Book List

Incomplete list of books that people on ##math have bought, sold, tried, read, taught, suffered through and would feel like suggesting.

If you are a self-learner and are looking for a few books to get started, the first section contains exclusively such books.

Other useful such lists on the web: [1 ] [2 ] [3 ]

and for physics: [4 ]

Introductory Books for Self Learners
So you don't have a mathematical education and want to get started somewhere?

These books are for you.

They have no prerequisites and provide a gentle, thorough introduction to the foundations you want to have in order to explore intermediate topics or get a head start on a degree.
 * Calculus: Guichard's free book: https://www.whitman.edu/mathematics/multivariable/
 * No prerequisites.
 * Kenneth Rosen, Discrete Mathematics and its Applications,
 * No prerequisites.
 * Especially relevant to programmers and aspiring computer scientists.
 * Provides, in a simple and progressive style, the basics of mathematical reasoning (proofs, etc) as well as an introduction to all the topics (such as logic and graph theory) that are imprescindible to computer science.
 * Continue with CLRS, Ben Ari and Cutland
 * Susanna Epp, Discrete Mathematics with Applications
 * An alternative if Rosen is not available.
 * Daniel Velleman, How To Prove It
 * No prerequisites
 * Logic, proofs, relations - basic mathematical tools.
 * A better, more in-depth treatment of a subset of the topics in Rosen. Probably still go with Rosen if have more than a passing interest in computer science.
 * Linear Algebra: Gilbert Strang's Introduction to Linear Algebra
 * Slightly "harder" than the previous entries.
 * Ross, First Course in Probability
 * Some familiarity with calculus and linear algebra (see previous entries) is useful


 * Stitz & Zeager, Precalculus
 * Intermediate Algebra is a prerequisite.

Linear Algebra

 * Axler, Linear Algebra Done Right, Elegant with its determinant-free proofs.
 * Hoffman & Kunze, Linear Algebra This is a classic book on linear algebra aimed for mathematicians.
 * Nicholson, Elementary Linear Algebra with Applications (again)
 * Strang, Linear Algebra and its Applications

Calculus and Mathematical Analysis

 * N.L. Carothers, Real Analysis Aimed for graduate students in mathematics.
 * Rudin, Principles of Mathematical Analysis ("Baby Rudin"): Classic. Rigorously structured (Theorem-lemma-proof).
 * Spivak. Classic.
 * Tom M. Apostol, One-Variable Calculus A little dry but one of its kind.

Probability

 * Achim Klenke, Probability Theory A great probability theory reference for graduate students.
 * Ross, First Course in Probability
 * Robert Ash, Basic Probability Theory Aimed for advanced undergraduate students in mathematics, this book has one of the best approaches in introducing probability theory.

Logic
Introductory:
 * Ben Ari, Mathematical Logic for Computer Science
 * Herbert B. Enderton, A Mathematical Introduction to Logic
 * Yiannis N. Moschovakis, Notes on Set Theory
 * Angelo Margaris, First Order Mathematical Logic. Intro to logic covering propositional logic, predicate logic, basic proofs, and the Deduction Theorem.

Combinatorics
Introductory:
 * Tucker, Applied Combinatorics. Very readable but IMHO poorly structured. You sometimes have to fish the definitions out of some very discursive text.

Computability

 * Cutland's Computability. Progressive, enjoyable but rigorous introduction. Recommended.
 * Sipser's Introduction To The Theory Of Computation

Algorithms

 * CLRS: Classic.

(Abstract) Algebra
Note that this refers to the "algebra" as found in university courses under this title. This is mostly disjoint from what is known as "algebra" in high school. Algebra courses


 * Herstein, Topics in Algebra
 * Dummit and Foote, Abstract Algebra
 * Aluffi, Algebra: Chapter 0
 * Michael Artin, Algebra
 * Gallian, Contemporary Abstract Algebra is a good entry into to the subject

Topology

 * Munkres is the standard text
 * Lee, Topological and Smooth Manifolds present the main principles

Algebraic Topology
Introductory:
 * Allen Hatcher, Algebraic Topology
 * Prerequisites: familiarity with what a topological space is, and basic group theory. Knowledge of Rings and Modules is helpful, especially in chapters 2 and 3.
 * Gentle book on algebraic topology. Free pdf is available on the author's website.

Advanced:
 * Massey, Algebraic Topology: An Introduction
 * Joseph Rotman, An Introduction to Algebraic Topology
 * A much shorter book compared to Hatcher's.
 * [I find this book to be a joy to read, except for the CW-complex chapter. Hatcher does it better.]
 * Spanier Algebraic Topology
 * Robert Switzer, Algebraic Topology - Homotopy and Homology
 * Prerequisites: Knowledge on the level of Hatcher
 * Very theory-focused book.

Machine Learning + Data Mining
Introductory: Intermediate: Notably absent from the list is Bishop's Pattern Recognition and Machine Learning. It is a ubiquitous applied reference text that is not good for reading sequentially.
 * Mitchell: very readable, gentle introduction to ML. Slightly outdated. No SVM, no clustering. If in doubt, read this first
 * Tibishirani, Hastie: Introduction to Statistical Learning, introductory, very much example-driven (in R), perhaps too much.
 * Scarpa, Azzalini: Data Analysis and Data Mining, roughly same league as ISL, but as the back cover blurb says, "More detailed than practically-oriented books". Can confirm.
 * Yaser S. Abu-Mostafa, Learning from Data A very readable introduction to machine learning.
 * Alpaydin, Introduction to Machine Learning
 * Tibishirani, Hastie: Elements of Statistical Learning