Difference between revisions of "Math books"
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− | These ''' | + | These '''math books''' are recommended by [[Art of Problem Solving]] administrators and members of the [http://aops.com/community AoPS Community]. |
Levels of reading and math ability are loosely defined as follows: | Levels of reading and math ability are loosely defined as follows: | ||
* Elementary is for elementary school students up through possibly early middle school. | * Elementary is for elementary school students up through possibly early middle school. | ||
− | * Getting Started is recommended for students grades | + | * Getting Started is recommended for students grades who are participating in contests like AMC 8/10 and Mathcounts. |
− | * Intermediate is recommended for students | + | * 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. | * Olympiad is recommended for high school students who are already studying math at an undergraduate level. | ||
* Collegiate is recommended for college and university students. | * Collegiate is recommended for college and university students. | ||
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− | == Books | + | == Books By Subject == |
+ | |||
+ | === General Introduction / Multiple Topics === | ||
+ | ==== Getting Started ==== | ||
+ | * [https://www.amazon.com/gp/product/B09PMLFHX2/ref=ox_sc_act_title_1?smid=ATVPDKIKX0DER&psc=1 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 [https://artofproblemsolving.com/community/user/243060 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 === | === Algebra === | ||
+ | |||
====Getting Started==== | ====Getting Started==== | ||
+ | * [https://www.amazon.com/After-School-Maths-100-Challenging-Problems-ebook/dp/B07QFWSTDD/ref=sr_1_2?crid=CB0XAM4P81WI&keywords=after+school+maths+kawasaki&qid=1581288606&sprefix=after+school+maths+%2Caps%2C268&sr=8-2 100 Challenging Maths Problems] | ||
* [[AoPS]] publishes [[Richard Rusczyk]]'s, [[David Patrick]]'s, and [[Ravi Boppana]]'s [http://www.artofproblemsolving.com/Store/viewitem.php?item=prealgebra Prealgebra] textbook, which is recommended for advanced elementary and middle school students. | * [[AoPS]] publishes [[Richard Rusczyk]]'s, [[David Patrick]]'s, and [[Ravi Boppana]]'s [http://www.artofproblemsolving.com/Store/viewitem.php?item=prealgebra Prealgebra] textbook, which is recommended for advanced elementary and middle school students. | ||
* [[AoPS]] publishes [[Richard Rusczyk]]'s [http://www.artofproblemsolving.com/Store/viewitem.php?item=intro:algebra Introduction to Algebra] textbook, which is recommended for advanced elementary, middle, and high school students. | * [[AoPS]] publishes [[Richard Rusczyk]]'s [http://www.artofproblemsolving.com/Store/viewitem.php?item=intro:algebra Introduction to Algebra] textbook, which is recommended for advanced elementary, middle, and high school students. | ||
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* [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]] | * [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]] | ||
− | |||
− | |||
− | === Calculus === | + | ===Abstract Algebra=== |
− | ==== | + | |
+ | ====Collegiate==== | ||
+ | * [https://www.amazon.com/Abstract-Algebra-3rd-David-Dummit/dp/0471433349/ref=sr_1_4dchild=1&keywords=abstract+algebra&qid=1634318876&s=books&sr=1-4 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. | ||
+ | * [https://www.amazon.com/Undergraduate-Algebra-Texts-Mathematics/dp/1441919597 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. | ||
+ | * [https://www.amazon.com/Abstract-Algebra-Applications-Thomas-Judson/dp/1944325131 Algebra: Theory and Applications] by [[Thomas Judson]]. One of the easiest books to get started with in the genre, and is very comprehensive. | ||
