Best products from r/math

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.

My big-picture recommendation would be to learn proofs. Can you read and understand proofs? If asked to justify a proposition, can you produce a coherent, rigorous proof that unambiguously communicates your understanding to others? Learning how to do proofs is, as I mentioned above, its own skillset, one that's a necessary condition for being able to do any kind of serious mathematics.

From my perspective, if you're interested in mathematics, then the specific content—i.e., analysis vs. abstract algebra, combinatorics vs. number theory—is secondary to two things:

1. Are you exploring parts of mathematics that you find interesting and accessible?
   
   After all, you don't want to burn yourself out with something you find boring, nor do you want to overwhelm yourself by trying to bite off more than you can chew (such as EGA).
   
   
2. Are you learning how to do proofs?
   
   Proof-writing is simply the lifeblood of doing mathematics, and the only effective way to learn this skill is by practice.
   
   One book I often recommend to people in your situation is Journey Through Genius: The Great Theorems of Mathematics by William Dunham. It's not really a textbook, but it's still a really interesting introduction to mathematics. It's a bit like a sampler plate, too, since it covers examples from all sorts of topics: number theory, set theory, calculus, and others, as I recall.
   
   One of the challenges of learning how to write proofs is that it can be difficult to do so as an autodidact; it really helps to get feedback from people who can help you sharpen both your thinking and your writing. This next recommendation may be more logistically challenging (or expensive) for you to pursue, but I'd nonetheless recommend that you look into summer math camps whose focus is teaching fluency with proofs. Three in particular include the following:
   
   

• The Ross Mathematics Program at Ohio State University
  
  
• PROMYS at Boston University
  
  
• Hampshire College Summer Studies in Mathematics at Hampshire College
  
  All three have good reputations, but you might have personal preferences that would lead you to prefer one over the other two. All three, as I understand it, use number theory as the entry point to teach you how to think about math abstractly, though Hampshire's program is a bit more eclectic than the number theory-specific focus of the Ross and PROMYS programs. Which program, if any, might be your best fit could turn as much on outside issues like when their sessions are held, how much you'd have to pay (US$3,800–4,000 for program costs alone, from what I could tell), and travel logistics (especially if you'd be an international student), separate from any narrowly mathematical considerations. Oh, and another advantage to attending one of these programs is that you're surrounded by fellow students like you who are really interested in mathematics. There's no way to replicate the value of that from any single textbook, no matter how inspired.
  
  Anyway, that's a starting point. If you have local, regional, or national mathematics competitions—e.g., AMC, ARML, as well as other assorted city, state or provincial, or regional competitions—then that's another good entry point into interesting math. From my experience, the main advantage of math contests is that they expose you early to concepts you might not otherwise see for years, and, again, you get to spend time with fellow math students like you.
  
  Competitions, whether individual or team-based, often have more of a proof-based focus than, say, typical the typical high school curriculum (with the exception of geometry), but "contest math" has the danger of students inferring a distorted picture of what it takes to become a mathematician. Namely: you do not have to be a prodigy in math competitions in order to become a good mathematician, let alone a mathematician. Separately, if there's a Math Circle near you, that might be another valuable resource.
  
  As a high school student who will have already completed Calculus BC before your senior year, you might be able to take college-level classes next year, assuming there's a nearby college or university that has some kind of arrangement with your high school. (Some public school districts even cover your tuition, too.) If that's an option for you, free or not, then I'd recommend coordinating with your school's guidance counselor and a professor in the math department to discuss your options.
  
  Oh, and as an obliquely-related topic: if you have time, now would be a good time to teach yourself how to use LaTeX (or one or more of its siblings) to typeset mathematics. (LaTeX may be useful to you if you pursue other scientific field, too, but it's especially useful in math.) If you're serious in pursuing math going forward, you'll inevitably be using LaTeX, and better to get a head start today on scaling its learning curve.
  
  ---
  
  I'm sure I will think of half a dozen more suggestions an hour from now, but I'll leave things here for now. I hope something above will help, and good luck!

Hello! I'm interested in trying to cultivate a better understanding/interest/mastery of mathematics for myself. For some context:

 




To be frank, Math has always been my least favorite subject. I do love learning, and my primary interests are Animation, Literature, History, Philosophy, Politics, Ecology & Biology. (I'm a Digital Media Major with an Evolutionary Biology minor) Throughout highschool I started off in the "honors" section with Algebra I, Geometry, and Algebra II. (Although, it was a small school, most of the really "excelling" students either doubled up with Geometry early on or qualified to skip Algebra I, meaning that most of the students I was around - as per Honors English, Bio, etc - were taking Math courses a grade ahead of me, taking Algebra II while I took Geometry, Pre-Calc while I took Algebra II, and AP/BC Calc/Calc I while I took Pre-Calc)

By my senior year though, I took a level down, and took Pre-Calculus in the "advanced" level. Not the lowest, that would be "College Prep," (man, Honors, Advanced, and College Prep - those are some really condescending names lol - of course in Junior & Senior year the APs open up, so all the kids who were in Honors went on to APs, and Honors became a bit lower in standard from that point on) but since I had never been doing great in Math I decided to take it a bit easier as I focused on other things.

So my point is, throughout High School I never really grappled with Math outside of necessity for completing courses, I never did all that well (I mean, grade-wise I was fine, Cs, Bs and occasional As) and pretty much forgot much of it after I needed to.

Currently I'm a sophmore in University. For my first year I kinda skirted around taking Math, since I had never done that well & hadn't enjoyed it much, so I wound up taking Statistics second semester of freshman year. I did okay, I got a C+ which is one of my worse grades, but considering my skills in the subject was acceptable. My professor was well-meaning and helpful outside of classes, but she had a very thick accent & I was very distracted for much of that semester.

Now this semester I'm taking Applied Finite Mathematics, and am doing alright. Much of the content so far has been a retread, but that's fine for me since I forgot most of the stuff & the presentation is far better this time, it's sinking in quite a bit easier. So far we've been going over the basics of Set Theory, Probability, Permutations, and some other stuff - kinda slowly tbh.

 




Well that was quite a bit of a preamble, tl;dr I was never all that good at or interested in math. However, I want to foster a healthier engagement with mathematics and so far have found entrance points of interest in discussions on the history and philosophy of mathematics. I think I could come to a better understanding and maybe even appreciation for math if I studied it on my own in some fashion.

So I've been looking into it, and I see that Dover publishes quite a range of affordable, slightly old math textbooks. Now, considering my background, (I am probably quite rusty but somewhat secure in Elementary Algebra, and to be honest I would not trust anything I could vaguely remember from 2 years ago in "Advanced" Pre-Calculus) what would be a good book to try and read/practice with/work through to make math 1) more approachable to me, 2) get a better and more rewarding understanding by attacking the stuff on my own, and/or 3) broaden my knowledge and ability in various math subjects?

Here are some interesting ones I've found via cursory search, I've so far just been looking at Dover's selections but feel free to recommend other stuff, just keep in mind I'd have to keep a rather small budget, especially since this is really on the side (considering my course of study, I really won't have to take any more math courses):
Prelude to Mathematics
A Book of Set Theory - More relevant to my current course & have heard good things about it
Linear Algebra
Number Theory
A Book of Abstract Algebra
Basic Algebra I
Calculus: An Intuitive and Physical Approach
Probability Theory: A Concise Course
A Course on Group Theory
Elementary Functional Analysis

Machine learning is largely based on the following chain of mathematical topics

Calculus (through Vector, could perhaps leave out a subsequent integration techniques course)

Linear Algebra (You are going to be using this all, a lot)

Abstract Algebra (This isn't always directly applicable but it is good to know for computer science and the terms of groups, rings, algebras etc will show up quite a bit)

General Topology (Any time we are going to deal with construction of a probability space on some non trivial manifold, we will need this. While most situations are based on just Borel sets in R^n or C^n things like computer vision, genomics, etc are going to care about Random Elements rather than Random Variables and those are constructed in topological spaces rather than metric ones. This is also helpful for understanding definitions in well known algorithms like Manifold Training)

Real Analysis (This is where you learn proper constructive formulations and a bit of measure theory as well as bounding theorems etc)

Complex Analysis (This is where you will get a proper treatment of Hilbert Spaces, Holomorphic functions etc, honestly unless you care about QM / QFT, P-chem stuff in general like molecular dynamics, you are likely not going to need a full course in this for most ML work, but I typically just tell people to read the full Rudin: Real and Complex Analysis. You'll get the full treatment fairly briefly that way)

Probability Theory (Now that you have your Measure theory out of the way from Real Analysis, you can take up a proper course on Measure Theoretic Probability Theory. Random Variables should be defined here as measurable functions etc, if they aren't then your book isn't rigorous enough imho.)

Ah, Statistics. Statistics sits atop all of that foundational mathematics, it is divided into two main philosophical camps. The Frequentists, and the Bayesians. Any self respecting statistician learns both.

After that, there are lots, and lots, and lots, of subfields and disciplines when it comes to statistical learning.

A sample of what is on my reference shelf includes:

Real and Complex Analysis by Rudin

Functional Analysis by Rudin

A Book of Abstract Algebra by Pinter

General Topology by Willard

Machine Learning: A Probabilistic Perspective by Murphy

Bayesian Data Analysis Gelman

Probabilistic Graphical Models by Koller

Convex Optimization by Boyd

Combinatorial Optimization by Papadimitriou

An Introduction to Statistical Learning by James, Hastie, et al.

The Elements of Statistical Learning by Hastie, et al.

Statistical Decision Theory by Liese, et al.

Statistical Decision Theory and Bayesian Analysis by Berger

I will avoid listing off the entirety of my shelf, much of it is applications and algorithms for fast computation rather than theory anyway. Most of those books, though, are fairly well known and should provide a good background and reference for a good deal of the mathematics you should come across. Having a solid understanding of the measure theoretic underpinnings of probability and statistics will do you a great deal--as will a solid facility with linear algebra and matrix / tensor calculus. Oh, right, a book on that isn't a bad idea either... This one is short and extends from your vector classes

Tensor Calculus by Synge

Anyway, hope that helps.

