Math books
These Math books are recommended by Art of Problem Solving administrators and members of the AoPS Community.
Levels of reading and math ability are loosely defined as follows:
- Elementary is for elementary school students up through possibly early middle school.
- Getting Started is recommended for students grades who are participating in contests like AMC 8/10 and Mathcounts.
- Intermediate is recommended for students who can expect to pass the AMC 10/12.
- Olympiad is recommended for high school students who are already studying math at an undergraduate level.
- Collegiate is recommended for college and university students.
More advanced topics are often left with the above levels unassigned.
Before adding any books to this page, please review the AoPSWiki:Linking books page.
Contents
Books By Subject
Algebra
Getting Started
- 100 Challenging Maths Problems
- AoPS publishes Richard Rusczyk's, David Patrick's, and Ravi Boppana's Prealgebra textbook, which is recommended for advanced elementary and middle school students.
- AoPS publishes Richard Rusczyk's Introduction to Algebra textbook, which is recommended for advanced elementary, middle, and high school students.
Intermediate
- Algebra by I.M. Gelfand and Alexander Shen.
- 101 Problems in Algebra from the Training of the US IMO Team by Titu Andreescu and Zuming Feng
- AoPS publishes Richard Rusczyk's and Mathew Crawford's Intermediate Algebra textbook, which is recommended for advanced middle and high school students.
- Complex Numbers from A to... Z by Titu Andreescu
Abstract Algebra
Collegiate
- Abstract Algebra by David S. Dummit and Richard M. Foote. This is a famous textbook, and is usually the go-to book for students wishing to learn about groups, rings, fields and their properties.
- Undergraduate Algebra by Serge Lang. Some compare it to being similar to Dummit and Foote with regards to rigor, although this text is slightly more terse.
- Algebra: Theory and Applications by Thomas Judson. One of the easiest books to get started with in the genre, and is very comprehensive.
Graduate
- Algebra by Serge Lang.
Calculus
Getting Started
Single Variable (Intermediate)
- AoPS publishes Dr. David Patrick's Calculus textbook, which is recommended for advanced middle and high school students.
- Calculus: Volume I by Tom M. Apostol. Provides a good transition into linear algebra which is uncommon in single variable calculus texts.
- Single Variable Calculus by James Stewart. Contains plenty of exercises for practice and focuses on application rather than rigor.
- Calculus by Michael Spivak. Top students swear by this book. Requires a high level proofing ability for most problems
- Honors Calculus by Charles R. MacCluer. Uses the topological definition of the limit rather than the traditional delta-epsilon approach.
Multivariable (Collegiate)
- Multivariable Calculus by James Stewart.
- Advanced Calculus by Frederick S. Woods. Became an iconic textbook because of how Richard Feynman learned calculus from it. Feynman later popularized a technique taught in the book in college, which is now called the "Feynman Integration Technique."
- Calculus: Volume II by Tom M. Apostol.
Analysis
Collegiate
- Understanding Analysis by Stephen Abbott.
- Principles of Mathematical Analysis by Walter Rudin. Affectionately called "Baby Rudin" by some, Principles of Mathematical Analysis is known to be very terse for the analysis layman.
- Analysis I by Terrence Tao. An easier first read than Rudin, and provides plenty of examples with thorough explanations.
- Analysis II by Terrence Tao. Continues off from where Volume I ended and finishes at the Lebesgue Integration.
- Real Analysis by Rami Shakarchi and Elias M. Stein.
- Complex Analysis by Rami Shakarchi and Elias M. Stein.
Graduate
- Real and Complex Analysis by Walter Rudin. Called "Papa Rudin" by some, and is typically used at the graduate level.
- Functional Analysis by Rami Shakarchi and Elias M. Stein.
Combinatorics
Getting Started
- AoPS publishes Dr. David Patrick's Introduction to Counting & Probability textbook, which is recommended for advanced middle and high school students.
Intermediate
- AoPS publishes Dr. David Patrick's Intermediate Counting & Probability textbook, which is recommended for advanced middle and high school students.
- Mathematics of Choice by Ivan Niven.
- 102 Combinatorial Problems by Titu Andreescu and Zuming Feng.
- A Path to Combinatorics for Undergraduates: Counting Strategies by Titu Andreescu and Zuming Feng.
Olympiad
Collegiate
- Enumerative Combinatorics, Volume 1 by Richard Stanley.
- Enumerative Combinatorics, Volume 2 by Richard Stanley.
- A First Course in Probability by Sheldon Ross
Geometry
Getting Started
- AoPS publishes Richard Rusczyk's Introduction to Geometry textbook, which is recommended for advanced middle and high school students.
Intermediate
- Challenging Problems in Geometry -- A good book for students who already have a solid handle on elementary geometry.
- Geometry Revisited -- A classic.
- 106 Geometry Problems from the AwesomeMath Summer Program by Titu Andreescu, Michal Rolinek, and Josef Tkadlec
Olympiad
- Euclidean Geometry in Mathematical Olympiads by Evan Chen
- Geometry Revisited -- A classic.
- Geometry of Complex Numbers by Hans Schwerfdtfeger.
