instant&ion=1&espv=2&ie=UTF-8#q=complex%20numbers%20from%20a%20to%20z.

That said if you can work through that analysis book it'll open the doors to tons of undergrad level math like Abstract Algebra, for example.

Just keep at it?

I don't want to say that it's impossible for you to get through Spivak, but I think it will be frustrating. Spivak is, I think, most useful for someone who already knows a little bit about what calculus is about. You might be better suited going with a gentler introduction to calculus like Stewart (pdfs exist on torrent sites if you don't want to drop $200), or even a proofs book like Chartrand, Polimeni, Zhang.

Stay away from Numberphile. Numberphile oversimplifies mathematical concepts to the point where they will give you misconceptions about common mathematical notions that will greatly impact your learning later on. I'm noticing this happening a lot with the "1+2+... = -1/12" video because it doesn't explain that they aren't using the standard partial sum definition of series convergence.

Not sure how "mathematical" it is, but Secrets of Mental Math is a great, useful book that will help you do really fast calculations in your head.

This is exactly right. It breaks down like this:

[; \sum_{i = 1}^{x} \frac{x}{i} = \frac{x}{1} + \frac{x}{2} + \ldots + \frac{x}{x} ;]
[; = x (\frac{1}{1} + \frac{1}{2} + \ldots + \frac{1}{x}) ;]
[; = x \sum_{i=1}^x \frac{1}{i} ;]

In other words, because [; x ;] is a constant inside the summand we can just pull it out to the front of the sum as a common factor. Then we just use the definition of the harmonic numbers:

[; H_x \equiv \sum_{i=1}^x \frac{1}{i} ;]

And we're done:

[; \sum_{i = 1}^{x} \frac{x}{i} = x H_x ;]

If you find yourself doing sums like this often, I HIGHLY recommend Concrete Mathematics. In fact even if you don't do sums like this often, you should probably read Concrete Mathematics anyway. Because it's great.

If you want to learn a modern (i.e., dynamical systems) approach, try Hirsch, Smale and Devaney for an intro-level book and Guckenheimer and Holmes for more advanced topics.

> a more Bourbaki-like approach

Unless you already have a lot of exposure to working with specific problems and examples in ODEs, it's much better to start with a well-motivated book with a lot of interesting examples instead of a dry, proof-theorem style book. I know it's tempting as a budding mathematician to have the "we are doing mathematics here after all" attitude and scoff at less-than-rigorous approaches, but you're really not doing yourself any favors. In light of that, I highly recommend starting with Strogatz which is my favorite math book of all time, and I'm not alone in that sentiment.

My favorite book on problem solving is Problem Solving Through Problems. There's an online copy, too. (I recommend you print it and get it bound at Kinkos if you intend to seriously work through it, though. This type of thing sucks on a screen.)

How To Solve It is another popular recommendation for that topic. Personally, I only read part of it. It's alright.

I can recommend other stuff if you tell me what level of math you're at, what you're interested in learning, etc.

I've had a similar experience with wanting to continue my math education and I've really enjoyed picking up Schaum's Outlines on topics I've been exposed to and ones that I have not. There's also a really fun textbook Non-Linear Dynamics and Chaos which I'm enjoying right now. I find looking up very advanced problems like the Clay Institute Millennium Prize Problems and trying to really understand the question can be very revealing.

The key thing that took me a while to realize about recreating that experience is forcing yourself to work as many problems as you have time to work, even (read: especially) when you don't really feel like it. You may not get the exact same experience and it's likely you won't be able to publish (remember, it takes a lot to really dig deeply enough into a field and understand what has already been written to be able to write something original), but you'll keep learning! And it will be really fun!

Understanding Analysis is a very nice book I used to get a good grasp on the concepts behind real analysis. It goes at a very nice pace, perfect for the analysis novice.

Personally, I would take the time to read them both. A strong linear algebra background will be very helpful in ML. Its especially useful if you want to expand out a little bit more into other areas of signal processing. Make sure you also spend some time getting a good background in probability and statistics.

EDIT: I haven't actually read Axler's book but me and some of my friends are partial to this book.

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.

Book of proof is a more gentle introduction to proofs then How to Prove it.

​

No bullshit guide to linear algebra is a gentle introduction to linear algebra when compared to the popular Linear Algebra Done Right.

​

An Illustrated Theory of Numbers is a fantastic introduction book to number theory in a similar style to the popular Visual Complex Analysis.

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.

