Mathematics | Definition, History, & Importance

<!doctype html> <html lang="en" class="topic-desktop ui-ie7 ui-ie"> <head prefix="og: https://ogp.me/ns# fb: https://ogp.me/ns/fb#"> <meta charset="utf-8"> <meta http-equiv="Content-Type" content="text/html; charset=UTF-8" /> <meta name="viewport" content="width=device-width, initial-scale=1.0" /> <link rel="dns-prefetch" href="https://cdn.britannica.com/mendel-resources/3-130"> <link rel="preconnect" href="https://cdn.britannica.com/mendel-resources/3-130"> <link rel="preload" as="script" href="https://www.googletagservices.com/tag/js/gpt.js" /> <link rel="icon" href="/favicon.png" /> <meta name="description" content="Mathematics, the science of structure, order, and relation that has evolved from counting, measuring, and describing the shapes of objects. Mathematics has been an indispensable adjunct to the physical sciences and technology and has assumed a similar role in the life sciences." /> <meta name="keywords" content="mathematics, encyclopedia, encyclopeadia, britannica, article" /> <link rel="canonical" href="https://www.britannica.com/science/mathematics" /> <title>Mathematics | Definition, History, & Importance | Britannica</title> <script type="text/javascript" data-type="Init Mendel"> window.$UI = {}; window.Constants = {"LICENSE_URL": "/bps/license","DEFAULT_TEST_VERSION": "A","DEFAULT_STATE": "XX","QUIZ_URL": "/quiz","SPOTLIGHT_BROWSE_URL": "/stories/spotlight","CONTENT_TYPE_TEXT": "text/plain;charset=UTF-8","TOPIC_FACTS_DATA_URL": "/facts","QUIZ_BROWSE_IMAGE_QUIZZES": "images","TOPIC_MEDIA_PATH": "/images-videos","USER_PROFILE_URL": "/user","DEBUG_URL": "/debug","ONE_GOOD_FACT_URL": "/one-good-fact","ERROR_404_URL": "/error404","PROCON_CITED_IN_THE_NEWS_URL": "/procon/ProCon-in-the-News","PROCON_URL": "/procon","TOPIC_PAGE_CONTENT_AJAX_URL": "/topic-content/page","INFINITE_SCROLL_PREFIX_URL": "/scroll","TOPIC_TOP_QUESTION_BROWSE_URL": "/questions","CC_USD": "USD","domain": "britannica.com","SURVEY_URL": "/survey","CATEGORY_BROWSE_URL": "/browse","STORY_BROWSE_URL": "/stories","COUNTRY_US": "US","OPEN_MEDIA_OVERLAY_PARAMETER": "/media","NEWSLETTER_SUBSCRIPTION_URL": "/newsletter-subscription","MAINTENANCE_ERROR_URL": "https://maintenance.eb.com","IMARS_EDITOR_ID": "12365882","PROFILE_EB_EDITOR_URL": "/editor","WEB_INF_RESOURCES_PATH": "WEB-INF/resources","AI_ABOUT_PAGE_URL": "/about-britannica-ai","TOPIC_ADDITIONAL_INFO_PATH": "/additional-info","SUDOKU_GAME_URL": "/games/sudoku","CC_INR": "INR","ARTICLE_PRINT_URL": "/print/article","FIRST_EDITION_URL": "/subscriber/firstedition","WW1_PORTAL_URL": "/discover/World-War-I","MENDEL_COOKIE": "__mendel","DEMYSTIFIED_BROWSE_URL": "/stories/demystified","LIST_BROWSE_URL": "/list/browse","PROFILE_EXPERT_URL": "/contributor","ASSEMBLY_IMAGE_URL": "/image/assembly","DAY_IN_HISTORY_URL": "/on-this-day","DEFAULT_CURRENCY": "USD","CONTENT_TYPE_XML": "text/xml;charset=UTF-8","PORTAL_FINANCE_BROWSE_URL_PREFIX": "/money/browse","MONEY_IMARS_CATEGORY": "13000","AJAX_PREFIX_URL": "/ajax","TOPIC_BROWSE_URL": "/topic-browse","MARKETING_CONTENT": "/marketing-content","ENV_RUNTIME": "runtime","GALLERY_URL": "/gallery","topicUrlClassesList": "topic|animal|art|biography|event|place|plant|science|sports|technology|procon","CONTENT_TYPE_HTML": "text/html;charset=UTF-8","ENV_LOCAL": "override","MEDIA_OVERLAY_URL": "/media-overlay","CHATBOT_PAGE_URL": "/chatbot","NEWSLETTER_PAGE_URL": "/newsletters","ENV_DEV": "development","MEDIA_URL": "/media","TOPIC_TOP_QUESTION_URL": "/question","PORTAL_FINANCE_URL_PREFIX": "/money","PODCASTS_URL": "/podcasts","STAND_ALONE_VIDEO_URL": "/video","MORE_ON_THIS_DAY_URL": "/more-on-this-day","TOPIC_QUOTES_URL": "/quotes","SEARCH_PAGE_URL": "/search","PROCON_CLASS": "PROCON","KUSTOM_MENDEL_APPLICATION_ID": "1","TOPIC_CONTENT_AJAX_URL": "/topic-content/topic","ENV_BRANCH": "branch","ERROR_URL": "/error","MAIN_VERSION": "mainVersion","DEFAULT_S3_REGION": "US_EAST_1","TOPIC_COLLECTION_URL": "/summary","LOGINBOX_URL": "/auth/loginbox","ONE_GOOD_FACT_BROWSE_URL": "/one-good-fact/all-good-facts","QUIZ_BROWSE_URL": "/quiz/browse","BIO_BROWSE_URL": "/browse/biographies","LIST_URL": "/list","TIGHTROPE_QUIZ_URL": "/quiz/tightrope","ALPHA_BROWSE_URL": "/sitemap","CONTENT_TYPE_JSON": "application/json","DICTIONARY_URL": "/dictionary","COBRAND_IMAGE_URL": "/image/cobrand","PROCON_IN_THE_NEWS_URL": "/procon/pro-and-con-issues-in-the-news","PROCON_BROWSE_URL": "/procon","QUIZ_BROWSE_VOCAB_QUIZZES": "vocabulary-quizzes","SUBMISSION_URL": "/submission","EB_LOG_OUT": "/auth2/logout","ENV_PRODUCTION": "production","EXPLORE_PORTAL_URL": "/explore","TOPIC_AJAX_URL": "/ajax/topic","TOPIC_SUMMARY_BROWSE_URL": "/summaries","WTFACT_BROWSE_URL": "/stories/wtfact","VIDEO_CHANNEL_URL": "/videos","GALLERY_BROWSE_URL": "/gallery/browse","CACHE_URL": "/cache","PROCON_ABOUT_URL": "/procon/About-ProCon","COMPANION_BROWSE_URL": "/stories/companion","MEDIA_FOLDER": "/eb-media","SHOW_ALL_CONTRIBUTORS": "/additional-info#contributors","BRITANNICA_EDITORS_ID": "4419","ENV_CACHE_DISABLED": "mendelCache","CALCULATORS_BROWSE_URL": "/calculators","STORY_URL": "/story","DEFAULT_COUNTRY": "US","NAVBAR_URL": "/ajax/navbar","EB_LOGIN_URL": "/auth/eb-login","NEW_ARTICLES_URL": "/new-articles",}; window.CDN = "https://cdn.britannica.com"; window.CAM_LOGIN_URL = "https://cam.britannica.com"; window.CAM_SIGN_UP_URL = "https://cam.britannica.com/registration" window.Mendel = { "config" : { "domain": "britannica.com", "page": "Topic", "videoPlayerId": "UyMCoK2v", "sharedUrl": "https://www.britannica.com/science/mathematics", "amuselabsUrl": "https://cdn3.amuselabs.com", "resourcesPrefixUrl": "https://cdn.britannica.com/mendel-resources/3-130/[url]?v=3.130.14", "date": 20241123, "userInfo": { "type": "ANONYMOUS" ,"currency": "AUUS" ,"country": "SG" ,"state": "XX" ,"timezone": "Asia/Singapore" ,"bcomId": "3002898432885867855" ,"hasAds": true ,"testVersion": "C" ,"adsTestVersion": "C" ,"consumerId": "" ,"instId": "" ,"consumerUserName": "" ,"instUserName": "" ,"cognito": null }, "tvs":{ "r":[25,25,25,25], "a": [25,25,25,25]}, "isLoggedInAsUser": false, "isPhone": false, "isDesktop": true, "logoutUrl": "/auth2/logout", "selfServiceUrl": "https://myaccount.britannica.com", "cdnUrl": "https://cdn.britannica.com", "chatbotApi": "https://www.britannica.com/chat-api", "fetchOffset": 800, "mendelCookieName": "__mendel", "mendelCookie": {"surveyShown":false,"visitedTopicId":369194,"currentDate":20241123}, "autocompleteToSearchPage": false,"topicUrl": "https://www.britannica.com/science/mathematics" ,"freeTopicReason": "NO_REFERRER" ,"topicId": 369194 ,"template": "DESKTOP" ,"type": "CORE" ,"hasToc": true ,"chatbotApi": "https://www.britannica.com/chat-api" ,"showPreview": false ,"infiniteScrollList": [{"p":27,"t":369194},{"p":1,"t":89161},{"p":10,"t":229851},{"p":11,"t":14885},{"p":10,"t":564172},{"p":1,"t":321794},{"p":7,"t":369237},{"p":3,"t":194901},{"p":2,"t":1947},{"p":8,"t":536159}] ,"sequence": 1 ,"topics": {} }, "GA": {"leg":"C","adLeg":"C","userType":"ANONYMOUS","pageType":"Topic","articleTemplateType":"PAGINATED","gisted":false,"pageNumber":1,"hasSummarizeButton":false,"hasAskButton":false} }; </script> <meta property="fb:app_id" content="1887621861548296"/ <meta name="twitter:card" content="summary_large_image" /> <meta name="twitter:site" content="@britannica" /> <meta name="twitter:image" content="https://cdn.britannica.com/54/2354-050-B47B3735/Babylonian-tablet.jpg" /> <meta name="twitter:description" content="Mathematics, the science of structure, order, and relation that has evolved from counting, measuring, and describing the shapes of objects. Mathematics has been an indispensable adjunct to the physical sciences and technology and has assumed a similar role in the life sciences."/> <meta property="og:type" content="ARTICLE"/> <meta property="og:title" content="Mathematics | Definition, History, & Importance | Britannica"/> <meta property="og:description" content="Mathematics, the science of structure, order, and relation that has evolved from counting, measuring, and describing the shapes of objects. Mathematics has been an indispensable adjunct to the physical sciences and technology and has assumed a similar role in the life sciences."/> <meta property="og:site_name" content="Encyclopedia Britannica" /> <meta property="og:url" content="https://www.britannica.com/science/mathematics"/> <meta property="og:image" content="https://cdn.britannica.com/54/2354-050-B47B3735/Babylonian-tablet.jpg" /> <meta property="og:image:type" content="image/jpeg" /> <script type="text/javascript" data-type="init opengraph"> Mendel.openGraph = {"type":"ARTICLE","title":"Mathematics | Definition, History, & Importance","description":"Mathematics, the science of structure, order, and relation that has evolved from counting, measuring, and describing the shapes of objects. Mathematics has been an indispensable adjunct to the physical sciences and technology and has assumed a similar role in the life sciences.","imageUrl":"https://cdn.britannica.com/54/2354-050-B47B3735/Babylonian-tablet.jpg","imageType":"image/jpeg","pageUrl":"https://www.britannica.com/science/mathematics"}</script> <link rel="preconnect" href="https://fonts.googleapis.com/"> <link rel="dns-prefetch" href="https://fonts.googleapis.com/" > <link rel="stylesheet" href="https://fonts.googleapis.com/icon?family=Material+Icons"> <link href="https://cdn.britannica.com/mendel-resources/3-130/dist/vendor-bundle.css?v=3.130.14" rel="stylesheet" /> <link href="https://cdn.britannica.com/mendel-resources/3-130/dist/mendel-css.css?v=3.130.14" rel="stylesheet" /> <link href="https://cdn.britannica.com/mendel-resources/3-130/dist/topic-page.css?v=3.130.14" rel="stylesheet" /> <script type="text/javascript"> if (self !