CINXE.COM

<!DOCTYPE html> <html class="client-nojs vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-toc-available" lang="zh" dir="ltr"> <head> <meta charset="UTF-8"> <title>數學證明 - 维基百科，自由的百科全书</title> <script>(function(){var className="client-js vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-toc-available";var cookie=document.cookie.match(/(?:^|; )zhwikimwclientpreferences=([^;]+)/);if(cookie){cookie[1].split('%2C').forEach(function(pref){className=className.replace(new RegExp('(^| )'+pref.replace(/-clientpref-\w+$|[^\w-]+/g,'')+'-clientpref-\\w+( |$)'),'$1'+pref+'$2');});}document.documentElement.className=className;}());RLCONF={"wgBreakFrames":false,"wgSeparatorTransformTable":["",""],"wgDigitTransformTable":["",""],"wgDefaultDateFormat":"zh", "wgMonthNames":["","1月","2月","3月","4月","5月","6月","7月","8月","9月","10月","11月","12月"],"wgRequestId":"e8bc637e-a3ae-466b-b087-a2b44c6ba83e","wgCanonicalNamespace":"","wgCanonicalSpecialPageName":false,"wgNamespaceNumber":0,"wgPageName":"數學證明","wgTitle":"數學證明","wgCurRevisionId":79648586,"wgRevisionId":79648586,"wgArticleId":64074,"wgIsArticle":true,"wgIsRedirect":false,"wgAction":"view","wgUserName":null,"wgUserGroups":["*"],"wgCategories":["有蓝链却未移除内部链接助手模板的页面","几何术语","數學推理"],"wgPageViewLanguage":"zh","wgPageContentLanguage":"zh","wgPageContentModel":"wikitext","wgRelevantPageName":"數學證明","wgRelevantArticleId":64074,"wgUserVariant":"zh","wgIsProbablyEditable":true,"wgRelevantPageIsProbablyEditable":true,"wgRestrictionEdit":[],"wgRestrictionMove":[],"wgNoticeProject":"wikipedia","wgCiteReferencePreviewsActive":true,"wgMediaViewerOnClick":true,"wgMediaViewerEnabledByDefault":true, "wgPopupsFlags":0,"wgVisualEditor":{"pageLanguageCode":"zh","pageLanguageDir":"ltr","pageVariantFallbacks":["zh-hans","zh-hant","zh-cn","zh-tw","zh-hk","zh-sg","zh-mo","zh-my"]},"wgMFDisplayWikibaseDescriptions":{"search":true,"watchlist":true,"tagline":true,"nearby":true},"wgWMESchemaEditAttemptStepOversample":false,"wgWMEPageLength":20000,"wgRelatedArticlesCompat":[],"wgCentralAuthMobileDomain":false,"wgEditSubmitButtonLabelPublish":true,"wgULSPosition":"interlanguage","wgULSisCompactLinksEnabled":false,"wgVector2022LanguageInHeader":true,"wgULSisLanguageSelectorEmpty":false,"wgWikibaseItemId":"Q11538","wgCheckUserClientHintsHeadersJsApi":["brands","architecture","bitness","fullVersionList","mobile","model","platform","platformVersion"],"GEHomepageSuggestedEditsEnableTopics":true,"wgGETopicsMatchModeEnabled":false,"wgGEStructuredTaskRejectionReasonTextInputEnabled":false,"wgGELevelingUpEnabledForUser":false};RLSTATE={"ext.gadget.large-font":"ready","ext.globalCssJs.user.styles": "ready","site.styles":"ready","user.styles":"ready","ext.globalCssJs.user":"ready","user":"ready","user.options":"loading","ext.cite.styles":"ready","ext.math.styles":"ready","skins.vector.search.codex.styles":"ready","skins.vector.styles":"ready","skins.vector.icons":"ready","jquery.makeCollapsible.styles":"ready","ext.wikimediamessages.styles":"ready","ext.visualEditor.desktopArticleTarget.noscript":"ready","ext.uls.interlanguage":"ready","wikibase.client.init":"ready","ext.wikimediaBadges":"ready"};RLPAGEMODULES=["ext.cite.ux-enhancements","mediawiki.page.media","site","mediawiki.page.ready","jquery.makeCollapsible","mediawiki.toc","skins.vector.js","ext.centralNotice.geoIP","ext.centralNotice.startUp","ext.gadget.edit0","ext.gadget.WikiMiniAtlas","ext.gadget.UnihanTooltips","ext.gadget.Difflink","ext.gadget.pseudonamespace-UI","ext.gadget.SpecialWikitext","ext.gadget.switcher","ext.gadget.VariantAlly","ext.gadget.AdvancedSiteNotices","ext.gadget.hideConversionTab", "ext.gadget.internalLinkHelper-altcolor","ext.gadget.noteTA","ext.gadget.NavFrame","ext.gadget.collapsibleTables","ext.gadget.scrollUpButton","ext.urlShortener.toolbar","ext.centralauth.centralautologin","mmv.bootstrap","ext.popups","ext.visualEditor.desktopArticleTarget.init","ext.visualEditor.targetLoader","ext.echo.centralauth","ext.eventLogging","ext.wikimediaEvents","ext.navigationTiming","ext.uls.interface","ext.cx.eventlogging.campaigns","ext.cx.uls.quick.actions","wikibase.client.vector-2022","ext.checkUser.clientHints","ext.growthExperiments.SuggestedEditSession","wikibase.sidebar.tracking"];</script> <script>(RLQ=window.RLQ||[]).push(function(){mw.loader.impl(function(){return["user.options@12s5i",function($,jQuery,require,module){mw.user.tokens.set({"patrolToken":"+\\","watchToken":"+\\","csrfToken":"+\\"}); }];});});</script> <link rel="stylesheet" href="/w/load.php?lang=zh&modules=ext.cite.styles%7Cext.math.styles%7Cext.uls.interlanguage%7Cext.visualEditor.desktopArticleTarget.noscript%7Cext.wikimediaBadges%7Cext.wikimediamessages.styles%7Cjquery.makeCollapsible.styles%7Cskins.vector.icons%2Cstyles%7Cskins.vector.search.codex.styles%7Cwikibase.client.init&only=styles&skin=vector-2022"> <script async="" src="/w/load.php?lang=zh&modules=startup&only=scripts&raw=1&skin=vector-2022"></script> <meta name="ResourceLoaderDynamicStyles" content=""> <link rel="stylesheet" href="/w/load.php?lang=zh&modules=ext.gadget.large-font&only=styles&skin=vector-2022"> <link rel="stylesheet" href="/w/load.php?lang=zh&modules=site.styles&only=styles&skin=vector-2022"> <meta name="generator" content="MediaWiki 1.44.0-wmf.4"> <meta name="referrer" content="origin"> <meta name="referrer" content="origin-when-cross-origin"> <meta name="robots" content="max-image-preview:standard"> <meta name="format-detection" content="telephone=no"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/8/8d/P._Oxy._I_29.jpg/1200px-P._Oxy._I_29.jpg"> <meta property="og:image:width" content="1200"> <meta property="og:image:height" content="731"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/8/8d/P._Oxy._I_29.jpg/800px-P._Oxy._I_29.jpg"> <meta property="og:image:width" content="800"> <meta property="og:image:height" content="487"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/8/8d/P._Oxy._I_29.jpg/640px-P._Oxy._I_29.jpg"> <meta property="og:image:width" content="640"> <meta property="og:image:height" content="390"> <meta name="viewport" content="width=1120"> <meta property="og:title" content="數學證明 - 维基百科，自由的百科全书"> <meta property="og:type" content="website"> <link rel="preconnect" href="//upload.wikimedia.org"> <link rel="alternate" media="only screen and (max-width: 640px)" href="//zh.m.wikipedia.org/wiki/%E6%95%B8%E5%AD%B8%E8%AD%89%E6%98%8E"> <link rel="alternate" type="application/x-wiki" title="编辑本页" href="/w/index.php?title=%E6%95%B8%E5%AD%B8%E8%AD%89%E6%98%8E&action=edit"> <link rel="apple-touch-icon" href="/static/apple-touch/wikipedia.png"> <link rel="icon" href="/static/favicon/wikipedia.ico"> <link rel="search" type="application/opensearchdescription+xml" href="/w/rest.php/v1/search" title="Wikipedia (zh)"> <link rel="EditURI" type="application/rsd+xml" href="//zh.wikipedia.org/w/api.php?action=rsd"> <link rel="canonical" href="https://zh.wikipedia.org/wiki/%E6%95%B8%E5%AD%B8%E8%AD%89%E6%98%8E"> <link rel="alternate" hreflang="zh" href="https://zh.wikipedia.org/wiki/%E6%95%B8%E5%AD%B8%E8%AD%89%E6%98%8E"> <link rel="alternate" hreflang="zh-Hans" href="https://zh.wikipedia.org/zh-hans/%E6%95%B8%E5%AD%B8%E8%AD%89%E6%98%8E"> <link rel="alternate" hreflang="zh-Hans-CN" href="https://zh.wikipedia.org/zh-cn/%E6%95%B8%E5%AD%B8%E8%AD%89%E6%98%8E"> <link rel="alternate" hreflang="zh-Hans-MY" href="https://zh.wikipedia.org/zh-my/%E6%95%B8%E5%AD%B8%E8%AD%89%E6%98%8E"> <link rel="alternate" hreflang="zh-Hans-SG" href="https://zh.wikipedia.org/zh-sg/%E6%95%B8%E5%AD%B8%E8%AD%89%E6%98%8E"> <link rel="alternate" hreflang="zh-Hant" href="https://zh.wikipedia.org/zh-hant/%E6%95%B8%E5%AD%B8%E8%AD%89%E6%98%8E"> <link rel="alternate" hreflang="zh-Hant-HK" href="https://zh.wikipedia.org/zh-hk/%E6%95%B8%E5%AD%B8%E8%AD%89%E6%98%8E"> <link rel="alternate" hreflang="zh-Hant-MO" href="https://zh.wikipedia.org/zh-mo/%E6%95%B8%E5%AD%B8%E8%AD%89%E6%98%8E"> <link rel="alternate" hreflang="zh-Hant-TW" href="https://zh.wikipedia.org/zh-tw/%E6%95%B8%E5%AD%B8%E8%AD%89%E6%98%8E"> <link rel="alternate" hreflang="x-default" href="https://zh.wikipedia.org/wiki/%E6%95%B8%E5%AD%B8%E8%AD%89%E6%98%8E"> <link rel="license" href="https://creativecommons.org/licenses/by-sa/4.0/deed.zh"> <link rel="alternate" type="application/atom+xml" title="Wikipedia的Atom feed" href="/w/index.php?title=Special:%E6%9C%80%E8%BF%91%E6%9B%B4%E6%94%B9&feed=atom"> <link rel="dns-prefetch" href="//meta.wikimedia.org" /> <link rel="dns-prefetch" href="//login.wikimedia.org"> </head> <body class="skin--responsive skin-vector skin-vector-search-vue mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject mw-editable page-數學證明 rootpage-數學證明 skin-vector-2022 action-view"><a class="mw-jump-link" href="#bodyContent">跳转到内容</a> <div class="vector-header-container"> <header class="vector-header mw-header"> <div class="vector-header-start"> <nav class="vector-main-menu-landmark" aria-label="站点"> <div id="vector-main-menu-dropdown" class="vector-dropdown vector-main-menu-dropdown vector-button-flush-left vector-button-flush-right" > <input type="checkbox" id="vector-main-menu-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-main-menu-dropdown" class="vector-dropdown-checkbox " aria-label="主菜单" > <label id="vector-main-menu-dropdown-label" for="vector-main-menu-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-menu mw-ui-icon-wikimedia-menu"></span> <span class="vector-dropdown-label-text">主菜单</span> </label> <div class="vector-dropdown-content"> <div id="vector-main-menu-unpinned-container" class="vector-unpinned-container"> <div id="vector-main-menu" class="vector-main-menu vector-pinnable-element"> <div class="vector-pinnable-header vector-main-menu-pinnable-header vector-pinnable-header-unpinned" data-feature-name="main-menu-pinned" data-pinnable-element-id="vector-main-menu" data-pinned-container-id="vector-main-menu-pinned-container" data-unpinned-container-id="vector-main-menu-unpinned-container" > <div class="vector-pinnable-header-label">主菜单</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-main-menu.pin">移至侧栏</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-main-menu.unpin">隐藏</button> </div> <div id="p-navigation" class="vector-menu mw-portlet mw-portlet-navigation" > <div class="vector-menu-heading"> 导航 </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-mainpage-description" class="mw-list-item"><a href="/wiki/Wikipedia:%E9%A6%96%E9%A1%B5" title="访问首页[z]" accesskey="z"><span>首页</span></a></li><li id="n-indexpage" class="mw-list-item"><a href="/wiki/Wikipedia:%E5%88%86%E7%B1%BB%E7%B4%A2%E5%BC%95" title="以分类索引搜寻中文维基百科"><span>分类索引</span></a></li><li id="n-Featured_content" class="mw-list-item"><a href="/wiki/Portal:%E7%89%B9%E8%89%B2%E5%85%A7%E5%AE%B9" title="查看中文维基百科的特色内容"><span>特色内容</span></a></li><li id="n-currentevents" class="mw-list-item"><a href="/wiki/Portal:%E6%96%B0%E8%81%9E%E5%8B%95%E6%85%8B" title="提供当前新闻事件的背景资料"><span>新闻动态</span></a></li><li id="n-recentchanges" class="mw-list-item"><a href="/wiki/Special:%E6%9C%80%E8%BF%91%E6%9B%B4%E6%94%B9" title="列出维基百科中的最近修改[r]" accesskey="r"><span>最近更改</span></a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/Special:%E9%9A%8F%E6%9C%BA%E9%A1%B5%E9%9D%A2" title="随机载入一个页面[x]" accesskey="x"><span>随机条目</span></a></li> </ul> </div> </div> <div id="p-help" class="vector-menu mw-portlet mw-portlet-help" > <div class="vector-menu-heading"> 帮助 </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-help" class="mw-list-item"><a href="/wiki/Help:%E7%9B%AE%E5%BD%95" title="寻求帮助"><span>帮助</span></a></li><li id="n-portal" class="mw-list-item"><a href="/wiki/Wikipedia:%E7%A4%BE%E7%BE%A4%E9%A6%96%E9%A1%B5" title="关于本计划、你可以做什么、应该如何做"><span>维基社群</span></a></li><li id="n-policy" class="mw-list-item"><a href="/wiki/Wikipedia:%E6%96%B9%E9%87%9D%E8%88%87%E6%8C%87%E5%BC%95" title="查看维基百科的方针和指引"><span>方针与指引</span></a></li><li id="n-villagepump" class="mw-list-item"><a href="/wiki/Wikipedia:%E4%BA%92%E5%8A%A9%E5%AE%A2%E6%A0%88" title="参与维基百科社群的讨论"><span>互助客栈</span></a></li><li id="n-Information_desk" class="mw-list-item"><a href="/wiki/Wikipedia:%E7%9F%A5%E8%AF%86%E9%97%AE%E7%AD%94" title="解答任何与维基百科无关的问题的地方"><span>知识问答</span></a></li><li id="n-conversion" class="mw-list-item"><a href="/wiki/Wikipedia:%E5%AD%97%E8%AF%8D%E8%BD%AC%E6%8D%A2" title="提出字词转换请求"><span>字词转换</span></a></li><li id="n-IRC" class="mw-list-item"><a href="/wiki/Wikipedia:IRC%E8%81%8A%E5%A4%A9%E9%A2%91%E9%81%93"><span>IRC即时聊天</span></a></li><li id="n-contact" class="mw-list-item"><a