Yet another lonely data scientist,

Tim.

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

Rasputin did give you good advice. But just so you know, there's an /r/mathbooks. It includes a textbook on set theory. Though, don't be intimidated if it's too advanced for you.

I'd recommend some of the more popular math books as well.

Flatterland is a fun, quirky adventure through some advanced geometric concepts. Very readable.

Euler's Gem is a book I've never read, but might some day. I thumbed through it, and it seemed like a good enough summary of topology for a laymen. The amazon reviews agree. Be warned: there are equations. But you're trying to discover the beauty of math, so equations are probably good!

Goedel Escher and Bach is . . . Famous. I'll leave it at that. I thought it was a good and simple, but it's too close to my primary area of interest for me to recommend in good faith.

That's about all of my experience with popular math books . . .

Euclid's elements (you can google it and find it online) is great for an introduction to mathematical proof using something highly diagramable (unlike set theory). But, I would definitely scope out my interests before putting down any money. Perhaps check out the library. And don't get intimidated if anything is over your head. When that happens to me, I tend to get a little excited in all honesty :).

I have not read many books explicitly devoted to the history of mathematics, such as those recommended in this math.stackechange post #31058, so I will refrain from recommending any of them. Instead, I'd like to mention a few books that do well discussing aspects of mathematical history, although this is not their main focus.

• Journey Through Genius, by William Dunham. This is a survey of some of math's creative "landmarks" throughout history, as well as the contexts in which they were achieved and the people who worked on them. (Ok, now that I write it out, this is clearly a "history of math" book. The others in this list, not as much...)
  
  
• Four Colors Suffice: How the Map Problem Was Solved, by Robin Wilson. Clear and (relatively) brief description of the development of the proof of the 4 color theorem, from the birth of graph theory to the computer-assisted proof and the discussions that has inspired. The newest edition is now in color, not black & white, and that may not sound like much, but the figures are genuinely awesome and make the concepts so much more understandable. Highly recommended.
  
  
• In Pursuit of the Traveling Salesman: Mathematics at the Limits of Computation, by William J. Cook. I lectured about the TSP briefly in a course I taught this past semester. I read this book in preparation and enjoyed it so thoroughly that I found myself quoting long passages from it in class and sharing many of its examples and figures.
  
  
• How to Lie With Statistics, by Darrell Huff (illustrations by Irving Geis). I recommend this because it's a modern classic. Written in 1954, the ideas are still relevant today. I believe this book should be a requirement in the high school curriculum. (Plus, available as free pdf.)
  
  
• The Emergence of Probability: A Philosophical Study of Early Ideas about Probability, Induction and Statistical Inference, by Ian Hacking. "A philosophical study of the early ideas about probability, induction and statistical inference, covering the period 1650-1705." Ok, this one is really specific and I often found myself rereading sentences 5 times to make sure I understood them which was frustrating. But, its specificity is what makes it so interesting. Worth checking out if it sounds cool, but not for everyone. (FWIW I found a copy at my public library.)
  
  
• Understanding Analysis, by Stephen Abbott. You mentioned you're learning real analysis. I taught a real analysis course this past semester using this book, and it's the one from which I learned the subject myself in college. Abbott writes amazingly well and makes the subject matter clear, inviting, and significant.
  
  
• I also recommend flipping through the volumes in the series The Best Writing on Mathematics. They have been published yearly since 2010. There are bound to be at least a few articles in each volume that will appeal to you. Moreover, they contain extensive lists of references and other recommended readings. I own a copy of each one and am nowhere near completion reading any of them because they always lead me elsewhere!
  
  Hope this is helpful!

> 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.

some amazing books I would suggest to you are:

• Godel Escher Bach
  
  
• Road to Reality By Roger Penrose.
  
  
• Code by
  Charles Petzold.
  
  
• Pi in the Sky by John Barrow.
  
  All of these I would love to read again, if I had the time, but none more so than Godel, Escher, Bach, which is one of the most beautiful books I have ever come across.
  
  Road to Reality is the most technical of these books, but gives a really clear outline of how mathematics is used to describe reality (in the sense of physics).
  
  Code, basically, teaches you how you could build a computer (minus, you know, all the engineering. But that's trivial surely? :) ). The last chapter on operating systems is pretty dated now but the rest of it is great.
  
  Pi in the Sky is more of a casual read about the philosophy of mathematics. But its very well written, good night time reading!
  
  You have a really good opportunity to get an intuitive understanding of the heart of mathematics, which even at a college level is somewhat glossed over, in my experience. Use it!

Escher's work with tessellation and other mathematical ideas are fairly well-known and documented so I'll try to mention a few examples of things I learned in an art history course a while ago.


DaVinci's Vitruvian Man used Phi in the calculation of ratios. Example: the ratio of your arm to your height or your eyes to your face is nearly always Phi. I'm not sure if I'm correct in the body parts mentioned, my art history class was nearly 6 years ago so I'm a bit rusty. I'll try to think of some more examples and post.


EDIT: a few more examples have come back from memory. DaVinci was a master of perspective as well. As you can see DaVinci used linear lines to draw attention to the subject of his works. In the case of The Last Supper, the lines from the structure of the building, to the eyes and gestures of the disciples aim towards Jesus.


Botticelli's Birth of Venus uses a triangle to bring the subject into the viewer's mind. The two subjects on the left and right form the lines that meet at the middle of the top and close off a triangle with the bottom of the work. Venus herself is in the middle of the triangle which brings your attention to her immediately upon viewing the work.


Michelangelo's Pieta also uses a triangle to highlight its subjects. Mary's figure creates a triangle (which is considered to be quite intentional based upon her size, both in relation to Jesus, a full grown man, and from her upper and obviously enlarged lower body). Her triangle makes the outline for the subject, Jesus. He is nearly in the center of both the horizontal and vertical axises. The way he is laying, from near the top of the left and then draping to the bottom of the right, depicts a very lifeless form because of the unnatural laying. Moving the viewer's gaze from the top to the bottom of the triangle strengthens the emotion of the scene.



Moving on to architecture, vaulted ceilings also use triangles to draw your eyes down a line also make an awe-inspiring impression.


In contrast to the European's love of straight lines and geometric figures, the traditional Japanese architectural style was opposed to using straight lines. As you can see, nearly every line in a traditional Japanese building is curved. The traditional belief was that straight lines were evil because they thought evil spirits could only travel in straight lines. This design criteria made for very interesting formations and building methods which I would encourage you to check out because of the sheer dedication to the matter.


The Duomo in Florence is a great example of Renaissance architecture and has a really cool octagonal shaped dome. I could go on and on about how awesome Brunelleschi's design was, but I'll just let you read about it here.


I could talk all day about this sort of stuff, just let me know if you want anything else or have any questions. Good luck with your class!


EDIT2: I've found some more links about the subject of mathematics in art and architecture. It looks like University of Singapore actually has a class on the subject. There's also a good Wikipedia page on it as well. This article is pretty lengthy and knowledgeable, but doesn't include pictures to illustrate the topics. Finally, as almost anybody in r/math will testify, Godel, Escher, Bach by Douglas Hofstadter is a fantastic read for anybody interested in mathematics and cool shit in general.



EDIT3: LITERATURE: I know we've all heard what a badass Shakespeare was, but it really hits you like a bus when you find out that how well the man (or for you Shakespeare conspiracy theorists, men) could use words in rhyme and meter. Here's a Wikipedia article about his use of iambic pentameter and style. Nothing else really comes to mind at the moment as far as writers using math (other than using rhyme and meter like I mentioned Shakespeare doing); however, I can think of a few ways to incorporate math. If you would like to go into any sort of programming during the class, you could show how to make an array out of a word. Once that concept is understood, you could make them solve anagrams or palindromes with arrays... a favorite of mine has always been making [ L , I , N , U , X ] into [ U , N , I , X ] ( [ 3 , 2 , 1 , 4 ] for the non-array folks ).

FWIW I had no fun with mathematics in school and didn't start studying it til I was in my thirties. I'm no genius, but I now teach the subject and still self-study it. You don't need any mysterious talent to get very competent at university-level maths, just to be interested enough in it to put the hours in.

Self-study is hard and frustrating. Be prepared for that. Reading one page can take a day. You can stare at a definition or theorem for hours and not understand it. Looking things up in multiple books can really help with that -- there are some good resources online as well. Also, some things just take a while to "cook" in the brain; keep at it. Take lots and lots of notes, preferably with pictures. Do plenty of exercises. When you're really stumped, post here.

I'll echo what others have said: add to Spivak a couple of other books so you can change it up. A book on group theory and one on linear algebra would be a nice combination -- maybe one on discrete maths, probability or something similar as well if that interests you. For group theory I think this book is fantastic, though it's expensive.

If you really want to make it through Spivak, make a plan. Break the book down into, say, 50-page chunks and make 50 pages your target for each week (I have no idea whether this is too ambitious for you -- try it and see). Track your progress. Celebrate when you hit milestones.

Good luck!

[EDIT: Also, be aware that maths books aren't really designed to be read like novels. Skim a chapter first looking for the highlights and general ideas, then drill into some of the details. Skip things that seem difficult and see if they become important later, then go back (with more motivation) etc.]

Copying my answer from another post:


I was literally in the bottom 14th percentile in math ability when i was 12.

One day, by pure chance, i stumbled across this (free and open) book written by Carl Stitz and Jeff Zeager, of Lakeland Community College

Precalculus

It covers everything from elementary algebra (think grade 5), all the way up to concepts used in Calculus and Linear Algebra (Partial fractions and matrix algebra, respectively.) The book is extremely well organized. Every sections starts with a dozen or so pages of proofs and derivations that show you the logic of why and how the formulas you'll be using work. This book, more than any other resource (and i've tried a lot of them), helped me build my math intuition from basically nothing.


Math is really, really intimidating when you've spent your whole life sucking at it. This book addresses that very well. The proofs are all really well explained, and are very long. You'll basically never go from one step to the next and be completely confused as to how they got there.