== top) { top.location = self.location; } // if ('scrollRestoration' in history) { history.scrollRestoration = 'manual'; } </script> <script src="https://cdn.britannica.com/mendel-resources/3-130/js/at.js?v=3.130.14" async ></script> <script> dataLayer = []; </script> <script type="text/javascript">(function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start': new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0], j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src= '//www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f); })(window,document,'script','dataLayer','GTM-5W6NC8'); </script> <meta name="last-modified" content="2024-11-15" /> <script type="application/ld+json"> {"headline":"Mathematics | Definition, History, & Importance","image":{"url":"https://cdn.britannica.com/54/2354-050-B47B3735/Babylonian-tablet.jpg","@type":"ImageObject"},"author":[{"name":"Wilbur R. Knorr","url":"https://www.britannica.com/contributor/Wilbur-R-Knorr/3576","@type":"Person"},{"name":"John L. Berggren","url":"https://www.britannica.com/contributor/John-L-Berggren/3484","@type":"Person"}],"keywords":"mathematics","wordcount":41449,"url":"https://www.britannica.com/science/mathematics","datePublished":"1999-07-26T00:00:00Z","dateModified":"2024-11-15T00:00:00Z","description":"Mathematics, the science of structure, order, and relation that has evolved from counting, measuring, and describing the shapes of objects. Mathematics has been an indispensable adjunct to the physical sciences and technology and has assumed a similar role in the life sciences.","publisher":{"name":"Encyclopedia Britannica","@type":"Organization","logo":{"url":"https://corporate.britannica.com/wp-content/themes/eb-corporate/_img/logo.png","@type":"ImageObject"}},"@context":"https://schema.org","@type":"article"} </script></head> <body data-leg="C" class="new-topic topic-desktop first-page-true user-ANONYMOUS user-ads md-desktop leg-cb-ie">  <script>Mendel.config.adProvider='ay';</script> <script async src="https://securepubads.g.doubleclick.net/tag/js/gpt.js"></script> <script> window.googletag = window.googletag || {cmd: []}; googletag.cmd.push(function() { googletag.defineSlot('/15510053/CMP_1x1', [1, 1], 'div-gpt-ad-1709766812090-0').addService(googletag.pubads()); googletag.pubads().enableSingleRequest(); googletag.enableServices(); }); </script> <script async defer src="https://launchpad-wrapper.privacymanager.io/0ccc6fe8-1870-4ad8-b47b-6d029ac116fc/launchpad-liveramp.js"></script> <script async src="https://JRyhoywLYXNLYMAhs.ay.delivery/manager/JRyhoywLYXNLYMAhs" type="text/javascript" referrerpolicy="no-referrer-when-downgrade" ></script><div class="ie-warning d-flex align-items-center align-self-center justify-content-center site-alert bg-orange"> <div> You are using an <strong>outdated</strong> browser. Please <a class="text-white text-underscore" href="https://browsehappy.com/">upgrade your browser</a> to improve your experience and security. </div> </div> <script id="json-navbar-info" type="application/json"> {"topSectionLinks":[{"title":"Ask the Chatbot","url":"/chatbot","navbarId":"CHATBOT"},{"title":"Games & Quizzes","url":"/quiz/browse","navbarId":"QUIZZES"},{"title":"ProCon","url":"/procon","navbarId":"PROCON"},{"title":"History & Society","url":"/History-Society","navbarId":"HISTORY"},{"title":"Science & Tech","url":"/Science-Tech","selected":true,"navbarId":"SCIENCE"},{"title":"Biographies","url":"/Biographies","navbarId":"BIOS"},{"title":"Animals & Nature","url":"/Animals-Nature","navbarId":"ANIMALS"},{"title":"Geography & Travel","url":"/Geography-Travel","navbarId":"GEOGRAPHY"},{"title":"Arts & Culture","url":"/Arts-Culture","navbarId":"ART"},{"title":"Money","url":"/money","navbarId":"MONEY"},{"title":"Videos","url":"/videos","navbarId":"VIDEOS"}],"selectedSuperCategory":{"id":6,"title":"Science & Tech","url":"Science-Tech","description":"Explore science and technology; astronomy; biology; chemistry; earth science; mathematics; physics; technology, agriculture, cars, computers, engineering, industry, inventions, communication","keywords":"astronomy; biology; chemistry; earth science; mathematics; physics; technology, agriculture, cars, computers, engineering, industry, inventions, communication","classId":"SCIENCE","sortOrder":2},"selectedNavbarLink":{"title":"Science & Tech","url":"/Science-Tech","selected":true,"navbarId":"SCIENCE"}} </script> <script id="json-hamburger-menu" type="application/json"> {"britannicaMenu1":[{"title":"Home","url":"/"},{"title":"ProCon","url":"/procon"},{"title":"History & Society","url":"/History-Society"},{"title":"Science & Tech","url":"/Science-Tech"},{"title":"Biographies","url":"/Biographies"},{"title":"Animals & Nature","url":"/Animals-Nature"},{"title":"Geography & Travel","url":"/Geography-Travel"},{"title":"Arts & Culture","url":"/Arts-Culture"},{"title":"Money","url":"/money"}],"britannicaMenu2":[{"title":"Games & Quizzes","url":"/quiz/browse"},{"title":"Videos","url":"/videos"},{"title":"On This Day","url":"/on-this-day"},{"title":"One Good Fact","url":"/one-good-fact"},{"title":"Dictionary","url":"/dictionary"},{"title":"New Articles","url":"/new-articles"}],"browseByCategory":[{"title":{"id":5,"title":"History & Society","url":"/History-Society"},"links":[{"title":"Lifestyles & Social Issues","url":"/browse/Lifestyles-Social-Issues"},{"title":"Philosophy & Religion","url":"/browse/Philosophy-Religion"},{"title":"Politics, Law & Government","url":"/browse/Politics-Law-Government"},{"title":"World History","url":"/browse/World-History"}]},{"title":{"id":6,"title":"Science & Tech","url":"/Science-Tech"},"links":[{"title":"Health & Medicine","url":"/browse/Health-Medicine"},{"title":"Science","url":"/browse/Science"},{"title":"Technology","url":"/browse/Technology"}]},{"title":{"id":3,"title":"Biographies","url":"/Biographies"},"links":[{"title":"Browse Biographies","url":"/browse/biographies"}]},{"title":{"id":1,"title":"Animals & Nature","url":"/Animals-Nature"},"links":[{"title":"Birds, Reptiles & Other Vertebrates","url":"/browse/Birds-Reptiles-Vertebrates"},{"title":"Bugs, Mollusks & Other Invertebrates","url":"/browse/Bugs-Mollusks-Invertebrates"},{"title":"Environment","url":"/browse/Environment"},{"title":"Fossils & Geologic Time","url":"/browse/Fossil-Geologic-Time"},{"title":"Mammals","url":"/browse/Mammals"},{"title":"Plants","url":"/browse/Plants"}]},{"title":{"id":4,"title":"Geography & Travel","url":"/Geography-Travel"},"links":[{"title":"Geography & Travel","url":"/browse/Geography-Travel"}]},{"title":{"id":2,"title":"Arts & Culture","url":"/Arts-Culture"},"links":[{"title":"Entertainment & Pop Culture","url":"/browse/Entertainment-Pop-Culture"},{"title":"Literature","url":"/browse/Literature"},{"title":"Sports & Recreation","url":"/browse/Sports-Recreation"},{"title":"Visual Arts","url":"/browse/Visual-Arts"}]}],"browseByFeature":[{"title":"Companions","url":"/stories/companion"},{"title":"Demystified","url":"/stories/demystified"},{"title":"Image Galleries","url":"/gallery/browse"},{"title":"Lists","url":"/list/browse"},{"title":"Podcasts","url":"/podcasts"},{"title":"Spotlight","url":"/stories/spotlight"},{"title":"Summaries","url":"/summary"},{"title":"The Forum","url":"/stories/the-forum"},{"title":"Top Questions","url":"/question"},{"title":"#WTFact","url":"/stories/wtfact"}],"moreFromBritannica":[{"title":"Britannica Kids","url":"https://kids.britannica.com/","newTab":true}],"menuType":"DEFAULT"} </script> <header id="header" class="bg-navy-dark"> <div class="global-nav-top-bar"> <div class="grid gx-0 h-100 justify-content-between align-items-center container-lg mx-auto p-0 position-relative"> <div class="d-flex align-items-center"> <button class="d-flex align-items-center justify-self-start js-toggle js-toggle-hamburger btn btn-link link-white btn-sm rounded-0 p-10"> <div class="hamburger-tooltip"> <em class="material-icons d-inline-block font-24" id="nav-toggle" data-icon="menu"></em> </div> <em class="material-icons d-inline-block font-24 global-nav-search-icon" id="nav-search-icon" data-icon="search" ></em> </button> <a href="/" class="d-flex align-items-center justify-content-center ml-10"> <img loading="lazy" src="https://cdn.britannica.com/mendel/eb-logo/MendelNewThistleLogo.png" alt="Encyclopedia Britannica" class="global-nav-logo global-nav-logo-left" /> </a> <div class="global-nav-top-search-bar global-nav-top-search-container global-nav-search-container" id="global-nav-top-search-bar"> <form method="get" action="/search" id="global-nav-search" class="md-search-form m-0 global-nav-search-bar-small"> <div class="search-box position-relative col-100"> <label class="sr-only" for="global-nav-search-query">Search Britannica</label> <input name="query" id="global-nav-search-query" placeholder="Search Britannica..." class="form-control form-control-lg rounded-lg font-16 search-query pl-20 pr-70 shadow-sm" maxlength="200" autocomplete="off" aria-label="Search Britannica" /> <button class="search-reset-btn btn btn-link px-10 position-absolute top-0 h-100 d-none" type="reset"> <em class="material-icons" data-icon="close"></em> </button> <button class="search-submit btn btn-link text-blue px-10 position-absolute top-0 right-0 h-100" type="submit" disabled> <span class="sr-only">Click here to search</span> <em class="material-icons search-icon" data-icon="search"></em> </button> </div> </form> </div> </div> <a href="/" class="d-flex align-items-center justify-content-center"> <img loading="lazy" src="https://cdn.britannica.com/mendel/eb-logo/MendelNewThistleLogo.png" alt="Encyclopedia Britannica" class="global-nav-center global-nav-logo non-homepage-logo" /> </a> <form method="get" action="/search" id="global-nav-search" class="md-search-form m-0 global-nav-search-bar-small global-nav-center search global-nav-center-search-container"> <div class="search-box position-relative col-100"> <label class="sr-only" for="global-nav-search-query">Search Britannica</label> <input name="query" id="global-nav-search-query" placeholder="Search Britannica..." class="form-control form-control-lg rounded-lg font-16 search-query pl-20 pr-70 shadow-sm" maxlength="200" autocomplete="off" aria-label="Search Britannica" /> <button class="search-reset-btn btn btn-link px-10 position-absolute top-0 h-100 d-none" type="reset"> <em class="material-icons" data-icon="close"></em> </button> <button class="search-submit btn btn-link text-blue px-10 position-absolute top-0 right-0 h-100" type="submit" disabled> <span class="sr-only">Click here to search</span> <em class="material-icons search-icon" data-icon="search"></em> </button> </div> </form> <div class="col-35 col-sm-auto text-right order-3 mr-lg-15 align-items-center d-flex justify-content-end"> <div class="d-none d-md-inline-block"> <SPAN class="marketing-HEADER_SUBSCRIPTION_DESKTOP2 marketing-content" data-marketing-id="HEADER_SUBSCRIPTION_DESKTOP2"><a href="https://subscription.britannica.com/subscribe?partnerCode=BP_Black_Friday_AUUS" class="subscribe-link btn btn-sm btn-outline-white-orange py-5 mr-10" target="_blank" rel="noopener"> Subscribe </a></SPAN></div> <div class="d-inline-block d-md-none mr-5 mr-sm-10"> <SPAN class="marketing-HEADER_SUBSCRIPTION_MOBILE marketing-content" data-marketing-id="HEADER_SUBSCRIPTION_MOBILE"><a href="https://subscription.britannica.com/subscribe?partnerCode=BP_Black_Friday_AUUS" class="subscribe-link btn btn-xs btn-orange-dark p-5" target="_blank" rel="noopener"> Subscribe </a></SPAN></div> <button class="js-toggle-user-dropdown js-toggle btn btn-sm btn-link link-white rounded-0 px-md-15 pl-5 pr-5"> <span class="d-none d-md-inline-block mr-5">Login</span> <em class="material-icons d-inline-block d-md-none font-16 font-sm-20" data-icon="account_circle"></em> <div class="d-none dropdown-menu-subscription-link">https://subscription.britannica.com/subscribe?partnerCode=BP_Black_Friday_AUUS</div> <em class="material-icons inactive-icon d-inline-block font-18" data-icon="keyboard_arrow_down"></em> <em class="material-icons active-icon d-inline-block font-18" data-icon="keyboard_arrow_up"></em> </button> </div> </div> </div> <div class="d-none hamburger-menu-subscription-link"><DIV class="marketing-HAMBURGER_MENU_CTA marketing-content" data-marketing-id="HAMBURGER_MENU_CTA"><a href="https://premium.britannica.com/premium-membership/?utm_source=premium&utm_medium=hamburger-menu&utm_campaign=black-friday-2024" class="subscribe-link btn btn-sm btn-orange py-5" target="_blank"> Subscribe <span class="d-none d-md-inline"> Now</span> </a></DIV></div> <div