href="/wiki/Wikipedia:%E8%81%94%E7%BB%9C%E6%88%91%E4%BB%AC" title="如何联络维基百科"><span>联络我们</span></a></li><li id="n-about" class="mw-list-item"><a href="/wiki/Wikipedia:%E5%85%B3%E4%BA%8E" title="查看维基百科的简介"><span>关于维基百科</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> <a href="/wiki/Wikipedia:%E9%A6%96%E9%A1%B5" class="mw-logo"> <img class="mw-logo-icon" src="/static/images/icons/wikipedia.png" alt="" aria-hidden="true" height="50" width="50"> <span class="mw-logo-container skin-invert"> <img class="mw-logo-wordmark" alt="维基百科" src="/static/images/mobile/copyright/wikipedia-wordmark-zh.svg" style="width: 6.5625em; height: 1.375em;"> <img class="mw-logo-tagline" alt="自由的百科全书" src="/static/images/mobile/copyright/wikipedia-tagline-zh.svg" width="103" height="14" style="width: 6.4375em; height: 0.875em;"> </span> </a> </div> <div class="vector-header-end"> <div id="p-search" role="search" class="vector-search-box-vue vector-search-box-collapses vector-search-box-show-thumbnail vector-search-box-auto-expand-width vector-search-box"> <a href="/wiki/Special:%E6%90%9C%E7%B4%A2" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only search-toggle" title="搜索维基百科[f]" accesskey="f"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>搜索</span> </a> <div class="vector-typeahead-search-container"> <div class="cdx-typeahead-search cdx-typeahead-search--show-thumbnail cdx-typeahead-search--auto-expand-width"> <form action="/w/index.php" id="searchform" class="cdx-search-input cdx-search-input--has-end-button"> <div id="simpleSearch" class="cdx-search-input__input-wrapper" data-search-loc="header-moved"> <div class="cdx-text-input cdx-text-input--has-start-icon"> <input class="cdx-text-input__input" type="search" name="search" placeholder="搜索维基百科" aria-label="搜索维基百科" autocapitalize="sentences" title="搜索维基百科[f]" accesskey="f" id="searchInput" > <span class="cdx-text-input__icon cdx-text-input__start-icon"></span> </div> <input type="hidden" name="title" value="Special:搜索"> </div> <button class="cdx-button cdx-search-input__end-button">搜索</button> </form> </div> </div> </div> <nav class="vector-user-links vector-user-links-wide" aria-label="个人工具"> <div class="vector-user-links-main"> <div id="p-vector-user-menu-preferences" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-userpage" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <nav class="vector-appearance-landmark" aria-label="外观"> <div id="vector-appearance-dropdown" class="vector-dropdown " title="更改页面字体大小、宽度和颜色的外观" > <input type="checkbox" id="vector-appearance-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-appearance-dropdown" class="vector-dropdown-checkbox " aria-label="外观" > <label id="vector-appearance-dropdown-label" for="vector-appearance-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-appearance mw-ui-icon-wikimedia-appearance"></span> <span class="vector-dropdown-label-text">外观</span> </label> <div class="vector-dropdown-content"> <div id="vector-appearance-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div id="p-vector-user-menu-notifications" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-overflow" class="vector-menu mw-portlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="https://donate.wikimedia.org/?utm_source=donate&utm_medium=sidebar&utm_campaign=spontaneous&uselang=zh-hans" class=""><span>资助维基百科</span></a> </li> <li id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Special:%E5%88%9B%E5%BB%BA%E8%B4%A6%E6%88%B7&returnto=%E6%95%B8%E5%AD%B8%E8%AD%89%E6%98%8E" title="我们推荐您创建账号并登录，但这不是强制性的" class=""><span>创建账号</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Special:%E7%94%A8%E6%88%B7%E7%99%BB%E5%BD%95&returnto=%E6%95%B8%E5%AD%B8%E8%AD%89%E6%98%8E" title="建议你登录，尽管并非必须。[o]" accesskey="o" class=""><span>登录</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="更多选项" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="个人工具" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">个人工具</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="用户菜单" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="https://donate.wikimedia.org/?utm_source=donate&utm_medium=sidebar&utm_campaign=spontaneous&uselang=zh-hans"><span>资助维基百科</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:%E5%88%9B%E5%BB%BA%E8%B4%A6%E6%88%B7&returnto=%E6%95%B8%E5%AD%B8%E8%AD%89%E6%98%8E" title="我们推荐您创建账号并登录，但这不是强制性的"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>创建账号</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:%E7%94%A8%E6%88%B7%E7%99%BB%E5%BD%95&returnto=%E6%95%B8%E5%AD%B8%E8%AD%89%E6%98%8E" title="建议你登录，尽管并非必须。[o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>登录</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> 未登录编辑者的页面 <a href="/wiki/Help:%E6%96%B0%E6%89%8B%E5%85%A5%E9%97%A8" aria-label="了解有关编辑的更多信息"><span>了解详情</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Special:%E6%88%91%E7%9A%84%E8%B4%A1%E7%8C%AE" title="来自此IP地址的编辑列表[y]" accesskey="y"><span>贡献</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Special:%E6%88%91%E7%9A%84%E8%AE%A8%E8%AE%BA%E9%A1%B5" title="对于来自此IP地址编辑的讨论[n]" accesskey="n"><span>讨论</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="站点"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="目录" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">目录</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">移至侧栏</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">隐藏</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">序言</div> </a> </li> <li id="toc-定義" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#定義"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>定義</span> </div> </a> <button aria-controls="toc-定義-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>开关定義子章节</span> </button> <ul id="toc-定義-sublist" class="vector-toc-list"> <li id="toc-形式化證明" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#形式化證明"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>形式化證明</span> </div> </a> <ul id="toc-形式化證明-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-常見的證明技巧" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#常見的證明技巧"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>常見的證明技巧</span> </div> </a> <button aria-controls="toc-常見的證明技巧-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>开关常見的證明技巧子章节</span> </button> <ul id="toc-常見的證明技巧-sublist" class="vector-toc-list"> <li id="toc-直接證明" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#直接證明"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>直接證明</span> </div> </a> <ul id="toc-直接證明-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-構造法" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#構造法"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>構造法</span> </div> </a> <ul id="toc-構造法-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-非构造性证明" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#非构造性证明"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>非构造性证明</span> </div> </a> <ul id="toc-非构造性证明-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-穷举法" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#穷举法"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>穷举法</span> </div> </a> <ul id="toc-穷举法-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-换质位法" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#换质位法"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.5</span> <span>换质位法</span> </div> </a> <ul id="toc-换质位法-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-個案分析" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#個案分析"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.6</span> <span>個案分析</span> </div> </a> <ul id="toc-個案分析-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-算兩次" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#算兩次"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.7</span> <span>算兩次</span> </div> </a> <ul id="toc-算兩次-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-間接證明" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#間接證明"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.8</span> <span>間接證明</span> </div> </a> <ul id="toc-間接證明-sublist" class="vector-toc-list"> <li id="toc-反證法" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#反證法"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.8.1</span> <span>反證法</span> </div> </a> <ul id="toc-反證法-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-數學歸納法" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#數學歸納法"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.8.2</span> <span>數學歸納法</span> </div> </a> <ul id="toc-數學歸納法-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-其他证明方式" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#其他证明方式"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>其他证明方式</span> </div> </a> <button aria-controls="toc-其他证明方式-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>开关其他证明方式子章节</span> </button> <ul id="toc-其他证明方式-sublist" class="vector-toc-list"> <li id="toc-直观证明" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#直观证明"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>直观证明</span> </div> </a> <ul id="toc-直观证明-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-计算机辅助证明" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#计算机辅助证明"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>计算机辅助证明</span> </div> </a> <ul id="toc-计算机辅助证明-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-證明的結尾" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#證明的結尾"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>證明的結尾</span> </div> </a> <ul id="toc-證明的結尾-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-參見" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#參見"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>參見</span> </div> </a> <ul id="toc-參見-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-参考资料" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#参考资料"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>参考资料</span> </div> </a> <ul id="toc-参考资料-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-外部連結" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#外部連結"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>外部連結</span> </div> </a> <ul id="toc-外部連結-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="目录" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="开关目录" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">开关目录</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">數學證明</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="前往另一种语言写成的文章。95种语言可用" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-95" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">95种语言</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Wiskundige_bewys" title="Wiskundige bewys – 南非荷兰语" lang="af" hreflang="af" data-title="Wiskundige bewys" data-language-autonym="Afrikaans" data-language-local-name="南非荷兰语" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Beweis_(Mathematik)" title="Beweis (Mathematik) – 瑞士德语" lang="gsw" hreflang="gsw" data-title="Beweis (Mathematik)" data-language-autonym="Alemannisch" data-language-local-name="瑞士德语" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Demostraci%C3%B3n_matematica" title="Demostración matematica – 阿拉贡语" lang="an" hreflang="an" data-title="Demostración matematica" data-language-autonym="Aragonés" data-language-local-name="阿拉贡语" class="interlanguage-link-target"><span>Aragonés</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%A8%D8%B1%D9%87%D8%A7%D9%86_%D8%B1%D9%8A%D8%A7%D8%B6%D9%8A" title="برهان رياضي – 阿拉伯语" lang="ar" hreflang="ar" data-title="برهان رياضي" data-language-autonym="العربية" data-language-local-name="阿拉伯语" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-as mw-list-item"><a href="https://as.wikipedia.org/wiki/%E0%A6%97%E0%A6%BE%E0%A6%A3%E0%A6%BF%E0%A6%A4%E0%A6%BF%E0%A6%95_%E0%A6%AA%E0%A7%8D%E0%A7%B0%E0%A6%AE%E0%A6%BE%E0%A6%A3" title="গাণিতিক প্ৰমাণ – 阿萨姆语" lang="as" hreflang="as" data-title="গাণিতিক প্ৰমাণ" data-language-autonym="অসমীয়া" data-language-local-name="阿萨姆语" class="interlanguage-link-target"><span>অসমীয়া</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Demostraci%C3%B3n_matem%C3%A1tica" title="Demostración matemática – 阿斯图里亚斯语" lang="ast" hreflang="ast" data-title="Demostración matemática" data-language-autonym="Asturianu" data-language-local-name="阿斯图里亚斯语" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Riyazi_isbat" title="Riyazi isbat – 阿塞拜疆语" lang="az" hreflang="az" data-title="Riyazi isbat" data-language-autonym="Azərbaycanca" data-language-local-name="阿塞拜疆语" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA_%D0%B8%D2%AB%D0%B1%D0%B0%D1%82%D0%BB%D0%B0%D1%83" title="Математик