Also, there is a metric shitload of exercises, ranging from trivial, to pretty difficult, to "it will literally take your entire class working together to solve this". Many of the questions follow sort of an "arc" through the chapters, where you revisit a previous problem in a new context, and solve it with different means (Also, Sasquatches. You'll understand when you read it.)


I spent 8 months reading this book an hour a day when i got home from work, and by the end of it i was ready for college. I'm now in my second year of computer science and holding my own (although it's hard as fuck) against Calculus II. I credit Stitz and Zeager entirely. Without this book, i would never have made it to college.


Edit: other resources

Khan Academy is good, and it definitely complements Stitz/Zeager, but Khan also lacks depth. Like, a lot of depth. Khan Academy is best used for the practice problems and the videos do a good job of walking you through application of math, but it doesn't teach you enough to really build off of it. I know this from experience, as i completed all of Khan's precalculus content. Trust me, Rely on the Stitz book, and use Khan to fill in the gaps.


Paul's Online Math Notes

This website is so good it's ridiculous. It has a ton of depth, and amazing reference sheets. Use this for when you need that little extra detail to understand a concept. It's still saving my ass even today (Damned integral trig substitutions...)

Stuff that's more important than you think (if you're interested in higher math after your GED)

Trigonometric functions: very basic in Algebra, but you gotta know the common values of all 6 trig functions, their domains and ranges, and all of their identities for calculus. This one bit me in the ass.

Matrix algebra: Linear algebra is p. cool. It's used extensively in computer science, particularly in graphics programming. It's relatively "easy", but there's more conceptual stuff to understand.


Edit 2: Electric Boogaloo

Other good, cheap math textbooks

/u/ismann has pointed out to me that Dover Publications has a metric shitload of good, cheap texts (~$25CAD on Amazon, as low as a few bucks USD from what i hear).

Search up Dover Mathematics on Amazon for a deluge of good, cheap math textbooks. Many are quite old, but i'm sure most will agree that math is a fairly mature discipline, so it's not like it makes a huge difference at the intro level. Here is a Math Overflow Exchange list of the creme de la creme of Dover math texts, all of which can be had for under $30, often much less. I just bought ~1,000 pages of Linear Algebra, Graph Theory, and Discrete Math text for $50. If you prefer paper to .pdf, this is probably a good route to go.

Also, How to Prove it is a very highly rated (and easy to read!) introduction to mathematical proofs. It introduces the basic logical constructs that mathematicians use to write rigorous proofs. It's very approachable, fairly short, and ~$30 new.

I've always enjoyed all types of math but all throughout (engineering) undergrad and grad school all I ever got to do was computational-based math, i.e. solving problems. This was enjoyable but it wasn't until I learned how to read and write proofs (by self-studying How to Prove It) that I really fell in love with it. Proofs are much more interesting because each one is like a logic puzzle, which I have always greatly enjoyed. I also love the duality of intuition and rigorous reasoning, both of which are often necessary to create a solid proof. Right now I'm going back and self-studying Control Theory (need it for my EE PhD candidacy but never took it because I was a CEG undergrad) and working those problems is just so mechanical and uninteresting relative to the real analysis I study for fun.

EDIT: I also love how math is like a giant logical structure resting on a small number of axioms and you can study various parts of it at various levels. I liken it to how a computer works, which levels with each higher level resting on those below it. There's the transistor level (loosely analogous to the axioms), the logic gate level, (loosely analogous set theory), and finally the high level programming language level (loosely analogous to pretty much everything else in math like analysis or algebra).

Saff and Snider is great for applied complex analysis. In my opinion it strikes a perfect balance between accessibility and rigor for a first course on the subject.

Visual Complex Analysis is another good choice, but it might be a little more advanced than what you're interested in.

The first half of Lang might also be a good choice, but Lang takes a slightly more formal, proof-based approach.

I've also skimmed through Brown and Churchill, which looks quite good but is prohibitively expensive.

Finally, you can find many cheap (~$10) books on the subject by Dover. The only one I've looked at is Knopp, which is quite formal and light on computation, but might be a good supplement. Here's another Dover book with outstanding Amazon reviews.

Complex analysis is both very elegant and very useful. Best of luck with your class!

I would guess that career prospects are a little worse than CS for undergrad degrees, but since my main concern is where a phd in math will take me, you should get a second opinion on that.

Something to keep in mind is that "higher" math (the kind most students start to see around junior level) is in many ways very different from the stuff before. I hated calculus and doing calculations in general, and was pursuing a math minor because I thought it might help with job prospects, but when I got to the more abstract stuff, I loved it. It's easily possible that you'll enjoy both, I'm just pointing out that enjoying one doesn't necessarily imply enjoying the other. It's also worth noting that making the transition is not easy for most of us, and that if you struggle a lot when you first have to focus a lot of time on proving things, it shouldn't be taken as a signal to give up if you enjoy the material.

This wouldn't be necessary, but if you like, here are some books on abstract math topics that are aimed towards beginners you could look into to get a basic idea of what more abstract math is like:

• theoretical computer science (essentially a math text)
  
  
• set theory
  
  
• linear algebra
  
  
• algebra
  
  
• predicate calculus
  
  Different mathematicians gravitate towards different subjects, so it's not easy to predict which you would enjoy more. I'm recommending these five because they were personally helpful to me a few years ago and I've read them in full, not because I don't think anyone can suggest better. And of course, you could just jump right into coursework like how most of us start. Best of luck!
  
  (edit: can't count and thought five was four)

Indeed; you may feel that you are at a disadvantage compared to your peers, and that the amount of work you need to pull off is insurmountable.

However, you have an edge. You realize you need help, and you want to catch up. Motivation and incentive is a powerful thing.

Indeed, being passionate about something makes you much more likely to remember it. Interestingly, the passion does not need to be a loving one.

A common pitfall when learning math is thinking it is like learning history, philosophy, or languages, where it doesn't matter if you miss out a bit; you will still understand everything later, and the missing bits will fall into place eventually. Math is nothing like that. Math is like building a house. A first step for you should therefore be to identify how much of the foundation of math you have, to know where to start from.

Khan Academy is a good resource for this, as it has a good overview of math, and how the different topics in math relate (what requires understanding of what). Khan Academy also has good exercises to solve, and ways to get help. There are also many great books on mathematics, and going through a book cover-to-cover is a satisfying experience. I have heard people speak highly of Serge Lang's "Basic Mathematics".

Finding sparetime activities to train your analytic and critical thinking skills will also help you immeasurably. Here I recommend puzzle books, puzzle games (I recommend Portal, Lolo, Lemmings, and The Incredible Machine), board/card games (try Eclipse, MtG, and Go), and programming (Scheme or Haskell).

It takes effort. But I think you will find your journey through maths to be a truly rewarding experience.

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.

You should absolutely not give up.

• Axler is fairly advanced for a freshman course in linear algebra. The fact that it's making more sense the second time you go over it is much more important than failing to understand it the first time.
  
• Nobody can learn sophisticated math from a lecture if they haven't seen it before. Well, maybe geniuses can, but my guess is that the majority of successful mathematicians reach a point where the lecture medium becomes much less important. You have to read the textbook with a pencil in hand, proving lemmas yourself. Digest proofs at your own pace, there's nothing wrong or unusual with not understanding it the way your Professor presented it.
  
• About talking math with people - this just takes time. Hold off on judging yourself. You can also get practice by getting involved with math subreddits or math.stackexchange.
  
• It's pretty unlikely that you are "too stupid" to study math. I've seen people with a variety of natural ability learn a tremendous amount about math and related disciplines, just by working hard.
  
  None of this is groundbreaking, and a lot of it is pretty cliché, but it's true. Everyone struggles with math at some point. Einstein said something like "whatever your struggles with math are, I assure you that mine are greater."
  
  As for specific recommendations, make the most of this summer. The most important factor in learning math in my experience is "time spent actively doing math." My favorite math quote is "you don't learn math, you get used to it." I might recommend a book like How to Prove It. I read it the summer before I entered college, and it helped immensely with proofs in real analysis and abstract algebra. Give that a read, and I bet you will be able to prove most lemmas in undergraduate algebra and topology books, and solve many of their problems. Just keep at it!

Here is an actual blog post that conveys the width of the text box better. Here is a Tufte-inspired LaTeX package that is nice for writing papers and displaying side-notes; it is not necessary for now but will be useful later on. To use it, create a tex file and type the following:

\documentclass{article}
\usepackage{tufte-latex}

\begin{document}
blah blah blah
\end{document}

But don't worry about it too much; for now, just look at the Sample handout to get a sense for what good design looks like.

I mention AoPS because they have good problem-solving books and will deepen your understanding of the material, plus there is an emphasis on proof-writing when solving USA(J)MO and harder problems. Their community and resources tabs have many useful things, including a LaTeX tutorial.

Free intro to proofs books/course notes are a google search away and videos on youtube/etc too. You can also get a free library membership as a community member at a nearby university to check out books. Consider Aluffi's notes, Chartrand, Smith et al, etc.

You can also look into Analysis with intro to proof, a student-friendly approach to abstract algebra, an illustrated theory of numbers, visual group theory, and visual complex analysis to get some motivation. It is difficult to learn math on your own, but it is fulfilling once you get it. Read a proof, try to break it down into your own words, then connect it with what you already know.

Feel free to PM me v2 of your proof :)

Hey! I am a math major at Harvey Mudd College (who went to high school in the Pacific NW!). I'll answer from what I've seen.

1. There seems to be tons. At least I keep being told there are tons! My school has a lot of recruiters come by who are interested in math people!
   
   
2. I can definitely recommend HMC, but I would also consider MIT, Caltech, Carnegie Melon, etc. I've heard UW is good, too!
   