id="global-nav-react"> <div class="d-none"> <ul> <li><a href="/">Home</a></li> <li><a href="/procon">ProCon</a></li> <li><a href="/History-Society">History & Society</a></li> <li><a href="/Science-Tech">Science & Tech</a></li> <li><a href="/Biographies">Biographies</a></li> <li><a href="/Animals-Nature">Animals & Nature</a></li> <li><a href="/Geography-Travel">Geography & Travel</a></li> <li><a href="/Arts-Culture">Arts & Culture</a></li> <li><a href="/money">Money</a></li> </ul> <ul> <li><a href="/quiz/browse">Games & Quizzes</a></li> <li><a href="/videos">Videos</a></li> <li><a href="/on-this-day">On This Day</a></li> <li><a href="/one-good-fact">One Good Fact</a></li> <li><a href="/dictionary">Dictionary</a></li> <li><a href="/new-articles">New Articles</a></li> </ul> <a href="/History-Society">History & Society</a> <ul> <li><a href="/browse/Lifestyles-Social-Issues">Lifestyles & Social Issues</a></li> <li><a href="/browse/Philosophy-Religion">Philosophy & Religion</a></li> <li><a href="/browse/Politics-Law-Government">Politics, Law & Government</a></li> <li><a href="/browse/World-History">World History</a></li> </ul> <a href="/Science-Tech">Science & Tech</a> <ul> <li><a href="/browse/Health-Medicine">Health & Medicine</a></li> <li><a href="/browse/Science">Science</a></li> <li><a href="/browse/Technology">Technology</a></li> </ul> <a href="/Biographies">Biographies</a> <ul> <li><a href="/browse/biographies">Browse Biographies</a></li> </ul> <a href="/Animals-Nature">Animals & Nature</a> <ul> <li><a href="/browse/Birds-Reptiles-Vertebrates">Birds, Reptiles & Other Vertebrates</a></li> <li><a href="/browse/Bugs-Mollusks-Invertebrates">Bugs, Mollusks & Other Invertebrates</a></li> <li><a href="/browse/Environment">Environment</a></li> <li><a href="/browse/Fossil-Geologic-Time">Fossils & Geologic Time</a></li> <li><a href="/browse/Mammals">Mammals</a></li> <li><a href="/browse/Plants">Plants</a></li> </ul> <a href="/Geography-Travel">Geography & Travel</a> <ul> <li><a href="/browse/Geography-Travel">Geography & Travel</a></li> </ul> <a href="/Arts-Culture">Arts & Culture</a> <ul> <li><a href="/browse/Entertainment-Pop-Culture">Entertainment & Pop Culture</a></li> <li><a href="/browse/Literature">Literature</a></li> <li><a href="/browse/Sports-Recreation">Sports & Recreation</a></li> <li><a href="/browse/Visual-Arts">Visual Arts</a></li> </ul> <ul> <li><a href="/stories/companion">Companions</a></li> <li><a href="/stories/demystified">Demystified</a></li> <li><a href="/gallery/browse">Image Galleries</a></li> <li><a href="/list/browse">Lists</a></li> <li><a href="/podcasts">Podcasts</a></li> <li><a href="/stories/spotlight">Spotlight</a></li> <li><a href="/summary">Summaries</a></li> <li><a href="/stories/the-forum">The Forum</a></li> <li><a href="/question">Top Questions</a></li> <li><a href="/stories/wtfact">#WTFact</a></li> </ul> <ul> <li><a href="https://kids.britannica.com/">Britannica Kids</a></li> </ul> </div> </div> </header> <div class="bg-navy-dark"> <div class="container-lg p-0 d-flex justify-content-center global-nav-categories-bar overflow-hidden"> <div class="slider js-slider position-relative d-inline-flex align-items-center mw-100 global-nav-slider category-snap-slider"> <div class="slider-container js-slider-container overflow-hidden d-flex font-14 overflow-hidden text-nowrap mx-5"> <a class="nav-bar-category mx-5 category-link-CHATBOT " href="/chatbot">Ask the Chatbot</a> <a class="nav-bar-category mx-5 category-link-QUIZZES " href="/quiz/browse">Games & Quizzes</a> <a class="nav-bar-category mx-5 category-link-PROCON " href="/procon">ProCon</a> <a class="nav-bar-category mx-5 category-link-HISTORY " href="/History-Society">History & Society</a> <a class="nav-bar-category mx-5 category-link-SCIENCE selected selected" href="/Science-Tech">Science & Tech</a> <a class="nav-bar-category mx-5 category-link-BIOS " href="/Biographies">Biographies</a> <a class="nav-bar-category mx-5 category-link-ANIMALS " href="/Animals-Nature">Animals & Nature</a> <a class="nav-bar-category mx-5 category-link-GEOGRAPHY " href="/Geography-Travel">Geography & Travel</a> <a class="nav-bar-category mx-5 category-link-ART " href="/Arts-Culture">Arts & Culture</a> <a class="nav-bar-category mx-5 category-link-MONEY " href="/money">Money</a> <a class="nav-bar-category mx-5 category-link-VIDEOS " href="/videos">Videos</a> </div> <button disabled class="prev-button js-prev-button position-absolute btn btn-circle shadow btn-blue " aria-label="Previous"> <span class="material-icons md-24" data-icon="keyboard_arrow_left"></span> </button> <button disabled class="next-button js-next-button position-absolute btn btn-circle shadow btn-blue " aria-label="Next"> <span class="material-icons md-24" data-icon="keyboard_arrow_right"></span> </button> </div> </div> </div> <main> <div class="md-page-wrapper"> <div id="content" class="md-content"> <div class="md-article-container template-desktop infinite-pagination"> <div class="infinite-scroll-container article last"> <article class="article-content container-lg qa-content px-0 pt-0 pb-40 py-lg-20 content md-expanded" data-topic-id="369194"> <div class="grid gx-0"> <div class="col-auto"> <div class="topic-left-rail md-article-drawer position-relative d-flex border-right-sm border-left-sm open"> <div class="drawer d-flex flex-column open"> <div class="left-rail-section-content"> <div class="topic-left-rail-header text-truncate bg-gray-50 position-relative text-right d-flex align-items-center"> <div class="tlr-title px-20 py-15 text-left"> <em class="material-icons text-gray-400 d-lg-none" data-icon="toc"></em> <a class="font-serif font-weight-bold text-black link-blue" href="https://www.britannica.com/science/mathematics">mathematics</a> </div> <button aria-label="Close" class="js-sections-close-button btn-link btn-sm btn d-lg-none position-absolute top-0 p-10 right-0" > <em class="material-icons font-26" data-icon="close"></em> </button> </div> <div class="section-content pl-10 pr-20 pl-sm-50 pr-sm-60 pl-lg-5 pr-lg-10 pt-10 pt-lg-0 bg-gray-50 clear-catfish-ad"> <div class="toc mb-20"> <div class="font-serif font-14 font-weight-bold mx-15 mb-15 mt-20"> Table of Contents </div> <ul class="list-unstyled my-0" data-level="h1"><li data-target="#ref1"><div class="pl-25"><a class="link-gray-900 w-100" href="/science/mathematics">Introduction</a></div><div class="ml-40 toc-drawer sub-toc-drawer"></div></li><li data-target="#ref253521"><div class="d-flex align-items-center"><div class="ml-25"></div><a class="w-100 link-gray-900" href="/science/mathematics/Ancient-mathematical-sources">Ancient mathematical sources</a></div><div class="ml-40 toc-drawer sub-toc-drawer"></div></li><li data-target="#ref65969"><div class="d-flex align-items-center"><button class="h1-link-drawer-button btn btn-xs btn-circle d-flex rounded" type="button" aria-label="Toggle Heading"><em class="material-icons font-18" data-icon="keyboard_arrow_right"></em></button><a class="w-100 link-gray-900" href="/science/mathematics/Ancient-mathematical-sources#ref65969">Mathematics in ancient Mesopotamia</a></div><div class="ml-40 toc-drawer sub-toc-drawer"><ul class="list-unstyled" data-level="h2"><li data-target="#ref65970"><a class="w-100 link-gray-900" href="/science/mathematics/Ancient-mathematical-sources#ref65970">The numeral system and arithmetic operations</a></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref65971"><a class="w-100 link-gray-900" href="/science/mathematics/Geometric-and-algebraic-problems">Geometric and algebraic problems</a></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref65972"><a class="w-100 link-gray-900" href="/science/mathematics/Geometric-and-algebraic-problems#ref65972">Mathematical astronomy</a></li></ul></div></li><li data-target="#ref65973"><div class="d-flex align-items-center"><button class="h1-link-drawer-button btn btn-xs btn-circle d-flex rounded" type="button" aria-label="Toggle Heading"><em class="material-icons font-18" data-icon="keyboard_arrow_right"></em></button><a class="w-100 link-gray-900" href="/science/mathematics/Mathematics-in-ancient-Egypt">Mathematics in ancient Egypt</a></div><div class="ml-40 toc-drawer sub-toc-drawer"><ul class="list-unstyled" data-level="h2"><li data-target="#ref65974"><a class="w-100 link-gray-900" href="/science/mathematics/Mathematics-in-ancient-Egypt#ref65974">The numeral system and arithmetic operations</a></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref65975"><a class="w-100 link-gray-900" href="/science/mathematics/Mathematics-in-ancient-Egypt#ref65975">Geometry</a></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref65976"><a class="w-100 link-gray-900" href="/science/mathematics/Assessment-of-Egyptian-mathematics">Assessment of Egyptian mathematics</a></li></ul></div></li><li data-target="#ref65977"><div class="d-flex align-items-center"><button class="h1-link-drawer-button btn btn-xs btn-circle d-flex rounded" type="button" aria-label="Toggle Heading"><em class="material-icons font-18" data-icon="keyboard_arrow_right"></em></button><a class="w-100 link-gray-900" href="/science/mathematics/Assessment-of-Egyptian-mathematics#ref65977">Greek mathematics</a></div><div class="ml-40 toc-drawer sub-toc-drawer"><ul class="list-unstyled" data-level="h2"><li data-target="#ref65978"><a class="w-100 link-gray-900" href="/science/mathematics/Assessment-of-Egyptian-mathematics#ref65978">The development of pure mathematics</a><ul class="list-unstyled" data-level="h3"><li data-target="#ref65979"><a class="w-100 link-gray-900" href="/science/mathematics/Assessment-of-Egyptian-mathematics#ref65979">The pre-Euclidean period</a></li></ul><ul class="list-unstyled" data-level="h3"><li data-target="#ref65980"><a class="w-100 link-gray-900" href="/science/mathematics/Assessment-of-Egyptian-mathematics#ref65980">The <em>Elements</em></a></li></ul><ul class="list-unstyled" data-level="h3"><li data-target="#ref65981"><a class="w-100 link-gray-900" href="/science/mathematics/The-three-classical-problems">The three classical problems</a></li></ul></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref65982"><a class="w-100 link-gray-900" href="/science/mathematics/The-three-classical-problems#ref65982">Geometry in the 3rd century <span class="text-smallcaps">bce</span></a><ul class="list-unstyled" data-level="h3"><li data-target="#ref65983"><a class="w-100 link-gray-900" href="/science/mathematics/The-three-classical-problems#ref65983">Archimedes</a></li></ul><ul class="list-unstyled" data-level="h3"><li data-target="#ref65984"><a class="w-100 link-gray-900" href="/science/mathematics/Apollonius">Apollonius</a></li></ul><ul class="list-unstyled" data-level="h3"><li data-target="#ref65985"><a class="w-100 link-gray-900" href="/science/mathematics/Applied-geometry">Applied geometry</a></li></ul></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref65986"><a class="w-100 link-gray-900" href="/science/mathematics/Applied-geometry#ref65986">Later trends in geometry and arithmetic</a><ul class="list-unstyled" data-level="h3"><li data-target="#ref65987"><a class="w-100 link-gray-900" href="/science/mathematics/Applied-geometry#ref65987">Greek