иҫбатлау – 巴什基尔语" lang="ba" hreflang="ba" data-title="Математик иҫбатлау" data-language-autonym="Башҡортса" data-language-local-name="巴什基尔语" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-bat-smg mw-list-item"><a href="https://bat-smg.wikipedia.org/wiki/Matemat%C4%97nis_iruod%C4%97ms" title="Matematėnis iruodėms – 薩莫吉希亞文" lang="sgs" hreflang="sgs" data-title="Matematėnis iruodėms" data-language-autonym="Žemaitėška" data-language-local-name="薩莫吉希亞文" class="interlanguage-link-target"><span>Žemaitėška</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D1%8D%D0%BC%D0%B0%D1%82%D1%8B%D1%87%D0%BD%D1%8B_%D0%B4%D0%BE%D0%BA%D0%B0%D0%B7" title="Матэматычны доказ – 白俄罗斯语" lang="be" hreflang="be" data-title="Матэматычны доказ" data-language-autonym="Беларуская" data-language-local-name="白俄罗斯语" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D1%8D%D0%BC%D0%B0%D1%82%D1%8B%D1%87%D0%BD%D1%8B_%D0%B4%D0%BE%D0%BA%D0%B0%D0%B7" title="Матэматычны доказ – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Матэматычны доказ" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%BE_%D0%B4%D0%BE%D0%BA%D0%B0%D0%B7%D0%B0%D1%82%D0%B5%D0%BB%D1%81%D1%82%D0%B2%D0%BE" title="Математическо доказателство – 保加利亚语" lang="bg" hreflang="bg" data-title="Математическо доказателство" data-language-autonym="Български" data-language-local-name="保加利亚语" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%97%E0%A6%BE%E0%A6%A3%E0%A6%BF%E0%A6%A4%E0%A6%BF%E0%A6%95_%E0%A6%AA%E0%A7%8D%E0%A6%B0%E0%A6%AE%E0%A6%BE%E0%A6%A3" title="গাণিতিক প্রমাণ – 孟加拉语" lang="bn" hreflang="bn" data-title="গাণিতিক প্রমাণ" data-language-autonym="বাংলা" data-language-local-name="孟加拉语" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Matemati%C4%8Dki_dokaz" title="Matematički dokaz – 波斯尼亚语" lang="bs" hreflang="bs" data-title="Matematički dokaz" data-language-autonym="Bosanski" data-language-local-name="波斯尼亚语" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Demostraci%C3%B3_(matem%C3%A0tiques)" title="Demostració (matemàtiques) – 加泰罗尼亚语" lang="ca" hreflang="ca" data-title="Demostració (matemàtiques)" data-language-autonym="Català" data-language-local-name="加泰罗尼亚语" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D8%A6%DB%8C%D8%B3%D9%BE%D8%A7%D8%AA_(%D9%85%D8%A7%D8%AA%D9%85%D8%A7%D8%AA%DB%8C%DA%A9)" title="ئیسپات (ماتماتیک) – 中库尔德语" lang="ckb" hreflang="ckb" data-title="ئیسپات (ماتماتیک)" data-language-autonym="کوردی" data-language-local-name="中库尔德语" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cs badge-Q17437798 badge-goodarticle mw-list-item" title="优良条目"><a href="https://cs.wikipedia.org/wiki/Matematick%C3%BD_d%C5%AFkaz" title="Matematický důkaz – 捷克语" lang="cs" hreflang="cs" data-title="Matematický důkaz" data-language-autonym="Čeština" data-language-local-name="捷克语" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%C4%94%D0%BD%D0%B5%D0%BD%D1%82%D0%B5%D1%80%D3%B3_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Ĕнентерӳ (математика) – 楚瓦什语" lang="cv" hreflang="cv" data-title="Ĕнентерӳ (математика)" data-language-autonym="Чӑвашла" data-language-local-name="楚瓦什语" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Prawf_mathemategol" title="Prawf mathemategol – 威尔士语" lang="cy" hreflang="cy" data-title="Prawf mathemategol" data-language-autonym="Cymraeg" data-language-local-name="威尔士语" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Bevis_(matematik)" title="Bevis (matematik) – 丹麦语" lang="da" hreflang="da" data-title="Bevis (matematik)" data-language-autonym="Dansk" data-language-local-name="丹麦语" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Beweis_(Mathematik)" title="Beweis (Mathematik) – 德语" lang="de" hreflang="de" data-title="Beweis (Mathematik)" data-language-autonym="Deutsch" data-language-local-name="德语" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%9C%CE%B1%CE%B8%CE%B7%CE%BC%CE%B1%CF%84%CE%B9%CE%BA%CE%AE_%CE%B1%CF%80%CF%8C%CE%B4%CE%B5%CE%B9%CE%BE%CE%B7" title="Μαθηματική απόδειξη – 希腊语" lang="el" hreflang="el" data-title="Μαθηματική απόδειξη" data-language-autonym="Ελληνικά" data-language-local-name="希腊语" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Mathematical_proof" title="Mathematical proof – 英语" lang="en" hreflang="en" data-title="Mathematical proof" data-language-autonym="English" data-language-local-name="英语" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo badge-Q17437796 badge-featuredarticle mw-list-item" title="典范条目"><a href="https://eo.wikipedia.org/wiki/Matematika_pruvo" title="Matematika pruvo – 世界语" lang="eo" hreflang="eo" data-title="Matematika pruvo" data-language-autonym="Esperanto" data-language-local-name="世界语" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Demostraci%C3%B3n_matem%C3%A1tica" title="Demostración matemática – 西班牙语" lang="es" hreflang="es" data-title="Demostración matemática" data-language-autonym="Español" data-language-local-name="西班牙语" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/T%C3%B5estus" title="Tõestus – 爱沙尼亚语" lang="et" hreflang="et" data-title="Tõestus" data-language-autonym="Eesti" data-language-local-name="爱沙尼亚语" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Froga_matematiko" title="Froga matematiko – 巴斯克语" lang="eu" hreflang="eu" data-title="Froga matematiko" data-language-autonym="Euskara" data-language-local-name="巴斯克语" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%A7%D8%AB%D8%A8%D8%A7%D8%AA_%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C" title="اثبات ریاضی – 波斯语" lang="fa" hreflang="fa" data-title="اثبات ریاضی" data-language-autonym="فارسی" data-language-local-name="波斯语" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Matemaattinen_todistus" title="Matemaattinen todistus – 芬兰语" lang="fi" hreflang="fi" data-title="Matemaattinen todistus" data-language-autonym="Suomi" data-language-local-name="芬兰语" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/D%C3%A9monstration_(logique_et_math%C3%A9matiques)" title="Démonstration (logique et mathématiques) – 法语" lang="fr" hreflang="fr" data-title="Démonstration (logique et mathématiques)" data-language-autonym="Français" data-language-local-name="法语" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Bewis" title="Bewis – 北弗里西亚语" lang="frr" hreflang="frr" data-title="Bewis" data-language-autonym="Nordfriisk" data-language-local-name="北弗里西亚语" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-gan mw-list-item"><a href="https://gan.wikipedia.org/wiki/%E6%95%B8%E5%AD%B8%E8%AD%89%E6%98%8E" title="數學證明 – 赣语" lang="gan" hreflang="gan" data-title="數學證明" data-language-autonym="贛語" data-language-local-name="赣语" class="interlanguage-link-target"><span>贛語</span></a></li><li class="interlanguage-link interwiki-gcr mw-list-item"><a href="https://gcr.wikipedia.org/wiki/D%C3%A9monstrasyon" title="Démonstrasyon – Guianan Creole" lang="gcr" hreflang="gcr" data-title="Démonstrasyon" data-language-autonym="Kriyòl gwiyannen" data-language-local-name="Guianan Creole" class="interlanguage-link-target"><span>Kriyòl gwiyannen</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Proba_matem%C3%A1tica" title="Proba matemática – 加利西亚语" lang="gl" hreflang="gl" data-title="Proba matemática" data-language-autonym="Galego" data-language-local-name="加利西亚语" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%94%D7%95%D7%9B%D7%97%D7%94" title="הוכחה – 希伯来语" lang="he" hreflang="he" data-title="הוכחה" data-language-autonym="עברית" data-language-local-name="希伯来语" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4%E0%A5%80%E0%A4%AF_%E0%A4%89%E0%A4%AA%E0%A4%AA%E0%A4%A4%E0%A5%8D%E0%A4%A4%E0%A4%BF" title="गणितीय उपपत्ति – 印地语" lang="hi" hreflang="hi" data-title="गणितीय उपपत्ति" data-language-autonym="हिन्दी" data-language-local-name="印地语" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hif mw-list-item"><a href="https://hif.wikipedia.org/wiki/Mathematical_proof" title="Mathematical proof – 斐濟印地文" lang="hif" hreflang="hif" data-title="Mathematical proof" data-language-autonym="Fiji Hindi" data-language-local-name="斐濟印地文" class="interlanguage-link-target"><span>Fiji Hindi</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Matemati%C4%8Dki_dokaz" title="Matematički dokaz – 克罗地亚语" lang="hr" hreflang="hr" data-title="Matematički dokaz" data-language-autonym="Hrvatski" data-language-local-name="克罗地亚语" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Matematikai_bizony%C3%ADt%C3%A1s" title="Matematikai bizonyítás – 匈牙利语" lang="hu" hreflang="hu" data-title="Matematikai bizonyítás" data-language-autonym="Magyar" data-language-local-name="匈牙利语" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%84%D5%A1%D5%A9%D5%A5%D5%B4%D5%A1%D5%BF%D5%AB%D5%AF%D5%A1%D5%AF%D5%A1%D5%B6_%D5%A1%D5%BA%D5%A1%D6%81%D5%B8%D6%82%D5%B5%D6%81" title="Մաթեմատիկական ապացույց – 亚美尼亚语" lang="hy" hreflang="hy" data-title="Մաթեմատիկական ապացույց" data-language-autonym="Հայերեն" data-language-local-name="亚美尼亚语" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hyw mw-list-item"><a href="https://hyw.wikipedia.org/wiki/%D5%84%D5%A1%D5%A9%D5%A5%D5%B4%D5%A1%D5%A9%D5%AB%D5%AF%D5%A1%D5%AF%D5%A1%D5%B6_%D5%A1%D5%BA%D5%A1%D6%81%D5%B8%D5%B5%D6%81" title="Մաթեմաթիկական ապացոյց – Western Armenian" lang="hyw" hreflang="hyw" data-title="Մաթեմաթիկական ապացոյց" data-language-autonym="Արեւմտահայերէն" data-language-local-name="Western Armenian" class="interlanguage-link-target"><span>Արեւմտահայերէն</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Prova_mathematic" title="Prova mathematic – 国际语" lang="ia" hreflang="ia" data-title="Prova mathematic" data-language-autonym="Interlingua" data-language-local-name="国际语" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Pembuktian_matematika" title="Pembuktian matematika – 印度尼西亚语" lang="id" hreflang="id" data-title="Pembuktian matematika" data-language-autonym="Bahasa Indonesia" data-language-local-name="印度尼西亚语" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/St%C3%A6r%C3%B0fr%C3%A6%C3%B0ileg_s%C3%B6nnun" title="Stærðfræðileg sönnun – 冰岛语" lang="is" hreflang="is" data-title="Stærðfræðileg sönnun" data-language-autonym="Íslenska" data-language-local-name="冰岛语" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Dimostrazione_matematica" title="Dimostrazione matematica – 意大利语" lang="it" hreflang="it" data-title="Dimostrazione matematica" data-language-autonym="Italiano" data-language-local-name="意大利语" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E8%A8%BC%E6%98%8E_(%E6%95%B0%E5%AD%A6)" title="証明 (数学) – 日语" lang="ja" hreflang="ja" data-title="証明 (数学)" data-language-autonym="日本語" data-language-local-name="日语" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-jam mw-list-item"><a href="https://jam.wikipedia.org/wiki/Matimatikal_pruuf" title="Matimatikal pruuf – 牙買加克里奧爾英文" lang="jam" hreflang="jam" data-title="Matimatikal pruuf" data-language-autonym="Patois" data-language-local-name="牙買加克里奧爾英文" class="interlanguage-link-target"><span>Patois</span></a></li><li class="interlanguage-link interwiki-jbo mw-list-item"><a href="https://jbo.wikipedia.org/wiki/cipra" title="cipra – 逻辑语" lang="jbo" hreflang="jbo" data-title="cipra" data-language-autonym="La .lojban." data-language-local-name="逻辑语" class="interlanguage-link-target"><span>La .lojban.