   
3. Most all of linear algebra is important later on. I will say that many texts treat linear algebra the same as "matrix algebra", which it is not. Linear algebra is much more general, and deals with things called vector spaces. Matrix algebra is a specific case of linear algebra. If you want a good linear algebra text (though it might be a bit difficult), check out http://www.amazon.com/Linear-Algebra-Right-Sheldon-Axler/dp/0387982582
   
   End: Also, if you wanna learn something cool, I'd check out Discrete math. It's usually required for both a math or CS major, and it's some of the coolest undergraduate math out there. Oh, and, unlike some other math, it's not terrible to self-teach. :)
   
   Good luck! Math is awesome!

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.

I highly recommend "Journey Through Genius" by William Dunham for people with an interest in math, but maybe with not much background yet.

Each chapter talks about one of the great theorems in math, starting with the ancient Greeks and ending with Cantor. The chapter explains some history behind the problem, and provides motivation for why the question is interesting. Then it actually presents a proof. It's a great way of getting exposure to new ideas, proofs, and is a nice survey of a wide range of math. Plus, it's well-written!

Personally, I don't think learning something like, say, category theory makes sense unless you've had some more higher math that will provide examples of where category theory is useful. I love abstraction as much as the next mathematician, but I've learned that it's usually useless unless you have a set of examples that help you understand the abstraction.

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.

Intoduction to Langlands Program you can find downloadable pdfs online. This is a fair overview of everything, not many rigorous proofs, but with something as ambiguous as Langlands the general idea is what is important.

To be able to tackle it, you're going to need to know pretty much all math up to second year graduate courses, especially Measure Theory, Representation Theory, Group Cohomology and Harmonic Analysis. You will then need, in addition to that, Class Field Theory and Tate's Thesis. These can both be found in Cassells and Frolich, though a more readable Tate's Thesis is also in the first book. This is just the standard reference book for Class Field Theory, I would recommend looking at Primes of the Form x^2 +ny^2 first. It is written at the advanced undergrad/beginning graduate level, but it is a very excellent read (probably my favorite book) and gives clear motivation for Class Field Theory which is where other books usually fail. It's good to have other sources too, like Neukirch or Lang.

TL;DR You need all of math to talk about Langlands.

I may be in the minority here, but I think that high school students should be exposed to statistics and probability. I don't think that it would be possible to exposed them to full mathematical statistics (like the CLT, regression, multivariate etc) but they should have a basic understanding of descriptive statistics. I would emphasize things like the normal distribution, random variables, chance, averages and standard deviations. This could improve numerical literacy, and help people evaluate news reports and polls critically. It could also cut down on some issues like the gambler's fallacy, or causation vs correlation.

It would be nice if we could teach everyone mathematical statistics, the CLT, and programming in R. But for the majority of the population a basic understanding of the key concepts would be an improvement, and would be useful.

Edit At the other end of the spectrum, I would like to see more access to an elective class that covers the basics of mathematical thinking. I would target this at upperclassmen who are sincerely interested in mathematics, and feel that the standard trig-precalculus-calculus is not enough. It would be based off of a freshman math course at my university, that strives to teach the basics of proofs and mathematical thinking using examples from different fields of math, but mostly set theory and discrete math. Maybe use Velleman's book or something similar as a text.

Depends what kind of math you're interested in. If you're looking for an introduction to higher (college) math, then How to Prove It is probably your best bet. It generally goes over how proofs work, different ways of proving stuff, and then some.

If you already know about proofs (i.e. you are comfortable with at least direct proofs, induction, and contradiction) then the world is kind of your oyster. Almost anything you pick up is at least accessible. I don't really know what to recommend in this case since it's highly dependent on what you like.

If you don't really know the basics about proofs and don't care enough to yet, then anything by Dover is around your speed. My favorites are Excursions in Number Theory and Excursions in Geometry. Those two books use pretty simple high school math to give a relatively broad look at each of those fields (both are very interesting, but the number theory one is much easier to understand).

If you're looking for high school math, then /u/ben1996123 is probably right that /r/learnmath is best for that.

If you want more specific suggestions, tell me what you have enjoyed learning about the most and I'd be happy to oblige.

I 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.

"A Book of Abstract Algebra" by Charles C. Pinter is nice, from what I've seen of it--which is about the first third. I'm going through it in an attempt to relearn the abstract algebra I've forgotten.

I was using Herstein (which was what I learned from the first time), and was doing fine, but saw the Pinter book at Barnes & Noble. I've found it is often helpful when relearning a subject to use a different book from the original, just to get a different approach, so gave it a try (it's a Dover, so was only ten bucks).

What is nice about the Pinter book is that it goes at a pretty relaxed pace, with a good variety of examples. A lot of the exercises apply abstract algebra to interesting things, like error correcting codes, and some of these things are developed over the exercises in several chapters.

You don't have to be a prodigy to be able to understand some real mathematics in middle school or early high school. By 9th grade, after a summer of reading calculus books from the local public library, I was able to follow things like Niven's proof that pi is irrational, for instance, and I was nowhere near a prodigy.

Best thing you can do is read your little heart out. Find a copy (electronic or library) of something like the Princeton Companion and browse it over the course of a few weeks/months and pick out a few fields that particularly interest you. Then hit up How To Become A Pure Mathematician and start on your reading.

Eventually - and by this I mean "in a year or two" - you want to be able to email the prof in your dept who shares your interest and say "Hey, I've read the foundational texts on xyz, what would you recommend next?" and from there develop a relationship that'll hopefully lead to some undergrad "research" and a glowing letter of recommendation in your final year.

The other, equally important thing is to be a likable, sociable person. Unless you're some kind of wunderkind, collaboration is the name of the game and it gives you a huge advantage over the smelly nerd that no-one really wants around.

e: also lol undergrad pure maths research hahahahaha. if you can read a contemporary research paper in most pure maths subfields by senior year, you're ahead of the game.

Hi there,

For all intents and purposes, for someone your level the following will be enough material to stick your teeth into for a while.

Mathematics: Its Content, Methods and Meaning https://www.amazon.com/Mathematics-Content-Methods-Meaning-Volumes/dp/0486409163

This is a monster book written by Kolmogorov, a famous probabilist and educator in maths. It will take you from very basic maths all the way to Topology, Analysis and Group Theory. It is however intended as an overview rather than an exhaustive textbook on all of the theorems, proofs and definitions you need to get to higher math.

For relearning foundations so that they're super strong I can only recommend:

Engineering Mathematics
https://www.amazon.co.uk/Engineering-Mathematics-K-Stroud/dp/1403942463

Engineering Mathematics is full of problems and each one is explained in detail. For getting your foundational, mechanical tools perfect, I'd recommend doing every problem in this book.

For low level problem solving I'd recommend going through the ENTIRE Art of Problem Solving curriculum (starting from Prealgebra).
https://www.artofproblemsolving.com/store/list/aops-curriculum

You might learn a thing or two about thinking about mathematical objects in new ways (as an example. When Prealgebra teaches you to think about inverses it forces you to consider 1/x as an object in its own right rather than 1 divided by x and to prove things. Same thing with -x. This was eye opening for me when I was making the transition from mechanical to more proof based maths.)


If you just want to know about what's going on in higher math then you can make do with:
The Princeton Companion to Mathematics
https://www.amazon.co.uk/Princeton-Companion-Mathematics-Timothy-Gowers/dp/0691118809

I've never read it but as far as I understand it's a wonderful book that cherry picks the coolest ideas from higher maths and presents them in a readable form. May require some base level of math to understand

EDIT: Further down the Napkin Project by Evan Chen was recommended by /u/banksyb00mb00m (http://www.mit.edu/~evanchen/napkin.html) which I think is awesome (it is an introduction to lots of areas of advanced maths for International Mathematics Olympiad competitors or just High School kids that are really interested in maths) but should really be approached post getting a strong foundation.

Like justrasputin says, there usually is quite a lot of work to be done before you start to really see the beauty everyone refers to. I'd like to suggest a few book about mathematics, written by mathematicians that explicitly try to capture the beauty -

By Marcus Du Sautoy (A group theorist at oxford)

1. Symmetry
   
2. The Music of the Primes
   
   By G.H. Hardy,
   
3. A Mathematician's Apology
   
   Also, a good collection of seminal works -
   God Created the Integers
   
   And a nice starter -
   What is Mathematics
   
   Good luck and don't give up!

I had a similar request to yours, except I wanted to go beyond Calculus to get a broad survey of mathematical topics, using a ground up approach. The Princeton Companion to Mathematics is exceptional, I can't recommend it enough! It covers all the topics you wish your mathematics teachers had instilled in you, all within a comprehensive & comprehensible form. It has been years since I studied math. I've long since forgotten a majority of what I was taught but, I can still easily progress in this book and I feel like I finally understand many of the ideas that were impenetrable before.

I'm not alone in my positive review. You'll note that people have been heaping praise onto this volume on Amazon and in more formal book reviews as well.

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.

I learned a lot from getting a copy of Rudin (however, this book is very challenging and probably not the best to self study from. I was able to get to about continuity before taking my analysis course and it was challenging, but worth while). You can probably find it online somewhere for free.

A teacher lent Introduction to Analysis to me and suggested I use it instead of the book by Rudin. It was a well written book and had exercises which were much more approachable (although it included very difficult ones as well). The layout of this book (and I'd bet many others) is quite similar to that of Rudin. It was nice to be able to read them together.

For linear algebra, I can't speak to the quality of many books, but there are plenty which can fairly easily be found online. You will likely be recommended Linear Algebra Done Right however I found it a bit challenging as a first introduction to linear algebra and never got quite far.

My university course used Larson, Falvo Linear Algebra and it was enjoyable and helps you learn the computations very well and gives a decent understanding of proofs.

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.

You would love Godel Escher Bach by Douglas R Hofstadter. It won the pullitzer prize and is basically just a really good popular math/computer science/art book. But a really excellent jumping off point. Yes it lacks mathematical rigor (of course) but if you are a bright clever person who likes these things, its a must read just for exposure to the inter-connectivity of all of these topics in a very artistic and philosophical way. But be prepared for computer code, musical staff notation, DNA sequences, paintings, and poetry (all themed around Godel, Escher and Bach).

