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There are <a class="external text" href="https://en.wikibooks.org/w/index.php?title=Famous_Theorems_of_Mathematics&oldid=3693221&diff=cur&diffonly=0">template/file changes</a> awaiting review.</div></div></div></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><table class="metadata plainlinks ambox ambox-notice" style=""> <tbody><tr> <td class="mbox-image"><div style="width: 52px;"> <span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1d/Information_icon4.svg/40px-Information_icon4.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1d/Information_icon4.svg/60px-Information_icon4.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1d/Information_icon4.svg/80px-Information_icon4.svg.png 2x" data-file-width="620" data-file-height="620" /></span></span></div></td> <td class="mbox-text" style=""> <b>A reader requests expansion of this book to include more material.</b><br />You can help by <a class="external text" href="https://en.wikibooks.org/w/index.php?title=Famous_Theorems_of_Mathematics&action=edit">adding new material</a> (<i><a href="/wiki/Using_Wikibooks" title="Using Wikibooks">learn how</a></i>) or ask for assistance in the <a href="/wiki/Wikibooks:PROJECTS" class="mw-redirect" title="Wikibooks:PROJECTS">reading room</a>. </td> </tr> </tbody></table> <p>Not all of mathematics deals with proofs, as mathematics involves a rich range of human experience, including ideas, problems, patterns, mistakes and corrections. However, proofs are a very big part of modern mathematics, and today, it is generally considered that whatever statement, remark, result etc. one uses in mathematics, it is considered meaningless until is accompanied by a rigorous mathematical proof. This book is intended to contain the proofs (or sketches of proofs) of many famous theorems in mathematics in no particular order. It should be used both as a learning resource, a good practice for acquiring the skill for writing your own proofs is to study the existing ones, and for general references. </p><p>It is not however intended as a companion to any other wikibook or wikipedia articles but can complement them by providing them with links to the proofs of the theorems they contain. </p><p>One note here. There are usually many ways to solve a problem. Many times the proof used comes down to the primary definitions of terms involved. We will follow the definition given by the first major contributor. </p> <div class="mw-heading mw-heading2"><h2 id="Table_of_contents">Table of contents</h2></div> <div class="mw-heading mw-heading4"><h4 id="High_school">High school</h4></div> <ul><li><a href="/wiki/Famous_Theorems_of_Mathematics/Pythagoras_theorem" title="Famous Theorems of Mathematics/Pythagoras theorem">Pythagoras theorem</a></li> <li><a href="/wiki/Famous_Theorems_of_Mathematics/Euclid%27s_proof_of_the_infinitude_of_primes" title="Famous Theorems of Mathematics/Euclid's proof of the infinitude of primes">Euclid's proof of the infinitude of primes</a></li> <li><a href="/wiki/Famous_Theorems_of_Mathematics/%E2%88%9A2_is_irrational" title="Famous Theorems of Mathematics/√2 is irrational">√2 is irrational</a></li> <li><a href="/wiki/Famous_Theorems_of_Mathematics/Pythagorean_trigonometric_identity" title="Famous Theorems of Mathematics/Pythagorean trigonometric identity">sin<sup>2</sup>Θ+cos<sup>2</sup>Θ=1</a></li></ul> <div class="mw-heading mw-heading4"><h4 id="Undergraduate">Undergraduate</h4></div> <ul><li><a href="/wiki/Famous_Theorems_of_Mathematics/e_is_irrational" title="Famous Theorems of Mathematics/e is irrational">e is irrational</a></li> <li><a href="/wiki/Famous_Theorems_of_Mathematics/%CF%80_is_irrational" title="Famous Theorems of Mathematics/π is irrational">π is irrational</a></li> <li><a href="/wiki/Famous_Theorems_of_Mathematics/Fermat%27s_little_theorem" title="Famous Theorems of Mathematics/Fermat's little theorem">Fermat's little theorem</a></li> <li><a href="/w/index.php?title=Famous_Theorems_of_Mathematics/Fermat%27s_theorem_on_sums_of_two_squares&action=edit&redlink=1" class="new" title="Famous Theorems of Mathematics/Fermat's theorem on sums of two squares (does not exist)">Fermat's theorem on sums of two squares</a></li> <li><a href="/w/index.php?title=Famous_Theorems_of_Mathematics/Sum_of_the_reciprocals_of_the_primes_diverges&action=edit&redlink=1" class="new" title="Famous Theorems of Mathematics/Sum of the reciprocals of the primes diverges (does not exist)">Sum of the reciprocals of the primes diverges</a></li> <li><a href="/w/index.php?title=Famous_Theorems_of_Mathematics/Bertrand%27s_postulate&action=edit&redlink=1" class="new" title="Famous Theorems of Mathematics/Bertrand's postulate (does not exist)">Bertrand's postulate</a></li> <li><a href="/wiki/Famous_Theorems_of_Mathematics/Law_of_large_numbers" title="Famous Theorems of Mathematics/Law of large numbers">Law of large numbers</a></li> <li><a href="/w/index.php?title=Famous_Theorems_of_Mathematics/Spectral_Theorem&action=edit&redlink=1" class="new" title="Famous Theorems of Mathematics/Spectral Theorem (does not exist)">Spectral Theorem</a></li> <li><a href="/wiki/Famous_Theorems_of_Mathematics/L%27H%C3%B4pital%27s_rule" title="Famous Theorems of Mathematics/L'Hôpital's rule">L'Hôpital's rule</a></li> <li><a href="/wiki/Famous_Theorems_of_Mathematics/Four_color_theorem" title="Famous Theorems of Mathematics/Four color theorem">Four color theorem</a></li> <li><a href="/wiki/Famous_Theorems_of_Mathematics/Euler%27s_equation" title="Famous Theorems of