</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%9B%E1%83%90%E1%83%97%E1%83%94%E1%83%9B%E1%83%90%E1%83%A2%E1%83%98%E1%83%99%E1%83%A3%E1%83%A0%E1%83%98_%E1%83%93%E1%83%90%E1%83%9B%E1%83%A2%E1%83%99%E1%83%98%E1%83%AA%E1%83%94%E1%83%91%E1%83%90" title="მათემატიკური დამტკიცება – 格鲁吉亚语" lang="ka" hreflang="ka" data-title="მათემატიკური დამტკიცება" data-language-autonym="ქართული" data-language-local-name="格鲁吉亚语" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%94%D3%99%D0%BB%D0%B5%D0%BB%D0%B4%D0%B5%D1%83" title="Дәлелдеу – 哈萨克语" lang="kk" hreflang="kk" data-title="Дәлелдеу" data-language-autonym="Қазақша" data-language-local-name="哈萨克语" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%A6%9D%EB%AA%85_(%EC%88%98%ED%95%99)" title="증명 (수학) – 韩语" lang="ko" hreflang="ko" data-title="증명 (수학)" data-language-autonym="한국어" data-language-local-name="韩语" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Demonstratio_mathematica" title="Demonstratio mathematica – 拉丁语" lang="la" hreflang="la" data-title="Demonstratio mathematica" data-language-autonym="Latina" data-language-local-name="拉丁语" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Matematinis_%C4%AFrodymas" title="Matematinis įrodymas – 立陶宛语" lang="lt" hreflang="lt" data-title="Matematinis įrodymas" data-language-autonym="Lietuvių" data-language-local-name="立陶宛语" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Matem%C4%81tisks_pier%C4%81d%C4%ABjums" title="Matemātisks pierādījums – 拉脱维亚语" lang="lv" hreflang="lv" data-title="Matemātisks pierādījums" data-language-autonym="Latviešu" data-language-local-name="拉脱维亚语" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D1%87%D0%BA%D0%B8_%D0%B4%D0%BE%D0%BA%D0%B0%D0%B7" title="Математички доказ – 马其顿语" lang="mk" hreflang="mk" data-title="Математички доказ" data-language-autonym="Македонски" data-language-local-name="马其顿语" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%A4%E0%B5%86%E0%B4%B3%E0%B4%BF%E0%B4%B5%E0%B5%8D_%E0%B4%97%E0%B4%A3%E0%B4%BF%E0%B4%A4%E0%B4%B6%E0%B4%BE%E0%B4%B8%E0%B5%8D%E0%B4%A4%E0%B5%8D%E0%B4%B0%E0%B4%A4%E0%B5%8D%E0%B4%A4%E0%B4%BF%E0%B5%BD" title="തെളിവ് ഗണിതശാസ്ത്രത്തിൽ – 马拉雅拉姆语" lang="ml" hreflang="ml" data-title="തെളിവ് ഗണിതശാസ്ത്രത്തിൽ" data-language-autonym="മലയാളം" data-language-local-name="马拉雅拉姆语" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4%E0%A5%80%E0%A4%AF_%E0%A4%AA%E0%A5%81%E0%A4%B0%E0%A4%BE%E0%A4%B5%E0%A4%BE" title="गणितीय पुरावा – 马拉地语" lang="mr" hreflang="mr" data-title="गणितीय पुरावा" data-language-autonym="मराठी" data-language-local-name="马拉地语" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Bukti_matematik" title="Bukti matematik – 马来语" lang="ms" hreflang="ms" data-title="Bukti matematik" data-language-autonym="Bahasa Melayu" data-language-local-name="马来语" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nds mw-list-item"><a href="https://nds.wikipedia.org/wiki/Bewies_(Mathematik)" title="Bewies (Mathematik) – 低地德语" lang="nds" hreflang="nds" data-title="Bewies (Mathematik)" data-language-autonym="Plattdüütsch" data-language-local-name="低地德语" class="interlanguage-link-target"><span>Plattdüütsch</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Wiskundig_bewijs" title="Wiskundig bewijs – 荷兰语" lang="nl" hreflang="nl" data-title="Wiskundig bewijs" data-language-autonym="Nederlands" data-language-local-name="荷兰语" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Matematisk_bevis" title="Matematisk bevis – 挪威尼诺斯克语" lang="nn" hreflang="nn" data-title="Matematisk bevis" data-language-autonym="Norsk nynorsk" data-language-local-name="挪威尼诺斯克语" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Matematisk_bevis" title="Matematisk bevis – 书面挪威语" lang="nb" hreflang="nb" data-title="Matematisk bevis" data-language-autonym="Norsk bokmål" data-language-local-name="书面挪威语" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Demostracion_matematica" title="Demostracion matematica – 奥克语" lang="oc" hreflang="oc" data-title="Demostracion matematica" data-language-autonym="Occitan" data-language-local-name="奥克语" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%97%E0%A8%A3%E0%A8%BF%E0%A8%A4%E0%A8%95_%E0%A8%B8%E0%A8%AC%E0%A9%82%E0%A8%A4" title="ਗਣਿਤਕ ਸਬੂਤ – 旁遮普语" lang="pa" hreflang="pa" data-title="ਗਣਿਤਕ ਸਬੂਤ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="旁遮普语" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Dow%C3%B3d_(matematyka)" title="Dowód (matematyka) – 波兰语" lang="pl" hreflang="pl" data-title="Dowód (matematyka)" data-language-autonym="Polski" data-language-local-name="波兰语" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D9%85%DB%8C%D8%AA%DA%BE%D9%85%DB%8C%D9%B9%DB%8C%DA%A9%D9%84_%D8%AB%D8%A8%D9%88%D8%AA" title="میتھمیٹیکل ثبوت – Western Punjabi" lang="pnb" hreflang="pnb" data-title="میتھمیٹیکل ثبوت" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target"><span>پنجابی</span></a></li><li class="interlanguage-link interwiki-ps mw-list-item"><a href="https://ps.wikipedia.org/wiki/%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C%DA%A9%D9%8A_%D8%AB%D8%A8%D9%88%D8%AA" title="ریاضیکي ثبوت – 普什图语" lang="ps" hreflang="ps" data-title="ریاضیکي ثبوت" data-language-autonym="پښتو" data-language-local-name="普什图语" class="interlanguage-link-target"><span>پښتو</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Prova_matem%C3%A1tica" title="Prova matemática – 葡萄牙语" lang="pt" hreflang="pt" data-title="Prova matemática" data-language-autonym="Português" data-language-local-name="葡萄牙语" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Demonstra%C8%9Bie_matematic%C4%83" title="Demonstrație matematică – 罗马尼亚语" lang="ro" hreflang="ro" data-title="Demonstrație matematică" data-language-autonym="Română" data-language-local-name="罗马尼亚语" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%BE%D0%B5_%D0%B4%D0%BE%D0%BA%D0%B0%D0%B7%D0%B0%D1%82%D0%B5%D0%BB%D1%8C%D1%81%D1%82%D0%B2%D0%BE" title="Математическое доказательство – 俄语" lang="ru" hreflang="ru" data-title="Математическое доказательство" data-language-autonym="Русский" data-language-local-name="俄语" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-rue mw-list-item"><a href="https://rue.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D1%96%D1%87%D0%BD%D0%B5_%D0%B4%D0%BE%D0%BA%D0%B0%D0%B7%D0%B0%D1%82%D0%B5%D0%BB%D1%8C%D1%81%D1%82%D0%B2%D0%BE" title="Математічне доказательство – 盧森尼亞文" lang="rue" hreflang="rue" data-title="Математічне доказательство" data-language-autonym="Русиньскый" data-language-local-name="盧森尼亞文" class="interlanguage-link-target"><span>Русиньскый</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Dimustrazzioni_matim%C3%A0tica" title="Dimustrazzioni matimàtica – 西西里语" lang="scn" hreflang="scn" data-title="Dimustrazzioni matimàtica" data-language-autonym="Sicilianu" data-language-local-name="西西里语" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Dokaz_(matematika)" title="Dokaz (matematika) – 塞尔维亚-克罗地亚语" lang="sh" hreflang="sh" data-title="Dokaz (matematika)" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="塞尔维亚-克罗地亚语" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B6%9C%E0%B6%AB%E0%B7%92%E0%B6%AD%E0%B6%B8%E0%B6%BA_%E0%B6%94%E0%B6%B4%E0%B7%8A%E0%B6%B4%E0%B7%94_%E0%B6%9A%E0%B7%92%E0%B6%BB%E0%B7%93%E0%B6%B8%E0%B7%8A" title="ගණිතමය ඔප්පු කිරීම් – 僧伽罗语" lang="si" hreflang="si" data-title="ගණිතමය ඔප්පු කිරීම්" data-language-autonym="සිංහල" data-language-local-name="僧伽罗语" class="interlanguage-link-target"><span>සිංහල</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Mathematical_proof" title="Mathematical proof – Simple English" lang="en-simple" hreflang="en-simple" data-title="Mathematical proof" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk badge-Q17437798 badge-goodarticle mw-list-item" title="优良条目"><a href="https://sk.wikipedia.org/wiki/D%C3%B4kaz_(matematika)" title="Dôkaz (matematika) – 斯洛伐克语" lang="sk" hreflang="sk" data-title="Dôkaz (matematika)" data-language-autonym="Slovenčina" data-language-local-name="斯洛伐克语" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Matemati%C4%8Dni_dokaz" title="Matematični dokaz – 斯洛文尼亚语" lang="sl" hreflang="sl" data-title="Matematični dokaz" data-language-autonym="Slovenščina" data-language-local-name="斯洛文尼亚语" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Provat_matematikore" title="Provat matematikore – 阿尔巴尼亚语" lang="sq" hreflang="sq" data-title="Provat matematikore" data-language-autonym="Shqip" data-language-local-name="阿尔巴尼亚语" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D1%87%D0%BA%D0%B8_%D0%B4%D0%BE%D0%BA%D0%B0%D0%B7" title="Математички доказ – 塞尔维亚语" lang="sr" hreflang="sr" data-title="Математички доказ" data-language-autonym="Српски / srpski" data-language-local-name="塞尔维亚语" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Matematiskt_bevis" title="Matematiskt bevis – 瑞典语" lang="sv" hreflang="sv" data-title="Matematiskt bevis" data-language-autonym="Svenska" data-language-local-name="瑞典语" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Thibitisho_la_kihisabati" title="Thibitisho la kihisabati – 斯瓦希里语" lang="sw" hreflang="sw" data-title="Thibitisho la kihisabati" data-language-autonym="Kiswahili" data-language-local-name="斯瓦希里语" class="interlanguage-link-target"><span>Kiswahili</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%95%E0%AE%A3%E0%AE%BF%E0%AE%A4_%E0%AE%A8%E0%AE%BF%E0%AE%B1%E0%AF%81%E0%AE%B5%E0%AE%B2%E0%AF%8D" title="கணித நிறுவல் – 泰米尔语" lang="ta" hreflang="ta" data-title="கணித நிறுவல்" data-language-autonym="தமிழ்" data-language-local-name="泰米尔语" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%81%E0%B8%B2%E0%B8%A3%E0%B8%9E%E0%B8%B4%E0%B8%AA%E0%B8%B9%E0%B8%88%E0%B8%99%E0%B9%8C%E0%B9%80%E0%B8%8A%E0%B8%B4%E0%B8%87%E0%B8%84%E0%B8%93%E0%B8%B4%E0%B8%95%E0%B8%A8%E0%B8%B2%E0%B8%AA%E0%B8%95%E0%B8%A3%E0%B9%8C" title="การพิสูจน์เชิงคณิตศาสตร์ – 泰语" lang="th" hreflang="th" data-title="การพิสูจน์เชิงคณิตศาสตร์" data-language-autonym="ไทย" data-language-local-name="泰语" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Patibay_pangmatematika" title="Patibay pangmatematika – 他加禄语" lang="tl" hreflang="tl" data-title="Patibay pangmatematika" data-language-autonym="Tagalog" data-language-local-name="他加禄语" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Matematiksel_ispat" title="Matematiksel ispat – 土耳其语" lang="tr" hreflang="tr" data-title="Matematiksel ispat" data-language-autonym="Türkçe" data-language-local-name="土耳其语" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA_%D0%B8%D1%81%D0%B1%D0%B0%D1%82%D0%BB%D0%B0%D1%83" title="Математик исбатлау – 鞑靼语" lang="tt" hreflang="tt" data-title="Математик исбатлау" data-language-autonym="Татарча / tatarça" data-language-local-name="鞑靼语" class="interlanguage-link-target"><span>Татарча / tatarça</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%94%D0%BE%D0%B2%D0%B5%D0%B4%D0%B5%D0%BD%D0%BD%D1%8F_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Доведення (математика) – 乌克兰语" lang="uk" hreflang="uk" data-title="Доведення (математика)" data-language-autonym="Українська" data-language-local-name="乌克兰语" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C%D8%A7%D8%AA%DB%8C_%D8%AB%D8%A8%D9%88%D8%AA" title="ریاضیاتی ثبوت – 乌尔都语" lang="ur" hreflang="ur" data-title="ریاضیاتی ثبوت" data-language-autonym="اردو" data-language-local-name="乌尔都语" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Ch%E1%BB%A9ng_minh_to%C3%A1n_h%E1%BB%8Dc" title="Chứng minh toán học – 越南语" lang="vi" hreflang="vi" data-title="Chứng minh toán học" data-language-autonym="Tiếng Việt" data-language-local-name="越南语" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Pruweba_panmatematika" title="Pruweba panmatematika – 瓦瑞语" lang="war" hreflang="war" data-title="Pruweba panmatematika" data-language-autonym="Winaray" data-language-local-name="瓦瑞语" class="interlanguage-link-target"><span>Winaray</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E8%AF%81%E6%98%8E" title="证明 – 吴语" lang="wuu" hreflang="wuu" data-title="证明" data-language-autonym="吴语" data-language-local-name="吴语" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%9E%D7%90%D7%98%D7%A2%D7%9E%D7%90%D7%98%D7%99%D7%A9%D7%A2%D7%A8_%D7%93%D7%A2%D7%A8%D7%95%D7%95%D7%99%D7%99%D7%96" title="מאטעמאטישער דערווייז – 意第绪语" lang="yi" hreflang="yi" data-title="מאטעמאטישער דערווייז" data-language-autonym="ייִדיש" data-language-local-name="意第绪语" class="interlanguage-link-target"><span>ייִדיש</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E8%AD%89%E6%98%8E" title="證明 – 文言文" lang="lzh" hreflang="lzh" data-title="證明" data-language-autonym="文言" data-language-local-name="文言文" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Ch%C3%A8ng-b%C3%AAng" title="Chèng-bêng – 闽南语" lang="nan" hreflang="nan" data-title="Chèng-bêng" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="闽南语" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E6%95%B8%E5%AD%B8%E8%AD%89%E6%98%8E" title="數學證明 – 粤语" lang="yue" hreflang="yue" data-title="數學證明" data-language-autonym="粵語" data-language-local-name="粤语" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q11538#sitelinks-wikipedia" title="编辑跨语言链接" class="wbc-editpage">编辑链接</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="命名空间"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/%E6%95%B8%E5%AD%B8%E8%AD%89%E6%98%8E" title="浏览条目正文[c]" accesskey="c"><span>条目</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Talk:%E6%95%B8%E5%AD%B8%E8%AD%89%E6%98%8E" rel="discussion" title="关于此页面的讨论[t]" accesskey="t"><span>讨论</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown " > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="更改语言变体" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">不转换</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-varlang-0" class="selected ca-variants-zh mw-list-item"><a href="/zh/%E6%95%B8%E5%AD%B8%E8%AD%89%E6%98%8E" lang="zh" hreflang="zh"><span>不转换</span></a></li><li id="ca-varlang-1" class="ca-variants-zh-Hans mw-list-item"><a href="/zh-hans/%E6%95%B8%E5%AD%B8%E8%AD%89%E6%98%8E" lang="zh-Hans" hreflang="zh-Hans"><span>简体</span></a></li><li id="ca-varlang-2" class="ca-variants-zh-Hant mw-list-item"><a href="/zh-hant/%E6%95%B8%E5%AD%B8%E8%AD%89%E6%98%8E" lang="zh-Hant" hreflang="zh-Hant"><span>繁體</span></a></li><li id="ca-varlang-3" class="ca-variants-zh-Hans-CN mw-list-item"><a href="/zh-cn/%E6%95%B8%E5%AD%B8%E8%AD%89%E6%98%8E" lang="zh-Hans-CN" hreflang="zh-Hans-CN"><span>大陆简体</span></a></li><li id="ca-varlang-4" class="ca-variants-zh-Hant-HK mw-list-item"><a href="/zh-hk/%E6%95%B8%E5%AD%B8%E8%AD%89%E6%98%8E" lang="zh-Hant-HK" hreflang="zh-Hant-HK"><span>香港繁體</span></a></li><li id="ca-varlang-5" class="ca-variants-zh-Hant-MO mw-list-item"><a href="/zh-mo/%E6%95%B8%E5%AD%B8%E8%AD%89%E6%98%8E" lang="zh-Hant-MO" hreflang="zh-Hant-MO"><span>澳門繁體</span></a></li><li id="ca-varlang-6" class="ca-variants-zh-Hans-MY mw-list-item"><a href="/zh-my/%E6%95%B8%E5%AD%B8%E8%AD%89%E6%98%8E" lang="zh-Hans-MY" hreflang="zh-Hans-MY"><span>大马简体</span></a></li><li id="ca-varlang-7" class="ca-variants-zh-Hans-SG mw-list-item"><a href="/zh-sg/%E6%95%B8%E5%AD%B8%E8%AD%89%E6%98%8E" lang="zh-Hans-SG" hreflang="zh-Hans-SG"><span>新加坡简体</span></a></li><li id="ca-varlang-8" class="ca-variants-zh-Hant-TW mw-list-item"><a href="/zh-tw/%E6%95%B8%E5%AD%B8%E8%AD%89%E6%98%8E" lang="zh-Hant-TW" hreflang="zh-Hant-TW"><span>臺灣正體</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="查看"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/%E6%95%B8%E5%AD%B8%E8%AD%89%E6%98%8E"><span>阅读</span></a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=%E6%95%B8%E5%AD%B8%E8%AD%89%E6%98%8E&action=edit" title="编辑该页面[e]" accesskey="e"><span>编辑</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=%E6%95%B8%E5%AD%B8%E8%AD%89%E6%98%8E&action=history" title="本页面的早前版本。