I recommend starting by pointing people at Nathan Carter’s book Visual Group Theory (site, amzn), which does a great job providing a number of examples of interesting groups, and motivating understanding of quotient groups etc.

If they start with a stable of nice nice concrete group theory examples, then a lot of the drier and more abstract parts of a “definition, theorem, proof, corollary, definition, ...” style presentation afterward will be much less cryptic.

The basic examples for ring theory / field theory are a bit more familiar to most students, so I expect the rest of abstract algebra usually turns out okay, though throwing in more concrete weird examples of rings and fields would be nice as well.

It would be interesting to start an introductory field theory course with a detailed concrete study of a few specific finite fields, and e.g. vector spaces and linear transformations and polynomials etc. worked out for those fields. It would give a very different flavor to the course.

Traditionally, a mathematical proof has one and only one job: convince other people that your proof is correct. (In this day and age, there is such a thing as a computer proof, but if you don't understand traditional proofs, you can't handle computer proofs either.)

Notice what I just said: "convince other people that your proof is correct." A proof is, in some sense, always an interactive undertaking, even if the interaction takes place across gulfs of space and time.

Because interaction is so central to the notion of a proof, it is rare for students to successfully self-study how to write proofs. That seems like what you're asking. Don't get me wrong. Self-study helps. But it is not the only thing you need. You need, at some point, to go through the process of presenting your proofs to others, answering questions about your proof, adjusting your proof to take into account new feedback, and using this experience to anticipate likely issues in future proofs.

What you're proposing to do, in most cases, is the wrong strategy. You need more interactive experience, not less. You should be beating down the doors of your professor or TA in your class during their office hours, asking for feedback on your proofs. (This implies that you should be preparing your proofs in advance for them to read before going to their office hours.) If your school has a tutorial center, that's a wonderful resource as well. A math tutor who knows math proofs is a viable source of help, but if you don't know how to do proofs, it's hard for you to judge whether or not your tutor knows how to do proofs.

If you do self-study anything, you should not be self-studying calculus, linear algebra, real analysis, or abstract algebra. You should be self-studying how to do proofs. Some people here say that How to Prove It is a useful resource. My own position is that while self-studying can be helpful, it needs to be balanced with some amount of external interactive feedback in order to really stick.

Well, Hardy & Wright is the classic book for elementary stuff. It has almost everything there is to know. There is also a nice book by Melvyn Nathanson called Elementary Methods in Number Theory which I really like and would probably be my first recommendation. Beyond that, you need to decide which flavour you like. Algebraic and analytic are the big branches.

For algebraic number theory you'll need a solid grounding in commutative algebra and Galois theory - say at the level of Dummit and Foote. Lang's book is pretty classic, but maybe a tough first read. I might try Number Fields by Marcus.

For analytic number theory, I think Davenport is the best option, although Montgomery and Vaughan is also popular.

Finally, Serre (who is often deemed the best math author ever) has the classic Course in Arithmetic which contains a bit of everything.

This may not exactly be an answer to your question but I would recommend buying this book: https://www.amazon.com/Mathematics-Elementary-Approach-Ideas-Methods/dp/0195105192

It's not quite a textbook nor it is a pop-sci book for the layperson. The blurb on the front says " "A lucid representation of the fundamental concepts and methods of the whole field of mathematics." - Albert Einstein"

In and of itself it is not a complete curriculum. It doesn't have anything about linear algebra for example but you could learn a lot of mathematics from it. It would be accessible to a reasonably intelligent and interested high-schooler, it touches on a variety of topics you may see in an undergraduate mathematics degree and it is a great introduction to thinking about mathematics in a slightly more creative and rigorous way. In fact I would say this book changed my life and I don't think I'm the only one. I'm not sure if i would be pursuing a degree in math if I had never encountered it. Also it's pretty cheap.

If you're still getting a handle on how to manipulate fractions and stuff like that you might not be ready for it but you will be soon enough.

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.

For real analysis I really enjoyed Understanding Analysis for how clear the material was presented for a first course. For abstract algebra I found A book of abstract algebra to be very concise and easy to read for a first course. Those two textbooks were a lifesaver for me since I had a hard time with those two courses using the notes and textbook for the class. We were taught out of rudin and dummit and foote as mainly a reference book and had to rely on notes primarily but those two texts were incredibly helpful to understand the material.

​

If any undergrads are struggling with those two courses I would highly recommend you check out those two textbooks. They are by far the easiest introduction to those two fields I have found. I also like that you can find solutions to all the exercises so it makes them very valuable for self study also. Both books also have a reasonable amount of excises so that you can in theory do nearly every problem in the book which is also nice compared to standard texts with way too many exercises to realistically go through.

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.

You might want to check out Stein and Shakarchi's book Complex Analysis http://press.princeton.edu/titles/7563.html. This book is a bit hard but iirc doesn't require you to have had real analysis before hand. I would highly recommend that you work through a proof based book before hand though. Often times this will be a course book but something like https://www.amazon.com/How-Prove-Structured-Approach-2nd/dp/0521675995?ie=UTF8&*Version*=1&*entries*=0 that should also get the job done.

Or you can go the traditional route like other people mentioned of getting about a semester's worth of real analysis under your belt. The reason why this is usually the suggested path is because it's not expected that you are 100% competent at writing proofs in the beginning of real but you are in complex.

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!

Linear algebra is about is about linear functions and is typically taken in the first or second year of college. College algebra normally refers to a remedial class that covers what most people do in high school. I highly recommend watching this series of videos for getting an intuitive idea of linear algebra no matter what book you go with. The book you should use depends on how comfortable you are with proofs and what your goal is. If you just want to know how to calculate and apply it, I've heard Strang's book with the accompanying MIT opencourseware course is good. This book also looks good if you're mostly interested in programming applications. A more abstract book like Linear Algebra Done Right or Linear Algebra Done Wrong would probably be more useful if you were familiar with mathematical proofs beforehand. How to Prove it is a good choice for learning this.

I haven't seen boolean algebra used to refer to an entire course, but if you want to learn logic and some proof techniques you could look at How to Prove it.

Most calculus books cover both differential and integral calculus. Differential calculus refers to taking derivatives. A derivative essentially tells you how rapidly a function changes at a certain point. Integral calculus covers finding areas under curves(aka definite integrals) and their relationship with derivatives. This series gives some excellent explanations for most of the ideas in calculus.

Analysis is more advanced, and is typically only done by math majors. You can think of it as calculus with complete proofs for everything and more abstraction. I would not recommend trying to learn this without having a strong understanding of calculus first. Spivak's Calculus is a good compromise between full on analysis and a standard calculus class. It's possible to use this as a first exposure to calculus, but it would be difficult.

There's a lot of ground to cover in math, but completely doable. I'm going to recommend a dense book, but I truly think it's worth the read.

Let me leave you with this. You understand how number work correct? 1 + 1 = 2. It's a matter of fact. It's not up for debate and to question it would see you insane.

This is all of math. You need to truly understand

1 + 1 = 2

a + a = b everything is a function. There are laws to everything, even if people wish to deny it. If we don't understand it, it's easier to state that there are no laws that govern it, but there are. You just don't know them yet. Math isn't overwhelming when you think of it that way, at least to me. It's whole.

Ask yourself, 'why does 1 + 1 = 2 ?' If you were given 1 + x = 2, how would you solve it? Why exactly would you solve it that way? What governing set of rules are you using to solve the equation? You don't need to memorize the names of the rules, but how to use them. Understand the terminology in math, or any language, and it's easier to grasp that language.

The book Mathematics

If desired, it is possible to make an elementary argument that (1+x/n)^n converges, for each x, to a function e(x) satisfying e(x)e(y) = e(x+y), using just inequalities to show convergence of the needed limits. This is outlined, for example, in the chapter on the AM-GM inequality in this book: https://www.amazon.com/Inequalities-Journey-into-Linear-Analysis/dp/0521876249

There's also an exercise in the first chapter of Baby Rudin outlining how to define exponentials using least upper bounds and monotonicity properties:
https://www.amazon.com/Principles-Mathematical-Analysis-International-Mathematics/dp/007054235X

Honestly though, while in general I support showing students the details, this is a case where I think that, pedagogically, it's right to pull the wool over students' eyes until the time is right. It's so much more elegant to define the exponential function as the solution of a differential equation, or as the sum of a power series, or as the inverse of the logarithm (defined as an integral), that one should simply put off a fully rigorous definition until it can be given in one of these forms.

The reasoning in doing so is not circular: The basic properties of integrals, power series, and solutions of differential equations are established through abstract theorems, and then one can use these tools to define the exponential and logarithmic functions and derive their properties. (See https://proofwiki.org/wiki/Definition:Exponential and https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Complex_Exponential)

Until then about all that needs to be mentioned is that a^m is a product of m copies of a, a^1/n is the nth root, a^m/n = (a^(1/n))^(m), and that this extends in a natural way to irrational exponents; as well as compound interest and the fact that (1+x/n)^n converges to a power of a special number e approx 2.718281827459, which is the "natural base" of the logarithm for reasons to be explained later.

>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!

I like Szekeres's A Course in Modern Mathematical Physics for referencing intro-grad-level material. It covers abstract linear algebra, differential geometry, measure theory, functional analysis, and Lie algebras, and teaches you some physics along the way.

More generally, the best "breadth" book on advanced mathematics is Princeton Companion to Mathematics by Gowers et al. and its slightly underachieving younger brother of a companion text, Princeton Companion to Applied Mathematics by Higham et al.. You won't properly learn advanced mathematics this way, but you'll get the bird's-eye view of modern research programs and the math underlying them.

If you want a more algebraic take on Szekeres's program to teach physicists all the math they need to know, check out Evan Chen's Napkin project, which is intended to introduce advanced undergrads (it's perfectly fine for grad students too) to a wide variety of advanced mathematics on the algebra side of things.

Since you're doing probability and statistics, check out Wasserman's All of Statistics and Knill's Probability Theory and Stochastic Processes for good, concise references for intro-grad-level material.