trigonometry and mensuration</a></li></ul><ul class="list-unstyled" data-level="h3"><li data-target="#ref65988"><a class="w-100 link-gray-900" href="/science/mathematics/Number-theory">Number theory</a></li></ul><ul class="list-unstyled" data-level="h3"><li data-target="#ref65989"><a class="w-100 link-gray-900" href="/science/mathematics/Number-theory#ref65989">Survival and influence of Greek mathematics</a></li></ul></li></ul></div></li><li data-target="#ref65990"><div class="d-flex align-items-center"><button class="h1-link-drawer-button btn btn-xs btn-circle d-flex rounded" type="button" aria-label="Toggle Heading"><em class="material-icons font-18" data-icon="keyboard_arrow_right"></em></button><a class="w-100 link-gray-900" href="/science/mathematics/Mathematics-in-the-Islamic-world-8th-15th-century">Mathematics in the Islamic world (8th–15th century)</a></div><div class="ml-40 toc-drawer sub-toc-drawer"><ul class="list-unstyled" data-level="h2"><li data-target="#ref65991"><a class="w-100 link-gray-900" href="/science/mathematics/Mathematics-in-the-Islamic-world-8th-15th-century#ref65991">Origins</a></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref65992"><a class="w-100 link-gray-900" href="/science/mathematics/Mathematics-in-the-Islamic-world-8th-15th-century#ref65992">Mathematics in the 9th century</a></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref65993"><a class="w-100 link-gray-900" href="/science/mathematics/Mathematics-in-the-Islamic-world-8th-15th-century#ref65993">Mathematics in the 10th century</a></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref65994"><a class="w-100 link-gray-900" href="/science/mathematics/Omar-Khayyam">Omar Khayyam</a></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref65995"><a class="w-100 link-gray-900" href="/science/mathematics/Omar-Khayyam#ref65995">Islamic mathematics to the 15th century</a></li></ul></div></li><li data-target="#ref65996"><div class="d-flex align-items-center"><button class="h1-link-drawer-button btn btn-xs btn-circle d-flex rounded" type="button" aria-label="Toggle Heading"><em class="material-icons font-18" data-icon="keyboard_arrow_right"></em></button><a class="w-100 link-gray-900" href="/science/mathematics/Omar-Khayyam#ref65996">European mathematics during the Middle Ages and Renaissance</a></div><div class="ml-40 toc-drawer sub-toc-drawer"><ul class="list-unstyled" data-level="h2"><li data-target="#ref65997"><a class="w-100 link-gray-900" href="/science/mathematics/The-transmission-of-Greek-and-Arabic-learning">The transmission of Greek and Arabic learning</a></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref65998"><a class="w-100 link-gray-900" href="/science/mathematics/The-transmission-of-Greek-and-Arabic-learning#ref65998">The universities</a></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref65999"><a class="w-100 link-gray-900" href="/science/mathematics/The-transmission-of-Greek-and-Arabic-learning#ref65999">The Renaissance</a></li></ul></div></li><li data-target="#ref66000"><div class="d-flex align-items-center"><button class="h1-link-drawer-button btn btn-xs btn-circle d-flex rounded" type="button" aria-label="Toggle Heading"><em class="material-icons font-18" data-icon="keyboard_arrow_right"></em></button><a class="w-100 link-gray-900" href="/science/mathematics/Mathematics-in-the-17th-and-18th-centuries">Mathematics in the 17th and 18th centuries</a></div><div class="ml-40 toc-drawer sub-toc-drawer"><ul class="list-unstyled" data-level="h2"><li data-target="#ref66001"><a class="w-100 link-gray-900" href="/science/mathematics/Mathematics-in-the-17th-and-18th-centuries#ref66001">The 17th century</a><ul class="list-unstyled" data-level="h3"><li data-target="#ref66002"><a class="w-100 link-gray-900" href="/science/mathematics/Mathematics-in-the-17th-and-18th-centuries#ref66002">Institutional background</a></li></ul><ul class="list-unstyled" data-level="h3"><li data-target="#ref66003"><a class="w-100 link-gray-900" href="/science/mathematics/Mathematics-in-the-17th-and-18th-centuries#ref66003">Numerical calculation</a></li></ul><ul class="list-unstyled" data-level="h3"><li data-target="#ref66004"><a class="w-100 link-gray-900" href="/science/mathematics/Analytic-geometry">Analytic geometry</a></li></ul><ul class="list-unstyled" data-level="h3"><li data-target="#ref66005" class="has-children"><a class="w-100 link-gray-900" href="/science/mathematics/The-calculus">The calculus</a><ul class="list-unstyled" data-level="h4"><li data-target="#ref66006"><a class="w-100 link-gray-900" href="/science/mathematics/The-calculus#ref66006">The precalculus period</a></li></ul><ul class="list-unstyled" data-level="h4"><li data-target="#ref66007"><a class="w-100 link-gray-900" href="/science/mathematics/Newton-and-Leibniz">Newton and Leibniz</a></li></ul></li></ul></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref66008"><a class="w-100 link-gray-900" href="/science/mathematics/Newton-and-Leibniz#ref66008">The 18th century</a><ul class="list-unstyled" data-level="h3"><li data-target="#ref66009"><a class="w-100 link-gray-900" href="/science/mathematics/Newton-and-Leibniz#ref66009">Institutional background</a></li></ul><ul class="list-unstyled" data-level="h3"><li data-target="#ref66010"><a class="w-100 link-gray-900" href="/science/mathematics/Analysis-and-mechanics">Analysis and mechanics</a></li></ul><ul class="list-unstyled" data-level="h3"><li data-target="#ref66011"><a class="w-100 link-gray-900" href="/science/mathematics/Analysis-and-mechanics#ref66011">History of analysis</a></li></ul></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref66012"><a class="w-100 link-gray-900" href="/science/mathematics/Analysis-and-mechanics#ref66012">Other developments</a><ul class="list-unstyled" data-level="h3"><li data-target="#ref66013"><a class="w-100 link-gray-900" href="/science/mathematics/Theory-of-equations">Theory of equations</a></li></ul><ul class="list-unstyled" data-level="h3"><li data-target="#ref66014"><a class="w-100 link-gray-900" href="/science/mathematics/Theory-of-equations#ref66014">Foundations of geometry</a></li></ul></li></ul></div></li><li data-target="#ref335980"><div class="d-flex align-items-center"><button class="h1-link-drawer-button btn btn-xs btn-circle d-flex rounded" type="button" aria-label="Toggle Heading"><em class="material-icons font-18" data-icon="keyboard_arrow_right"></em></button><a class="w-100 link-gray-900" href="/science/mathematics/Mathematics-in-the-19th-century">Mathematics in the 19th century</a></div><div class="ml-40 toc-drawer sub-toc-drawer"><ul class="list-unstyled" data-level="h2"><li data-target="#ref337058"><a class="w-100 link-gray-900" href="/science/mathematics/Mathematics-in-the-19th-century#ref337058">Projective geometry</a></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref337059"><a class="w-100 link-gray-900" href="/science/mathematics/Mathematics-in-the-19th-century#ref337059">Making the calculus rigorous</a></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref337060"><a class="w-100 link-gray-900" href="/science/mathematics/Fourier-series">Fourier series</a></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref337061"><a class="w-100 link-gray-900" href="/science/mathematics/Fourier-series#ref337061">Elliptic functions</a></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref337062"><a class="w-100 link-gray-900" href="/science/mathematics/Fourier-series#ref337062">The theory of numbers</a></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref337063"><a class="w-100 link-gray-900" href="/science/mathematics/The-theory-of-equations">The theory of equations</a></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref337064"><a class="w-100 link-gray-900" href="/science/mathematics/The-theory-of-equations#ref337064">Gauss</a></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref337065"><a class="w-100 link-gray-900" href="/science/mathematics/The-theory-of-equations#ref337065">Non-Euclidean geometry</a></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref337066"><a class="w-100 link-gray-900" href="/science/mathematics/Riemann">Riemann</a></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref337067"><a class="w-100 link-gray-900" href="/science/mathematics/Riemann#ref337067">Riemann’s influence</a></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref337068"><a class="w-100 link-gray-900" href="/science/mathematics/Differential-equations">Differential equations</a></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref337069"><a class="w-100 link-gray-900" href="/science/mathematics/Differential-equations#ref337069">Linear algebra</a></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref337070"><a class="w-100 link-gray-900" href="/science/mathematics/Differential-equations#ref337070">The foundations of geometry</a></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref337071"><a class="w-100 link-gray-900" href="/science/mathematics/Differential-equations#ref337071">The foundations of mathematics</a></li></ul></div></li><li data-target="#ref337072"><div class="d-flex align-items-center"><button class="h1-link-drawer-button btn btn-xs btn-circle d-flex rounded" type="button" aria-label="Toggle Heading"><em class="material-icons font-18" data-icon="keyboard_arrow_right"></em></button><a class="w-100 link-gray-900" href="/science/mathematics/Mathematics-in-the-20th-and-21st-centuries">Mathematics in the 20th and 21st centuries</a></div><div class="ml-40 toc-drawer sub-toc-drawer"><ul class="list-unstyled" data-level="h2"><li data-target="#ref66030"><a class="w-100 link-gray-900" href="/science/mathematics/Mathematics-in-the-20th-and-21st-centuries#ref66030">Cantor</a></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref66031"><a class="w-100 link-gray-900" href="/science/mathematics/Mathematics-in-the-20th-and-21st-centuries#ref66031">Mathematical physics</a></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref66032"><a class="w-100 link-gray-900" href="/science/mathematics/Algebraic-topology">Algebraic topology</a></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref66033"><a class="w-100 link-gray-900" href="/science/mathematics/Algebraic-topology#ref66033">Developments in pure mathematics</a></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref66034"><a class="w-100 link-gray-900" href="/science/mathematics/Mathematical-physics-and-the-theory-of-groups">Mathematical physics and the theory of groups</a></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref335981"><a class="w-100 link-gray-900" href="/science/mathematics/Probabilistic-mathematics">Probabilistic mathematics</a></li></ul></div></li></ul> <a class="toc-extra-link link-gray-900" href="https://www.britannica.com/science/mathematics/additional-info">References & Edit History</a> <a class="toc-extra-link link-gray-900" href="/facts/mathematics">Quick Facts & Related Topics</a> </div> <div class="tlr-media-slider pb-10 mb-30"> <a class="section-header link-gray-900 font-serif font-14 font-weight-bold mb-10 mx-10" href="https://www.britannica.com/science/mathematics/images-videos">Images & Videos</a> <div class="slider js-slider position-relative d-inline-flex align-items-center mw-100 "> <div class="slider-container js-slider-container overflow-hidden d-flex overflow-hidden text-nowrap ml-15"> <a href="https://cdn.britannica.com/54/2354-050-B47B3735/Babylonian-tablet.jpg" data-href="/media/1/369194/2213" class="media-overlay-link d-inline-block mr-5"> <img loading="lazy" src="https://cdn.britannica.com/54/2354-004-969C8844/Babylonian-tablet.jpg" alt="Babylonian mathematical tablet" height="50" /> </a> <a href="https://cdn.britannica.com/92/74892-050-8F76B0AA/Egyptians-right-left-system-power-symbols.jpg" data-href="/media/1/369194/68823" class="media-overlay-link d-inline-block mr-5"> <img loading="lazy" src="https://cdn.britannica.com/92/74892-004-4EFCA3ED/Egyptians-right-left-system-power-symbols.jpg" alt="Ancient Egyptian numerals" height="50" /> </a> <a href="https://cdn.britannica.com/93/74893-050-AF385FBC/Egyptian-numerals.jpg" data-href="/media/1/369194/68835" class="media-overlay-link d-inline-block mr-5"> <img loading="lazy" src="https://cdn.britannica.com/93/74893-004-93F0C348/Egyptian-numerals.jpg" alt="Egyptian hieratic numerals" height="50" /> </a> <a href="https://cdn.britannica.com/02/69702-050-1C1B19A6/reciprocal-seked-Egyptian-run-ratio-Egyptians-definition.jpg" data-href="/media/1/369194/59963" class="media-overlay-link d-inline-block mr-5"> <img loading="lazy" src="https://cdn.britannica.com/02/69702-004-599357AC/reciprocal-seked-Egyptian-run-ratio-Egyptians-definition.jpg" alt="Egyptian seked" height="50" /> </a> <a href="https://cdn.britannica.com/61/67361-050-F48C1CC9/map-millennium-mathematicians-Greco-Roman-Thales-Hypatia-Alexandria.jpg" data-href="/media/1/369194/57444" class="media-overlay-link d-inline-block mr-5"> <img loading="lazy" src="https://cdn.britannica.com/61/67361-004-46CBD13A/map-millennium-mathematicians-Greco-Roman-Thales-Hypatia-Alexandria.jpg" alt="mathematicians of the Greco-Roman world" height="50" /> </a> <a href="https://cdn.britannica.com/06/80206-004-F6463F0F/cube-Menaechmus-volume-solution-problem-two-curves.jpg" data-href="/media/1/369194/90066" class="media-overlay-link d-inline-block mr-5"> <img loading="lazy" src="https://cdn.britannica.com/06/80206-004-F6463F0F/cube-Menaechmus-volume-solution-problem-two-curves.jpg" alt="doubling the volume of a cube" height="50" /> </a> <a href="https://cdn.britannica.com/11/66811-050-C6924436/circumscribing-cylinder-Sphere-volume-sphere-surface-area.jpg" data-href="/media/1/369194/57320" class="media-overlay-link d-inline-block mr-5"> <img loading="lazy" src="https://cdn.britannica.com/11/66811-004-96E09D0B/circumscribing-cylinder-Sphere-volume-sphere-surface-area.jpg" alt="sphere with circumscribing cylinder" height="50" /> </a> <a href="https://cdn.britannica.com/40/840-004-543CF3D4/sections-plane-figure-cone-families-ellipse-parabola.jpg" data-href="/media/1/369194/1743" class="media-overlay-link d-inline-block mr-5"> <img loading="lazy" src="https://cdn.britannica.com/40/840-004-543CF3D4/sections-plane-figure-cone-families-ellipse-parabola.jpg" alt="conic sections" height="50" /> </a> <a href="https://cdn.britannica.com/08/80208-004-0CEE43E7/curve-Conchoid-lines-P-distance-line-points.jpg" data-href="/media/1/369194/74591" class="media-overlay-link d-inline-block mr-5"> <img loading="lazy" src="https://cdn.britannica.com/08/80208-004-0CEE43E7/curve-Conchoid-lines-P-distance-line-points.jpg" alt="conchoid curve" height="50" /> </a> <a href="https://cdn.britannica.com/09/80209-004-F51212B2/Nicomedes-angle-conchoid-Angle-trisection-curve-side.jpg" data-href="/media/1/369194/90801" class="media-overlay-link d-inline-block mr-5"> <img loading="lazy" src="https://cdn.britannica.com/09/80209-004-F51212B2/Nicomedes-angle-conchoid-Angle-trisection-curve-side.jpg" alt="angle trisection using a conchoid" height="50" /> </a> </div> <button disabled class="prev-button js-prev-button position-absolute btn btn-circle shadow btn-blue " aria-label="Previous"> <span class="material-icons md-24" data-icon="keyboard_arrow_left"></span> </button> <button disabled class="next-button js-next-button position-absolute btn btn-circle shadow btn-blue " aria-label="Next"> <span class="material-icons md-24" data-icon="keyboard_arrow_right"></span> </button> </div> </div> <div class="mb-30 tlr-student-links"> <div class="text-gray-900 p-5 font-serif font-14 font-weight-bold mx-10 mb-10"> For Students </div> <div class="imagelink-with-image-on-the-side card card-horizontal tlr-img-with-side-link ml-15 link-gray-900 mb-10" > <div class="position-relative card-media" style="flex: 0;"> <a class="ilf-image position-relative" href="/summary/mathematics"> <img loading="lazy" src="https://cdn.britannica.com/mendel-resources/3-130/images/shared/default3.png?v=3.130.14" class="default " height="200" width="200"/> </a> </div> <div class="card-body ilf-content"> <a class="font-weight-semi-bold d-block mb-5 font-16 ilf-title" href="/summary/mathematics" >mathematics summary</a> </div> </div> </div> <div class="mb-30 tlr-related-quizzes"> <div class="text-gray-900 p-5 font-serif font-14 font-weight-bold mx-10 mb-10"> Quizzes </div> <div class="imagelink-with-image-on-the-side card card-horizontal tlr-img-with-side-link ml-15 link-gray-900 mb-10" > <div class="position-relative card-media" style="flex: 0;"> <a class="ilf-image position-relative" href="/quiz/Numbers-and-mathematics"> <img loading="lazy" src="https://cdn.britannica.com/86/94086-131-0BAE374D/Equations-blackboard.jpg?w=200&h=200&c=crop" alt="Equations written on blackboard" width="200" height="200" /> </a> </div> <div class="card-body ilf-content"> <a class="font-weight-semi-bold d-block mb-5 font-16 ilf-title" href="/quiz/Numbers-and-mathematics" >Numbers and Mathematics</a> </div> </div> <div class="imagelink-with-image-on-the-side card card-horizontal tlr-img-with-side-link ml-15 link-gray-900 mb-10" > <div class="position-relative card-media" style="flex: 0;"> <a class="ilf-image position-relative" href="/quiz/fun-facts-of-measurement-math"> <img loading="lazy" src="https://cdn.britannica.com/68/166068-131-03976F7B/barometer-Barometer-Technology-measurement-readout-pressure-mathematics.jpg?w=200&h=200&c=crop" alt="barometer. Antique Barometer with readout. Technology measurement, mathematics, measure atmospheric pressure" width="200" height="200" /> </a> </div> <div class="card-body ilf-content"> <a class="font-weight-semi-bold d-block mb-5 font-16 ilf-title" href="/quiz/fun-facts-of-measurement-math" >Fun Facts of Measurement & Math</a> </div> </div> <div class="imagelink-with-image-on-the-side card card-horizontal tlr-img-with-side-link ml-15 link-gray-900 mb-10" > <div class="position-relative card-media" style="flex: 0;"> <a class="ilf-image position-relative" href="/quiz/define-it-math-terms"> <img loading="lazy" src="https://cdn.britannica.com/01/115001-131-7278E518/Enrico-Fermi-Italian-problem-physics-1950.jpg?w=200&h=200&c=crop" alt="Italian-born physicist Dr. Enrico Fermi draws a diagram at a blackboard with mathematical equations. circa 1950." width="200" height="200" /> </a> </div> <div class="card-body ilf-content"> <a class="font-weight-semi-bold d-block mb-5 font-16 ilf-title" href="/quiz/define-it-math-terms" >Define It: Math Terms</a> </div> </div> </div> <div class="mb-30 tlr-read-next"> <div class="text-gray-900 p-5 font-serif font-14 font-weight-bold mx-10 mb-10"> Read Next </div> <div class="imagelink-with-image-on-the-side card card-horizontal tlr-img-with-side-link ml-15 link-gray-900 mb-10" > <div class="position-relative card-media" style="flex: 0;"> <a class="ilf-image position-relative" href="/story/unusual-counting-systems"> <img loading="lazy" src="https://cdn.britannica.com/41/191041-131-7ECE668A/bunch-numbers.jpg?w=200&h=200&c=crop" alt="bunch of numbers" width="200" height="200" /> </a> </div> <div class="card-body ilf-content"> <a class="font-weight-semi-bold d-block mb-5 font-16 ilf-title" href="/story/unusual-counting-systems" >Unusual Counting Systems</a> </div> </div> <div class="imagelink-with-image-on-the-side card card-horizontal tlr-img-with-side-link ml-15 link-gray-900 mb-10" > <div class="position-relative card-media" style="flex: 0;"> <a class="ilf-image position-relative" href="/story/is-zero-an-even-or-an-odd-number"> <img loading="lazy" src="https://cdn.britannica.com/49/191949-131-3E2AC277/balloon.jpg?w=200&h=200&c=crop" alt="number zero, 0 balloon" width="200" height="200" /> </a> </div> <div class="card-body ilf-content"> <a class="font-weight-semi-bold d-block mb-5 font-16 ilf-title" href="/story/is-zero-an-even-or-an-odd-number" >Is Zero an Even or an Odd Number?</a> </div> </div> <div class="imagelink-with-image-on-the-side card card-horizontal tlr-img-with-side-link ml-15 link-gray-900 mb-10" > <div class="position-relative card-media" style="flex: 0;"> <a class="ilf-image position-relative" href="/story/how-fast-is-the-universe-expanding"> <img loading="lazy" src="https://cdn.britannica.com/27/238127-131-9A610712/galaxy-cluster-Abell-2744.jpg?w=200&h=200&c=crop" alt="Galaxy clusters like Abell 2744 can act as a natural cosmic lens, magnifying light from more distant, background objects through gravity. NASA's James Webb Space Telescope may be able to detect light from the first stars in the universe if they are gravitationally lensed by such clusters. (astronomy, space exploration, galaxies)" width="200" height="200" /> </a> </div> <div class="card-body ilf-content"> <a class="font-weight-semi-bold d-block mb-5 font-16 ilf-title" href="/story/how-fast-is-the-universe-expanding" >How Fast Is the Universe Expanding?</a> </div> </div> <div class="imagelink-with-image-on-the-side card card-horizontal tlr-img-with-side-link ml-15 link-gray-900 mb-10" > <div class="position-relative card-media" style="flex: 0;"> <a class="ilf-image position-relative" href="/list/5-incredible-women-in-stem-you-need-to-know"> <img loading="lazy" src="https://cdn.britannica.com/82/222682-131-2FB267DE/American-mathematician-Gladys-West.jpg?w=200&h=200&c=crop" alt="Mathematician Dr. Gladys West was inducted into the Air Force Space and Missile Pioneers Hall of Fame during a ceremony in her honor at the Pentagon, Dec. 6, 2018. GPS (Global Positioning System); computing; women in STEM." width="200" height="200" /> </a> </div> <div class="card-body ilf-content"> <a class="font-weight-semi-bold d-block mb-5 font-16 ilf-title" href="/list/5-incredible-women-in-stem-you-need-to-know" >5 Incredible Women in STEM You Need to Know</a> </div> </div> </div> <div class="mb-30 tlr-discover"> <div class="text-gray-900 p-5 font-serif font-14 font-weight-bold mx-10 mb-10"> Discover </div> <div class="imagelink-with-image-on-the-side card card-horizontal tlr-img-with-side-link ml-15 link-gray-900 mb-10" > <div class="position-relative card-media" style="flex: 0;"> <a class="ilf-image position-relative" href="/list/11-egyptian-gods-and-goddesses"> <img loading="lazy" src="https://cdn.britannica.com/52/196952-131-0665E4EE/Egyptians-hieroglyphics-carvings.jpg?w=200&h=200&c=crop" alt="An ancient egyptian hieroglyphic painted carving showing the falcon headed god Horus seated on a throne and holding a golden fly whisk. Before him are the Pharoah Seti and the goddess Isis. Interior wall of the temple to Osiris at Abydos, Egypt." width="200" height="200" /> </a> </div> <div class="card-body ilf-content"> <a class="font-weight-semi-bold d-block mb-5 font-16 ilf-title" href="/list/11-egyptian-gods-and-goddesses" >11 Egyptian Gods and Goddesses</a> </div> </div> <div class="imagelink-with-image-on-the-side card card-horizontal tlr-img-with-side-link ml-15 link-gray-900 mb-10" > <div class="position-relative card-media" style="flex: 0;"> <a class="ilf-image position-relative" href="/story/where-do-anti-semitic-conspiracy-theories-about-the-rothschild-family-come-from"> <img loading="lazy" src="https://cdn.britannica.com/67/200267-131-4F79E2AC/Mayer-Amschel-Rothschild.jpg?w=200&h=200&c=crop" alt="Mayer Amschel Rothschild (1744-1855), founder of the house of Rothschild; lithograph by Friedrich Lieder, c. 1830." width="200" height="200" /> </a> </div> <div class="card-body ilf-content"> <a class="font-weight-semi-bold d-block mb-5 font-16 ilf-title" href="/story/where-do-anti-semitic-conspiracy-theories-about-the-rothschild-family-come-from" >Where Do Anti-Semitic Conspiracy Theories About the Rothschild Family Come From?