[h]" accesskey="h"><span>查看历史</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="页面工具"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="工具" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">工具</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">工具</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">移至侧栏</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">隐藏</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="更多选项" > <div class="vector-menu-heading"> 操作 </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/%E6%95%B8%E5%AD%B8%E8%AD%89%E6%98%8E"><span>阅读</span></a></li><li id="ca-more-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=%E6%95%B8%E5%AD%B8%E8%AD%89%E6%98%8E&action=edit" title="编辑该页面[e]" accesskey="e"><span>编辑</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=%E6%95%B8%E5%AD%B8%E8%AD%89%E6%98%8E&action=history"><span>查看历史</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> 常规 </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Special:%E9%93%BE%E5%85%A5%E9%A1%B5%E9%9D%A2/%E6%95%B8%E5%AD%B8%E8%AD%89%E6%98%8E" title="列出所有与本页相链的页面[j]" accesskey="j"><span>链入页面</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Special:%E9%93%BE%E5%87%BA%E6%9B%B4%E6%94%B9/%E6%95%B8%E5%AD%B8%E8%AD%89%E6%98%8E" rel="nofollow" title="页面链出所有页面的更改[k]" accesskey="k"><span>相关更改</span></a></li><li id="t-upload" class="mw-list-item"><a href="/wiki/Project:%E4%B8%8A%E4%BC%A0" title="上传图像或多媒体文件[u]" accesskey="u"><span>上传文件</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Special:%E7%89%B9%E6%AE%8A%E9%A1%B5%E9%9D%A2" title="全部特殊页面的列表[q]" accesskey="q"><span>特殊页面</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=%E6%95%B8%E5%AD%B8%E8%AD%89%E6%98%8E&oldid=79648586" title="此页面该修订版本的固定链接"><span>固定链接</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=%E6%95%B8%E5%AD%B8%E8%AD%89%E6%98%8E&action=info" title="关于此页面的更多信息"><span>页面信息</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Special:%E5%BC%95%E7%94%A8%E6%AD%A4%E9%A1%B5%E9%9D%A2&page=%E6%95%B8%E5%AD%B8%E8%AD%89%E6%98%8E&id=79648586&wpFormIdentifier=titleform" title="有关如何引用此页面的信息"><span>引用此页</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Special:URL%E7%BC%A9%E7%9F%AD%E7%A8%8B%E5%BA%8F&url=https%3A%2F%2Fzh.wikipedia.org%2Fwiki%2F%25E6%2595%25B8%25E5%25AD%25B8%25E8%25AD%2589%25E6%2598%258E"><span>获取短链接</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Special:QrCode&url=https%3A%2F%2Fzh.wikipedia.org%2Fwiki%2F%25E6%2595%25B8%25E5%25AD%25B8%25E8%25AD%2589%25E6%2598%258E"><span>下载二维码</span></a></li> </ul> </div> </div> <div id="p-electronpdfservice-sidebar-portlet-heading" class="vector-menu mw-portlet mw-portlet-electronpdfservice-sidebar-portlet-heading" > <div class="vector-menu-heading"> 打印/导出 </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="electron-print_pdf" class="mw-list-item"><a href="/w/index.php?title=Special:DownloadAsPdf&page=%E6%95%B8%E5%AD%B8%E8%AD%89%E6%98%8E&action=show-download-screen"><span>下载为PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="javascript:print();" rel="alternate" title="本页面的可打印版本[p]" accesskey="p"><span>打印页面</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> 在其他项目中 </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="wb-otherproject-link wb-otherproject-commons mw-list-item"><a href="https://commons.wikimedia.org/wiki/Category:Mathematical_proof" hreflang="en"><span>维基共享资源</span></a></li><li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q11538" title="链接到连接的数据仓库项目[g]" accesskey="g"><span>维基数据项目</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="页面工具"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="外观"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">外观</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">移至侧栏</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">隐藏</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> <div id="mw-indicator-noteTA-307af9e9" class="mw-indicator"><div class="mw-parser-output"><span class="skin-invert" typeof="mw:File"><span title="本页使用了标题或全文手工转换"><img alt="本页使用了标题或全文手工转换" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Zh_conversion_icon_m.svg/35px-Zh_conversion_icon_m.svg.png" decoding="async" width="35" height="22" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Zh_conversion_icon_m.svg/53px-Zh_conversion_icon_m.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Zh_conversion_icon_m.svg/70px-Zh_conversion_icon_m.svg.png 2x" data-file-width="32" data-file-height="20" /></span></span></div></div> </div> <div id="siteSub" class="noprint">维基百科，自由的百科全书</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="zh" dir="ltr"><figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:P._Oxy._I_29.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8d/P._Oxy._I_29.jpg/250px-P._Oxy._I_29.jpg" decoding="async" width="250" height="152" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8d/P._Oxy._I_29.jpg/375px-P._Oxy._I_29.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8d/P._Oxy._I_29.jpg/500px-P._Oxy._I_29.jpg 2x" data-file-width="1694" data-file-height="1032" /></a><figcaption><a href="/wiki/%E5%B9%BE%E4%BD%95%E5%8E%9F%E6%9C%AC" class="mw-redirect" title="幾何原本">幾何原本</a>中有許多證明的技巧，圖中是Book II, Proposition 5.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></figcaption></figure> <div id="noteTA-307af9e9" class="noteTA"><div class="noteTA-group"><div data-noteta-group-source="module" data-noteta-group="Math"></div></div></div> <p>在<a href="/wiki/%E6%95%B8%E5%AD%B8" class="mw-redirect" title="數學">數學</a>上，<b>數學證明</b>（mathematical proof）是在一個特定的<a href="/wiki/%E5%85%AC%E7%90%86%E7%B3%BB%E7%BB%9F" title="公理系统">公理系統</a>中，根据一定的规则或标准，由<a href="/wiki/%E5%85%AC%E7%90%86" title="公理">公理</a>和<a href="/wiki/%E5%AE%9A%E7%90%86" title="定理">定理</a>推導出某些<a href="/wiki/%E5%91%BD%E9%A2%98" title="命题">命題</a>的過程。比起证据，数学证明一般依靠<a href="/wiki/%E6%BC%94%E7%BB%8E%E6%8E%A8%E7%90%86" title="演绎推理">演绎推理</a>，而不是依靠<a href="/wiki/%E5%BD%92%E7%BA%B3%E6%8E%A8%E7%90%86" title="归纳推理">自然归纳</a>和经验性的理据。這樣推導出來的命題也叫做該系統中的<a href="/wiki/%E5%AE%9A%E7%90%86" title="定理">定理</a>。 </p><p>數學證明建立在<a href="/wiki/%E9%80%BB%E8%BE%91" title="逻辑">逻辑</a>之上，但通常會包含若干程度的<a href="/wiki/%E8%87%AA%E7%84%B6%E8%AA%9E%E8%A8%80" class="mw-redirect" title="自然語言">自然語言</a>，因此可能會產生一些含糊的部分。 </p><p>實際上，用文字形式寫成的數學證明，在大多數情況都可以視為<a href="/wiki/%E9%9D%9E%E5%BD%A2%E5%BC%8F%E9%82%8F%E8%BC%AF" class="mw-redirect" title="非形式邏輯">非形式邏輯</a>的應用。在<a href="/wiki/%E8%AD%89%E6%98%8E%E8%AB%96" class="mw-redirect" title="證明論">證明論</a>的範疇內，則考慮那些用純形式化的语言写出的證明。這個区别导致了对過往到現在的<a href="/w/index.php?title=%E6%95%B8%E5%AD%B8%E5%AE%9E%E8%B7%B5&action=edit&redlink=1" class="new" title="數學实践（页面不存在）">數學实践</a>、<span class="ilh-all" data-orig-title="數學上的擬經驗論" data-lang-code="en" data-lang-name="英语" data-foreign-title="Quasi-empiricism in mathematics"><span class="ilh-page"><a href="/w/index.php?title=%E6%95%B8%E5%AD%B8%E4%B8%8A%E7%9A%84%E6%93%AC%E7%B6%93%E9%A9%97%E8%AB%96&action=edit&redlink=1" class="new" title="數學上的擬經驗論（页面不存在）">數學上的擬經驗論</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Quasi-empiricism_in_mathematics" class="extiw" title="en:Quasi-empiricism in mathematics"><span lang="en" dir="auto">Quasi-empiricism in mathematics</span></a></span>）</span></span>和<span class="ilh-all" data-orig-title="民俗数学" data-lang-code="en" data-lang-name="英语" data-foreign-title="Folk mathematics"><span class="ilh-page"><a href="/w/index.php?title=%E6%B0%91%E4%BF%97%E6%95%B0%E5%AD%A6&action=edit&redlink=1" class="new" title="民俗数学（页面不存在）">民俗数学</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Folk_mathematics" class="extiw" title="en:Folk mathematics"><span lang="en" dir="auto">Folk mathematics</span></a></span>）</span></span>的大部分检验。 </p><p><a href="/wiki/%E6%95%B8%E5%AD%B8%E5%93%B2%E5%AD%B8" class="mw-redirect" title="數學哲學">數學哲學</a>就關注<a href="/wiki/%E8%AA%9E%E8%A8%80" title="語言">語言</a>和邏輯在數學證明中的角色，和<a href="/w/index.php?title=%E6%95%B8%E5%AD%B8%E8%AA%9E%E8%A8%80&action=edit&redlink=1" class="new" title="數學語言（页面不存在）">作為語言的數學</a>。 </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="定義"><span id=".E5.AE.9A.E7.BE.A9"></span>定義</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%95%B8%E5%AD%B8%E8%AD%89%E6%98%8E&action=edit&section=1" title="编辑章节：定義"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>数学上的证明包括两个不同的概念。 </p><p>首先是非形式化的证明：一种以<a href="/wiki/%E8%87%AA%E7%84%B6%E8%AF%AD%E8%A8%80" title="自然语言">自然语言</a>寫成的严密論證，用来说服听众或读者去接受某个定理或论断的真確性。由于这种证明使用了自然语言，因此對於非形式化证明在严謹性上的標準，将取决于听众或读者对課題的理解程度。非形式化证明出现在大多数的应用场合中，例如<a href="/wiki/%E7%A7%91%E6%99%AE" class="mw-redirect" title="科普">科普</a>讲座、口头辩论、初等教育或高等教育的某些部分。有时候非形式化的证明被称作“正式的”，但這只是強調其中論證的嚴謹性。 </p><p>而當逻辑学家使用“正式证明”一詞时，指的是另一种完全不同的证明——形式化证明。 </p> <div class="mw-heading mw-heading4"><h4 id="形式化證明"><span id=".E5.BD.A2.E5.BC.8F.E5.8C.96.E8.AD.89.E6.98.8E"></span>形式化證明</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%95%B8%E5%AD%B8%E8%AD%89%E6%98%8E&action=edit&section=2" title="编辑章节：形式化證明"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>在<a href="/wiki/%E6%95%B0%E7%90%86%E9%80%BB%E8%BE%91" title="数理逻辑">数理逻辑</a>中，形式化证明并不是以自然语言书写，而是以形式化的语言书写：这种语言包含了由一个給定的<a href="/wiki/%E5%AD%97%E6%AF%8D%E8%A1%A8" class="mw-redirect" title="字母表">字母表</a>中的字符所构成的字符串。而证明则是一种由該些字符串組成的有限长度的序列。这种定义使得人們可以談論嚴格意義上的「证明」，而不涉及任何逻辑上的模糊之处。 </p><p>研究证明的形式化和公理化的理论称为<a href="/wiki/%E8%AF%81%E6%98%8E%E8%AE%BA" title="证明论">证明论</a>。 </p><p>尽管理论上来说，每个非形式化的证明都可以转化为形式化证明，但实际中很少會這樣做。对形式化证明的研究主要应用在探討關於可证明性的一般性质，或说明某些命題的不可证明性等等。 </p> <div class="mw-heading mw-heading2"><h2 id="常見的證明技巧"><span id=".E5.B8.B8.E8.A6.8B.E7.9A.84.E8.AD.89.E6.98.8E.E6.8A.80.E5.B7.A7"></span>常見的證明技巧</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%95%B8%E5%AD%B8%E8%AD%89%E6%98%8E&action=edit&section=3" title="编辑章节：常見的證明技巧"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="直接證明"><span id=".E7.9B.B4.E6.8E.A5.E8.AD.89.E6.98.8E"></span>直接證明</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%95%B8%E5%AD%B8%E8%AD%89%E6%98%8E&action=edit&section=4" title="编辑章节：直接證明"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><span class="ilh-all" data-orig-title="直接证明" data-lang-code="en" data-lang-name="英语" data-foreign-title="Direct proof"><span class="ilh-page"><a href="/w/index.php?title=%E7%9B%B4%E6%8E%A5%E8%AF%81%E6%98%8E&action=edit&redlink=1" class="new" title="直接证明（页面不存在）">直接证明</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Direct_proof" class="extiw" title="en:Direct proof"><span lang="en" dir="auto">Direct proof</span></a></span>）</span></span>也称为逻辑演绎，是指从公认的事实或者<a href="/wiki/%E5%85%AC%E7%90%86" title="公理">公理</a>出发，运用逻辑<a href="/wiki/%E6%BC%94%E7%BB%8E%E6%8E%A8%E7%90%86" title="演绎推理">推演</a>而导出需要证明的<a href="/wiki/%E5%91%BD%E9%A2%98" title="命题">命题</a>的真伪的方法。