I will second what /u/Ovationification said, though. I didn't really learn anything with the above books, I just use them occasionally for reference or to think about pedagogy.

Intro Calculus, in American sense, could as well be renamed "Physics 101" or some such since it's not a very mathematical course. Since Intro Calculus won't teach you how to think you're gonna need a book like How to Solve Word Problems in Calculus by Eugene Don and Benay Don pretty soon.

Aside from that, try these:

Excursions In Calculus by Robert Young.

Calculus:A Liberal Art by William McGowen Priestley.

Calculus for the Ambitious by T. W. KORNER.

Calculus: Concepts and Methods by Ken Binmore and Joan Davies

You can also start with "Calculus proper" = Analysis. The Bible of not-quite-analysis is:

[Calculus by Michael Spivak] (http://www.amazon.com/Calculus-4th-Michael-Spivak/dp/0914098918/ref=sr_1_1?s=books&ie=UTF8&qid=1413311074&sr=1-1&keywords=spivak+calculus).

Also, Analysis is all about inequalities as opposed to Algebra(identities), so you want to be familiar with them:

Introduction to Inequalities by Edwin F. Beckenbach, R. Bellman.

Analytic Inequalities by Nicholas D. Kazarinoff.

As for Linear Algebra, this subject is all over the place. There is about a million books of all levels written every year on this subject, many of which is trash.

My plan would go like this:

1. Learn the geometry of LA and how to prove things in LA:

Linear Algebra Through Geometry by Thomas Banchoff and John Wermer.

Linear Algebra, Third Edition: Algorithms, Applications, and Techniques
by Richard Bronson and Gabriel B. Costa.

2. Getting a bit more sophisticated:

Linear Algebra Done Right by Sheldon Axler.

Linear Algebra: An Introduction to Abstract Mathematics by Robert J. Valenza.

Linear Algebra Done Wrong by Sergei Treil.

3. Turn into the LinAl's 1% :)

Advanced Linear Algebra by Steven Roman.

Good Luck.

Hey mathit.

I'm 32, and just finished a 3 year full-time adult education school here in Germany to get the Abitur (SAT-level education) which allows me to study. I'm collecting my graduation certificate tomorrow, woooo!

Now, I'm going to study math in october and wanted to know what kind of extra prep you might recommend.

I'm currently reading How to Prove It and The Haskell Road to Logic, Maths and Programming.
Both overlap quite a bit, I think, only that the latter is more focused on executing proofs on a computer.

Now, I've just been looking into books that might ease the switch to uni-level math besides the 2 already mentioned and the most promising I found are these two:
How to Study for a Mathematics Degree and Bridging the Gap to University Mathematics.

Do you agree with my choices? What else do you recommend?

I found online courses to be ineffective, I prefer books.

What's your opinion, mathit?

Cheers and many thanks in advance!

I am surprised no one has mentioned M. Spivak's very well known text Calculus. I thought this book was a pleasure to read. His writing was very fun and lighthearted and the book certainly teaches the material very well. In my opinion this is the best introductory calculus text there is.

I am a master's student with interests in algebraic geometry and number theory. And I have a good collection of textbooks on various topics in these two fields. Also, as part of my undergraduate curriculum, I learnt abstract algebra from the books by Dummit-Foote, Hoffman-Kunze, Atiyah-MacDonald and James-Liebeck; analysis from the books by Bartle-Sherbert, Simmons, Conway, Bollobás and Stein-Shakarchi; topology from the books by Munkres and Hatcher; and discrete mathematics from the books by Brualdi and Clark-Holton. I also had basic courses in differential geometry and multivariable calculus but no particular textbook was followed. (Please note that none of the above-mentioned textbooks was read from cover to cover).

As you can see, I didn't learn much geometry during my past 4 years of undergraduate mathematics. In high school, I learnt a good amount of Euclidean geometry but after coming to university geometry appears very mystical to me. I keep hearing terms like hyperbolic/spherical geometry, projective geometry, differential geometry, Riemannian manifold etc. and have read general maths books on them, like the books by Hartshorne, Ueno-Shiga-Morita-Sunada and Thorpe.

I will be grateful if you could suggest a series of books on geometry (like Stein-Shakarchi's Princeton Lectures in Analysis) or a book discussing various flavours of geometry (like Dummit-Foote for algbera). I am aware that Coxeter has written a series of textbooks in geometry, and I have read Geometry Revisited in high school (which I enjoyed). If these are the ideal textbooks, then where to start? Also, what about the geometry books by Hilbert?

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.

I am currently reading a fantastic book which might be interesting for you. It is called Journey through Genius. The book starts from the beginning of math and presents hand picked theorems in a very engaging way. Background information on the great mathematicians and what drove them to come up with these proofs in the first place makes the information stick long after reading. I also second PuTongHua who recommended Better Explained.

You've been posting lots of vague and confused questions about sequences, derivatives, and cardinality to /r/math. You also have a habit of inventing our own terminology without motivating it or even acting as if folks should naturally understand it.

• https://www.reddit.com/r/math/comments/3yfkns/analysis_of_sequencesseries/
  
• https://www.reddit.com/r/math/comments/3ytox0/cardinality_and_fourier_series/
  
• https://www.reddit.com/r/math/comments/3yuf3h/understanding_taylor_expansion/
  
• https://www.reddit.com/r/math/comments/3z5j6r/recurring_sequence/
  
• https://www.reddit.com/r/math/comments/408sx6/going_from_countable_to_uncountable/
  
  It seems clear from these threads (including this one) that you're confused about some fundamental ideas surrounding sequences, cardinality, (un)countability, and differentiation. You're getting so-so responses because folks have to nail down what exactly you're asking/thinking before they can attempt to answer and that requires a ton of effort.
  
  If you really want to learn this stuff, I'm of the mind that you need to spend less time posting ill-formed questions on /r/math and instead make sure you understand the fundamentals. For example, here some of the first things one typically learns when studying countability:
  
  

1. The union of any finite number of countable sets is itself countable
   
2. The union of a countable number of finite sets is itself countable
   
3. The union of a countable number of countable sets is itself countable
   
   Obviously (3) implies the first two, but each is progressively more difficult to prove for someone approaching these ideas for the first time. The latter two require some version of the axiom of countable choice, for example, which isn't something most newcomers would think to deploy unless they had encountered it before.
   
   They do, however, answer your question: if we have a countable set and "glue on" a countable number of countable collections of new numbers, the resulting set will still be countable.
   
   I strongly recommend you buy and read Daniel Velleman's How to Prove It. It will help you organize your thoughts better and help you get comfortable with the "standard" mathematical terminology and notation. Topic-wise it covers basic set theory and the last chapter is all about infinite sets, cardinality, (un)countability, and so on.
   
   Here are some screenshots from Amazon's "Search Inside the Book" to show you what you can expect by the end of the book:
   
   

• http://cl.ly/370g072S2a3i
  
• http://cl.ly/013u1w0g0H39

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 books that will help you appreciate math, I recommend Journey Through Genius by William Dunham for a general historical approach, and Love and Math by Edward Frenkel and Prime Obsession by John Derbyshire for specific focuses in "modern" mathematics (in these cases, the Langlands program and the Riemann Hypothesis).

There's a lot of mathematical lore that you'll find really interesting the first time you read it, but then it becomes more and more grating each subsequent time you come across it. (The example that springs most readily to mind is how the Pythagorean theorem rocked the Greeks' socks about their belief in numbers and what the brotherhood supposedly did to the guy who proved that irrational numbers exist). For that reason, I recommend reading only one or two books that summarize the historical developments in math up to the present, and then finding books that focus on one mathematician or one theorem that is relatively modern. In addition to the books I mentioned above, there are also some good ones on the Poincare Conjecture and Fermat's Last Theorem, and given that you're a computer science guy, I'm sure you can find a good one about P = NP.

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.

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.

One of the most fun things I did when I was first learning about proofs was proving the basic facts about algebra from axioms. Where I first read about these ideas was the first chapter of Spivak's Calculus. This would be a very high level book for an 18 year old, but if you decide to look at it, don't be afraid to take your time a little.

Another option is just picking up an introduction to proof, like Velleman's How to Prove It. This wil lteach you the basics for proving anything, really, and is a great start if you want to do more math.

If you want a free alternative to that last one, you can look at The Book of Proof by Richard Hammack. It's well-written although I think it's shorter than How to Prove It.

The Mis-Education of Mathematics Teachers made a huge impression on me, in particular its emphasis on content knowledge and the fundamental principles of mathematics. More recently, the following comment by Ian Stewart has persuaded me to put more emphasis on the visual aspects of the subjects I teach:

> One of the saddest developments in school mathematics has been the downgrading of the visual for the formal. I'm not lamenting the loss of traditional Euclidean geometry, despite its virtues, because it too emphasised stilted formalities. But to replace our rich visual tradition by silly games with 2x2 matrices has always seemed to me to be the height of folly. It is therefore a special pleasure to see Tristan Needham's Visual Complex Analysis with its elegantly illustrated visual approach. Yes, he has 2x2 matrices―but his are interesting. (Ian Stewart, New Scientist, 11 October 1997) (source)

I think everyone is on point for the most part, but I'd like to be the devil's advocate and suggest a different route.

Learn logic, proof techniques and set theory as early as possible. It will aid you in further study of all 'types' of math and broaden your mind in a general sense. This book is a perfect place to start.

http://www.amazon.com/How-Prove-It-Structured-Approach/dp/0521675995

The best part is, when you start doing proofs you realize you've been thinking about math all wrong (at least I did). It's an exercise in creativity, not calculation.

In my mind, set theory & calculus are necessary pre-requisites to probability anyway, and linear algebra means much more once you have been introduced to inductive proofs, as well.

I unfortunately don't know of a good generally accessible source on this. It's really one specific (and particularly nice looking) example of a more general theory. As such it's the sort of thing you usually learn as an example while you're learning about the more arithmetic side of modular forms, and I don't know of any good self contained sources for it.

