Number theory

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1. steppiJan 2023
   Some favorites below. Books 0-3 are accessible. The remaining books are more difficult but I'd highly recommend them to math students.

   0. Jan Gullberg, Mathematics, From the Birth of Numbers. A highly accessible popular survey on different branches of higher mathematics. I read this over the Summer between high school and starting my undergraduate degree. It's what made me want to study math. Previously I'd wanted to be a guitar player, but had to find a new ambition after an injury left me unable to play.

   1. The high school mathematics series by Israel Gelfand. Algebra, Trigonometry, The Method of Coordinates, and Functions and Graphs. I didn't have much mathematics background in high school, but working through these really solidified my grasp on the basics.

   2. George Polya. How to Solve it. A short book giving excellent high level advice on mathematical problem solving.

   3. George E. Andrews, Number Theory. I worked through this freshman year contemporaneously with my first proof based class on simple logic and set theory. A very beautiful and accessible introduction to basic number theory. The combinatorial/geometric proofs of Fermat's Little Theorem and Wilson's Theorem are lovely. It also includes a very nice proof of Chebyshev's theorem on the asymptotic density of primes and even the Rogers-Ramanujan identities for integer partitions.

   4. Vladimir Arnold, Ordinary Differential Equations: Undergrad ODE classes are often taught in a cookbook fashion and if so, don't offer much enlightenment. This book explains what's going on at geometrical level. I didn't appreciate ODEs until I read this. See https://www.uni-muenster.de/Physik.TP/~munsteg/arnold.html for Arnold's views on teaching mathematics.

   5. E.C. Titchmarsh The Theory of Functions: Recommended by my undergraduate advisor because he noticed that I liked reading older books. It contains sections on complex analysis and real analysis with measure theory, but I've only read the complex analysis sections. It's not for everyone, if I recall correctly, there is not a single picture, but it is very lively and has a lot of material you won't find in a standard complex analysis book, including Dirichlet series. Excellent as a supplement to a standard complex analysis book.

   6. George Polya. Mathematics and Plausible Reasoning. An excellent expansion on Polya's ideas on How to Solve it. While the goal is to seek rigorous proofs, to get there it's powerful to be able to think based on intuition, heuristics, and plausible reasoning. A lot of math exposition is theorem/proof based and doesn't help develop these skills. In a similar vein, see also Terence Tao's classic post There's more to mathematics than rigour and proofs https://terrytao.wordpress.com/career-advice/theres-more-to-....

   7. H.S.M Coxeter, An Introduction to Geometry. A book of very beautiful classical geometry. Something typically not touched on at all in a typical mathematics curriculum.

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2. imranqSep 2022
   Very cool.

   I'd also recommend Number Theory by Andrews. It's a great comprehensive introduction.

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3. csenseJul 2020
   At that age, I enjoyed Number Theory by George E. Andrews

   https://www.amazon.com/Number-Theory-Dover-Books-Mathematics...

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4. trengrjMar 2018
   I would select books based on your interest. I find the Dover publications good because they are both cheap and slightly older, this means they are less focused on undergraduate monetisation (version hopping, not supply answers to problems, glossy print), and more focused on proofs and algorithms. You can see them here http://store.doverpublications.com/by-subject-mathematics.ht.... I particularly liked Probability: A concise course, and Number Theory by George Andrews.

   Amazon used to have a great number of graduate preparation book lists which always included books such as Rudin's Principles of Mathematics Analysis, and Halmos' Finite Dimensional Vector Spaces. These classic maths books are brilliant but usually easier to understand if you already have some experience with the material.

   Final advice is to find a study partner as it can be hard to track how you are going and keep motivated, especially without the instant feedback loop you get with programming.

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5. phllipSep 2008
   Steinhaus, Mathematical Snapshots Polya, Mathematics and Plausible Reasoning Pappas, The Joy of Mathematics Gardner, The Colossal Book of Short Puzzles and Problems Winkler, Mathematical Puzzles: A Connoisseur's Collection Andrews, Number Theory (Dover Books on Advanced Mathematics)

   - The last book is not really that advanced, It can be understood by most with an understanding through high school algebra II.

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6. uptownfunkMay 2023
   Chartrand - mathematical proofs (if you don’t have background with proofs)

   Then-

   Baby Rudin - analysis (first 8 chapters is good)

   Dummit Foote - algebra (as much as you can read, groups, rings, fields, even more if you can)

   Needham - complex analysis

   Friedberg - linear algebra

   After that you can go in many directions

   Many other interesting courses

   Number theory

   Representation theory

   Probability -> stochastic processes

   Statistics

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