Calculus

spivak calculus opened my eyes to what exactly truth is and how to prove truth by build proofs from parts.

* The Art of Probability by Hamming. An opinionated, slightly quirky text on probability. Unlike the text used in my university course its explanations were clear and rigourous without being pedantic. The exercises were both interesting and enlightening. The only book in this list that taught skills I've actually used in the real world. * Calculus by Spivak. This was used in my intro calculus course in university. It's very much a bottom-up, first-principles construction of calculus. Very proof-based, so you have to be into that. Tons of exercises, including some that sneakily introduce pretty advanced concepts not explicitly covered in the main text. This book, along with the course, rearranged by brain. Not sure how useful it would be for self-study though. * Measurement by Lockhart. I haven't read the whole thing, but have enjoyed working through some of the exercises. A good book for really grokking geometric proofs and understanding "mathematical beauty", rather than just cranking through algebraic proofs step by step.…

Seconding all of Spivak's books. My favorite treatment of differential geometry.

The other benefit is that all of the AoPS books have extremely complete solutions manuals so you'll get immediate feedback on the quality of your solution, or you can just look if you've been banging your head against something for a while. Solutions manuals are the autodidact's best friend. Other high quality books with complete solutions included or in separate solutions manuals include Spivak's Calculus, Hubbard and Hubbard's Vector Calculus, Tenenbaum and Pollard's Ordinary Differential Equations (a really cheap Dover text from the 60s!), and Knuth et. al's Concrete Mathematics. Going through the AoPS books and the above will give you an extremely solid foundation to learn any further math you might need as a software engineer as needed. If you have a career spanning 3-4 decades, spending a few years mastering this material can be extremely useful.

Supposing you have at least high school mathematics under the belt: 1. Book of Proof by Hammack [1] 2. Tom Apostol Calculus Vol 1 (stop at Linear Algebra) 3. Linear Algebra by Insel, Spence & Friedberg 4. Understanding Analysis by Stephen Abbott 5. Tom Apostol Calculus Vol 2 These are books with excellent exposition, and solution manuals are available when needed. You are absolutely required to do all the exercises. To learn mathematics is to do mathematics. Two other excellent books are Spivak - Calculus, the problems in this book are very good, but much more difficult than Apostol. For more advanced analysis there's Walter Rudin's Principles of Mathematical Analysis (baby Rudin), which on its own is too difficult as a first exposure to analysis, but there's a cool professor that recorded an entire semester of baby Rudin lecture videos [2]. This is as close as it gets without being enrolled in a university. [1] https://www.people.vcu.edu/~rhammack/BookOfProof/ [2] https://www.youtube.com/watch?v=ab41LEw9oiI&list=PLun8-Z_lTk... (https://www.youtube.com/watch?v=ab41LEw9oiI&list=PLun8-Z_lTkC6qJF1sVh3_Hx7aL6FPd0IN) (the first video is a bit blurry, it gets better)

Spivak's Calculus! (It's really a book about real analysis.) It's extremely well written and starts by rebuilding your understanding of your fundamental mathematical building blocks, only using things you can prove. It also teaches you how to prove them.

Spivak Calculus. Fantastic book.

If you are young and stupid enough, you don't even realize when a book is too difficult. You just plow through it if you can make any sense of it at all. So for me it was TAOCP, a few different theory books by Aho and Ullman, and Spivak's Calculus. I can't say I understood everything in them, but they got me going.

Since you mention analysis, I recommend Yet Another Introduction to Analysis and Metric Spaces by Victor Bryant. They should be at the right level and a lot of fun. Another analysis suggestion is Creative Mathematics by H.S. Wall. It is a book that walks a high school level student through creating the proofs themselves. The topic covered is a stripped down version of analysis, calculus, and later even differential geometry. It's really brilliant. Going slow and having fun when you're a teen would be much more productive than going fast and burning out. The Spivak book is a good suggestion and might be a little difficult depending on their actual background. The books by Gelfand mentioned by someone else (there's actually a series of them that cover algebra, functions, coordinates, trigonometry, etc.) would help provide the needed background. The book Conceptual Mathematics is claimed to be aimed at high school students. Maybe give it a whirl and see what happens. If they know calculus, then Advanced Calculus: A Differential Forms Approach by Harold Edwards is a gem. The first three chapters should be readable, as they give heuristic discussions of the topic.

There are many books on that front, particularly the ones related to recreational math or intro higher math (see https://mathvault.ca/books for instance). Spivak's Calculus as an intro would be an interesting start, though Stewart's Calculus is dense but more accessible.