+ | * [https://www.amazon.com/Algebra-Graduate-Texts-Mathematics-Serge/dp/038795385X Algebra] by [[Serge Lang]] -- Extends undergraduate Abstract Algebra to the graduate level by studying homological algebra and more. | ||
+ | * [https://www.amazon.com/Basic-Algebra-Second-Dover-Mathematics/dp/0486471896 Basic Algebra I] by [[Nathan Jacobson ]] -- Contains harder and more interesting problems than Dummit and Foote. Assumes a decent coverage of Linear Algebra | ||
+ | |||
+ | ===Calculus=== | ||
+ | |||
+ | ==== Getting Started ==== | ||
+ | * [http://www.amazon.com/exec/obidos/ASIN/0883858126/artofproblems-20 The Hitchhiker's Guide to Calculus] by [[Michael Spivak]]. | ||
+ | * [https://www.amazon.com/Calculus-Made-Easy-Very-Simplest-Introduction/dp/1409724670 Calculus Made Easy] by [[Silvanus P. Thompson]]. | ||
+ | |||
+ | ==== Single Variable (Intermediate) ==== | ||
* [[AoPS]] publishes Dr. [[David Patrick]]'s [http://www.artofproblemsolving.com/Store/viewitem.php?item=calculus Calculus] textbook, which is recommended for advanced middle and high school students. | * [[AoPS]] publishes Dr. [[David Patrick]]'s [http://www.artofproblemsolving.com/Store/viewitem.php?item=calculus Calculus] textbook, which is recommended for advanced middle and high school students. | ||
+ | * [https://www.amazon.com/Calculus-Vol-One-Variable-Introduction-Algebra/dp/0471000051 Calculus: Volume I] by [[Tom M. Apostol]] -- Provides a good transition into linear algebra which is uncommon in single variable calculus texts. | ||
+ | * [https://www.amazon.com/Single-Variable-Calculus-James-Stewart-dp-1305266633/dp/1305266633/ref=mt_other?_encoding=UTF8&me=&qid= Single Variable Calculus] by [[James Stewart]] -- Contains plenty of exercises for practice and focuses on application rather than rigor. | ||
* [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/0914098896/artofproblems-20 Calculus] by [[Michael Spivak]]. Top students swear by this book. | ||
− | * [ | + | * [https://press.princeton.edu/books/hardcover/9780691125336/honors-calculus Honors Calculus] by [[Charles R. MacCluer]] -- Uses the topological definition of the limit rather than the traditional delta-epsilon approach. |
− | * [ | + | |
+ | ==== Multivariable (Collegiate) ==== | ||
+ | * [https://www.amazon.com/dp/1305266641/?_encoding=UTF8&pd_rd_w=dgDsf&pf_rd_p=f0565570-f67b-4783-ab26-5a1f2c0bb3fd&pf_rd_r=7Y23GMHWH3DGTT7ZYJQF&pd_rd_r=a9ba1496-356e-4cbd-8e81-6d00bf440a1e&pd_rd_wg=OeBPr&ref_=bd_tags_dp_rec Multivariable Calculus] by [[James Stewart]]. | ||
+ | * [https://www.amazon.com/Advanced-calculus-Frederick-S-Woods/dp/B0006AMNBI 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." | ||
+ | * [https://www.amazon.com/Calculus-Vol-Multi-Variable-Applications-Differential/dp/0471000078/ref=sr_1_1?dchild=1&keywords=apostol+calculus+volume+2&qid=1634316891&sr=8-1 Calculus: Volume II] by [[Tom M. Apostol]]. | ||
+ | |||
+ | |||
+ | |||
+ | === Analysis === | ||
==== Collegiate ==== | ==== Collegiate ==== | ||
− | * [ | + | * [https://www.amazon.com/Understanding-Analysis-Undergraduate-Texts-Mathematics-ebook/dp/B00XWDQUH4/ref=reads_cwrtbar_4/141-5921801-5552153?pd_rd_w=qfNPT&pf_rd_p=0285128d-50e0-4388-acba-48a4a1f64720&pf_rd_r=KKVZB6CTYFYZZBXTX003&pd_rd_r=a64cf661-9db5-4d77-82ad-10b31b05dc41&pd_rd_wg=5WaBc&pd_rd_i=B00XWDQUH4&psc=1 Understanding Analysis] by [[Stephen Abbott]]. |
+ | * [https://www.amazon.com/Principles-Mathematical-Analysis-International-Mathematics/dp/007054235X 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. | ||
+ | * [https://www.amazon.com/Analysis-Third-Texts-Readings-Mathematics/dp/9380250649 Analysis I] by [[Terrence Tao]] -- An easier first read than Rudin, and provides plenty of examples with thorough explanations. | ||
+ | * [https://www.amazon.com/Analysis-II-Third-Readings-Mathematics/dp/9380250657/ref=pd_bxgy_img_1/141-5921801-5552153?pd_rd_w=uYcOn&pf_rd_p=c64372fa-c41c-422e-990d-9e034f73989b&pf_rd_r=G13XQBEGM3PWH1RT97BC&pd_rd_r=32b7fa0f-65e8-4d8c-ad45-9f1a1c3c592a&pd_rd_wg=cbkGA&pd_rd_i=9380250657&psc=1 Analysis II] by [[Terrence Tao]] -- Continues off from where Volume I ended and finishes at the Lebesgue Integral. | ||