What's your current background in math? If you haven't already taken algebraic number theory, you'd definitely want to start there (although you'd need abstract algebra, and especially Galois theory, as a prerequisite). The most important concept to pick up there would be the notion of a "Frobenius element".

If you're already familiar with algebraic number theory, then (as other people have said here), primes of the form x^(2)+ny^(2) by Cox is a good place to start, although if I remember correctly they don't actually do the n = 23 example there. That combined with a book on modular forms, such as Diamond and Shurman should give you a pretty good understanding of this.

You might want to try "What is Mathematics?" by R.Courant and H.Robbins. The book is written for people new to the field of theoretical mathematics and is intended for those who wish to develop a solid foundation on the topic.

I had started college as an engineer, switched to English, and now work as an ESL instructor. However, my love of math never died (despite my university professors' best attempts). So, I picked up that book a little while ago. It's a good read (albeit a dense one), and it covers a little bit of what you have listed.

[Amazon link here] (http://www.amazon.com/Mathematics-Elementary-Approach-Ideas-Methods/dp/0195105192)

Edit: some words

Foolproof is a good example of this. Lots of self-contained chapters on random fun problems. (My only large critique is that the first chapter is very out of place; being basically a history schpiel. Mischaracterizes the book.)

Then there’s math adjacent stuff like Zero: the history of a dangerous idea that look at the history of math development.
(Side note: the first chapter of Pinter’s A Book of Abstract Algebra is a top knotch example of that. And very much in place, unlike the foolproof chapter I mentioned.

Then there are things that aren’t quite “pop”, but make themselves more accessible. Like An Illustrated Guide to Number Theory, which is both a legitimate intro to number theory and a reasonably sexy coffee table book that guests can leaf through. (Though I’d like to see a book that pushes the coffee table style accessibility further.)

This is one of the best books of abstract algebra I've seen, very well explained, favoring clear explanations over rigor, highly recommended (take your time to read the reviews, the awesomeness of this book is real :P): http://www.amazon.com/Book-Abstract-Algebra-Edition-Mathematics/dp/0486474178/ref=sr_1_6?ie=UTF8&qid=1345229432&sr=8-6&keywords=introduction+to+abstract+algebra

On a side note, trust me, Dummit or Fraileigh are not what you want.

I like Journey Through Genius. It is completely elementary, requiring nothing beyond perhaps a semester of basic algebra. It presents some amazing theorems and emphasizes both the creativity and the logical rigor required to achieve them. I can't remember every theorem, but I know it includes Pythagorus' Theorem, the irrationality of the square root of two, Euclids geometry, the infinitude of the primes, some number theory of Fermat, Isaac Newton on the Binomial Theorem, the quadratic equation and the solution of the third and fourth degree polynomials by radicals and why this requires complex numbers, an exploration of complex numbers, and some non-Euclidean geometry. All that whilst requiring, as I said, no mathematical maturity whatsoever, and being quite easy and enjoyable to read. I highly recommend it.

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...

> I'm not looking for a book to help me become a set theory pro, I'm literally just looking for a book that will give me some challenging, enjoyable bedtime reading.

Are you sure that you want to read a book on axiomatic set theory or are you happy with any math subject and it's just that set theory is the only one that comes to your mind?

In the latter case, I would recommend Mathematics and its history by John Stillwell for bedtime reading (and it does have a bit of set theory, too). Also, the The Princeton Companion to Mathematics is highly recommended.

And in any case, the mathematics section of your local library provides more low cost bedtime reading than I could ever note here. :-)

Proofs: Hammack's Book of Proof. Free and contains solutions to odd-numbered problems. Covers basic logic, set theory, combinatorics, and proof techniques. I think the third edition is perfect for someone who is familiar with calculus because it covers proofs in calculus (and analysis).

Calculus: Spivak's Calculus. A difficult but rewarding book on calculus that also introduces analysis. Good problems, and a solution manual is available. Another option is Apostol's Calculus which also covers linear algebra. Knowledge of proofs is recommended.

Number Theory: Hardy and Wright's An Introduction to the Theory of Numbers. As he explains in a foreword to the sixth edition, Andrew Wiles received this book from his teacher in high school and was a starting point for him. It also covers the zeta function. However, it may be too difficult for absolute beginners as it doesn't contain any problems. Another book is Stark's An Introduction to Number Theory which has a great section on continued fractions. You should have familiarity with proof before learning number theory.

Journey Through Genius: Exploring the Great Theorems of Mathematics. - William Dunham

Journey through Genius: The Great Theorems of Mathematics https://www.amazon.com/dp/014014739X/ref=cm_sw_r_cp_api_i_uOd3CbD8DH8CN

A great read that does walkthroughs of proofs and breakthroughs. Highly recommended.

Congratulations are in order, to you as well as lysa_m, shizzy0 and all the other helpful redditors here. It must feel really great to get over this hurdle!

I just wanted to add a link to the book of Tristram Needham, Visual Complex Analysis. As lysa_m pointed out, you are not the first person in history to find "imaginary" numbers baffling. You can read the first 5 or 6 pages of Needham's book online at the Amazon page above. There he outlines the history of the subject and explains some of the same points made in the comments here.

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.

You're not really doing higher math right now as much as you're learning tricks to solve problems. Once you start proving stuff that'll be a big jump. Usually people start doing that around Real Analysis like your father said. Higher math classes almost entirely consist of proofs. It's a lot of fun once you get the hang of it, but if you've never done it much before it can be jarring to learn how. The goal is to develop mathematical maturity.

Start learning some geometry proofs or pick up a book called "Calculus" by Spivak if you want to start proving stuff now. The Spivak book will give you a massive head start if you read it before college. Differential equations will feel like a joke after this book. It's called calculus but it's really more like real analysis for beginners with a lot of the harder stuff cut out. If you can get through the first 8 chapters or so, which are the hardest ones, you'll understand a lot of mathematics much more deeply than you do now. I'd also look into a book called Linear Algebra done right. This one might be harder to jump into at first but it's overall easier than the other book.

If you like stuff like this you may be interested in my favorite book: Godel, Echer, Bach: The Eternal Golden Braid:

http://amzn.com/dp/0465026567

Edit: Also see the great MIT course with video lectures:

http://ocw.mit.edu/OcwWeb/hs/geb/geb/

If you're looking for other texts, I would suggest Spivak's Calculus and Calculus on Manifolds. At first the text may seem terse, and the exercises difficult, but it will give you a huge advantage for later (intermediate-advanced) undergraduate college math.

It may be a bit obtuse to recommend you start with these texts, so maybe your regular calculus texts, supplemented with linear algebra and differential equations, should be approached first. When you start taking analysis and beyond, though, these books are probably the best way to return to basics.

What Is Mathematics?: An Elementary Approach to Ideas and Methods

"Succeeds brilliantly in conveying the intellectual excitement of mathematical inquiry and in communicating the essential ideas and methods." Journal of Philosophy

https://www.amazon.ca/What-Mathematics-Elementary-Approach-Methods/dp/0195105192

Linear algebra is an essential tool in many areas of mathematics. Computations with matrices aren't always that important; far more important are the concepts of vector space and linear transformation. Pretty much any time you work with coordinates, dimension, changes of coordinates, vectors, linear relations, or anything like that, you're going to need some linear algebra.

If you're interested, I recommend taking a look at Axler's Linear Algebra Done Right. Axler has very clear exposition and proofs, and if you've only seen the computational aspect of linear algebra, it'll provide a different, more abstract and conceptual perspective.

You'll remember and forget formulae as you use them. It's the using them that makes things concrete in your head.

Once you're comfortable with algebra, trig. I'm assuming you've had geometry, since you were taking algebra 2; if not, geometry as well.

Once you're comfortable with those topics, you'll have enough of the basics to start branching out. Calculus is one obvious direction; a lot people have recommended Spivak's book for that. Introductory statistics is another (far too few people are even basically statistically literate.) Discrete math is yet another possibility. You can also start playing with "problem math", like the Green Book or Red Book. Algebraic structures is yet another possibility (I found Herstein's abstract algebra book pretty easy to read when we used it in school).

Edit: added Amazon links.

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.

there's a lot going on here, so i'll try to take it a few steps at a time.

> how many REAL operators do we have?

you might be careful about your language here, as the word "real" has implications in the world of mathematics to mean "takes values in the real numbers", i.e., is non-complex. also, "real" in the normal sense of real or fake doesn't have a lot of meaning in mathematics. a better question might be "how many unique operators do we have?", but even that isn't quite good enough. you need to define context. a blanket answer to your question is that there are uncountably infinite amount of operators in mathematics that take all kinds of forms: linear operators, functional operators, binary operators, etc.

> taking a number to the power of another is just defined in terms of multiplication

similar to /u/theowoll's response, how would you define 2^(4.18492) in terms of multiplication? i know you're basing this question off of the interesting fact that 2^1 = 2, 2^2 = 2 2, 2^3 = 2 2 * 2, etc. and similarly for other certain classes of numbers, but how do you multiply 2 by itself 4.18492 times? it gets even more tricky to think of exponents like this if the base and power are non-rational (4.18492=418492/100000 is rational). what about the power of e^X, where e is the normal exponential and X is a matrix? take a look at wikipedia's article on exponentiation to see what a can of worms this discussion opens.

> So am I just plain wrong about all this, or there is some truth to it?

although there is a lot of incorrect things in your description when you consider general classes of "things you can multiply and add", what you are sort of getting at is what the theory of abstract algebra covers. in such a theory, it explores what it means to add, multiply, have inverses, etc. for varying collections of things called groups, rings, fields, vector spaces, modules, etc. and the relationships and properties of such things. you might take a look at a book of abstract algebra by charles pinter. you should be able to follow it, as it is an excellent book.

If you really want to get a feel for what (pure) mathematics is and how it all interrelates, you should read the Princeton Companion to Mathematics. This book is kind of like wikipedia for pure math, though extremely well written (and by prominent mathematicians). I think the best audience for the book are undergraduate students in math, but anyone really interested in learning what "real" mathematics is about should enjoy it.