Mathematics/Euler's equation">e<sup>π<i>i</i></sup>+1=0</a></li> <li><a href="/w/index.php?title=Famous_Theorems_of_Mathematics/Gaussian_integral&action=edit&redlink=1" class="new" title="Famous Theorems of Mathematics/Gaussian integral (does not exist)">Gaussian integral</a></li> <li><a href="/w/index.php?title=Famous_Theorems_of_Mathematics/Gamma_function&action=edit&redlink=1" class="new" title="Famous Theorems of Mathematics/Gamma function (does not exist)">Gamma function</a></li> <li><a href="/w/index.php?title=Famous_Theorems_of_Mathematics/Zeta_function&action=edit&redlink=1" class="new" title="Famous Theorems of Mathematics/Zeta function (does not exist)">Zeta function</a></li></ul> <div class="mw-heading mw-heading4"><h4 id="Postgraduate">Postgraduate</h4></div> <ul><li><a href="/wiki/Famous_Theorems_of_Mathematics/Fermat%27s_last_theorem" title="Famous Theorems of Mathematics/Fermat's last theorem">Fermat's last theorem</a></li> <li><a href="/wiki/Famous_Theorems_of_Mathematics/Brouwer_fixed-point_theorem" title="Famous Theorems of Mathematics/Brouwer fixed-point theorem">Brouwer fixed-point theorem</a></li> <li><a href="/w/index.php?title=Famous_Theorems_of_Mathematics/Prime_number_theorem&action=edit&redlink=1" class="new" title="Famous Theorems of Mathematics/Prime number theorem (does not exist)">Prime number theorem</a></li> <li><a href="/w/index.php?title=Famous_Theorems_of_Mathematics/Riemann_hypothesis&action=edit&redlink=1" class="new" title="Famous Theorems of Mathematics/Riemann hypothesis (does not exist)">Riemann hypothesis</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Old_table_of_contents">Old table of contents</h2></div> <style data-mw-deduplicate="TemplateStyles:r4440582">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}html.client-js body.skin-minerva .mw-parser-output .mbox-text-span{margin-left:23px!important}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-Warning plainlinks metadata ambox ambox-content" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Ambox_important.svg/40px-Ambox_important.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Ambox_important.svg/60px-Ambox_important.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Ambox_important.svg/80px-Ambox_important.svg.png 2x" data-file-width="40" data-file-height="40" /></span></span></div></td><td class="mbox-text" style="text-align: left;"><div class="mbox-text-span">This section contains the table of content of the book as according to its original intentions. The material here should be either incorporated in the existing book or discarded.</div></td></tr></tbody></table> <p>Proofs and definitions will be arranged according to the fields of mathematics: </p> <ul><li><a href="/wiki/Famous_Theorems_of_Mathematics/Algebra" title="Famous Theorems of Mathematics/Algebra">Algebra</a></li> <li><a href="/wiki/Famous_Theorems_of_Mathematics/Analysis" title="Famous Theorems of Mathematics/Analysis">Analysis</a></li> <li><a href="/wiki/Famous_Theorems_of_Mathematics/Applied_Mathematics" title="Famous Theorems of Mathematics/Applied Mathematics">Applied Mathematics</a></li> <li><a href="/wiki/Famous_Theorems_of_Mathematics/Geometry" title="Famous Theorems of Mathematics/Geometry">Geometry</a></li> <li><a href="/wiki/Famous_Theorems_of_Mathematics/Logic" title="Famous Theorems of Mathematics/Logic">Logic</a></li> <li><a href="/wiki/Famous_Theorems_of_Mathematics/Number_Theory" title="Famous Theorems of Mathematics/Number Theory">Number Theory</a></li> <li><a href="/wiki/Famous_Theorems_of_Mathematics/Set_Theory" title="Famous Theorems of Mathematics/Set Theory">Set Theory</a></li> <li><a href="/wiki/Famous_Theorems_of_Mathematics/Boy%27s_surface" title="Famous Theorems of Mathematics/Boy's surface">Boy's surface</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div> <ul><li><a href="/wiki/Mathematical_Proof" title="Mathematical Proof">Mathematical Proof</a> - about the theory and techniques of proving mathematical theorems</li></ul> <div class="mw-heading mw-heading2"><h2 id="Resources">Resources</h2></div> <div class="mw-heading mw-heading3"><h3 id="Manual_of_style">Manual of style</h3></div> <ul><li><a href="/wiki/Famous_Theorems_of_Mathematics/Proof_style" title="Famous Theorems of Mathematics/Proof style">Proof style</a> - Style guide for proofs.</li></ul> <div class="mw-heading mw-heading3"><h3 id="External_links_etc">External links etc</h3></div> <ul><li><a class="external text" href="https://en.wikipedia.org/wiki/Category:Article_proofs">Proofs on the wikipedia project</a></li> <li><a href="https://en.wikipedia.org/wiki/List_of_mathematical_proofs" class="extiw" title="w:List of mathematical proofs">List of mathematical proofs</a></li> <li><a href="https://en.wikipedia.org/wiki/List_of_theorems" class="extiw" title="w:List of theorems">List of theorems</a></li> <li><a rel="nofollow" class="external text" href="https://www.amazon.com/Mathematical-Handbook-Scientists-Engineers-Definitions/dp/0486411478/ref=sr_1_8/102-5853617-9460969?ie=UTF8&s=books&qid=1189341426&sr=1-8">Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review</a></li> <li><a rel="nofollow" class="external text" href="http://web.archive.org/web/20090330133636/http://personal.stevens.edu/~nkahl/Top100Theorems.html">The Hundred Greatest Theorems</a></li> <li><a rel="nofollow" class="external text" href="https://www.proofwiki.org">ProofWiki.org - The online compendium of mathematical proofs</a></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; 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