> This is an amazing book, but it mostly covers ODEs sadly. Both the style and the material covered are great. It might not be exactly what you're looking for, but it's a great read nonetheless.

This book changed my life. I was all set to become an experimental condensed matter physicist. Then I took a course based on Strogatz... and now I've been a mathematical physicist for the last ten years instead.

The Turing Omnibus has a bit of that sort of thing. It is mainly focused on computer science, and features some anecdotes about the uses of the techniques explained. This book has a lot of contributors, so the tone varies a bit from chapter to chapter, but it introduces a lot of topics.

In Code examines the RSA (and goes into a bit of depth about Modular Arithmetic) as well as the author's exploration of an alternative encryption.

Aha! Insight and The Number Devil are good books too. They're both aimed at younger readers, and feature lots of illustrations but focus more on thinking about numbers (and problems) than the mechanics of doing calculations.

Many thanks for the suggestions!

For the interested, I bought this book for GT:

http://www.amazon.com/Introductory-Graph-Theory-Gary-Chartrand/dp/0486247759

I also was tempted by the following book:

http://www.amazon.com/Concepts-Modern-Mathematics-Ian-Stewart/dp/0486284247



I think buying a book feels better than sex. (I can compare.)

Not sure what level you're approaching it from, but Steve Strogatz's Nonlinear Dynamics and Chaos is a pretty good upper-level undergraduate introduction to the topic.

I am not a big fan of Rudin. The tone is incredibly stuffy and his style is fairly loose.

I would recommend the small Dover book Introduction to Analysis by Rosenlicht. It's a very small book, hardly 200 pages, but the style is much nicer. It doesn't cover nearly as much (there is no introduction to Fourier Analysis, differential forms, or the gamma function), but that's a good thing for an introductory book, since you can expect to master everything in it.

We used Abbott in a class I audited. I skimmed bits of it, and it seemed pretty nice. Very expository, which is always nice to have when self-studying.

I would eventually pick up a copy of Rudin, just because it's a cultural icon. But it's just very brutal for an introduction to the subject.

Jan Gullberg's Mathematics: From the birth of numbers is a great book I'd recommend: https://www.amazon.com/Mathematics-Birth-Numbers-Jan-Gullberg/dp/039304002X

It introduces a lot of mathematical topics starting from the "simplest" (numbers you asked about) and advances to common stuff found in university studies (although not going extremely far), but what might be the biggest feat and useful to your case is that tells as a non-fictional story while at it, explaining mathematical tools, their history and how they relate to each other extremely well in a way a normal college textbook doesn't, and it doesn't assume you already know everything from school.

I really enjoyed Godel's Proof by Nagel + Newman. It's a layman's guide to Godel incompleteness theorem. It avoids some of the more finnicky details, while still giving the overall impression.

https://www.amazon.com/Gödels-Proof-Ernest-Nagel/dp/0814758371/

If you like that, it's edited by Hofstadter, who wrote Godel-Escher-Bach, a famous book about recurrence.

Finally, I would recommend Nonzero: The Logic of Human Destiny by Robert Wright. It's a life-changing book that dives into the relevance of game theory, evolutionary biology and information technology. (Warning that the first 80 pages are very dry.)

https://www.amazon.com/Nonzero-Logic-Destiny-Robert-Wright/dp/0679758941/

I know this is not exactly what you had in mind, but one of the most significant proofs of the 20th century has an entire book written about it:

http://www.amazon.com/G%C3%B6dels-Proof-Ernest-Nagel/dp/0814758371

The proof they cover is long and complicated, but the book is nonetheless intended for the educated layperson. It is very, very well written and goes to great lengths to avoid unnecessary mathematical abstraction. Maybe check it out.

I read this book in high school when it was originally published as "Mathemagics." https://www.amazon.com/Secrets-Mental-Math-Mathemagicians-Calculation/dp/0307338401/ref=pd_lpo_sbs_14_t_2?_encoding=UTF8&psc=1&refRID=WQYSFNW9WRJY77M30PZG

Its a collection of tips and shortcuts to make mental math easier. I really enjoyed it and found it very useful.

The following easy to read book teaches kids (and adults) you how to do it. Its actually really easy:

Secrets of Mental Math: The Mathemagician's Guide to Lightning Calculation and Amazing Math Tricks

His love of math is the most important thing to preserve. Do look for local math circles and places he can play with math, rather than simply doing it. It is not simply about going to the next level of the school progression. Get him math toys if you can. I have some suggestions for resources.