I haven't read the book you say you are working through, inducing, but I can tell you that based on the description on Amazon, it sounds like OP would be better served by picking up Velleman's How to Prove It which is, as far as I know, currently considered the gold standard for learning basic mathematical logic and proof techniques. It is also much, much cheaper.

Also, I can vouch for it. I used it in undergrad several years ago. It sounds like it does everything inducing's book does based on his/her post and the amazon description.

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!

I recommend this:

https://www.amazon.com/Mathematics-Content-Methods-Meaning-Volumes/dp/0486409163

Unlike most professional mathematical literature, it is aimed at novices and attempts to communicate ideas, not details. Unlike most popular treatments of mathematics, and in particular unlike the YouTubers you mention, it is written by expert mathematicians and is about advanced mathematical topics. I got a hardcover set from a used bookstore when I was young and enjoyed it very much. It's well worth your time.

Take my recommendation as a grain of salt as i didn't take my formal math education further than where you're currently at, but I felt the same way after similar classes learning the mechanics but not the motivations. Mathematics: Its Content, Methods and Meaning was recommended to me by a friend and I think it help fills the gaps in motivation and historical context/connecting different fields not covered in classes.

Since you hope to study mathematics more seriously, I would look into this book link.

It's an excellent book that treats high school/basic college mathematics in an "adult" way. By adult I mean in the way that mathematicians think about it.
(The fun thing about Lang is that you can read only his books and get pretty much a high school through advanced graduate education).

The problem you are having is that math education is shitty.

> What I want is to have a concrete understanding [...]

If you want to actually understand anything you learn in class, you'll have to seek it out yourself. Actual mathematics isn't taught until you get to college, and even then, only to students majoring in the subject.

"Why the fuck calculus works" typically goes under the name "analysis." You can look up a popular textbook, Baby Rudin, although I've never used it. I had this cheap-o Dover book. You can't beat it for $12. There's also this nice video series from Harvey Mudd.

The general pattern you see in actual, real mathematics isn't method-problem-problem-problem-problem, but rather definition-theorem-proof. The definitions tell you what you're working with. The theorems tell you what is true. The proofs give a strong technical reason to believe it.

> I know that to grasp mathematical concepts, it is advisable to do lots of problems from your textbook.

For some reason, schools are notorious for drilling exercises until you're just about to bleed from the fucking skull. Once you understand how an exercise is done, don't waste your time with another exercise of the same type. If you can correctly take the derivative of three different polynomials, then you probably understand it.

Just a heads up, analysis is built on the foundations of set theory and the real numbers. What you work with in high school are an intuitive notion of what a real number is. However, to do proper mathematics with them, it's better to have a proper understanding of how they are defined. Any good book on analysis will start off by giving a full, rigorous definition of what a real number is. This is typically done either in terms of cauchy sequences (sequences that seem like they deserve to converge), in terms of dedekind cuts (splitting the rational numbers up into two sets), or axiomatically (giving you a characterization involving least upper bounds of bounded sets). (No good mathematical book would ever talk about decimals. Decimals are a powerful tool, but pure mathematicians avoid them whenever possible).

Calculus and analysis can both be summed up shortly as "the cool things you can do with limits". Limits are the primary way we work with infinities in analysis. Their technical definition is often confusing the first time you see it, but the idea behind them is straightforward. Imagining a world where you can't measure things exactly, you have to rely on approximations. You want accuracy, though, and so you only have so much room for error. Suppose you want to make a measurement with a very small error. (We use ε for denoting the maximum allowable error). If the equipment you're using to make the measurement is calibrated well enough, then you can do this just fine. (The calibration of your machine is denoted δ, and so, these definitions commonly go by the name of "ε-δ definitions").

I would recommend the following two books:

1. "How to Prove It" by Daniel Velleman.
   
   
2. "Understanding Analysis" by Stephen Abbott.
   
   The first book introduces most of the topics in the book that you linked, and was what was used in my Foundations of Mathematics class (essentially the same thing as your class).
   
   Understanding Analysis, on the other hand, is probably the perfect book to follow up with, since it is such a well-motivated, yet rigorous book on the analysis of one real variable, that you may, in fact, become too accustomed to such lucid and entertaining prose for your own good.

I'm going to shamelessly plug this book which I consider to be one of my favorite books ever. For the price it is definitely worth keeping a copy and reading it on the side if you're learning abstract algebra for the first time and it reads like a novel. It's definitely a small treasure I feel I discovered.

I've never taught the course, but a couple of my colleagues are very fond of Linear Algebra Done Wrong and would willingly teach from it if (1) the title wouldn't immediately turn students off of it and (2) the school would be okay with sacrificing some income from students having to purchase a book.

If you're curious, the book title is a play on the title of another well-known linear algebra book.

Yeah. Part of me feels like I've just been lucky in finding easy problems that the "real" scientists in my field hadn't bothered to try yet.

I still don't really understand linear algebra or vector calculus, for instance. I have Linear Algebra Done Right, Div, Grad, Curl, and all that, and the Princeton Companion to Mathematics on my wish list, which may help.

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.

I think looking at the history of math is a great starting point. Where did all the ideas come from? How were they formed? Who were these people that came up with them? What inspired them?

A good read (I thought) on this subject was Journey through Genius:
http://www.amazon.com/Journey-through-Genius-Theorems-Mathematics/dp/014014739X

I've always liked Journey Through Genius. It's pretty small, ~280 pages of paperback novel size, but it covers a nice selection of mathematical history and thinking. It's not comprehensive, but it's a very good introduction to math history. It starts in 440 BCE (Hippocrates) and ends in 1891 CE (Cantor).

Paperback version is only $12: http://www.amazon.com/Journey-through-Genius-Theorems-Mathematics/dp/014014739X

My two cents

1. Maths is difficult. There isn't one of us who at some point has not struggled with it
   
2. Maths should be difficult. The moment you find it easy you are not pushing yourself!
   
   If you want to improve your skills you can do two things in the short term -- read and practice.
   
   I would recommend Basic Mathematics by Lang (it gets mentioned a lot around here). Or if you are interested in higher math look at How to Prove It by Velleman
   
   The great thing is that both include exercises.

If she's bright and interested enough you might want to consider getting her an entry level college calculus book such as Spivak's.

It won't pose a replacement to the technical approach of high school, but it will illuminate a lot.

I think this is a better approach than trying to tie connections between calculus and other areas of math, because calculus has an inherent beauty of its own which could be very compelling when taught with the right philosophical approach.

If you are getting your degree in math or computer science, you will probably have to take a course on "Discrete math" (or maybe an "introduction to proofs") in your first year or two (it should be by your 3rd semester). Unfortunately, this will probably be the first time you will take a course that is more about the why than the how. (On the bright side, almost everything after this will focus on why instead of how.) Depending on how linear algebra is taught at your university, and the order you take classes in, linear algebra may be also be such a class.

If your degree is anything else, you may have no formal requirement to learn the why.

For the math you are learning right now, analysis is the "why". I'm not sure of a good analysis book, but there are two calculus books which treat the subject more like a gentle introduction to analysis-- Apostol's and Spivak's. Your library might have a copy you can check out. If not, you can probably find pdfs (which are probably[?] legal) online.

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.)

If you're interested in doing mathematical biology later on I'd recommend keeping dynamical systems stuff fresh in your memory. Maybe read and do some exercises from Tenenbaum and Pollard once in a while? Also, looks like you haven't taken Linear Algebra yet, so maybe self-study from Linear Algebra Done Right by Axler.

The first step is to become comfortable with proofs. It is extremely different than the type of math you likely did in calculus, linear, and diffeq. There is very little "carry out this set of steps until you have computed the answer". This is not what proofs are like, and it is not what mathematics is really about. I've heard this is a very good book for learning about proofs and proving: http://www.amazon.com/How-Prove-It-Structured-Approach/dp/0521675995

After that, you can begin studying specific topics. I would recommend starting with abstract algebra or analysis. To do this, just get the textbooks. There are lots of resources to answer the question "which textbook should I use" (see sidebar).

Well for one, if you haven't, you should talk to the disability services people at your school. At the very least, you can probably get extra time for exams.

Look up the stuff you're doing on Khan Academy. I was a math hater for 30 years before realizing that I might actually like it if I tried to learn it. One day of Khan Academy and I was hooked. I even quit my job to go back to school. If you don't like his style, just browse YouTube and find someone whose style you do like. Khan focuses a lot on intuition which is what you need to solve exam problems that are a little different than your homework.

If you have specific questions about how things work, you have reddit. Check the sidebar for links to a couple different math subreddits. There are answers to all of your why questions and very few of them are difficult to understand, although you may have to learn a few other things first. I still shit myself when I read a confusing definition or theorem in my math classes, but they're never out of reach. Usually there are like 2 sentences of knowledge missing in my brain and it almost feels silly once I figure out what they are.

What you're doing now probably won't impact your life outside of letting you finish your degree, but the fun barrier is breakable. You just need to fill in those knowledge gaps so that you have the necessary tools to solve all of those little puzzles that you're given. That means asking specific questions about specific topics.

If you actually want to like math, check out this book: http://www.amazon.com/How-Prove-Structured-Approach-2nd/dp/0521675995/

That's what math is actually like and it's around your level.

Not quite encyclopaedic, but this gives a good overview of most topics you might encounter in an undergraduate course. The first section also gives a very good defense of the need for basic research into mathematics.

If there is something close to an Encyclopaedia Mathematica, but you can read it like a novel, it is these three volumes from Aleksandrov/Kolmogorov/Laurentiev. Amazon

Edit: Ahem, but after reading carefully post0, I would recommend you simply to begin with the textbooks of secondary school or so.

I'm going to recommend the book How to Prove It. Its all about learning the logic for proofs and strategies for writing proofs. Its one of those books that you work through slowly and complete all the exercises. Its recommended around here a-lot. I'd also suggest using the search feature if you ever want to look for other recommended books because those threads come up often around here.

Best wishes.