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These '''Math books''' are recommended by [[Art of Problem Solving]] administrators and members of the <url>index.php AoPS-MathLinks Community</url>.

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 6 to 9.
* Intermediate is recommended for students grades 9 to 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====
* [[AoPS]] publishes [[Richard Rusczyk]]'s [http://www.artofproblemsolving.com/Books/AoPS_B_Item.php?page_id=11 Introduction to Algebra] textbook, which is recommended for advanced elementary, middle, and high school students.
==== Intermediate ====
* [http://www.amazon.com/exec/obidos/ASIN/0817636773/artofproblems-20 Algebra] by I.M. Gelfand and Alexander Shen.
* [http://www.amazon.com/Problems-Algebra-Training-Team-Enrichment/dp/187642012X/ref=sr_1_2?ie=UTF8&s=books&qid=1204029534&sr=8-2 101 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 [http://www.artofproblemsolving.com/Books/AoPS_B_Item.php?item_id=300 Intermediate Algebra] textbook, which is recommended for advanced middle and high school students.
* [http://www.amazon.com/Complex-Numbers-Z-Titu-Andreescu/dp/0817643265/ref=sr_1_1?ie=UTF8&s=books&qid=1204029652&sr=1-1 Complex Numbers from A to... Z] by [[Titu Andreescu]]

=== Analysis ===
* [http://www.amazon.com/exec/obidos/ASIN/0486428753/artofproblems-20 Counterexamples in Analysis] by Bernard R. Gelbaum and John M. H. Olmsted.


=== Calculus ===
==== High School ====
* [http://www.amazon.com/exec/obidos/ASIN/0914098896/artofproblems-20 Calculus] by [[Michael Spivak]].  Top students swear by this book.
* [http://www.amazon.com/exec/obidos/ASIN/0883858126/artofproblems-20 The Hitchhiker's Guide to Calculus] by [[Michael Spivak]].
* [http://www.artofproblemsolving.com/Books/AoPS_B_Item.php?page_id=8 AP Calculus Problems and Solutions Part II AB and BC] -- A fantastic resource for students mastering the material required for the AP exam.

==== Collegiate ====
* [http://www.amazon.com/exec/obidos/ASIN/0805390219/artofproblems-20 Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus] by [[Michael Spivak]].


=== Combinatorics ===
==== Getting Started ====
* [[AoPS]] publishes Dr. [[David Patrick]]'s [http://www.artofproblemsolving.com/Books/AoPS_B_Item.php?page_id=3 Introduction to Counting & Probability] textbook, which is recommended for advanced middle and high school students.

==== Intermediate ====
* [[AoPS]] publishes Dr. [[David Patrick]]'s [http://www.artofproblemsolving.com/Books/AoPS_B_Item.php?item_id=302 Intermediate Counting & Probability] textbook, which is recommended for advanced middle and high school students.
* [http://www.amazon.com/exec/obidos/ASIN/0883856158/artofproblems-20 Mathematics of Choice] by Ivan Niven.
* [http://www.amazon.com/exec/obidos/ASIN/0817643176/artofproblems-20 102 Combinatorial Problems] by [[Titu Andreescu]] and [[Zuming Feng]].
* [http://www.amazon.com/Path-Combinatorics-Undergraduates-Counting-Strategies/dp/0817642889/ref=sr_1_2?ie=UTF8&s=books&qid=1219586040&sr=1-2 A Path to Combinatorics for Undergraduates: Counting Strategies] by [[Titu Andreescu]] and [[Zuming Feng]].