直接证明法一般使用<a href="/wiki/%E8%B0%93%E8%AF%8D%E9%80%BB%E8%BE%91" title="谓词逻辑">谓词逻辑</a>，运用<a href="/wiki/%E5%AD%98%E5%9C%A8%E9%87%8F%E8%AF%8D" class="mw-redirect" title="存在量词">存在量词</a>或<a href="/wiki/%E5%85%A8%E7%A7%B0%E9%87%8F%E8%AF%8D" class="mw-redirect" title="全称量词">全称量词</a>。主要的证明方式有<a href="/wiki/%E8%82%AF%E5%AE%9A%E5%89%8D%E4%BB%B6" title="肯定前件">肯定前件</a>论式、<a href="/wiki/%E5%90%A6%E5%AE%9A%E5%BE%8C%E4%BB%B6" title="否定後件">否定后件</a>论式、<a href="/wiki/%E5%81%87%E8%A8%80%E4%B8%89%E6%AE%B5%E8%AE%BA" title="假言三段论">假言三段论</a>式以及<a href="/wiki/%E9%80%89%E8%A8%80%E4%B8%89%E6%AE%B5%E8%AE%BA" title="选言三段论">选言三段论</a>式等等。比如说要证明命题：“任何<a href="/wiki/%E5%A5%87%E6%95%B0" class="mw-redirect" title="奇数">奇数</a>乘以另一个<a href="/wiki/%E5%A5%87%E6%95%B0" class="mw-redirect" title="奇数">奇数</a>仍然是奇数”，可以直接证明如下： </p> <dl><dd>任何奇数都可以写成<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2a+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>a</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2a+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/082d2a14e82762143fffb8ea5be469aa82f9a6fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.395ex; height:2.343ex;" alt="{\displaystyle 2a+1}"></span>的形式，其中<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span>是<a href="/wiki/%E6%95%B4%E6%95%B0" title="整数">整数</a>。任取两个奇数，它们分别可以写作<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2a+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>a</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2a+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/082d2a14e82762143fffb8ea5be469aa82f9a6fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.395ex; height:2.343ex;" alt="{\displaystyle 2a+1}"></span>和<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2b+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>b</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2b+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9edea4e387c0980bc2f2cf7bfc7e9110ccee796" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.163ex; height:2.343ex;" alt="{\displaystyle 2b+1}"></span>，其中<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span>和<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span>是整数。它们的乘积为<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (2a+1)\times (2b+1)=4ab+2a+2b+1=2(2ab+a+b)+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>a</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>×</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>b</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>4</mn> <mi>a</mi> <mi>b</mi> <mo>+</mo> <mn>2</mn> <mi>a</mi> <mo>+</mo> <mn>2</mn> <mi>b</mi> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mn>2</mn> <mo stretchy="false">(</mo> <mn>2</mn> <mi>a</mi> <mi>b</mi> <mo>+</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (2a+1)\times (2b+1)=4ab+2a+2b+1=2(2ab+a+b)+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5384dd9fecca7fb2fa8bf75609688e313d9bc7b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:61.112ex; height:2.843ex;" alt="{\displaystyle (2a+1)\times (2b+1)=4ab+2a+2b+1=2(2ab+a+b)+1}"></span>。所有能写成整数的两倍加1的数都是奇数。<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2ab+a+b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>a</mi> <mi>b</mi> <mo>+</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2ab+a+b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9294f076e8f5ff1e1010ca08b697c378fc55dfe8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.298ex; height:2.343ex;" alt="{\displaystyle 2ab+a+b}"></span>是整数，所以<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (2a+1)\times (2b+1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>a</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>×</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>b</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (2a+1)\times (2b+1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/180d00769ae6a4ba0841f110bb7cc115bdf35f6b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.017ex; height:2.843ex;" alt="{\displaystyle (2a+1)\times (2b+1)}"></span>是奇数。证明完毕。</dd></dl> <div class="mw-heading mw-heading3"><h3 id="構造法"><span id=".E6.A7.8B.E9.80.A0.E6.B3.95"></span>構造法</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%95%B8%E5%AD%B8%E8%AD%89%E6%98%8E&action=edit&section=5" title="编辑章节：構造法"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/%E6%9E%84%E9%80%A0%E6%B3%95" class="mw-redirect" title="构造法">构造法</a>一般用于证明<a href="/wiki/%E5%AD%98%E5%9C%A8%E6%80%A7%E5%AE%9A%E7%90%86" title="存在性定理">存在性定理</a>，运用构造法的证明称为<a href="/wiki/%E6%9E%84%E9%80%A0%E6%80%A7%E8%AF%81%E6%98%8E" title="构造性证明">构造性证明</a>。具体做法是構造一個帶有命题裡所要求的特定性質的實例，以顯示具有該性質的物體或概念的存在性。也可以构造一个<a href="/wiki/%E5%8F%8D%E4%BE%8B" title="反例">反例</a>，来证明命题是错误的<sup id="cite_ref-gzfjt_2-0" class="reference"><a href="#cite_note-gzfjt-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>。例如证明命题“2的<a href="/wiki/%E8%B4%A8%E6%95%B0" title="质数">质数</a><a href="/wiki/%E5%B9%82" class="mw-redirect" title="幂">次幂</a>减一后不总是质数”，便可用构造法： </p> <dl><dd>只需证明存在某个质数<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span>，使得2的<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span>次幂减一后不是质数。为此，考察质数11。2的<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 11}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>11</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 11}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da6aabe7c6af49fe640b2d401cb2dbe909bb7475" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle 11}"></span>次幂减一等于<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2047}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2047</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2047}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5463813e085c63b34618bbc6ed8c657b2e8e5c42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.65ex; height:2.176ex;" alt="{\displaystyle 2047}"></span>。<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2047=23\times 89}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2047</mn> <mo>=</mo> <mn>23</mn> <mo>×</mo> <mn>89</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2047=23\times 89}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf342e2dde558b0c3235a5fd9709a56188183f29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.239ex; height:2.176ex;" alt="{\displaystyle 2047=23\times 89}"></span>不是质数。因此命题得证。</dd></dl> <p>有些构造法证明中并不直接构造满足命题要求的例子，而是构造某些辅助性的工具或对象，使得问题更容易解决。一个典型的例子是<a href="/wiki/%E5%B8%B8%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B" title="常微分方程">常微分方程</a>稳定性理论中的<a href="/wiki/%E6%9D%8E%E9%9B%85%E6%99%AE%E8%AF%BA%E5%A4%AB%E5%87%BD%E6%95%B0" class="mw-redirect" title="李雅普诺夫函数">李亚普诺夫函数</a>的构造<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>。又如许多几何证明题中常常用到的添加辅助线或辅助图形的办法。 </p> <div class="mw-heading mw-heading3"><h3 id="非构造性证明"><span id=".E9.9D.9E.E6.9E.84.E9.80.A0.E6.80.A7.E8.AF.81.E6.98.8E"></span>非构造性证明</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%95%B8%E5%AD%B8%E8%AD%89%E6%98%8E&action=edit&section=6" title="编辑章节：非构造性证明"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>与构造法证明相对的是<a href="/wiki/%E9%9D%9E%E6%9E%84%E9%80%A0%E6%80%A7%E8%AF%81%E6%98%8E" title="非构造性证明">非构造性证明</a>，即不给出具体的构造而证明命题所要求对象的存在性的证明方法。比如下面例子： </p> <dl><dd>命题：存在两个<a href="/wiki/%E6%97%A0%E7%90%86%E6%95%B0" class="mw-redirect" title="无理数">无理数</a><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>和<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span>，使得<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{y}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{y}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8561c712e86598255e8434a70affa18ffd7e0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.379ex; height:2.343ex;" alt="{\displaystyle x^{y}}"></span>是<a href="/wiki/%E6%9C%89%E7%90%86%E6%95%B0" title="有理数">有理数</a>。</dd> <dd>证明：考虑<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2}}^{\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}^{\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50fb2611d11dbc1741aefae7aea2bd2b317a768f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.521ex; height:3.676ex;" alt="{\displaystyle {\sqrt {2}}^{\sqrt {2}}}"></span>，若它是有理数，则命题得证。若<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2}}^{\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}^{\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50fb2611d11dbc1741aefae7aea2bd2b317a768f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.521ex; height:3.676ex;" alt="{\displaystyle {\sqrt {2}}^{\sqrt {2}}}"></span>不是有理数，则一定是无理数。考虑它的<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4afc1e27d418021bf10898eb44a7f5f315735ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}}"></span>次幂： <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\sqrt {2}}^{\sqrt {2}})^{\sqrt {2}}={\sqrt {2}}^{{\sqrt {2}}\cdot {\sqrt {2}}}={\sqrt {2}}^{2}=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </msup> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mrow> </msup> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\sqrt {2}}^{\sqrt {2}})^{\sqrt {2}}={\sqrt {2}}^{{\sqrt {2}}\cdot {\sqrt {2}}}={\sqrt {2}}^{2}=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/729ae66c9098e3c5cd37576eb31b7c48c8154065" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.534ex; height:3.843ex;" alt="{\displaystyle ({\sqrt {2}}^{\sqrt {2}})^{\sqrt {2}}={\sqrt {2}}^{{\sqrt {2}}\cdot {\sqrt {2}}}={\sqrt {2}}^{2}=2}"></span>为有理数，命题仍然正确。</dd></dl></dd> <dd>于是无论如何，都存在满足命题要求的无理数。</dd></dl> <p>在这个证明里并没有给出使得<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{y}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{y}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8561c712e86598255e8434a70affa18ffd7e0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.379ex; height:2.343ex;" alt="{\displaystyle x^{y}}"></span>是<a href="/wiki/%E6%9C%89%E7%90%86%E6%95%B0" title="有理数">有理数</a>的两个具体的无理数<sup id="cite_ref-gzfjt_2-1" class="reference"><a href="#cite_note-gzfjt-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>。 </p> <div class="mw-heading mw-heading3"><h3 id="穷举法"><span id=".E7.A9.B7.E4.B8.BE.E6.B3.95"></span>穷举法</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%95%B8%E5%AD%B8%E8%AD%89%E6%98%8E&action=edit&section=7" title="编辑章节：穷举法"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><span class="ilh-all" data-orig-title="穷举法 (数学)" data-lang-code="en" data-lang-name="英语" data-foreign-title="Proof by exhaustion"><span class="ilh-page"><a href="/w/index.php?title=%E7%A9%B7%E4%B8%BE%E6%B3%95_(%E6%95%B0%E5%AD%A6)&action=edit&redlink=1" class="new" title="穷举法 (数学)（页面不存在）">穷举法</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Proof_by_exhaustion" class="extiw" title="en:Proof by exhaustion"><span lang="en" dir="auto">Proof by exhaustion</span></a></span>）</span></span>是一种列举出命题所包含的所有情况从而证明命题的方法<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>。例如证明“所有两位数中只有25和76的平方是以自己作为尾数”，只需计算所有两位数：10至99的平方，一一验证即可。显然，使用穷举法的条件是命题所包含的可能情况为有限种，否则无法一一罗列。 </p> <div class="mw-heading mw-heading3"><h3 id="换质位法"><span id=".E6.8D.A2.E8.B4.A8.E4.BD.8D.E6.B3.95"></span>换质位法</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%95%B8%E5%AD%B8%E8%AD%89%E6%98%8E&action=edit&section=8" title="编辑章节：换质位法"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>在<a href="/wiki/%E8%B0%93%E8%AF%8D%E9%80%BB%E8%BE%91" title="谓词逻辑">谓词逻辑</a>裡，若同时否定一个命题的主词和谓词，则其结果称为原命题的<b>换质</b>。若交换主词和谓词的位置，则其结果被称作<b>换位</b>。先换质再换位则被称为<b>换质位</b>，同理先换位再换质则被称为<b><a href="/wiki/%E6%8F%9B%E8%B3%AA%E6%8F%9B%E4%BD%8D%E5%BE%8B" title="換質換位律">换位质</a></b>。例如“所有的S是P”的换质位是“所有的不是P的不是S”。<a href="/wiki/%E5%AF%B9%E4%BD%8D%E8%AF%81%E6%98%8E%E6%B3%95" title="对位证明法">换质位法</a>是指利用换质及换位，将一个命题改为一个与其逻辑<a href="/wiki/%E7%AD%89%E5%83%B9" class="mw-redirect" title="等價">等价</a>的命题，因此只要证明了后者就证明了原来的命题<sup id="cite_ref-jinyuelin_5-0" class="reference"><a href="#cite_note-jinyuelin-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup>。例如，要证明<a href="/wiki/%E9%B8%BD%E7%AC%BC%E5%8E%9F%E7%90%86" class="mw-redirect" title="鸽笼原理">鸽笼原理</a>：“如果n个鸽笼裡装有多于n隻<a href="/wiki/%E9%B8%BD%E5%AD%90" class="mw-redirect" title="鸽子">鸽子</a>，那么至少有一个笼子裡有两隻或者两隻以上鸽子”，可以转证与其等价的<a href="/wiki/%E9%80%86%E5%90%A6%E5%91%BD%E9%A2%98" title="逆否命题">逆否命题</a>：“如果n个鸽笼的每一个中至多装有一隻鸽子，那么n个鸽笼裡至多装有n隻鸽子”。而后者是明顯的。 </p> <div class="mw-heading mw-heading3"><h3 id="個案分析"><span id=".E5.80.8B.E6.A1.88.E5.88.86.E6.9E.90"></span>個案分析</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%95%B8%E5%AD%B8%E8%AD%89%E6%98%8E&action=edit&section=9" title="编辑章节：個案分析"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>个案分析或分类讨论，是指將結論分成有限的個案，然後逐個證明的方法。 </p> <div class="mw-heading mw-heading3"><h3 id="算兩次"><span id=".E7.AE.97.E5.85.A9.E6.AC.A1"></span>算兩次</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%95%B8%E5%AD%B8%E8%AD%89%E6%98%8E&action=edit&section=10" title="编辑章节：算兩次"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/%E7%AE%97%E5%85%A9%E6%AC%A1" title="算兩次">算两次</a>是一种對同一個量進行兩種雖不同但都正確的分析，得到兩個雖不同但相等的表達式的方法，常用于证明<a href="/wiki/%E6%81%92%E7%AD%89%E5%BC%8F" title="恒等式">恒等式</a><sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup>。 </p> <div class="mw-heading mw-heading3"><h3 id="間接證明"><span id=".E9.96.93.E6.8E.A5.E8.AD.89.E6.98.8E"></span><a href="/w/index.php?title=%E9%96%93%E6%8E%A5%E8%AD%89%E6%98%8E&action=edit&redlink=1" class="new" title="間接證明（页面不存在）">間接證明</a></h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%95%B8%E5%AD%B8%E8%AD%89%E6%98%8E&action=edit&section=11" title="编辑章节：間接證明"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading4"><h4 id="反證法"><span id=".E5.8F.8D.E8.AD.89.E6.B3.95"></span>反證法</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%95%B8%E5%AD%B8%E8%AD%89%E6%98%8E&action=edit&section=12" title="编辑章节：反證法"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/%E5%8F%8D%E8%AF%81%E6%B3%95" class="mw-redirect" title="反证法">反证法</a>是一种古老的证明方法，其思想为：欲證明某命題是假命題，则反过来假設該命題為真。在这种情况下，若能通过正确有效的推理導致逻辑上的<a href="/wiki/%E7%9F%9B%E7%9B%BE" title="矛盾">矛盾</a>（如导出该命题自身为假，于是陷入命题既真且假的矛盾），則能證明原来的命題為假。<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><a href="/wiki/%E7%84%A1%E7%9F%9B%E7%9B%BE%E5%BE%8B" class="mw-redirect" title="無矛盾律">無矛盾律</a>和<a href="/wiki/%E6%8E%92%E4%B8%AD%E5%BE%8B" title="排中律">排中律</a>是反證法的邏輯基礎。反证法的好处是在反过来假设该命题为真的同时，等于多了一个已知条件，这样对题目的证明常有帮助<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup>。 </p><p>例子：证明命题“<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4afc1e27d418021bf10898eb44a7f5f315735ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}}"></span>不是有理数”。 </p> <dl><dd>命题：<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4afc1e27d418021bf10898eb44a7f5f315735ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}}"></span>不是有理数。</dd> <dd>证明：假设<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4afc1e27d418021bf10898eb44a7f5f315735ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}}"></span>是有理数，那么存在正整数<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span>使得<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p{\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p{\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23812378cac1d325a38bab3e362f59164f905c8d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:4.357ex; height:3.009ex;" alt="{\displaystyle p{\sqrt {2}}}"></span>为整数。不妨设<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span>为其中最小的（根据<span class="ilh-all ilh-blue" data-orig-title="良序原理" data-lang-code="en" data-lang-name="英语" data-foreign-title="Well-ordering principle"><span class="ilh-page"><a href="/wiki/%E8%89%AF%E5%BA%8F%E5%8E%9F%E7%90%86" class="mw-redirect" title="良序原理">良序原理</a></span></span>，必然存在最小的<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span>）。考虑<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b=a{\sqrt {2}}-a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>=</mo> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>−</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b=a{\sqrt {2}}-a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae8d888b42865a20484829d9dcbfbed845156b2b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.494ex; height:3.009ex;" alt="{\displaystyle b=a{\sqrt {2}}-a}"></span>。<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span>是一个比<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span>小的正整数，但<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ b{\sqrt {2}}=2a-a{\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>=</mo> <mn>2</mn> <mi>a</mi> <mo>−</mo> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ b{\sqrt {2}}=2a-a{\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/973fc613286908dd45a969d9ba5acb0c9cf599e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.336ex; height:3.009ex;" alt="{\displaystyle \ b{\sqrt {2}}=2a-a{\sqrt {2}}}"></span>也是整数。这与<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span>的最小性矛盾。所以根号2不是有理数<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup>。</dd></dl> <div class="mw-heading mw-heading4"><h4 id="數學歸納法"><span id=".E6.95.B8.E5.AD.B8.E6.AD.B8.E7.B4.8D.E6.B3.95"></span>數學歸納法</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%95%B8%E5%AD%B8%E8%AD%89%E6%98%8E&action=edit&section=13" title="编辑章节：數學歸納法"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Dominoeffect.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/92/Dominoeffect.png/200px-Dominoeffect.png" decoding="async" width="200" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/92/Dominoeffect.png/300px-Dominoeffect.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/92/Dominoeffect.png/400px-Dominoeffect.png 2x" data-file-width="800" data-file-height="600" /></a><figcaption>骨牌一个接一个倒下，就如同一个值到下一个值的过程。</figcaption></figure> <p><a href="/wiki/%E6%95%B8%E5%AD%B8%E6%AD%B8%E7%B4%8D%E6%B3%95" class="mw-redirect" title="數學歸納法">數學歸納法</a>是一種证明<a href="/wiki/%E5%8F%AF%E6%95%B0" class="mw-redirect" title="可数">可數</a><a href="/wiki/%E7%84%A1%E7%AA%AE" class="mw-redirect" title="無窮">無窮</a>個命題的技巧。欲證明以自然數<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>編號的一串命題，先證明命題1成立，並證明當命題<i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span></i>成立時命題<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a135e65a42f2d73cccbfc4569523996ca0036f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\displaystyle n+1}"></span>也成立，则对所有的命題都成立<sup id="cite_ref-jinyuelin_5-1" class="reference"><a href="#cite_note-jinyuelin-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup>。在<a href="/wiki/%E7%9A%AE%E4%BA%9A%E8%AF%BA%E5%85%AC%E7%90%86" title="皮亚诺公理">皮亚诺公理</a>系统中，自然数集合的公理化定义就包括了数学归纳法。数学归纳法有不少变体，比如从0以外的自然数开始归纳，证明当命題对小于等于<i>n</i>的自然数成立时命題<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a135e65a42f2d73cccbfc4569523996ca0036f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\displaystyle n+1}"></span>也成立，反向归纳法，递降归纳法等等。广义上的数学归纳法也可以用于证明一般<a href="/wiki/%E8%89%AF%E5%9F%BA" class="mw-redirect" title="良基">良基</a>结构，例如<a href="/wiki/%E9%9B%86%E5%90%88%E8%AE%BA" title="集合论">集合论</a>中的<a href="/wiki/%E6%A0%91_(%E5%9B%BE%E8%AE%BA)" title="树 (图论)">树</a>。另外，<a href="/wiki/%E8%B6%85%E9%99%90%E6%AD%B8%E7%B4%8D%E6%B3%95" class="mw-redirect" title="超限歸納法">超限歸納法</a>提供了一種處理<a href="/wiki/%E4%B8%8D%E5%8F%AF%E6%95%B8" class="mw-redirect" title="不可數">不可數</a>無窮個命題的技巧，是數學歸納法的<a href="/wiki/%E5%B9%BF%E4%B9%89%E5%8C%96" title="广义化">推廣</a><sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup>。 </p><p>例子：證明对所有自然数<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>，命题<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(n):\;\;1^{2}+2^{2}+3^{2}+...+n^{2}={\frac {n(n+1)(2n+1)}{6}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>:</mo> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>+</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mn>6</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(n):\;\;1^{2}+2^{2}+3^{2}+...+n^{2}={\frac {n(n+1)(2n+1)}{6}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5090689049f829cdf78e06a883e26fa80abf99da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:50.58ex; height:5.676ex;" alt="{\displaystyle P(n):\;\;1^{2}+2^{2}+3^{2}+...+n^{2}={\frac {n(n+1)(2n+1)}{6}}}"></span> </p><p>當<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9ec7e1edc2e6d98f5aec2a39ae5f1c99d1e1425" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=1}"></span>，左邊=1，右邊=<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(1):\;\;{\frac {1(1+1)(2+1)}{6}}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>:</mo> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mn>6</mn> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(1):\;\;{\frac {1(1+1)(2+1)}{6}}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9fb7abd1ca14874cd720f27e3634c023c009068" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:28.153ex; height:5.676ex;" alt="{\displaystyle P(1):\;\;{\frac {1(1+1)(2+1)}{6}}=1}"></span> </p><p>假設对某个自然数<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span>，命题<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(k)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(k)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b41614fb84549b21f2c7f2793bbd8a87a2105027" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.766ex; height:2.843ex;" alt="{\displaystyle P(k)}"></span>成立：<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1^{2}+2^{2}+3^{2}+...+k^{2}={\frac {k(k+1)(2k+1)}{6}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>+</mo> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>k</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mn>6</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1^{2}+2^{2}+3^{2}+...+k^{2}={\frac {k(k+1)(2k+1)}{6}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/045161e2e54b37b1cd283fa8279b135753fe4308" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:41.669ex; height:5.676ex;" alt="{\displaystyle 1^{2}+2^{2}+3^{2}+...+k^{2}={\frac {k(k+1)(2k+1)}{6}}}"></span>，以下證明<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(k+1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(k+1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c82953120fe42f14025a07cd09ba7eaee65fcee1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.769ex; height:2.843ex;" alt="{\displaystyle P(k+1)}"></span>成立，即：<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1^{2}+2^{2}+3^{2}+...+k^{2}+(k+1)^{2}={\frac {(k+1)(k+2)(2k+3)}{6}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>+</mo> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> <mn>6</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1^{2}+2^{2}+3^{2}+...+k^{2}+(k+1)^{2}={\frac {(k+1)(k+2)(2k+3)}{6}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0c0b51358cbdbfda431f6ff8b5c7b1dff760cf6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:58.4ex; height:5.676ex;" alt="{\displaystyle 1^{2}+2^{2}+3^{2}+...+k^{2}+(k+1)^{2}={\frac {(k+1)(k+2)(2k+3)}{6}}}"></span>： </p> <dl><dd>左邊<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =1^{2}+2^{2}+3^{2}+...+k^{2}+(k+1)^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>+</mo> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =1^{2}+2^{2}+3^{2}+...+k^{2}+(k+1)^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/323b2199664612ea17e07843a1ea6268d1b0b02e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.686ex; height:3.176ex;" alt="{\displaystyle =1^{2}+2^{2}+3^{2}+...+k^{2}+(k+1)^{2}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ={\frac {k(k+1)(2k+1)}{6}}+(k+1)^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>k</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mn>6</mn> </mfrac> </mrow> <mo>+</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ={\frac {k(k+1)(2k+1)}{6}}+(k+1)^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a40fed74dbed908d8c94c03a3892d398a715448e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:30.628ex; height:5.676ex;" alt="{\displaystyle ={\frac {k(k+1)(2k+1)}{6}}+(k+1)^{2}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ={\frac {(k+1)(2k^{2}+7k+6)}{6}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>2</mn> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>7</mn> <mi>k</mi> <mo>+</mo> <mn>6</mn> <mo stretchy="false">)</mo> </mrow> <mn>6</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ={\frac {(k+1)(2k^{2}+7k+6)}{6}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb89ef0cce9767b023d6cb1af94a01a608b4bfd0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:24.767ex; height:5.843ex;" alt="{\displaystyle ={\frac {(k+1)(2k^{2}+7k+6)}{6}}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ={\frac {(k+1)(k+2)(2k+3)}{6}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> <mn>6</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ={\frac {(k+1)(k+2)(2k+3)}{6}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60b583ba782080cf0d6a30028bdc7238dc7e9e1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:25.522ex; height:5.676ex;" alt="{\displaystyle ={\frac {(k+1)(k+2)(2k+3)}{6}}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/505a4ceef454c69dffd23792c84b90f488543743" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle =}"></span>右邊</dd></dl> <p>所以，对任意自然数<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>，都有<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1^{2}+2^{2}+3^{2}+...