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.

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.
• Algebra by Serge Lang -- Extends undergraduate Abstract Algebra to the graduate level by studying homological algebra and more.

Calculus

Getting Started

• The Hitchhiker's Guide to Calculus by Michael Spivak.
• Calculus Made Easy by Silvanus P. Thompson.

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.
• 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. Advanced Calculus 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 Integral.
• Real Analysis by Rami Shakarchi and Elias M. Stein.
• Complex Analysis by Rami Shakarchi and Elias M. Stein.
• Real and Complex Analysis by Walter Rudin. Called "Papa Rudin" by some, Real and Complex Analysis 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

• 102 Combinatorial Problems by Titu Andreescu and Zuming Feng.
• Generatingfunctionology

Collegiate

• Enumerative Combinatorics, Volume 1 by Richard Stanley.
• Enumerative Combinatorics, Volume 2 by Richard Stanley.
• A First Course in Probability by Sheldon Ross
• Introductory Combinatorics by Kenneth P. Bogart

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.

Topology

Collegiate

• Topology by James Munkres. Topology is arguably the most renowned topology textbook of all time. It also contains an excellent introduction to set theory and logic.

Inequalities

Intermediate

• Introduction to Inequalities
• Geometric Inequalities

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
• Modern Olympiad Number Theory by Aditya Khurmi.

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

• 103 Trigonometry Problems by Titu Andreescu and Zuming Feng.

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:
  • ARML-NYSML 1989-1994 (see ARML).
  • ARML 1995-2004
• 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.

See also

• Math textbooks
• Resources for mathematics competitions
• Olympiad Books