Reddit mentions of Understanding Analysis (Undergraduate Texts in Mathematics)

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

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

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

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

Understanding Analysis by Abbott is a book that is more gentle than most.

You are missing Abstract Algebra that usually comes before or after Real Analysis. As for that 4chan post, Rudin's book will hand anyone their ass if they havent seen proofs and dont have a proper foundation (Logic/Proofs/Sets/Functions). Transition to Higher Math courses usually cover such matters. Covering Rudin in 4 months is a stretch. It has to be the toughest intro to Real Analysis. There are tons of easier going alternatives:

Real Mathematical Analysis by Charles Pugh

Understanding Analysis by Stephen Abbot

A Primer of Real Functions by Ralph Boas

Yet Another Introduction to Analysis

Elementary Analysis: The Theory of Calculus

Real Analysis: A Constructive Approach

Introduction to Topology and Modern Analysis by George F. Simmons

...and tons more.

The 1st Course In Math Analysis by Brannan

Analysis I by Terrence Tao

Yet Another Intro To Real Analysis by Bryant

Understanding Analysis Stephen Abbott

Elementary Analysis: The Theory of Calculus Kenneth A. Ross


Metric Spaces by Robert Reisel

A Problem Text in Advanced Calculus by John Erdman. PDF

Advanced Calculus by Shlomo Sternberg and Lynn Loomis.PDF

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

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

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

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

Just keep at it?

I 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.

Here's my radical idea that might feel over-the-top and some here might disagree but I feel strongly about it:

In order to be a grad student in any 'mathematical science', it's highly recommended (by me) that you have the mathematical maturity of a graduated math major. That also means you have to think of yourself as two people, a mathematician, and a mathematical-scientist (machine-learner in your case).

AFAICT, your weekends, winter break and next summer are jam-packed if you prefer self-study. Or if you prefer classes then you get things done in fall, and spring.

Step 0 (prereqs): You should be comfortable with high-school math, plus calculus. Keep a calculus text handy (Stewart, old edition okay, or Thomas-Finney 9th edition) and read it, and solve some problem sets, if you need to review.

Step 0b: when you're doing this, forget about machine learning, and don't rush through this stuff. If you get stuck, seek help/discussion instead of moving on (I mean move on, attempt other problems, but don't forget to get unstuck). As a reminder, math is learnt by doing, not just reading. Resources:

• math subreddit
  
• math.stackexchange.com
  

• math on irc.freenode.net
  

• the math department of your college (don't forget that!)
  
  
  Here are two possible routes, one minimal, one less-minimal:
  
  Minimal
  
  
• Get good with proofs/math-thinking. Texts: One of Velleman or Houston (followed by Polya if you get a chance).
  
• Elementary real analysis. Texts: One of Spivak (3rd edition is more popular), Ross, Burkill, Abbott. (If you're up for two texts, then Spivak plus one of the other three).
  
  
  Less-minimal:
  
  
• Two algebras (linear, abstract)
  
• Two analyses (real, complex)
  
• One or both of geometry, and topology.
  
  
  NOTE: this is pure math. I'm not aware of what additional material you'd need for machine-learning/statistical math. Therefore I'd suggest to skip the less-minimal route.

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

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

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

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

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

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

I'd suggest taking a look at Understanding Analysis by Stephen Abbot.

Analysis is typically skipped in favor of calculus because it's so theoretical that its usefulness is not immediately apparent, but once you get a handle on it, it can be quite nice. It's also, incidentally, the entire theoretical groundwork calculus is based upon. So I think it's probably the field you're looking for.

The Amazon page I linked to lets you preview the first few pages of the book. It essentially takes you from algebra and a very small amount of set theory (opening with a proof that sqrt(2) is irrational). Abbott gives you pretty much everything you need to know if you know algebra. Plus, his proofs are very, very rigorous while at the same time being readable (something missing in a lot of texts).

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

You can start self-learning, but read this: http://www.amazon.com/Understanding-Analysis-Undergraduate-Texts-Mathematics/dp/0387950605 before trying Rudin. It's the same stuff but more introductory;it treats you like you haven't done proofs. Also there's a full solutions manual.

>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

This book.

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

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

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

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

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