For your son's age a couple of things that might also be useful are the books Math Circles for 3-7 year olds and The Number devil.

(I am a math professor, but have worked with bright kids in this age group in a variety of ways)

Here are some suggestions :

https://www.coursera.org/course/maththink

https://www.coursera.org/course/intrologic

Also, this is a great book :

http://www.amazon.com/Mathematics-Birth-Numbers-Jan-Gullberg/dp/039304002X/ref=sr_1_5?ie=UTF8&qid=1346855198&sr=8-5&keywords=history+of+mathematics

It covers everything from number theory to calculus in sort of brief sections, and not just the history. Its pretty accessible from what I've read of it so far.


EDIT : I read what you are taking and my recommendations are a bit lower level for you probably. The history of math book is still pretty good, as it gives you an idea what people were thinking when they discovered/invented certain things.

For you, I would suggest :

http://www.amazon.com/Principles-Mathematical-Analysis-Third-Edition/dp/007054235X/ref=sr_1_1?ie=UTF8&qid=1346860077&sr=8-1&keywords=rudin

http://www.amazon.com/Invitation-Linear-Operators-Matrices-Bounded/dp/0415267994/ref=sr_1_4?ie=UTF8&qid=1346860052&sr=8-4&keywords=from+matrix+to+bounded+linear+operators

http://www.amazon.com/Counterexamples-Analysis-Dover-Books-Mathematics/dp/0486428753/ref=sr_1_5?ie=UTF8&qid=1346860077&sr=8-5&keywords=rudin

http://www.amazon.com/DIV-Grad-Curl-All-That/dp/0393969975

http://www.amazon.com/Nonlinear-Dynamics-Chaos-Applications-Nonlinearity/dp/0738204536/ref=sr_1_2?s=books&ie=UTF8&qid=1346860356&sr=1-2&keywords=chaos+and+dynamics

http://www.amazon.com/Numerical-Analysis-Richard-L-Burden/dp/0534392008/ref=sr_1_5?s=books&ie=UTF8&qid=1346860179&sr=1-5&keywords=numerical+analysis

This is from my background. I don't have a strong grasp of topology and haven't done much with abstract algebra (or algebraic _____) so I would probably recommend listening to someone else there. My background is mostly in graduate numerical analysis / functional analysis. The Furata book is expensive, but a worthy read to bridge the link between linear algebra and functional analysis. You may want to read a real analysis book first however.

One thing to note is that topology is used in some real analysis proofs. After going through a real analysis book you may also want to read some measure theory, but I don't have an excellent recommendation there as the books I've used were all hard to understand for me.

How about selected chapters from Stewart's Concepts of Modern Mathematics? It has a pretty wide range of jumping off points and is a relatively affordable Dover book. You could go into more or lesser detail on these topics based on the students' backgrounds.

Another idea would be to focus on foundations like set theory, logic, construction/progression of number systems from ℕ -> ℤ -> ℚ -> ℝ -> ℂ , and then maybe move into some philosophy of math. There could be some fun and accessible class discussion, such as having them argue for or against Platonism. [Edit: You could throw in some Smullyan puzzle book stuff for the logic portion of this for further entertainment value.]

I would enumerate on the various techniques I've used over the years, which drove my early math teachers somewhat mad, but, well, those little tricks and more are readily available in the book The Secrets Of Mental Math. I never finished the book, but it's got quite a few very useful tips, just in the opening couple of chapters, and it builds on them to add other neat things.

I don't know much about AI, though I do know that (there's a theme, here) linear algebra gets a starring role. So, if you're currently enjoying linear algebra, continue with that. Axler is frequently recommended, if you want a textbook to go through.

After that it's really up to you what you want to go for next, since you have many paths available. Sipser is a great intro to theoretical CS, but, again, don't spend $200 on it. Try to find it in a library, or use something like this to find a much cheaper international edition.