+ | * [https://www.amazon.com/Real-Analysis-Integration-Princeton-Lectures-ebook/dp/B007BOK6PW Real Analysis] by [[Rami Shakarchi]] and [[Elias M. Stein]]. | ||
+ | * [https://www.amazon.com/Complex-Analysis-Elias-M-Stein-ebook/dp/B007K1BYD4/ref=reads_cwrtbar_1/141-5921801-5552153?pd_rd_w=hTDD7&pf_rd_p=0285128d-50e0-4388-acba-48a4a1f64720&pf_rd_r=VCM2JFE523FGVQE2HZ85&pd_rd_r=67a8e84d-8c73-4f2b-a3f1-463517afb999&pd_rd_wg=zaS7Z&pd_rd_i=B007K1BYD4&psc=1 Complex Analysis] by [[Rami Shakarchi]] and [[Elias M. Stein]]. | ||
+ | * [https://www.amazon.com/Real-Complex-Analysis-Higher-Mathematics/dp/0070542341/ref=pd_bxgy_img_1/141-5921801-5552153?pd_rd_w=bf70N&pf_rd_p=c64372fa-c41c-422e-990d-9e034f73989b&pf_rd_r=T14A8XPXYK2XY7TCTSAC&pd_rd_r=2c7958c3-a431-4714-a756-12927e7267f1&pd_rd_wg=hbV0A&pd_rd_i=0070542341&psc=1 Real and Complex Analysis] by [[Walter Rudin]]. Called "Papa Rudin" by some, Real and Complex Analysis is typically used at the graduate level. | ||
+ | * [https://www.amazon.com/gp/product/B005HDOLUK?notRedirectToSDP=1&ref_=dbs_mng_calw_2&storeType=ebooks Functional Analysis] by [[Rami Shakarchi]] and [[Elias M. Stein]]. | ||
+ | |||
=== Combinatorics === | === Combinatorics === | ||
+ | |||
==== Getting Started ==== | ==== Getting Started ==== | ||
* [[AoPS]] publishes Dr. [[David Patrick]]'s [http://www.artofproblemsolving.com/Store/viewitem.php?item=intro:counting Introduction to Counting & Probability] textbook, which is recommended for advanced middle and high school students. | * [[AoPS]] publishes Dr. [[David Patrick]]'s [http://www.artofproblemsolving.com/Store/viewitem.php?item=intro:counting Introduction to Counting & Probability] textbook, which is recommended for advanced middle and high school students. | ||
+ | |||
+ | |||
+ | |||
+ | https://www.awesomemath.org/product/112-combinatorial-problems-from-amsp/.112 problems is a great discrete math book covering topics ranging from permutations and combinations to using creativity to count to doing proofs and then gives exposure to advanced topics like probability theory.Great for AMC 8 /10/12 | ||
==== Intermediate ==== | ==== Intermediate ==== | ||
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* [http://www.amazon.com/exec/obidos/ASIN/0521789877/artofproblems-20 Enumerative Combinatorics, Volume 2] 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 | * [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 | ||
+ | * [https://www.amazon.com/Introductory-Combinatorics-Kenneth-P-Bogart/dp/0121108309 Introductory Combinatorics] by [[Kenneth P. Bogart]] | ||
=== Geometry === | === Geometry === | ||
+ | |||
==== Getting Started ==== | ==== Getting Started ==== | ||
* [[AoPS]] publishes [[Richard Rusczyk]]'s [http://www.artofproblemsolving.com/Store/viewitem.php?item=intro:geometry Introduction to Geometry] textbook, which is recommended for advanced middle and high school students. | * [[AoPS]] publishes [[Richard Rusczyk]]'s [http://www.artofproblemsolving.com/Store/viewitem.php?item=intro:geometry Introduction to Geometry] textbook, which is recommended for advanced middle and high school students. | ||
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==== Olympiad ==== | ==== Olympiad ==== | ||
− | * Euclidean Geometry in | + | * [https://www.amazon.com/gp/product/0883858398?%2AVersion%2A=1&%2Aentries%2A=0&pldnSite=1 Euclidean Geometry in Mathematical Olympiads] by Evan Chen |
+ | * [https://www.amazon.com/Solving-Problems-Geometry-Mathematical-Competitions/dp/981458374X/ref=sr_1_1?crid=2ZR4GP9R2R7KG&keywords=Solving+Problems+in+Geometry&qid=1646794260&sprefix=solving+problems+in+g%2Caps%2C1809&sr=8-1 Solving Problems In Geometry: Insights And Strategies For Mathematical Olympiad And Competitions] by Kim Hoo Hang and Haibin Wang | ||