</a> </div> </div> <div class="imagelink-with-image-on-the-side card card-horizontal tlr-img-with-side-link ml-15 link-gray-900 mb-10" > <div class="position-relative card-media" style="flex: 0;"> <a class="ilf-image position-relative" href="/list/secret-service-code-names-of-11-us-presidents"> <img loading="lazy" src="https://cdn.britannica.com/45/189145-131-45FF672E/Secret-Service-Agent-Earpiece.jpg?w=200&h=200&c=crop" alt="Secret Service Agent Listens To Earpiece" width="200" height="200" /> </a> </div> <div class="card-body ilf-content"> <a class="font-weight-semi-bold d-block mb-5 font-16 ilf-title" href="/list/secret-service-code-names-of-11-us-presidents" >Secret Service Code Names of 11 U.S. Presidents</a> </div> </div> <div class="imagelink-with-image-on-the-side card card-horizontal tlr-img-with-side-link ml-15 link-gray-900 mb-10" > <div class="position-relative card-media" style="flex: 0;"> <a class="ilf-image position-relative" href="/story/why-do-we-eat-turkey-on-thanksgiving"> <img loading="lazy" src="https://cdn.britannica.com/72/201772-131-C576ED87/Turkey-sides.jpg?w=200&h=200&c=crop" alt="Thanksgiving Turkey with all the sides" width="200" height="200" /> </a> </div> <div class="card-body ilf-content"> <a class="font-weight-semi-bold d-block mb-5 font-16 ilf-title" href="/story/why-do-we-eat-turkey-on-thanksgiving" >Why Do We Eat Turkey on Thanksgiving?</a> </div> </div> <div class="imagelink-with-image-on-the-side card card-horizontal tlr-img-with-side-link ml-15 link-gray-900 mb-10" > <div class="position-relative card-media" style="flex: 0;"> <a class="ilf-image position-relative" href="/story/was-santa-claus-a-real-person"> <img loading="lazy" src="https://cdn.britannica.com/92/202092-131-ACFF5B40/Santa-Claus-sleigh.jpg?w=200&h=200&c=crop" alt="Santa Claus flying in his sleigh, christmas, reindeer" width="200" height="200" /> </a> </div> <div class="card-body ilf-content"> <a class="font-weight-semi-bold d-block mb-5 font-16 ilf-title" href="/story/was-santa-claus-a-real-person" >Was Santa Claus a Real Person?</a> </div> </div> <div class="imagelink-with-image-on-the-side card card-horizontal tlr-img-with-side-link ml-15 link-gray-900 mb-10" > <div class="position-relative card-media" style="flex: 0;"> <a class="ilf-image position-relative" href="/list/12-novels-considered-the-greatest-book-ever-written"> <img loading="lazy" src="https://cdn.britannica.com/55/142355-131-EFF621AF/books-Stack-literature-pile-reading-entertainment-society-2010.jpg?w=200&h=200&c=crop" alt="Close up of books. Stack of books, pile of books, literature, reading. Homepage 2010, arts and entertainment, history and society" width="200" height="200" /> </a> </div> <div class="card-body ilf-content"> <a class="font-weight-semi-bold d-block mb-5 font-16 ilf-title" href="/list/12-novels-considered-the-greatest-book-ever-written" >12 Novels Considered the “Greatest Book Ever Written”</a> </div> </div> <div class="imagelink-with-image-on-the-side card card-horizontal tlr-img-with-side-link ml-15 link-gray-900 mb-10" > <div class="position-relative card-media" style="flex: 0;"> <a class="ilf-image position-relative" href="/list/12-greek-gods-and-goddesses"> <img loading="lazy" src="https://cdn.britannica.com/23/176123-131-94DCF6F8/Aphrodite.jpg?w=200&h=200&c=crop" alt="Aphrodite. Greek mythology. Sculpture. Aphrodite is the Greek goddess of love and beauty." width="200" height="200" /> </a> </div> <div class="card-body ilf-content"> <a class="font-weight-semi-bold d-block mb-5 font-16 ilf-title" href="/list/12-greek-gods-and-goddesses" >12 Greek Gods and Goddesses</a> </div> </div> </div> </div> </div> </div> <button class="drawerToggle btn position-sticky border btn-xs btn-white btn-circle rounded-sm d-none d-lg-flex " type="button" aria-label="Toggle Drawer"> <em class="material-icons font-18 text-blue" data-icon="keyboard_arrow_left"></em> </button> </div> </div> <div class="col"> <div class="h-100 ml-0 pr-lg-0 "> <div class="h-100 grid gx-0 gx-lg-20"> <div class="h-100 col-sm"> <div class="h-100 infinite-pagination-container d-flex flex-column position-relative"> <div class="position-absolute top-0 h-100 w-100"> <div class="toc-sticky-header d-none d-lg-none bg-gray-50 px-10 px-sm-30 position-sticky w-100 "> <div class="toc-sticky-header-inner-container align-items-center d-flex mx-auto h-100 w-100"> <button class="d-flex d-lg-none btn btn-sm btn-white text-blue border-2 border-gray-100 gtm-mobile-toc-header-button js-sections-button d-lg-none p-10"> <em class="material-icons my-n5 md-icon" data-icon="toc"></em> Contents </button> <div class="header-ai-ask-button-placeholder"></div> <div class="header-ai-summarize-button-placeholder"></div> </div> </div> </div> <div class="grey-box w-100 grey-box-top"> <div class="grey-box-content mx-auto w-100"> <script type="application/ld+json"> { "@context" : "https://schema.org", "@type" : "BreadcrumbList", "itemListElement" : [ { "@type" : "ListItem", "position" : 1, "item" : { "@id" : "https://www.britannica.com/browse/Science", "name": "Science" } } , { "@type" : "ListItem", "position" : 2, "item" : { "@id" : "https://www.britannica.com/browse/Mathematics", "name": "Mathematics" } } ] } </script> <nav class="breadcrumb mt-20"> <span class="breadcrumb-item "> <a class="link-gray-600" href="/browse/Science">Science</a> </span> <span class="breadcrumb-item "> <a class="link-gray-600" href="/browse/Mathematics">Mathematics</a> </span> </nav> <div class="page2ref-true topic-content topic-type-REGULAR" data-student-article="true"> <script class="page-description-json" type="application/json"> { "url": "/science/mathematics", "shareUrl": "https://www.britannica.com/science/mathematics", "browserTitle": "Mathematics | Definition, History, & Importance", "firstTopicPage": true, "topicId":369194 } </script> <div class="reading-channel"> <div class="topic-header"> <div class="d-flex align-items-top justify-content-between"> <div class="d-flex flex-column"> <div> <div> <h1>mathematics</h1></div> </div> </div> </div> <div class="d-none d-sm-flex flex-row"> <div class="mr-10 mb-15"> <button class="ai-ask-button btn border-2 btn-sm js-inline-ai-ask-button btn-outline-red-400 border-red-400"> Ask the Chatbot a Question </button> </div> <div class="d-none d-sm-block md-topic-tools qa-action-buttons mb-15" data-topic-id="369194"> <button class="js-tooltip btn btn-sm btn-outline-blue border pr-10 border-2" > <em class="material-icons md-icon ml-n10 my-n5 mr-5" data-icon="more_vert"></em> More Actions </button> <div class="md-more-popover popover popover-sm p-0 font-14 z-1"> <div> <button class="js-print-modal-button js-modal gtm-topic-tool btn btn-sm btn-link gtm-topic-tool font-weight-bold btn-link" data-modal="[data-topic-id=369194] .md-print-modal" > <em class="material-icons mr-5 ml-n10 my-n5 md-icon" data-icon="print"></em> Print </button> <div class="md-print-modal size-lg d-none"> <div class="md-modal-body"> <div class="h2 font-serif d-flex align-items-center pb-15 border-bottom"> <em class="material-icons text-blue mr-10">print</em> Print </div> <div class="mt-20 mb-10"> Please select which sections you would like to print: </div> <form action="/print/article/369194" method="post" target="_blank" rel="noopener"> <div class="print-box-items"> <ul class="list-unstyled"> <li><label><input class="mr-10" type="checkbox" name="sequence[]" value="0">Table Of Contents</label></li> </ul> </div> <input type="submit" class="btn btn-blue md-disabled" value="Print" /> </form> </div> </div> </div> <div> <button class="js-modal qa-cite-modal-button btn btn-sm btn-link gtm-topic-tool font-weight-bold btn-link" data-modal="[data-topic-id=369194] .md-cite-modal"> <em class="material-icons mr-5 ml-n10 my-n5 md-icon" data-icon="verified"></em> Cite </button> <div class="md-cite-modal size-lg d-none"> <div class="md-modal-body"> <div class="h2 font-serif d-flex align-items-center pb-15 border-bottom mb-15"> <em class="material-icons text-blue mr-10">verified</em>Cite </div> <div class="font-serif"> While every effort has been made to follow citation style rules, there may be some discrepancies. Please refer to the appropriate style manual or other sources if you have any questions. </div> <div class="label mt-20 mb-10">Select Citation Style</div> <select class="js-citation-format-select form-select"> <option selected value="mla">MLA</option> <option value="apa">APA</option> <option value="chicago">Chicago Manual of Style</option> </select> <div class="citation font-serif border rounded p-15 mt-20" data-authors="Wilbur R. Knorr, John L. Berggren, Menso Folkerts, Jeremy John Gray, Craig G. Fraser" data-title="mathematics" data-published-date="15 Nov. 2024" data-url="https://www.britannica.com/science/mathematics" > <div class="citation-text"></div> </div> <button class="js-copy-citation-button mt-20 btn btn-xs btn-outline-blue border shadow-sm pr-10" > <em class="material-icons md-icon ml-n10 my-n5 mr-5" data-icon="file_copy"></em> <span class="js-citation-status-text">Copy Citation</span> </button> </div> </div> </div> <div> <button class="js-share-modal-button js-modal btn btn-sm btn-link gtm-topic-tool font-weight-bold btn-link" data-modal="[data-topic-id=369194] .md-share-modal"> <em class="material-icons mr-5 ml-n10 my-n5 md-icon" data-icon="share"></em> Share </button> <div class="md-share-modal size-lg d-none qa-share-modal"> <div class="md-modal-body"> <div class="h2 font-serif d-flex align-items-center pb-15 border-bottom"> <em class="material-icons text-blue mr-10" data-icon="share"></em> Share </div> <div class="label my-20">Share to social media</div> <div class="md-social-toolbar-circle d-flex align-items-start inverted" data-value="share" title="mathematics" data-url="https://www.britannica.com/science/mathematics" > <a class="social-icon facebook justify-content-center d-flex align-items-center