Edit: Forgot to mention, CLRS is the standard for algorithms, but I'm not sure how useful it is as a primary source for learning. Maybe try to borrow a copy to see if you like it.

take the most advance math courses you can. do undergraduate research...summer programs, independent studies. make sure to write a math research paper. it doesn't have to be published, but a published paper would look great. give a talk about your research at an undergraduate math conference. go to many math conferences. many schools require the math subject test gre, which is difficult and requires a fair amount of study outside of coursework.

that being said, since you are still a beginner, be warned that upper level math is very different than high school math. after a certain point, computations are no longer of use and all math is theoretical and abstract. you will be focusing on "proofs" and generally these are much more logic based and theoretical than any math you do before university. any proofs you did proofs in a highschool geometry class are also not relevant. to get a better idea, look at an elementary proof-writing book. for example http://www.amazon.com/Introduction-Mathematical-Reasoning-Peter-Eccles/dp/0521597188/ref=sr_1_2?s=books&ie=UTF8&qid=1320289226&sr=1-2#reader_0521597188

more specifically, once you are enrolled in a phd program, you will have to take at least 2 years of coursework. you will also need to pass one or two sets of "qualifying exams", the number and style of testing is based on the university. these test you on your basic knowledge of math, and also on the subject of your research. to obtain a phd you have to do NEW mathematical research and then write a dissertation about it. the research part of the phd can take 2-4 years on average.

Paul Nahin has published many good historical math books that don't skimp on the mathematical underpinnings. I particularly enjoyed An Imaginary Tale: http://www.amazon.com/An-Imaginary-Tale-Princeton-Science/dp/0691146004

Regarding Spivaks: I'm also working on it, and found that my proof technique was lacking. An Introduction to Mathematical Reasoning (Eccles) was helpful for me: http://www.amazon.com/Introduction-Mathematical-Reasoning-Peter-Eccles/dp/0521597188

Strogatz is probably the best introductory book on the subject.

When studying nonlinear ODEs, analytical solutions are not always helpful and rarely necessary to understand the behavior of the dynamical system. If you absolutely need an answer (ie for a measured quantity) using RKF 4-5 (adaptive) for anything nonstiff is usually what you would do. There are no real good general tricks besides understanding system behavior without solving the ODE.

If you really want a close approximation, the only other option is to use perturbation theory (multiple scales, WKB, etc) to come up with an approximated solution. But it really isn't worth it in most cases (unless you have some eqution which is singularly perturbed). A good example of this is how to deal with the Schrodinger equation.

As for your example: it is separable, so separate and integrate. But if you have something remotely complicated you either won't get an analytical solution, or it will be such a pain that it isn't useful.

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

First, to get a sense as to the world of math and what it encompasses, and what different sub-subjects are about, watch this: https://www.youtube.com/watch?v=OmJ-4B-mS-Y

Ok, now that's out of the way -- I'd recommend doing some grunt work, and have a basic working knowledge of algebra + calculus. My wife found this book useful to do just that after having been out of university for a while: https://www.amazon.com/No-bullshit-guide-math-physics/dp/0992001005

At this point, you can tackle most subjects brought up from first video without issue -- just find a good introductory book! One that I recommend that is more on computer science end of things is a discrete math
book.

https://www.amazon.com/Concrete-Mathematics-Foundation-Computer-Science/dp/0201558025

And understanding proofs is important: https://www.amazon.com/Book-Proof-Richard-Hammack/dp/0989472108

He's my major advisor, and he loves occasionally showing off (who wouldn't?). I find it very entertaining. As far as I can tell, it's just a lot of practice plus some pattern recognition. For multiplying large numbers he just uses the distributive property combined with a certain method of remembering numbers in his head he uses.

I also read his book Secrets of Mental Math back in high school. He outlines some of the techniques there although its more basic.

Proofs from The Book is a great collection of easy to understand and accessible proofs. As someone who majored in math, but who will be not pursuing mathematics at the graduate level for a while, I've enjoyed working through them.

How about The Number Devil? It might be a bit below the reading (and mathematical) level of a 15-year-old, but it brings up some really insightful ideas that highlight how basic principles can lead to really exciting results.

If you are serious about learning, Linear Algebra by Friedberg Insel and Spence, or Linear Algebra by Greub are your best bets. I love both books, but the first one is a bit easier to read.

I 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?

Definitely Strogatz! Out of the books I have read, IMO, here is the order of readability: Strogatz, Meiss, Perko, Hartman. The italicized books are the ones I would recommend, but Perko and Hartman are really good too.