- Geometry: A Comprehensive Course by Dan Pedoe.
- Non-Euclidean Geometry by H.S.M. Coxeter.
- Projective Geometry by H.S.M. Coxeter.
- Geometric Transformations I, Geometric Transformations II, and Geometric Transformations III by I. M. Yaglom.
- 107 Geometry Problems from the AwesomeMath Year-Round Program Titu Andreescu, Michal Rolinek, and Josef Tkadlec
Collegiate
- Geometry of Complex Numbers by Hans Schwerfdtfeger.
- Geometry: A Comprehensive Course by Dan Pedoe.
- Non-Euclidean Geometry by H.S.M. Coxeter.
- Projective Geometry by H.S.M. Coxeter.
Inequalities
Intermediate
Olympiad
- Advanced Olympiad Inequalities by Alijadallah Belabess.
- The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities by J. Michael Steele.
- Problem Solving Strategies by Arthur Engel contains significant material on inequalities.
- Titu Andreescu's Book on Geometric Maxima and Minima
- Topics in Inequalities by Hojoo Lee
- Olympiad Inequalities by Thomas Mildorf
- A<B (A is less than B) by Kiran S. Kedlaya
- Secrets in Inequalities vol 1 and 2 by Pham Kim Hung
Collegiate
- Inequalities by G. H. Hardy, J. E. Littlewood, and G. Polya.
Number Theory
Getting Started
- The AoPS Introduction to Number Theory by Mathew Crawford.
- Number Theory by George E. Andrews.
Olympiad
- Number Theory: A Problem-Solving Approach by Titu Andreescu and Dorin Andrica.
- 104 Number Theory Problems from the Training of the USA IMO Team by Titu Andreescu, Dorin Andrica and Zuming Feng.
- Problems in Elementary Number Theory by Hojoo Lee.
- Olympiad Number Theory through Challenging Problems by Justin Stevens.
- Elementary Number theory by David M. Burton
Collegiate
- An Introduction to the Theory of Numbers by G. H. Hardy, Edward M. Wright, and Andrew Wiles (6th Edition).
Trigonometry
Getting Started
- Trigonometry by I.M. Gelfand and Mark Saul.
Intermediate
- Trigonometry by I.M. Gelfand and Mark Saul.
- 103 Trigonometry Problems by Titu Andreescu and Zuming Feng.
Olympiad
Problem Solving
Getting Started
- the Art of Problem Solving Volume 1 by Sandor Lehoczky and Richard Rusczyk is recommended for avid math students in grades 7-9.
- Mathematical Circles -- A wonderful peak into Russian math training.
- 100 Great Problems of Elementary Mathematics by Heinrich Dorrie.
Intermediate
- the Art of Problem Solving Volume 2 by Sandor Lehoczky and Richard Rusczyk is recommended for avid math students in grades 9-12.
- The Art and Craft of Problem Solving by Paul Zeitz, former coach of the U.S. math team.
- How to Solve It by George Polya.
- A Mathematical Mosaic by Putnam Fellow Ravi Vakil.
- Proofs Without Words, Proofs Without Words II
- Sequences, Combinations, Limits
- 100 Great Problems of Elementary Mathematics by Heinrich Dorrie.
Olympiad
- Mathematical Olympiad Challenges
- Problem Solving Strategies by Arthur Engel.
- Problem Solving Through Problems by Loren Larson.
General Interest
- The Code Book by Simon Singh.
- Count Down by Steve Olson.
- Fermat's Enigma by Simon Singh.
- Godel, Escher, Bach
- Journey Through Genius by William Dunham.
- A Mathematician's Apology by G. H. Hardy.
- The Music of the Primes by Marcus du Sautoy.
- Proofs Without Words by Roger B. Nelsen.
- What is Mathematics?by Richard Courant, Herbert Robbins and Ian Stewart.
Math Contest Problem Books
Elementary School
- Mathematical Olympiads for Elementary and Middle Schools (MOEMS) publishes two excellent contest problem books.
Getting Started
- MATHCOUNTS books -- Practice problems at all levels from the MATHCOUNTS competition.
- Contest Problem Books from the AMC.
- More Mathematical Challenges by Tony Gardiner. Over 150 problems from the UK Junior Mathematical Olympiad, for students ages 11-15.
Intermediate
- The Mandelbrot Competition has two problem books for sale at AoPS.
- ARML books:
- Five Hundred Mathematical Challenges -- An excellent collection of problems (with solutions).
- The USSR Problem Book
- Leningrad Olympiads (Published by MathProPress.com)
Olympiad
- USAMO 1972-1986 -- Problems from the United States of America Mathematical Olympiad.
- The IMO Compendium: A Collection of Problems Suggested for The International Mathematical Olympiads: 1959-2004
- Mathematical Olympiad Challenges
- Problem Solving Strategies by Arthur Engel.
- Problem Solving Through Problems by Loren Larson.
- Hungarian Problem Book III
- Mathematical Miniatures
- Mathematical Olympiad Treasures
- Collections of Olympiads (APMO, China, USSR to name the harder ones) published by MathProPress.com.
Collegiate
- Three Putnam competition books are available at AoPS.