align-self-center" data-provider="facebook" href="https://www.facebook.com/BRITANNICA/" target="_blank" rel="noopener"><span>Facebook</span></a> <a class="social-icon x justify-content-center d-flex align-items-center align-self-center" data-provider="x" href="https://x.com/britannica" target="_blank" rel="noopener"><span>X</span></a> </div> <div class="label pt-20 mt-20 mb-5 border-top">URL</div> <a class="font-serif text-truncate d-inline-block" href="https://www.britannica.com/science/mathematics">https://www.britannica.com/science/mathematics</a> </div> </div> </div> <div> <button class="js-feedback-modal-button js-modal btn btn-sm btn-link gtm-topic-tool font-weight-bold btn-link" data-modal=".md-feedback-modal"> <em class="material-icons mr-5 ml-n10 my-n5 md-icon" data-icon="message"></em> Feedback </button> </div> <div> <button class="qa-external-website-modal-button js-modal btn btn-sm btn-link gtm-topic-tool font-weight-bold btn-link" data-modal="[data-topic-id=369194] .md-websites-modal"> <em class="material-icons md-icon ml-n10 mr-5" data-icon="link"></em> External Websites </button> </div> </div> <div class="md-feedback-modal size-lg d-none"> <div class="md-modal-body"> <div class="h2 font-serif pb-15 border-bottom"> Feedback </div> <form method="post" action="/submission/feedback/369194"> <div class="my-20"> Corrections? Updates? Omissions? Let us know if you have suggestions to improve this article (requires login). </div> <div class="type-menu"> <label for="feedback-type" class="label mb-10">Feedback Type</label> <select id="feedback-type" class="form-select mb-30" name="feedbackTypeId" required> <option value="" selected="selected">Select a type (Required)</option> <option value="1">Factual Correction</option> <option value="2">Spelling/Grammar Correction</option> <option value="3">Link Correction</option> <option value="4">Additional Information</option> <option value="5">Other</option> </select> </div> <label for="feedback" class="label mb-10">Your Feedback</label> <textarea id="feedback" class="form-control mb-30" name="feedback" maxlength="3000" rows="7" required></textarea> <button class="btn btn-blue" type="submit">Submit Feedback</button> </form> <div class="success-messaging d-none mt-30"> <div class="title">Thank you for your feedback</div> <p>Our editors will review what you’ve submitted and determine whether to revise the article.</p> </div> </div> </div> <div class="md-websites-modal size-lg d-none"> <div class="md-modal-body"> <div class="h2 font-serif pb-15 border-bottom font-weight-bold"> External Websites </div> <div class="pb-20"> <ul class="list-unstyled mt-20 lh-lg"> <li><a class="external" href="https://www.livescience.com/38936-mathematics.html" target="_blank" rel="noopener ">LiveScience - What is Mathematics?</a></li> <li><a class="external" href="https://socialsci.libretexts.org/Courses/Arapahoe_Community_College/Introduction_to_Curriculum_for_Early_Childhood_Education/07%3A_Mathematics" target="_blank" rel="noopener ">Social Sci LibreTexts - Mathematics</a></li> <li><a class="external" href="https://abelprize.no/" target="_blank" rel="noopener nofollow">Official Site of The Abel Prize</a></li> <li><a class="external" href="https://illinois.pbslearningmedia.org/resource/nvmm-math-mathmusic/ancient-math-music/" target="_blank" rel="noopener ">PBS LearningMedia - Ancient Math & Music</a></li> <li><a class="external" href="https://mathweb.ucsd.edu/~harel/What%20Is%20Mathematics.pdf" target="_blank" rel="noopener ">University of California, San Diego - What is Mathematics?</a></li> </ul> </div> <div class="md-websites-ebk-title">Britannica Websites</div> <div class="md-websites-ebk-subtitle">Articles from Britannica Encyclopedias for elementary and high school students.</div> <ul class="list-unstyled bps-topic-web-sites lh-lg"> <li><a class="external" href="https://kids.britannica.com/kids/article/mathematics/353443" target="_blank" rel="noopener">mathematics - Children's Encyclopedia (Ages 8-11)</a></li> <li><a class="external" href="https://kids.britannica.com/students/article/mathematics/275734" target="_blank" rel="noopener">mathematics - Student Encyclopedia (Ages 11 and up)</a></li> </ul> </div> </div> </div> </div> <div class="toc-header-marker"></div> <button class="ai-ask-button btn border-2 js-header-ai-ask-button d-none btn-sm btn-outline-red-400 border-red-400 mr-0 mr-lg-10 ml-5 ml-sm-10 ml-lg-0 p-10"> Ask the Chatbot a Question </button> <div class="caption alternate-titles">Also known as: math</div> <div class="md-byline module-spacing "> <div class="font-serif font-12"> <span class="written-by text-gray-700"> Written by </span> <div class="editor-popover popover p-0"> <a class="d-block p-20 gtm-byline font-12 byline-contributor" href="/contributor/Wilbur-R-Knorr/3576" > <div class="editor-title font-16 font-weight-bold">Wilbur R. Knorr</div> <div class="editor-description font-12 font-serif mt-5 clamp-description text-black">Professor of the History of Science, Stanford University, California. Author of <i>Ancient Tradition of Geometric problems </i>and others.</div> </a> <div data-popper-arrow></div> </div> <span class="btn btn-link editor-link p-0 qa-byline-link gtm-byline font-12 byline-contributor text-decoration-underline"> Wilbur R. Knorr</span>, <div class="editor-popover popover p-0"> <a class="d-block p-20 gtm-byline font-12 byline-contributor" href="/contributor/John-L-Berggren/3484" > <div class="editor-title font-16 font-weight-bold">John L. Berggren</div> <div class="editor-description font-12 font-serif mt-5 clamp-description text-black">Professor of Mathematics, Simon Fraser University, Burnaby, British Columbia. Author of <i>Episodes in the Mathematics of Medieval Islam.</i></div> </a> <div data-popper-arrow></div> </div> <span class="btn btn-link editor-link p-0 qa-byline-link gtm-byline font-12 byline-contributor text-decoration-underline"> John L. Berggren</span><span class="text-gray-700 mx-5">•</span><a class="see-all border-gray-700 gtm-byline" rel="nofollow" href="https://www.britannica.com/science/mathematics/additional-info#contributors">All</a> </div> <div class="font-serif font-12 text-gray-700"> <span class="qa-fact-checked-by">Fact-checked by</span> <div class="editor-popover popover p-0"> <a class="d-block p-20 font-12" href="/editor/The-Editors-of-Encyclopaedia-Britannica/4419" > <div class="editor-title font-16 font-weight-bold">The Editors of Encyclopaedia Britannica</div> <div class="editor-description font-12 font-serif mt-5 text-black">Encyclopaedia Britannica's editors oversee subject areas in which they have extensive knowledge, whether from years of experience gained by working on that content or via study for an advanced degree. They write new content and verify and edit content received from contributors.</div> </a> <div data-popper-arrow></div> </div> <span class="btn btn-link editor-link p-0 qa-byline-link font-12 "> The Editors of Encyclopaedia Britannica</span></div> <div class="last-updated font-12 font-serif"> <span class="text-gray-700"> Last Updated: <time datetime="2024-11-15T00:00:00CST" >Nov 15, 2024</time> •</span> <a class="byline-edit-history" href="https://www.britannica.com/science/mathematics/additional-info#history" rel="nofollow">Article History</a> </div></div> </div> <button class="d-flex d-lg-none btn btn-outline-blue border rounded-sm shadow-sm mobile-toc-button gtm-mobile-toc-inline-button d-none d-sm-block js-sections-inline-button module-spacing btn d-lg-none"> <em class="material-icons mr-5 ml-n10 my-n5 md-icon" data-icon="toc"></em> Table of Contents </button> <div class="d-flex d-sm-none flex-row"> <button class="d-flex d-lg-none btn btn-outline-blue border rounded-sm shadow-sm mobile-toc-button gtm-mobile-toc-inline-button js-sections-inline-button module-spacing"> <em class="material-icons mr-5 ml-n10 my-n5 md-icon" data-icon="toc"></em> Table of Contents </button> <button class="ai-ask-button btn border-2 ai-ask-button btn border-2 module-spacing btn-sm js-inline-ai-ask-button btn-outline-red-400 border-red-400 p-10 ml-5"> Ask the Chatbot a Question </button> </div> <div class="js-qf-module qf-module px-40 px-sm-20 py-15 mx-auto module-spacing font-14 bg-gray-50 rounded"> <div class="facts-list mt-10"> <div class=""> <div class="js-fact mb-10 line-clamp clamp-3"> <dl> <dt>Key People: </dt> <dd><a href="/biography/Mary-Cartwright" topicid="2252584">Mary Cartwright</a></dd> <dd><a href="/biography/Gladys-West" topicid="2188733">Gladys West</a></dd> <dd><a href="/biography/Isaac-Newton" topicid="413189">Isaac Newton</a></dd> <dd><a href="/biography/Galileo-Galilei" topicid="224058">Galileo</a></dd> <dd><a href="/biography/Bertrand-Russell" topicid="513124">Bertrand Russell</a></dd> </dl> <button class="js-more-btn d-none btn btn-unstyled font-12 bg-gray-50" aria-label="Toggle more/less fact data"> <em class="js-content link-blue">(Show more)</em> </button> </div> </div> <div class=""> <div class="js-fact mb-10 line-clamp clamp-3"> <dl> <dt>Related Topics: </dt> <dd><a href="/science/analysis-mathematics" topicid="22486">analysis</a></dd> <dd><a href="/science/probability-theory" topicid="477530">probability theory</a></dd> <dd><a href="/topic/cryptology" topicid="145058">cryptology</a></dd> <dd><a href="/science/computer-science" topicid="130675">computer science</a></dd> <dd><a href="/topic/percentage" topicid="2124609">percentage</a></dd> </dl> <button class="js-more-btn d-none btn btn-unstyled font-12 bg-gray-50" aria-label="Toggle more/less fact data"> <em class="js-content link-blue">(Show more)</em> </button> </div> <div class="text-center"> <a class="btn btn-sm btn-link p-0" href="/facts/mathematics"> See all related content </a> </div> </div> </div> </div><div class="bg-gray-50 p-15 rounded module-spacing recent-news d-flex flex-column float-false"> <div> <h2 class="font-weight-bold font-14 m-0 d-inline"> News <span class="text-gray-600">•</span> </h2> <div class="recent-news-item first-recent-news-item d-inline"> <a class="font-14 gtm-ap-news-link" href="https://www.nytimes.com/2024/11/14/arts/design/math-museum-new-location.html" rel="nofollow">Math Museum Announces Move to New Location</a> <span class="font-14 text-gray-600"> <span>•</span> Nov. 14, 2024, 11:49 AM ET (New York Times) </span> </div> </div> <div class="rest-of-recent-news-items"> <button class="js-toggle-recent-news d-flex btn btn-unstyled font-14 pr-10 rounded-sm mt-10" aria-label="Toggle additional news items"> Show less <em class="material-icons" data-icon="expand_less"></em> </button> </div> </div><span class="marker before-article"></span><section data-level="1" id="ref1"><span class="marker PREMOD1 mod-inline"></span><p class="topic-paragraph"><strong><span id="ref536020"></span>mathematics</strong>, the <a href="https://www.britannica.com/science/science" class="md-crosslink autoxref " data-show-preview="true">science</a> of structure, order, and relation that has evolved from elemental practices of counting, measuring, and describing the shapes of objects. It deals with logical reasoning and quantitative calculation, and its development has involved an increasing degree of idealization and abstraction of its subject matter. Since the 17th century, mathematics has been an <a class="md-dictionary-link md-dictionary-tt-off eb" data-term="indispensable" href="https://www.britannica.com/dictionary/indispensable" data-type="EB">indispensable</a> adjunct to the physical sciences and technology, and in more recent times it has assumed a similar role in the quantitative aspects of the life sciences.