Strogatz: http://www.amazon.co.uk/Nonlinear-Dynamics-Chaos-Applications-nonlinearity/dp/0738204536

Meiss: http://www.amazon.co.uk/Differential-Dynamical-Systems-Mathematical-Computation/dp/0898716357/ref=sr_1_1?s=books&ie=UTF8&qid=1369504121&sr=1-1&keywords=meiss+dynamical+systems

Perko: http://www.amazon.co.uk/Differential-Equations-Dynamical-Systems-Mathematics/dp/0387951164/ref=sr_1_1?s=books&ie=UTF8&qid=1369504149&sr=1-1&keywords=perko+dynamical+systems

Hartman: http://www.amazon.co.uk/Ordinary-Differential-Equations-Classics-Mathematics/dp/0898715105/ref=sr_1_1?s=books&ie=UTF8&qid=1369504196&sr=1-1&keywords=hartman+differential+equations

I'm currently in my first year of undergraduate Maths and our course uses the book 'An Introduction to Mathematical Reasoning: Numbers, Sets, and Functions' by Peter J Eccles. It's such a helpful book aimed at introducing first year university students to pure mathematics, the book has definitely helped me feel confident in my pure module.

It states propositions and theorems and proves them and gives problems for you to solve or prove with the solutions at the back.

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

Hey I'm a physics BSc turned mathematician.

I would suggest starting with topology and functional analysis. Functional analysis is the foundation of quantum mechanics, and topology is necessary to properly understand manifolds, which are the foundation of relativity.

I would suggest Kreyszig for functional analysis. It's probably the most gentle functional analysis book out there.

For topology, I would suggest John Lee. This topology text is unique because it teaches general topology with a view towards manifolds. This makes it ideal for a physicist. If you want to know about Lie algebras and Lie groups, the sequel to this text discusses them.

I'm a huge fan of linear algebra. My favorite book for a theoretical understanding is this book. A pdf copy of the solutions manual can be found here.

Concrete mathematics is basically an expansion of knuths mathematical preliminary chapter in TAOCP, although it would probably be a bit much for typical engineering students at non elite colleges.

http://www.amazon.com/Concrete-Mathematics-Foundation-Computer-Science/dp/0201558025

For generating functions, the following is probably the best book.

http://www.amazon.com/generatingfunctionology-Second-Edition-Herbert-Wilf/dp/0127519564/ref=pd_sxp_f_pt/186-0047988-3920906

I had a combinatorics class this semester that covered the aforementioned topics but I wouldn't really recommend the book: it's a typical hand wavey undergrad cookbook type of presentation.

Amazon reviews are generally on point I would say just search combinatorics and delve in.

Differential Equations, Linear Nonlinear, Ordinary, Partial is a really decent book, he explains loads of details in it and gives a fair few examples, I would also strongly recommend Strogatz, he gives really decent explanations on dynamical systems.

This was the class as it was last quarter (Spring 2012), they used this textbook. I live off-campus and only go to the UW to drop off homework and almost never talk to anyone, so you almost definitely don't know me, but perhaps we walked right by each other one time without ever knowing it. EXCITING!

Kreyszig is the best first book on functional analysis IMO. For measure theory I liked Royden, specifically the 3rd edition.

It really depends which direction in mathematics you want to go. Even as a math major, I didn't really understand how vast it was until I got into abstract math.

My favorite way to learn is browse Amazon for "Dover Books on Mathematics." They are generally had for a penny + shipping if you don't mind buying used.

A good intro into modern mathematics: https://www.amazon.com/Concepts-Modern-Mathematics-Dover-Books/dp/0486284247

Steven Strogatz is a great one too:

• Nonlinear Dynamics and Chaos
  
  
• Sync: The Emerging Science of Spontaneous Order

I wouldn't recommend reading research papers that early. The are usually awfully specific and tend to use incoherent notation.

If you want to read some nice proofs, check out Proofs from THE BOOK. It's a collection of beautiful proofs covering many topics.

This is meant for younger children, probably, but The Number Devil is still an excellent children's book on many mathematical topics.

Read the book by Arthur Benjamin. He's one of my role models. :D The book has the most amazing mental math tricks ever, and I can square 2, 3, and even 4 digit numbers in my head. Getting to 5 digits soon. There are a lot of other cool tricks in there as well.

Almost forgot to reply. Linear Algebra by Friedberg is one of the more mathematically rigorous texts I've seen for undergraduates. My school used it in the honors linear algebra course. I think you'll find that it covers most of what you need. Hope it helps (if you can find it at the library or something).

It depends on where they are and what the purpose is. If you are trying to discourage them (and there might be valid reasons to do that), I'd say try measure theory.

Maybe use the Bartle book.

That would give them a taste for how abstract things can get and also drive home the point tiny books can require a lot o work.

On the other hand, if you want to do something that will help them, they An Introduction to Mathematical Reasoning.