* [http://www.amazon.com/exec/obidos/ASIN/0883856190/artofproblems-20 Geometry Revisited] -- A classic. | * [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/0486638308/artofproblems-20 Geometry of Complex Numbers] by Hans Schwerfdtfeger. | ||
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* [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/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/exec/obidos/ASIN/0387406239/artofproblems-20 Projective Geometry] by [[H.S.M. Coxeter]]. | ||
+ | |||
+ | |||
+ | |||
+ | ===Topology=== | ||
+ | |||
+ | ====Collegiate==== | ||
+ | * [https://www.amazon.com/Topology-James-Munkres-January-2000/dp/B015X4YE2M/ref=pd_sbs_14/141-5921801-5552153?pd_rd_w=u2xlH&pf_rd_p=690958f6-2825-419e-9c16-73ffd4055b65&pf_rd_r=JK1FK4KKJ1DSRW3V75C5&pd_rd_r=4c987b1e-0d6e-43ae-90ab-4cf445052ae5&pd_rd_wg=o2PH5&pd_rd_i=B015X4YE2M&psc=1 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 === | === Inequalities === | ||
+ | |||
==== Intermediate ==== | ==== Intermediate ==== | ||
* [http://www.amazon.com/exec/obidos/ASIN/0883856034/artofproblems-20 Introduction to Inequalities] | * [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] | * [http://www.amazon.com/exec/obidos/ASIN/0883856042/artofproblems-20 Geometric Inequalities] | ||
− | ==== Olympiad ==== | + | ==== Olympiad ==== |
+ | * [https://www.amazon.co.uk/Advanced-Olympiad-Inequalities-Algebraic-Geometric/dp/1794193928/ref=sr_1_fkmrnull_1?crid=XVQYS8R7NOL9&keywords=advanced+olympiad+inequalities&qid=1555930111&s=gateway&sprefix=advanced+ol%2Caps%2C165&sr=8-1-fkmrnull Advanced Olympiad Inequalities] by Alijadallah Belabess. | ||
* [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/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. | * [http://www.amazon.com/exec/obidos/ASIN/0387982191/artofproblems-20 Problem Solving Strategies] by Arthur Engel contains significant material on inequalities. | ||
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* [http://ultrametric.googlepages.com/tin2007.pdf Topics in Inequalities] by Hojoo Lee | * [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/MildorfInequalities.pdf Olympiad Inequalities] by Thomas Mildorf | ||
− | * [ | + | * [https://artofproblemsolving.com/articles/files/KedlayaInequalities.pdf A<B (A is less than B)] by Kiran S. Kedlaya |
* [http://can-hang2007.blogspot.com/2009/12/secrets-in-inequalities-volume-1-basic.html Secrets in Inequalities vol 1 and 2] by Pham Kim Hung | * [http://can-hang2007.blogspot.com/2009/12/secrets-in-inequalities-volume-1-basic.html Secrets in Inequalities vol 1 and 2] by Pham Kim Hung | ||
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=== Number Theory === | === Number Theory === | ||
− | ==== | + | |
+ | ==== Getting Started ==== | ||
* The AoPS [http://www.artofproblemsolving.com/Store/viewitem.php?item=intro:nt Introduction to Number Theory] by [[Mathew Crawford]]. | * The AoPS [http://www.artofproblemsolving.com/Store/viewitem.php?item=intro:nt Introduction to Number Theory] by [[Mathew Crawford]]. | ||
+ | *[https://www.amazon.com/Number-Theory-Dover-Books-Mathematics/dp/0486682528 Number Theory] by [[George E. Andrews]]. | ||
+ | |||
==== Olympiad ==== | ==== 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/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.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. | * [http://www.problem-solving.be/pen/published/pen-20070711.pdf Problems in Elementary Number Theory] by Hojoo Lee. | ||
+ | * [https://numbertheoryguy.com/publications/olympiad-number-theory-book/ Olympiad Number Theory through Challenging Problems] by Justin Stevens. | ||
+ | *[https://www.amazon.in/Elementary-Number-Theory-David-Burton/dp/1259025764 Elementary Number theory] by David M. Burton | ||
+ | *[https://drive.google.com/file/d/1BcJTLjQaelZ4w_70oHKyImC2I8zLfyrt/view Modern Olympiad Number Theory] by [[Aditya Khurmi]]. | ||