==== Olympiad ====
* [http://www.amazon.com/exec/obidos/ASIN/0817643176/artofproblems-20 102 Combinatorial Problems] by [[Titu Andreescu]] and [[Zuming Feng]].
* [http://www.math.upenn.edu/~wilf/DownldGF.html Generatingfunctionology]

==== Collegiate ====
* [http://www.amazon.com/exec/obidos/ASIN/0521663512/artofproblems-20 Enumerative Combinatorics, Volume 1] by Richard Stanley.
* [http://www.amazon.com/exec/obidos/ASIN/0521789877/artofproblems-20 Enumerative Combinatorics, Volume 2] by Richard Stanley.
* [http://www.amazon.com/First-Course-Probability-Sheldon-Ross/dp/0131856626/ref=pd_bbs_sr_1/103-7161656-8805468?ie=UTF8&s=books&qid=1190719501&sr=8-1 A First Course in Probability] by Sheldon Ross



=== Geometry ===
==== Getting Started ====
* [[AoPS]] publishes [[Richard Rusczyk]]'s [http://www.artofproblemsolving.com/Books/AoPS_B_Item.php?page_id=9 Introduction to Geometry] textbook, which is recommended for advanced middle and high school students.

==== Intermediate ====
* [http://www.amazon.com/exec/obidos/ASIN/0486691543/artofproblems-20 Challenging Problems in Geometry] -- A good book for students who already have a solid handle on elementary geometry.
* [http://www.amazon.com/exec/obidos/ASIN/0883856190/artofproblems-20 Geometry Revisited] -- A classic.

==== Olympiad ====
* [http://www.amazon.com/exec/obidos/ASIN/0883856190/artofproblems-20 Geometry Revisited] -- A classic.
* [http://www.amazon.com/exec/obidos/ASIN/0486638308/artofproblems-20 Geometry of Complex Numbers] by Hans Schwerfdtfeger.
* [http://www.amazon.com/exec/obidos/ASIN/0486658120/artofproblems-20 Geometry: A Comprehensive Course] by Dan Pedoe.
* [http://www.amazon.com/exec/obidos/ASIN/0883855224/artofproblems-20 Non-Euclidean Geometry] by [[H.S.M. Coxeter]].
* [http://www.amazon.com/exec/obidos/ASIN/0387406239/artofproblems-20 Projective Geometry] by [[H.S.M. Coxeter]].
* [http://www.amazon.com/Geometric-Transformations-I-Number-8/dp/0883856085/ref=sr_1_6?ie=UTF8&s=books&qid=1199807141&sr=1-6 Geometric Transformations I], [http://www.amazon.com/Geometric-Transformations-New-Mathematical-Library/dp/0883856212/ref=sr_1_5?ie=UTF8&s=books&qid=1199807203&sr=1-5 Geometric Transformations II], and [http://www.amazon.com/Geometric-Transformations-III-Mathematical-Library/dp/0883856247/ref=sr_1_1?ie=UTF8&s=books&qid=1199807249&sr=1-1 Geometric Transformations III] by I. M. Yaglom.

==== Collegiate ====
* [http://www.amazon.com/exec/obidos/ASIN/0486638308/artofproblems-20 Geometry of Complex Numbers] by Hans Schwerfdtfeger.
* [http://www.amazon.com/exec/obidos/ASIN/0486658120/artofproblems-20 Geometry: A Comprehensive Course] by Dan Pedoe.
* [http://www.amazon.com/exec/obidos/ASIN/0883855224/artofproblems-20 Non-Euclidean Geometry] by [[H.S.M. Coxeter]].
* [http://www.amazon.com/exec/obidos/ASIN/0387406239/artofproblems-20 Projective Geometry] by [[H.S.M. Coxeter]].



=== Inequalities ===
==== Intermediate ====
* [http://www.amazon.com/exec/obidos/ASIN/0883856034/artofproblems-20 Introduction to Inequalities]
* [http://www.amazon.com/exec/obidos/ASIN/0883856042/artofproblems-20 Geometric Inequalities]

==== Olympiad ====
* [http://www.amazon.com/exec/obidos/ASIN/052154677X/artofproblems-20 The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities] by J. Michael Steele.
* [http://www.amazon.com/exec/obidos/ASIN/0387982191/artofproblems-20 Problem Solving Strategies] by Arthur Engel contains significant material on inequalities.
*Titu Andreescu's Book on Geometric Maxima and Minima
* [http://ultrametric.googlepages.com/tin2007.pdf Topics in Inequalities] by Hojoo Lee
* [http://www.artofproblemsolving.com/Resources/Papers/MildorfInequalities.pdf Olympiad Inequalities] by Thomas Mildorf
* [http://www.artofproblemsolving.com/Resources/Papers/KedlayaInequalities.pdf A<B (A is less than B)] by Kiran S. Kedlaya

==== Collegiate ====

* [http://www.amazon.com/exec/obidos/ASIN/0521358809/artofproblems-20 Inequalities] by [[G. H. Hardy]], [[J. E. Littlewood]], and [[G. Polya]].