It won't break the bank and, despite a few small typos, covers a lot material fairly gently.

http://www.amazon.com/Understanding-Analysis-Undergraduate-Texts-Mathematics/dp/0387950605

You can thank me later- it's really good. Also, a full solutions manual can be found with some googlefu.

The easiest one that I know of is the one by Epp. She doesn't go into the history as much, but her writing style is extremely easy on the brain.

You'll usually find the following recommended:

• Axler's Linear Algebra Done Right
  
• Friedberg's Linear Algebra
  
  I've personally used Friedberg's text, and I found it to be pretty well written.

Here's another one that's pretty good

Hmm I'm surprised you've had point-set topology, linear algebra, and basic functional analysis but have yet to encounter locally convex topological vector spaces! No worries, you have most likely developed all oft the machinery to understand them. I agree with G-Brain, Rudin's function analysis will do. Most functional analysis books should cover this at some point. The only I use is Kreyszig. Hope that helps!

It mentions Rosenlicht at the bottom. Lucky you, that book's only 8 bucks! It's a good book, too.

I read Mathematics: From the Birth of Numbers in high school / early college. It's a long book, but it's definitely worth checking out.

You might like Rosenlicht's book, Introduction to Analysis. Google Books will show you the first 2 chapters for free. It's a Dover book, so it's good and also cheap. I believe that it is often used as the text for the first "serious" real analysis course.

This is a pretty good book too. http://www.amazon.com/Introduction-Analysis-Dover-Books-Mathematics/dp/0486650383/ref=sr_1_1?ie=UTF8&qid=1323212337&sr=8-1

I don't know why more people on here don't recommend it, especially considering how cheap it is.

I don't think it contains any group theory, but everything else is there:

Discrete Mathematics with Applications by Susanna S. Epp (


This one below contains some algebra(groups):

Mathematics: A Discrete Introduction by Edward A. Scheinerman

Both are pretty elementary.

I also really struggled with real analysis in the beginning. Stephen Abbot's Understanding Analysis saved my ass, I went from "reconsidering my career choice" to passing the course with a pretty good grade thanks to that book.

http://www.amazon.com/Understanding-Analysis-Undergraduate-Texts-Mathematics/dp/0387950605/ref=sr_1_1?ie=UTF8&qid=1426932693&sr=8-1&keywords=understanding+analysis

Gödel's Proof is a good starting point for the incompleteness theorem. Covers the basics of the theorem and its impacts. Unless you are prepping for coursework in logic than this book likely has the right amount of depth for you.

I don't have a recommendation for Tarski. Hopefully someone else has something for you.

What's up, man. I failed geometry twice (sophomore and junior year) in high school. I barely graduated high school with a 2.0 gpa. I am now a senior studying math and computer science (going to be getting masters in math). I am at the top of my class, and I will be graduating with a ~3.91 GPA.

Math, just like anything else, is about practice and perseverance. I thought I sucked at math (and basically everyone told me I was more of an "english" kind of guy). But when I got to college, I found that I really enjoyed the challenge, and I found the material interesting as hell. So I worked my ass off at it.

If you work hard (some may need to work harder than others!) and persevere, then you will be fine. There will definitely be challenges, but that's what makes math so fun.

edit: Also, unless you are a math major, I can't imagine you will be getting into too much rigorous theory. You will likely continue mostly just be doing calculations (Calc 1, Calc 2 and Calc 3). That is how it is at my university, at least. However, if you are a math major, it can't hurt to get a head start on writing simple proofs. For that, I recommend the following book: https://www.amazon.com/Mathematical-Proofs-Transition-Advanced-Mathematics/dp/0321797094/ref=sr_1_4?s=books&ie=UTF8&qid=1474297774&sr=1-4&keywords=a+transition+to+advanced+mathematics

Seriously, that book is such a fucking good introductory text. It helped me so much.

For a general overview of everything to do with the history of math, which might be what you're looking for, I recommend Mathematics: From the Birth of Numbers. Very inspiring with a little bit of "how to do everything."

Nearly everyone on this subreddit recommends Strogatz. However, I've never read this book myself. The one I'm familiar with is Jordan and Smith, which I definitely can recommend, with the caveat that there are a lot of typos in it.

The Nature of Computation

(I don't care for people who say this is computer science, not real math. It's math. And it's the greatest textbook ever written at that.)

Concrete Mathematics

Understanding Analysis

An Introduction to Statistical Learning

Numerical Linear Algebra

Introduction to Probability

I think there could do some really cool analysis tracing lines of thought and how they developed or comparing what was in vogue in math to world developments at the time. This book might be a good overview for modern developments and this one has a overview of the development of math through history

Mathematics: From the Birth of Numbers

It's gigantic, but really entertaining to flip around in.