A group of us maintain an annotated bibliography of math textbooks used at the University of Chicago. The entry level ones (like Spivak's Calculus) would be good to check out: https://github.com/ystael/chicago-ug-math-bib

The princeton companion is nice to have around, but you do not really read it end to end. Spivak's calculus you bring to the beach and read it between swim and swim. EDIT: Also, some books by Hilbert are breathtakingly beautiful: Geometry and the Imagination (just the chapter on synthetic differential geometry is worth more than 10 other great books), and the Methods of Mathematical Physics is also great. It begins by giving three proofs of cauchy-schwartz inequality, and then goes on to give several different definitions of the eigenvectors of a matrix. Both of those make great beach readings for this summer.

I'm neither bright nor a teenager but I have been enjoying working through Spivak's Calculus. I picked it up because it was recommended as a good intro to pure mathematical thinking for someone who knows calculus. I've found it challenging but it has delivered on that promise. There are many good recommendations here but I do think it will be good for them to gain some exposure to pure mathematics. It's different than what's typically taught in high school so they can start to get an idea whether they actually want to be a mathematician or instead focus on applied math in an engineering discipline. Also you're probably going to get a computing bias here. I found the threads on physicsforums.com helpful so you might ask there as well if you want a different bias. (https://www.physicsforums.com/forums/science-and-math-textbo... (https://www.physicsforums.com/forums/science-and-math-textbooks.21/))

There are some really good suggestions here. I would like to add that learning calculus properly is of the utmost importance. It's no exaggeration when I say that this is unquestionably the single best, most profitable action you can take. Any one of these books can change your life. 1. Spivak Calculus 2. Apostol Calculus vol. I and II 3. Courant and Johns, Introduction to Calculus and Analysis vol. I, IIA and IIB For the more casual computer science or physics major, I'd go with choice 3 which resembles the Feynmann lectures the most. All the rigour of the other two is there but in a more digestible form. It's hard for someone not accustomed to hard maths to digest long proofs, it could give you a bad case of indigestion. It's more a function of patience. Courant and Johns get to the point much more quickly while Spivak and Apostol take their time to do everything thoroughly. Courant and Johns does do everything thoroughly but they are kinder to the reader and delay lengthy rigoourous proofs as long as possible while giving plenty of motivation and intuition. Also I strongly recommend any books by Ray Smullyan particularly his introduction to mathematical logic.…

Spivak's Calculus. It's "just" calculus... but it's also everything else leading up to it. It's a wonderful book, written in a very engaging style, and it shows you how mathematicians think and how they play. It shows you why we have proofs, why things go wrong, and all that had to happen before we came up with a definition of derivatives and integrals that we're happy with (and of course, all of the things we can do with our newfound definitions).

seconded. this book is fantastic!

Calculus by Michael Spivak

Focusing on mathematics, I think about two kinds of foundational books. 1. Historically foundational: books that brought important ideas, introduced a new branch, progressed the maths significantly. You can read them these days, but their content, proofs, language, etc. is not up to date. Examples: Elements by Euclid, Philosophiae Naturalis Principia Mathematica by Newton, Disquisitiones Arithmeticae by Gauss. 2. Modern foundational books: The ones that can be used for studying now. Books that are clearly written, popular, cover their subject well. Examples: Visual Complex Analysis, Spivak’s Calculus, Linear Algebra Done Right, Hardy’s An Introduction to the Theory of Numbers.

Spivak's "Calculus".

Spivak's Calculus is like SICP.

I clicked on this topic in order to make this suggestion. To expand on it a little bit, I was a math major and read Spivak's 'Calculus' after I had already taken real analysis. I found it delightful - it really approaches the topics from first principles and unlike many calculus textbooks actually goes through the effort of presenting proofs of the theorems. Highly recommended. As some recreational reading, less suiting the original request, I very much enjoyed David Foster Wallace's 'Everything and More: A Compact History of Infinity' (http://www.amazon.com/Everything-More-Compact-History-Infini... (http://www.amazon.com/Everything-More-Compact-History-Infinity/dp/0393339289/)). DFW is not for everyone, but I enjoyed it a lot. Maybe just check it out of the library first to see if it's for you.

It's difficult to find what you are looking for, since maths is a much older field. However, Spivak's 'Calculus' is pretty close: http://www.amazon.com/gp/aw/d/0914098918.

I'm pretty at Mathematics and doing that takes a lot of work. And the Mathematics books for noobs are no good. The better approach is to get a theoretical book, something like Spivak's Calculus or Linear Algebra by Friedberg, Insel and Spence. And then from there whenever you have difficulty with the material spread out laterally and you really start to gradually grow an understanding of mathematics. And then perhaps, one day you'll be up for Spivak's Calculus on Manifolds!