# 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

### General Introduction / Multiple Topics

#### Getting Started

- Getting Started with Competition Math, a textbook meant for true beginners (on-target middle school students, or advanced elementary school students). It is written by AoPS Community Member cargeek9, currently a junior in high school. It covers the basics of algebra, geometry, combinatorics, and number theory, along with sets of accompanying practice problems at the end of every section.

### 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

#### 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

#### 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
- Solving Problems In Geometry: Insights And Strategies For Mathematical Olympiad And Competitions by Kim Hoo Hang and Haibin Wang
- 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

#### 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

### 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.