Ah yeah you're at a more advanced stage than I thought. In that case an analysis text might appeal -- I like Abbot's Understanding Analysis but, again, it's quite pricey.

I suspect you'd love Galois theory, but I can't recommend a good text for self-study offhand.

I had a copy of The Number Devil when I was a kid and it was wonderful.

Nonlinear Dynamics and Chaos by Strogatz is supposed to be good.

I'm also planning on doing a Masters in Math or CS. What do you plan to write for your masters?


> Anybody else feels like this?

I think its natural to doubt yourself, sometimes. I dont know what else to say, but just try to be objective and emotionless about it (when you get stuck in a problem).

The following books that helped me improve my math problem solving skills when I was an undergrad:

• How to Think Like a Mathematician: A Companion to Undergraduate Mathematics
  
  
• Thinking Mathematically
  
  
• How to Solve It: A New Aspect of Mathematical Method
  
  
• The Art of Problem Posing
  
  
• How to Study as a Mathematics Major
  
  
• How to Read and Do Proofs: An Introduction to Mathematical Thought Processes

Nonlinear Dynamics And Chaos: With Applications To Physics, Biology, Chemistry, And Engineering by Steven H. Strogatz is an excellent book on nonlinear dynamical systems and you definitely don't need any probability or statistics to study it, just a good knowledge of multivariable calculus and linear algebra. Chaos theory actually doesn't have anything to do with randomness since one of the defining features of a chaotic system is that it is deterministic.

Edit: There is a freely available course by Strogatz on YouTube.

Proofs from THE BOOK. This is basically a showcase of mathematicians at their most clever. The book could be read by anyone with a solid grounding in calculus, linear algebra and discrete math.

To lean from a book like that of Artin's, you need to get a few basics down:

How to Prove It: A Structured Approach by Daniel Velleman

Mathematical Proofs: A Transition to Advanced Mathematics by Gary Chartrand et al

If you're looking for a concise introductory level reference, I don't know of any at only the high-school level; additionally most undergrad level textbooks are gonna assume a certain level of sophistication w.r.t. the student.


However, if you are interested, the book "Godel's Proof" by Nagel, offers many accessible insights into the workings of mathemical logic

https://www.amazon.com/Gödels-Proof-Ernest-Nagel/dp/0814758371

For real analysis, I would avoid Rudin. I think it's overrated as a good book to learn from, especially for people who aren't math majors. I'd go with Introduction to Analysis by Rosenlicht. It's basically a friendlier version of Rudin, and a heck of a lot cheaper.

I strongly recommend Proofs from THE BOOK.

If you really want to understand probability then you'll need to learn measure theory, which will require some background knowledge in real analysis. This is the book I used, which I highly recommend (and it's cheap!): http://www.amazon.com/Introduction-Analysis-Dover-Books-Mathematics/dp/0486650383/ref=sr_1_1?ie=UTF8&qid=1414974523&sr=8-1&keywords=introduction+to+analysis

As for an actual book on probability, I'm not too sure since my probability course was based on lecture notes provided by the professor, although I just ordered this book because it looked decent: http://www.amazon.com/Graduate-Course-Probability-Dover-Mathematics-ebook/dp/B00I17XTXY/ref=sr_1_1?ie=UTF8&qid=1414974533&sr=8-1&keywords=graduate+book+on+probability

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 highly recommend Polya's How to Solve it too.

You can read the "Highlights of the Fourth Edition" on Page xvii through Amazon's preview.

Edit: This Amazon review is also relevant to you:

> I've taught discrete math from the 3rd Edition of this book at least 6 times, and struggled with several issues. (The textbook for our Discrete Math course is chosen by a committee in our department.) Much of a discrete math course involves looking closely at some very simple mathematics. Most of the mathematics is already known to a typical university freshman; what a set is, what a prime is, what an ordered pair is, etc. Of course they have had little rigor in these elementary topics, but still, they have the notions and vocabulary. The 3rd Edition pretended that sets, e.g., did not exist until one finally arrived at the chapter on sets. It's unnatural to lecture one's way through two chapters on logic and a chapter on techniques of proof, without being able to draw on simple examples from set theory. One gets tired quickly of examples of dogs and cats in highly artificial situations, and would like to say something about primes or the set of even integers.