</p><span class="marker MOD1 mod-inline"></span><span class="marker PREMOD2 mod-inline"></span><p class="topic-paragraph">In many cultures—under the stimulus of the needs of practical pursuits, such as commerce and agriculture—mathematics has developed far beyond basic counting. This growth has been greatest in societies complex enough to sustain these activities and to provide leisure for contemplation and the opportunity to build on the achievements of earlier mathematicians.</p><span class="marker MOD2 mod-inline"></span><span class="marker PREMOD3 mod-inline"></span><p class="topic-paragraph">All mathematical systems (for example, <a href="https://www.britannica.com/science/Euclidean-geometry" class="md-crosslink " data-show-preview="true">Euclidean geometry</a>) are combinations of sets of axioms and of theorems that can be logically deduced from the axioms. <a class="md-dictionary-link md-dictionary-tt-off eb" data-term="Inquiries" href="https://www.britannica.com/dictionary/Inquiries" data-type="EB">Inquiries</a> into the logical and philosophical basis of mathematics reduce to questions of whether the axioms of a given system ensure its completeness and its consistency. For full treatment of this aspect, <em>see</em> <a href="https://www.britannica.com/science/foundations-of-mathematics" class="md-crosslink " data-show-preview="true">mathematics, foundations of</a>.</p><span class="marker MOD3 mod-inline"></span><span class="marker PREMOD4 mod-inline"></span><p class="topic-paragraph">This article offers a history of mathematics from ancient times to the present. As a consequence of the exponential growth of science, most mathematics has developed since the 15th century <span class="text-smallcaps">ce</span>, and it is a historical fact that, from the 15th century to the late 20th century, new developments in mathematics were largely concentrated in Europe and <a href="https://www.britannica.com/place/North-America" class="md-crosslink autoxref " data-show-preview="true">North America</a>. For these reasons, the bulk of this article is devoted to European developments since 1500.</p><span class="marker MOD4 mod-inline"></span><span class="marker PREMOD5 mod-inline"></span><p class="topic-paragraph">This does not mean, however, that developments elsewhere have been unimportant. Indeed, to understand the history of mathematics in Europe, it is necessary to know its history at least in ancient Mesopotamia and Egypt, in ancient Greece, and in Islamic civilization from the 9th to the 15th century. The way in which these civilizations influenced one another and the important direct contributions Greece and Islam made to later developments are discussed in the first parts of this article.</p><a class="link-module shadow-sm d-block qa-quiz-module" href="/quiz/define-it-math-terms" data-link-module-iframe-link=""> <img loading="lazy" src="https://cdn.britannica.com/01/115001-131-7278E518/Enrico-Fermi-Italian-problem-physics-1950.jpg" alt="Italian-born physicist Dr. Enrico Fermi draws a diagram at a blackboard with mathematical equations. circa 1950." class="rounded-sm mr-15" width="70" /> <div class="line-clamp clamp-5"> <div class="module-title bg-green">Britannica Quiz</div> <div class="font-weight-semi-bold mt-5">Define It: Math Terms</div> </div> </a><span class="marker MOD5 mod-inline"></span><span class="marker PREMOD6 mod-inline"></span><p class="topic-paragraph">India’s contributions to the development of contemporary mathematics were made through the considerable influence of Indian achievements on Islamic mathematics during its formative years. A separate article, <a href="https://www.britannica.com/science/Indian-mathematics" class="md-crosslink " data-show-preview="true">South Asian mathematics</a>, focuses on the early history of mathematics in the <a href="https://www.britannica.com/place/Indian-subcontinent" class="md-crosslink autoxref " data-show-preview="true">Indian subcontinent</a> and the development there of the modern decimal place-value <a href="https://www.britannica.com/science/numeral-system" class="md-crosslink autoxref " data-show-preview="true">numeral system</a>. The article <a href="https://www.britannica.com/science/East-Asian-mathematics" class="md-crosslink " data-show-preview="true">East Asian mathematics</a> covers the mostly independent development of mathematics in China, Japan, Korea, and Vietnam.</p><div class="one-good-fact-module"> </div><span class="marker MOD6 mod-inline"></span> <span class="marker PREMOD7 mod-inline"></span><p class="topic-paragraph">The <a class="md-dictionary-link md-dictionary-tt-off mw" data-term="substantive" href="https://www.merriam-webster.com/dictionary/substantive" data-type="MW">substantive</a> branches of mathematics are treated in several articles. <em>See</em> <a href="https://www.britannica.com/science/algebra" class="md-crosslink " data-show-preview="true">algebra</a>; <a href="https://www.britannica.com/science/analysis-mathematics" class="md-crosslink " data-show-preview="true">analysis</a>; <a href="https://www.britannica.com/science/arithmetic" class="md-crosslink " data-show-preview="true">arithmetic</a>; <a href="https://www.britannica.com/science/combinatorics" class="md-crosslink " data-show-preview="true">combinatorics</a>; <a href="https://www.britannica.com/science/game-theory" class="md-crosslink " data-show-preview="true">game theory</a>; <a href="https://www.britannica.com/science/geometry" class="md-crosslink " data-show-preview="true">geometry</a>; <a href="https://www.britannica.com/science/number-theory" class="md-crosslink " data-show-preview="true">number theory</a>; <a href="https://www.britannica.com/science/numerical-analysis" class="md-crosslink " data-show-preview="true">numerical analysis</a>; <a href="https://www.britannica.com/science/optimization" class="md-crosslink " data-show-preview="true">optimization</a>; <a href="https://www.britannica.com/science/probability-theory" class="md-crosslink " data-show-preview="true">probability theory</a>; <a href="https://www.britannica.com/science/set-theory" class="md-crosslink " data-show-preview="true">set theory</a>; <a href="https://www.britannica.com/science/statistics" class="md-crosslink " data-show-preview="true">statistics</a>; <a href="https://www.britannica.com/science/trigonometry" class="md-crosslink " data-show-preview="true">trigonometry</a>.</p><div class="module-spacing"> <DIV class="marketing-INLINE_SUBSCRIPTION marketing-content" data-marketing-id="INLINE_SUBSCRIPTION"><style> .student-promo-banner-wrapper { container-type: inline-size; margin-bottom: 15px; } @container (min-width: 475px) { .student-promo-banner { flex-direction: row; } .student-promo-banner-img-wrapper { margin-bottom: 0; margin-right: 10px; justify-content: flex-start; } .student-promo-banner-text-wrapper { text-align: left; margin-bottom: 0px; margin-left: 10px; } .student-promo-banner-button-wrapper { margin-right: 0; } }</style> <div class="student-promo-banner-wrapper"> <div class="student-promo-banner d-flex flex-column align-items-center bg-blue rounded p-20"> <div class="student-promo-banner-img-wrapper mb-20 mr-0 d-flex justify-content-center"> <img class="rounded" style="max-width: 100px; min-width: 80px" src="https://cdn.britannica.com/marketing/BlueThistle.webp" /> </div> <div class="student-promo-banner-text-wrapper ml-0 mb-10 text-center text-white"> <div class="h2 mb-10">Get Unlimited Access</div> <div class="h4 font-weight-semi-bold">Try Britannica Premium for free and discover more.</div> </div> <div class="student-promo-banner-button-wrapper d-flex justify-content-center align-items-center ml-auto mr-auto"> <a class="btn btn-m btn-orange" href="https://premium.britannica.com/premium-membership/?utm_source=premium&utm_medium=inline-cta&utm_campaign=black-friday-2024">Subscribe</a> </div> </div> </div> </DIV></div><span class="marker MOD7 mod-inline"></span> </section> <span class="marker end-of-content"></span><span class="marker after-article"></span></div> <div id="chatbot-root"></div> </div> </div> </div> <div class="ai-dialog-placeholder"></div> </div> </div> <aside class="col-md-da-320"></aside> </div> </div> </div> </div> </article></div> </div></div> </div> </main> <div id="md-footer"></div> <noscript><iframe src="//www.googletagmanager.com/ns.html?id=GTM-5W6NC8" height="0" width="0" style="display:none;visibility:hidden"></iframe></noscript> <script type="text/javascript" id="_informizely_script_tag"> var IzWidget = IzWidget || {}; (function (d) { var scriptElement = d.createElement('script'); scriptElement.type = 'text/javascript'; scriptElement.async = true; scriptElement.src = "https://insitez.blob.core.windows.net/site/f780f33e-a610-4ac2-af81-3eb184037547.js"; var node = d.getElementById('_informizely_script_tag'); node.parentNode.insertBefore(scriptElement, node); } )(document); </script>  <script> window.ap3c = window.ap3c || {}; var ap3c = window.ap3c; ap3c.cmd = ap3c.cmd || []; ap3c.cmd.push(function() { ap3c.init('ZO4siT4cLwnykPnzZWJtd3Byb2Q', 'https://engage.email.britannica.com/'); ap3c.track({v: 0}); }); ap3c.activity = function(act) { ap3c.act = (ap3c.act || []); ap3c.act.push(act); }; var s, t; s = document.createElement('script'); s.type = 'text/javascript'; s.src = "https://engage.email.britannica.com/app.js"; t = document.getElementsByTagName('script')[0]; t.parentNode.insertBefore(s, t); </script> <script class="marketing-page-info" type="application/json"> {"pageType":"Topic","templateName":"DESKTOP","pageNumber":1,"pagesTotal":27,"pageId":369194,"pageLength":487,"initialLoad":true,"lastPageOfScroll":false} </script> <script class="marketing-content-info" type="application/json"> [] </script> <script src="https://cdn.britannica.com/mendel-resources/3-130/js/libs/jquery-3.5.0.min.js?v=3.130.14"></script> <script type="text/javascript" data-type="Init Mendel Code Splitting"> (function() { $.ajax({ dataType: 'script', cache: true, url: 'https://cdn.britannica.com/mendel-resources/3-130/dist/topic-page.js?v=3.130.14' }); })(); </script> <script class="analytics-metadata" type="application/json"> {"leg":"C","adLeg":"C","userType":"ANONYMOUS","pageType":"Topic","pageSubtype":null,"articleTemplateType":"PAGINATED","gisted":false,"pageNumber":1,"hasSummarizeButton":false,"hasAskButton":false} </script> <script type="text/javascript"> EBStat={accountId:-1,hostnameOverride:'webstats.eb.com',domain:'www.britannica.com', json:''}; </script> <script type="text/javascript"> ( function() { $.ajax( { dataType: 'script', cache: true, url: '//www.britannica.com/webstats/mendelstats.js?v=1' } ) .done( function() { try {writeStat(null,EBStat);} catch(err){} } ); })(); </script> <div id="bc-fixed-dialogue"></div> </body> </html>

CINXE.COM

Mathematics | Definition, History, & Importance | Britannica