+ | |||
+ | ==== Collegiate ==== | ||
+ | * [https://www.amazon.com/Introduction-Theory-Numbers-G-Hardy/dp/0199219869 An Introduction to the Theory of Numbers] by [[G. H. Hardy]], [[Edward M. Wright]], and [[Andrew Wiles]] (6th Edition). | ||
+ | |||
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=== Problem Solving === | === Problem Solving === | ||
+ | |||
==== Getting Started ==== | ==== 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. | * 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. | ||
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* 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. | * 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/0471135712/artofproblems-20 The Art and Craft of Problem Solving] by [[Paul Zeitz]], former coach of the U.S. math team. | ||
− | * [ | + | * [https://www.amazon.com/How-Solve-Mathematical-Princeton-Science-dp-069111966X/dp/069111966X/ref=dp_ob_title_bk 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/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/0883857006/artofproblems-20 Proofs Without Words], [http://www.amazon.com/exec/obidos/ASIN/0883857219/artofproblems-20 Proofs Without Words II] | ||
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* [http://www.amazon.com/exec/obidos/ASIN/0387961712/artofproblems-20 Problem Solving Through Problems] by Loren Larson. | * [http://www.amazon.com/exec/obidos/ASIN/0387961712/artofproblems-20 Problem Solving Through Problems] by Loren Larson. | ||
− | + | == General Interest == | |
− | |||
− | == General | ||
* [http://www.amazon.com/exec/obidos/ASIN/0385495323/artofproblems-20 The Code Book] by Simon Singh. | * [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/0618251413/artofproblems-20 Count Down] by Steve Olson. | ||
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* [http://www.amazon.com/exec/obidos/ASIN/0195105192/artofproblems-20 What is Mathematics?]by Richard Courant, Herbert Robbins and Ian Stewart. | * [http://www.amazon.com/exec/obidos/ASIN/0195105192/artofproblems-20 What is Mathematics?]by Richard Courant, Herbert Robbins and Ian Stewart. | ||
− | == Math | + | == Math Contest Problem Books == |
=== Elementary School === | === 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]. | * [[Mathematical Olympiads for Elementary and Middle Schools]] ([[MOEMS]]) publishes [http://www.artofproblemsolving.com/Books/AoPS_B_CP_MOEMS.php two excellent contest problem books]. | ||
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=== Getting Started === | === Getting Started === | ||
− | * [http://www.artofproblemsolving.com/Books/AoPS_B_CP_MC.php | + | * [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.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. | * [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. | ||
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=== Collegiate === | === Collegiate === | ||
* Three [[Putnam competition]] books are [http://www.artofproblemsolving.com/Books/AoPS_B_CP_Putnam.php available at AoPS]. | * Three [[Putnam competition]] books are [http://www.artofproblemsolving.com/Books/AoPS_B_CP_Putnam.php available at AoPS]. | ||
− | |||
== See also == | == See also == |
Latest revision as of 11:10, 31 August 2024
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.
- Basic Algebra I by Nathan Jacobson -- Contains harder and more interesting problems than Dummit and Foote. Assumes a decent coverage of Linear Algebra
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.
https://www.awesomemath.org/product/112-combinatorial-problems-from-amsp/.112 problems is a great discrete math book covering topics ranging from permutations and combinations to using creativity to count to doing proofs and then gives exposure to advanced topics like probability theory.Great for AMC 8 /10/12
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.