>The 4th Edition corrects this problem by the addition of an introductory chapter which fixes the vocabulary and notation. This was a needed change. The 3rd Edition required considerable acrobatics in avoiding words like "is an element of" until Chapter 5 (Set Theory.) Really? I'm supposed to cover the proof technique of "division into cases" and I can't say "the set of integers of the form 4k+1?" So good change.

>Every semester, I get e-mails from my students asking if the previous edition of the text will suffice for my course. Usually, I say yes. In the case of my discrete math course, I'll have to say no. The modifications of this text are substantial. Besides the above, the old Chapter 8 (Recursion) is now incorporated into the new (much expanded) Chapter 5 (Sequences and Induction.) That is also a sensible change.

>My remaining complain about the text is that it's a bit condescending. I think it's bad form to always present mathematics apologetically. "There, there now, I know it's difficult, but we'll go extremely slowly and take tiny, tiny bites covered in catsup so you can scarcely taste them." There's no need for us at the university level to re-enforce the bad attitudes the students learned in high school. It's math. It's hard. You can do it, not because math is made easy, but because you are, in fact, clever enough.

>I would not have recommended the 3rd Edition to anyone, but I would recommend the 4th. I'm very happy with the changes.

You could try Abbott's Understanding Analysis. Quite a few students seem to like this book.

One concrete suggestion I can give you is when faced with a theorem or definition, try first to understand what it means in 'words' and then try to reason why it may be true, again in 'words'. I've noticed that often what trips students up is the symbolism -- often when I see incorrect answers from bright students, 10 to 1, its because they've got caught up in symbols and are now mentally running around in circles. This, I feel, is the unfortunate transition-pangs from school math to real math.

Remember math is not about symbols, formulas or equations, its about the concepts and ideas that hide behind those things.

A First Course in Graph Theory by Chartrand and Zhang

Combinatorics: A Guided Tour by Mazur

Discrete Math by Epp

For Linear Algebra I like these below:

Lecture Notes by Tao

Linear Algebra: An Introduction to Abstract Mathematics by Robert Valenza

Linear Algebra Done Right by Axler

Linear Algebra by Friedberg, Insel and Spence

Proofs from THE BOOK
It's "the bible" of the most elegant mathematical proofs, which Paul Erdös always talked about.

> Calculus has a huge foundation in mathematical analysis that at most universities takes roughly half a year to a year of graduate/upper-undergrad study to develop (at least this is how it is at my university).

Graduate/upper undergrad? At Copenhagen University (KU) material corresponding roughly to Abbott's Understanding Analysis is covered in the first year. Plus some linear algebra and other stuff.

KU does have the advantage that it doesn't have to teach any engineers. They are all over at DTU in Lyngby learning to use maths to compute things leaving the mathematics department at KU to focus on teaching maths students to prove things.

this one is pretty good.

This one is pretty good

Yes. However, you should probably read something that introduces you to proofs. My Intro to Higher Math classes (commonly called Intro to Proof-Writing or Intro to Analysis, the class or series of classes that introduce you to higher math and proofwriting skills) used this book alongside a prepackaged set of detailed lecture notes. I'd say that'd be a good place to start before reading about Abstract Algebra, plus the book is dirt cheap.

An Introduction to Mathematical Reason - Peter Eccles. Very good book.

http://www.amazon.com/Introduction-Mathematical-Reasoning-Peter-Eccles/dp/0521597188

On a more serious note, this book by Polya is wonderful.

It's cool that you're interested in complex systems, but your post is a bit vague. I liked Nonlinear Dynamics and Chaos (Strogatz). It is a very easy/friendly intro to the field. Another good book, depending on what you're wanting to do, might be Daniel Gillespie's book on markov processes. In general, you basically need to read some papers, find a type of problem/approach that interests you and then fill in the blanks with supplementary material. Most of what you need to know is in a journal somewhere. Google that shit. If you want to code stuff, learn python & C.



http://www.amazon.com/Nonlinear-Dynamics-And-Chaos-Applications/dp/0738204536/ref=sr_1_2?s=books&ie=UTF8&qid=1335215605&sr=1-2

Aside from Khan, The Secrets of Mental Math was extremely helpful in this endeavor.

If you're coming from a more applied background (or physics / engineering) https://www.amazon.com/Introductory-Functional-Analysis-Applications-Kreyszig/dp/0471504599 is pretty easy to follow. Obviously it goes into the infinite dim too but it covers all the finite stuff first.

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.