=== Number Theory ===
==== Introductory ====
* The AoPS [http://www.artofproblemsolving.com/Books/AoPS_B_Item.php?page_id=10 Introduction to Number Theory] by [[Mathew Crawford]].
==== Olympiad ====
* [http://www.amazon.com/exec/obidos/ASIN/081763245X/artofproblems-20 Number Theory: A Problem-Solving Approach] by [[Titu Andreescu]] and Dorin Andrica.
* [http://www.amazon.com/104-Number-Theory-Problems-Training/dp/0817645276/ref=pd_bbs_sr_1?ie=UTF8&s=books&qid=1199806669&sr=8-1 104 Number Theory Problems from the Training of the USA IMO Team] by [[Titu Andreescu]], Dorin Andrica and Zuming Feng.
* [http://www.problem-solving.be/pen/published/pen-20070711.pdf Problems in Elementary Number Theory] by Hojoo Lee.


=== Trigonometry ===
==== Getting Started ====
* [http://www.amazon.com/exec/obidos/ASIN/0817639144/artofproblems-20 Trigonometry] by I.M. Gelfand and Mark Saul.

==== Intermediate ====
* [http://www.amazon.com/exec/obidos/ASIN/0817639144/artofproblems-20 Trigonometry] by I.M. Gelfand and Mark Saul.
* [http://www.amazon.com/exec/obidos/ASIN/0817643346/artofproblems-20 103 Trigonometry Problems] by [[Titu Andreescu]] and [[Zuming Feng]].

==== Olympiad ====
* [http://www.amazon.com/exec/obidos/ASIN/0817643346/artofproblems-20 103 Trigonometry Problems] by [[Titu Andreescu]] and [[Zuming Feng]].



=== Problem Solving ===
==== Getting Started ====
* the [http://www.artofproblemsolving.com/Books/AoPS_B_Item.php?page_id=1 Art of Problem Solving Volume 1] by Sandor Lehoczky and Richard Rusczyk is recommended for avid math students in grades 7-9.
* [http://www.amazon.com/exec/obidos/ASIN/0821804308/artofproblems-20 Mathematical Circles] -- A wonderful peak into Russian math training.
* [http://www.amazon.com/exec/obidos/ASIN/0821804308/artofproblems-20 100 Great Problems of Elementary Mathematics] by Heinrich Dorrie.

==== Intermediate ====
* the [http://www.artofproblemsolving.com/Books/AoPS_B_Item.php?page_id=2 Art of Problem Solving Volume 2] by Sandor Lehoczky and Richard Rusczyk is recommended for avid math students in grades 9-12.
* [http://www.amazon.com/exec/obidos/ASIN/0471135712/artofproblems-20 The Art and Craft of Problem Solving] by [[Paul Zeitz]], former coach of the U.S. math team.
* [http://www.amazon.com/exec/obidos/ASIN/0691023565/artofproblems-20 How to Solve It] by [[George Polya]].
* [http://www.amazon.com/exec/obidos/ASIN/1895997046/artofproblems-20 A Mathematical Mosaic] by [[Putnam Fellow]] [[Ravi Vakil]].
* [http://www.amazon.com/exec/obidos/ASIN/0883857006/artofproblems-20 Proofs Without Words], [http://www.amazon.com/exec/obidos/ASIN/0883857219/artofproblems-20 Proofs Without Words II]
* [http://www.amazon.com/exec/obidos/ASIN/0486425665/artofproblems-20 Sequences, Combinations, Limits]
* [http://www.amazon.com/exec/obidos/ASIN/0821804308/artofproblems-20 100 Great Problems of Elementary Mathematics] by Heinrich Dorrie.

