Ask HN: Math books that made you significantly better at math?

“Mathematical Notation: A Guide for Engineers and Scientists”[0] really changed my abilities with being able to read papers and decipher what was going on. I had university math experience but it was a long time ago. When I started reading papers for algorithms later in my career I couldn’t get past the notation. Once the symbols are explained, as a programmer, I was able to grok so much more. This should be on everyone’s shelf.

[0] https://a.co/d/gQmDIo7

I'm coming from an applied math perspective. A few of my favorites (and ones I find myself regularly referring to) are:

Matrix Analysis by Horn and Johnson (perhaps the best end-of-chapter problem sets of any math book I've encountered!)

Matrix Computations by Golub and Van Loan

Elements of Statistical Learning by Hastie, Friedman, Tibshirani

Functional Analysis by Reed and Simon

I think you guys might find this list I found long ago very useful when deciding on a mathematics book you want to read.

This is an introduction written by the original author of the list:

"Somehow I became the canonical undergraduate source for bibliographical references, so I thought I would leave a list behind before I graduated. I list the books I have found useful in my wanderings through mathematics (in a few cases, those I found especially unuseful), and give short descriptions and comparisons within each category. I hope that this list may serve as a useful “road map” to other undergraduates picking their way through Eckhart Library. In the end, of course, you must explore on your own; but the list may save you a few days wasted reading books at the wrong level or with the wrong emphasis.

The list is biased in two senses. One, it is light on foundations and applied areas, and heavy (especially in the advanced section) on geometry and topology; this is a consequence of my interests. I welcome additions from people interested in other fields. Two, and more seriously, I am an honors-track student and the list reflects that. I don't list any “regular” analysis or algebra texts, for instance, because I really dislike the ones I've seen. If you are a 203 student looking for an alternative to the awful pink book (Marsden/Hoffman), you will find a few here; they are all much clearer, better books, but none are nearly as gentle. I know that banging one's head against a more difficult text is not a realistic option for most students in this position. On the other hand, reading mathematics can't be taught, and it has to be learned sometime. Maybe it's better to get used to frustration as a way of life sooner, rather than later. I don't know." - by original author.

[List] https://www.ocf.berkeley.edu/~abhishek/chicmath.htm

Rudin's Principles of Mathematical Analysis has a really special place in my heart. Chapter 3 is great- it's a great reference for derivations of a lot of fundamental identities about limits used in undergrad calculus.

Chapter 4 is a great place to learn about topology for the first time.

In general, it kicks up the mathematical rigor you're used to a notch. Seeing ">" defined as "not <" really blew my mind when I first read it! "<" is just something that satisfies some axioms, like anything else in math.

Plus one for this! I bought two copies of the referenced book…and for the exact same reasons; I’m a programmer and being able to explain my algorithms using mathematical notation helps validate a program as well as troubleshoot a program… An oldie but goodie is “Mathematics for the million”