==== Olympiad ====
* [http://www.amazon.com/exec/obidos/ASIN/0817641556/artofproblems-20 Mathematical Olympiad Challenges]
* [http://www.amazon.com/exec/obidos/ASIN/0387982191/artofproblems-20 Problem Solving Strategies] by Arthur Engel.
* [http://www.amazon.com/exec/obidos/ASIN/0387961712/artofproblems-20 Problem Solving Through Problems] by Loren Larson.



== General interest ==
* [http://www.amazon.com/exec/obidos/ASIN/0385495323/artofproblems-20 The Code Book] by Simon Singh.
* [http://www.amazon.com/exec/obidos/ASIN/0618251413/artofproblems-20 Count Down] by Steve Olson.
* [http://www.amazon.com/exec/obidos/ASIN/0385493622/artofproblems-20 Fermat's Enigma] by Simon Singh.
* [http://www.amazon.com/exec/obidos/ASIN/0465026567/artofproblems-20 Godel, Escher, Bach]
* [http://www.amazon.com/exec/obidos/ASIN/014014739X/artofproblems-20 Journey Through Genius] by William Dunham.
* [http://www.amazon.com/exec/obidos/ASIN/0521427061/artofproblems-20 A Mathematician's Apology] by [[G. H. Hardy]]. 
* [http://www.amazon.com/exec/obidos/ASIN/0066210704/artofproblems-20 The Music of the Primes] by Marcus du Sautoy.
* [http://www.amazon.com/exec/obidos/ASIN/0883857006/artofproblems-20 Proofs Without Words] by Roger B. Nelsen. 
* [http://www.amazon.com/exec/obidos/ASIN/0195105192/artofproblems-20 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 [http://www.artofproblemsolving.com/Books/AoPS_B_CP_MOEMS.php two excellent contest problem books].



=== Getting Started ===
* [http://www.artofproblemsolving.com/Books/AoPS_B_CP_MC.php MathCounts books] -- Practice problems at all levels from the [[MathCounts]] competition.
* [http://www.artofproblemsolving.com/Books/AoPS_B_CP_AMC.php Contest Problem Books] from the [[AMC]].
* [http://www.amazon.com/exec/obidos/ASIN/0521585686/artofproblems-20 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 [http://www.artofproblemsolving.com/Books/AoPS_B_CP_Mand.php two problem books for sale] at [[AoPS]].
* ARML books:
**[http://www.amazon.com/exec/obidos/ASIN/0962640166/artofproblems-20 ARML-NYSML 1989-1994] (see [[ARML]]).
** [http://www.arml.com/books.htm ARML 1995-2004]
* [http://www.amazon.com/exec/obidos/ASIN/0883855194/artofproblems-20 Five Hundred Mathematical Challenges] -- An excellent collection of problems (with solutions).
* [http://www.amazon.com/exec/obidos/ASIN/0486277097/artofproblems-20 The USSR Problem Book]
*Leningrad Olympiads (Published by MathProPress.com)

=== Olympiad ===
* [http://www.artofproblemsolving.com/Books/AoPS_B_Item.php?page_id=50 USAMO 1972-1986] -- Problems from the [[United States of America Mathematical Olympiad]].
* [http://www.amazon.com/exec/obidos/ASIN/0387242996/artofproblems-20 The IMO Compendium: A Collection of Problems Suggested for The International Mathematical Olympiads: 1959-2004]
* [http://www.amazon.com/exec/obidos/ASIN/0817641556/artofproblems-20 Mathematical Olympiad Challenges]
* [http://www.amazon.com/exec/obidos/ASIN/0387982191/artofproblems-20 Problem Solving Strategies] by Arthur Engel.
* [http://www.amazon.com/exec/obidos/ASIN/0387961712/artofproblems-20 Problem Solving Through Problems] by Loren Larson.
* [http://www.amazon.com/exec/obidos/ASIN/0883856441/artofproblems-20 Hungarian Problem Book III]
* [http://www.amazon.com/exec/obidos/ASIN/088385645X/artofproblems-20 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 [http://www.artofproblemsolving.com/Books/AoPS_B_CP_Putnam.php available at AoPS].


== See also ==
* [[Math textbooks]]
* [[Resources for mathematics competitions]]

[[Category:Books]]