+n^{2}={\frac {n(n+1)(2n+1)}{6}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>+</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mn>6</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1^{2}+2^{2}+3^{2}+...+n^{2}={\frac {n(n+1)(2n+1)}{6}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffba1599c9900849eef5c017323479efe6951cd9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:42.403ex; height:5.676ex;" alt="{\displaystyle 1^{2}+2^{2}+3^{2}+...+n^{2}={\frac {n(n+1)(2n+1)}{6}}}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="其他证明方式"><span id=".E5.85.B6.E4.BB.96.E8.AF.81.E6.98.8E.E6.96.B9.E5.BC.8F"></span>其他证明方式</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%95%B8%E5%AD%B8%E8%AD%89%E6%98%8E&action=edit&section=14" title="编辑章节：其他证明方式"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="直观证明"><span id=".E7.9B.B4.E8.A7.82.E8.AF.81.E6.98.8E"></span>直观证明</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%95%B8%E5%AD%B8%E8%AD%89%E6%98%8E&action=edit&section=15" title="编辑章节：直观证明"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r84833064">.mw-parser-output .hatnote{font-size:small}.mw-parser-output div.hatnote{padding-left:2em;margin-bottom:0.8em;margin-top:0.8em}.mw-parser-output .hatnote-notice-img::after{content:"\202f \202f \202f \202f "}.mw-parser-output .hatnote-notice-img-small::after{content:"\202f \202f "}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}body.skin-minerva .mw-parser-output .hatnote-notice-img,body.skin-minerva .mw-parser-output .hatnote-notice-img-small{display:none}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">主条目：<a href="/wiki/%E6%97%A0%E5%AD%97%E8%AF%81%E6%98%8E" title="无字证明">无字证明</a></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:GouguAnim.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e6/GouguAnim.gif/120px-GouguAnim.gif" decoding="async" width="120" height="123" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e6/GouguAnim.gif/180px-GouguAnim.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/e/e6/GouguAnim.gif 2x" data-file-width="231" data-file-height="237" /></a><figcaption>勾股定理的一个图示证明</figcaption></figure> <p><a href="/w/index.php?title=%E7%9B%B4%E8%A7%82%E8%AF%81%E6%98%8E&action=edit&redlink=1" class="new" title="直观证明（页面不存在）">直观证明</a>或可视化证明是指用图像或表格等直观的手段证明命题的方法。这类证明可以达到不借助语言而证明的效果。如右图是<a href="/wiki/%E5%8B%BE%E8%82%A1%E5%AE%9A%E7%90%86" title="勾股定理">勾股定理</a>的一个图示证明。 </p> <div class="mw-heading mw-heading3"><h3 id="计算机辅助证明"><span id=".E8.AE.A1.E7.AE.97.E6.9C.BA.E8.BE.85.E5.8A.A9.E8.AF.81.E6.98.8E"></span>计算机辅助证明</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%95%B8%E5%AD%B8%E8%AD%89%E6%98%8E&action=edit&section=16" title="编辑章节：计算机辅助证明"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84833064"><div role="note" class="hatnote navigation-not-searchable">主条目：<a href="/wiki/%E9%9B%BB%E8%85%A6%E5%8D%94%E5%8A%A9%E8%AD%89%E6%98%8E" title="電腦協助證明">電腦協助證明</a></div> <p><a href="/wiki/%E9%9B%BB%E8%85%A6%E5%8D%94%E5%8A%A9%E8%AD%89%E6%98%8E" title="電腦協助證明">電腦協助證明</a>是二十世紀出現的證明方式。直到二十世纪中，人们一直认为任何的数学证明都应当能够被一个水平足够的<a href="/wiki/%E6%95%B0%E5%AD%A6%E5%AE%B6" title="数学家">数学家</a>检验，以证实其正确性。然而，今天的数学家已经能够运用计算机来证明定理，并且完成人类難以做到的计算<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup>。1976年<a href="/wiki/%E5%9B%9B%E8%89%B2%E5%AE%9A%E7%90%86" title="四色定理">四色定理</a>的证明是计算机辅助证明的经典例子<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup>。证明的方法是将地图上的无限种可能情况减少为1936种状态，并由计算机对每个可能的情况进行验证。有不少数学家对于计算机证明持谨慎态度，因为很多证明太长，不能由人手直接验证。此外，<a href="/wiki/%E7%AE%97%E6%B3%95" title="算法">算法</a>上的错误，输入时的失误甚至计算机运行期间出现的错误都有可能导致错误的结果。 </p> <div class="mw-heading mw-heading2"><h2 id="證明的結尾"><span id=".E8.AD.89.E6.98.8E.E7.9A.84.E7.B5.90.E5.B0.BE"></span>證明的結尾</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%95%B8%E5%AD%B8%E8%AD%89%E6%98%8E&action=edit&section=17" title="编辑章节：證明的結尾"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84833064"><div role="note" class="hatnote navigation-not-searchable">主条目：<a href="/wiki/%E8%AD%89%E6%98%8E%E5%AE%8C%E7%95%A2" title="證明完畢">證明完畢</a></div> <p>有時在證明的結尾會加上Q.E.D.三個字母，這是拉丁文Quod Erat Demonstrandum的縮寫，意思是「證明完畢」。現在的證明完畢符號，通常是<span class="Unicode">■</span>（實心黑色正方形），稱之為「<a href="/wiki/%E5%A2%93%E7%A2%91_(%E6%95%B8%E5%AD%B8)" class="mw-redirect" title="墓碑 (數學)">墓碑</a>」或「哈爾莫斯（Halmos symbol）」（因<a href="/wiki/%E4%BF%9D%E7%BE%85%C2%B7%E5%93%88%E7%88%BE%E8%8E%AB%E6%96%AF" title="保羅·哈爾莫斯">保羅·哈爾莫斯</a>最先採用此做法）。墓碑有時是空心的<span class="Unicode">□</span>。另一個簡單方法是寫「proven」、「shown」或「證畢」之類的文字。 </p> <div class="mw-heading mw-heading2"><h2 id="參見"><span id=".E5.8F.83.E8.A6.8B"></span>參見</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%95%B8%E5%AD%B8%E8%AD%89%E6%98%8E&action=edit&section=18" title="编辑章节：參見"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/%E8%AD%89%E6%98%8E%E8%AB%96" class="mw-redirect" title="證明論">證明論</a></li> <li><a href="/wiki/%E6%A8%A1%E5%9E%8B%E8%AB%96" class="mw-redirect" title="模型論">模型論</a></li> <li><a href="/wiki/%E8%87%AA%E5%8B%95%E5%AE%9A%E7%90%86%E8%AD%89%E6%98%8E" class="mw-redirect" title="自動定理證明">自動定理證明</a></li> <li><a href="/wiki/%E7%84%A1%E6%95%88%E8%AD%89%E6%98%8E" title="無效證明">無效證明</a></li> <li><a href="/wiki/Q.E.D." class="mw-redirect" title="Q.E.D.">Q.E.D.</a></li> <li><a href="/wiki/%E5%A2%93%E7%A2%91%E7%AC%A6%E5%8F%B7" title="墓碑符号">墓碑符号</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="参考资料"><span id=".E5.8F.82.E8.80.83.E8.B5.84.E6.96.99"></span>参考资料</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%95%B8%E5%AD%B8%E8%AD%89%E6%98%8E&action=edit&section=19" title="编辑章节：参考资料"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><cite class="citation web">Bill Casselman. <a rel="nofollow" class="external text" href="http://www.math.ubc.ca/~cass/Euclid/papyrus/papyrus.html">One of the Oldest Extant Diagrams from Euclid</a>. University of British Columbia. <span class="reference-accessdate"> [<span class="nowrap">2008-09-26</span>]</span>. （原始内容<a rel="nofollow" class="external text" href="https://archive.today/20120604095737/http://www.math.ubc.ca/~cass/Euclid/papyrus/papyrus.html">存档</a>于2012-06-04）.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E6%95%B8%E5%AD%B8%E8%AD%89%E6%98%8E&rft.au=Bill+Casselman&rft.btitle=One+of+the+Oldest+Extant+Diagrams+from+Euclid&rft.genre=unknown&rft.pub=University+of+British+Columbia&rft_id=http%3A%2F%2Fwww.math.ubc.ca%2F~cass%2FEuclid%2Fpapyrus%2Fpapyrus.html&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-gzfjt-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-gzfjt_2-0"><sup><b>2.0</b></sup></a> <a href="#cite_ref-gzfjt_2-1"><sup><b>2.1</b></sup></a></span> <span class="reference-text"><cite class="citation book">余红兵; 严镇军. <a rel="nofollow" class="external text" href="http://books.google.com.tw/books?id=d_gPSQAACAAJ&dq=%E6%9E%84%E9%80%A0%E6%B3%95%E8%A7%A3%E9%A2%98&hl=en&sa=X&ei=HivRUq3ZLsXTkQW-0YCYDA&redir_esc=y">《构造法解题》</a>. 中国科学技术大学出版社. 2009 <span class="reference-accessdate"> [<span class="nowrap">2014-01-11</span>]</span>. <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/7312024831" title="Special:网络书源/7312024831"><span title="国际标准书号">ISBN</span> 7312024831</a>. （原始内容<a rel="nofollow" class="external text" href="https://web.archive.org/web/20190503015148/https://books.google.com.tw/books?id=d_gPSQAACAAJ&dq=%E6%9E%84%E9%80%A0%E6%B3%95%E8%A7%A3%E9%A2%98&hl=en&sa=X&ei=HivRUq3ZLsXTkQW-0YCYDA&redir_esc=y">存档</a>于2019-05-03）.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E6%95%B8%E5%AD%B8%E8%AD%89%E6%98%8E&rft.au=%E4%B8%A5%E9%95%87%E5%86%9B&rft.au=%E4%BD%99%E7%BA%A2%E5%85%B5&rft.btitle=%E3%80%8A%E6%9E%84%E9%80%A0%E6%B3%95%E8%A7%A3%E9%A2%98%E3%80%8B&rft.date=2009&rft.genre=book&rft.isbn=7312024831&rft.pub=%E4%B8%AD%E5%9B%BD%E7%A7%91%E5%AD%A6%E6%8A%80%E6%9C%AF%E5%A4%A7%E5%AD%A6%E5%87%BA%E7%89%88%E7%A4%BE&rft_id=http%3A%2F%2Fbooks.google.com.tw%2Fbooks%3Fid%3Dd_gPSQAACAAJ%26dq%3D%25E6%259E%2584%25E9%2580%25A0%25E6%25B3%2595%25E8%25A7%25A3%25E9%25A2%2598%26hl%3Den%26sa%3DX%26ei%3DHivRUq3ZLsXTkQW-0YCYDA%26redir_esc%3Dy&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite class="citation book">王高雄; 周之铭; 朱思铭; 王寿松. <a rel="nofollow" class="external text" href="http://books.google.com.tw/books?id=2eF_AQAACAAJ&dq=9787040193664&hl=en&sa=X&ei=HizRUrCcGo6hlQXMnYC4Bw&redir_esc=y">《常微分方程》（第三版）</a>. 高等教育出版社. <span class="reference-accessdate"> [<span class="nowrap">2014-01-11</span>]</span>. <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/9787040193664" title="Special:网络书源/9787040193664"><span title="国际标准书号">ISBN</span> 9787040193664</a>. （原始内容<a rel="nofollow" class="external text" href="https://web.archive.org/web/20190503043216/https://books.google.com.tw/books?id=2eF_AQAACAAJ&dq=9787040193664&hl=en&sa=X&ei=HizRUrCcGo6hlQXMnYC4Bw&redir_esc=y">存档</a>于2019-05-03）.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E6%95%B8%E5%AD%B8%E8%AD%89%E6%98%8E&rft.au=%E5%91%A8%E4%B9%8B%E9%93%AD&rft.au=%E6%9C%B1%E6%80%9D%E9%93%AD&rft.au=%E7%8E%8B%E5%AF%BF%E6%9D%BE&rft.au=%E7%8E%8B%E9%AB%98%E9%9B%84&rft.btitle=%E3%80%8A%E5%B8%B8%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B%E3%80%8B%EF%BC%88%E7%AC%AC%E4%B8%89%E7%89%88%EF%BC%89&rft.genre=book&rft.isbn=9787040193664&rft.pub=%E9%AB%98%E7%AD%89%E6%95%99%E8%82%B2%E5%87%BA%E7%89%88%E7%A4%BE&rft_id=http%3A%2F%2Fbooks.google.com.tw%2Fbooks%3Fid%3D2eF_AQAACAAJ%26dq%3D9787040193664%26hl%3Den%26sa%3DX%26ei%3DHizRUrCcGo6hlQXMnYC4Bw%26redir_esc%3Dy&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text">Reid, D. A. & Knipping, C. (2010).<a rel="nofollow" class="external text" href="https://www.amazon.com/Proof-Mathematics-Education-Research-Learning/dp/9460912443"><i>Proof in Mathematics Education: Research, Learning, and Teaching</i></a> （<a rel="nofollow" class="external text" href="//web.archive.org/web/20190502145156/http://www.amazon.com/Proof-Mathematics-Education-Research-Learning/dp/9460912443">页面存档备份</a>，存于<a href="/wiki/%E4%BA%92%E8%81%94%E7%BD%91%E6%A1%A3%E6%A1%88%E9%A6%86" title="互联网档案馆">互联网档案馆</a>） Sense Publishers, p. 133.</span> </li> <li id="cite_note-jinyuelin-5"><span class="mw-cite-backlink">^ <a href="#cite_ref-jinyuelin_5-0"><sup><b>5.0</b></sup></a> <a href="#cite_ref-jinyuelin_5-1"><sup><b>5.1</b></sup></a></span> <span class="reference-text"><cite class="citation book">金岳霖; 杨宝星. <a rel="nofollow" class="external text" href="http://books.google.com.tw/books?id=6y-CnQEACAAJ&dq=9787010002033&hl=en&sa=X&ei=AC3RUq6iBYXkkAXC64HgBw&ved=0CCwQ6AEwAA">《形式逻辑》</a>. 辽宁人民出版社. 1979 <span class="reference-accessdate"> [<span class="nowrap">2014-01-11</span>]</span>. <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/7010002037" title="Special:网络书源/7010002037"><span title="国际标准书号">ISBN</span> 7010002037</a>. （原始内容<a rel="nofollow" class="external text" href="https://web.archive.org/web/20140111125947/http://books.google.com.tw/books?id=6y-CnQEACAAJ&dq=9787010002033&hl=en&sa=X&ei=AC3RUq6iBYXkkAXC64HgBw&ved=0CCwQ6AEwAA">存档</a>于2014-01-11）.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E6%95%B8%E5%AD%B8%E8%AD%89%E6%98%8E&rft.au=%E6%9D%A8%E5%AE%9D%E6%98%9F&rft.au=%E9%87%91%E5%B2%B3%E9%9C%96&rft.btitle=%E3%80%8A%E5%BD%A2%E5%BC%8F%E9%80%BB%E8%BE%91%E3%80%8B&rft.date=1979&rft.genre=book&rft.isbn=7010002037&rft.pub=%E8%BE%BD%E5%AE%81%E4%BA%BA%E6%B0%91%E5%87%BA%E7%89%88%E7%A4%BE&rft_id=http%3A%2F%2Fbooks.google.com.tw%2Fbooks%3Fid%3D6y-CnQEACAAJ%26dq%3D9787010002033%26hl%3Den%26sa%3DX%26ei%3DAC3RUq6iBYXkkAXC64HgBw%26ved%3D0CCwQ6AEwAA&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><cite class="citation book">单壿. <a rel="nofollow" class="external text" href="http://books.google.com.tw/books?id=GIR_SQAACAAJ&dq=%E7%AE%97%E4%B8%A4%E6%AC%A1&hl=en&sa=X&ei=hy3RUtGGHsejlQXT9oDQBg&ved=0CCwQ6AEwAA">《算两次》</a>. 中国科学技术大学出版社. 2009 <span class="reference-accessdate"> [<span class="nowrap">2014-01-11</span>]</span>. <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/7312024823" title="Special:网络书源/7312024823"><span title="国际标准书号">ISBN</span> 7312024823</a>. （原始内容<a rel="nofollow" class="external text" href="https://web.archive.org/web/20140111130025/http://books.google.com.tw/books?id=GIR_SQAACAAJ&dq=%E7%AE%97%E4%B8%A4%E6%AC%A1&hl=en&sa=X&ei=hy3RUtGGHsejlQXT9oDQBg&ved=0CCwQ6AEwAA">存档</a>于2014-01-11）.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E6%95%B8%E5%AD%B8%E8%AD%89%E6%98%8E&rft.au=%E5%8D%95%E5%A3%BF&rft.btitle=%E3%80%8A%E7%AE%97%E4%B8%A4%E6%AC%A1%E3%80%8B&rft.date=2009&rft.genre=book&rft.isbn=7312024823&rft.pub=%E4%B8%AD%E5%9B%BD%E7%A7%91%E5%AD%A6%E6%8A%80%E6%9C%AF%E5%A4%A7%E5%AD%A6%E5%87%BA%E7%89%88%E7%A4%BE&rft_id=http%3A%2F%2Fbooks.google.com.tw%2Fbooks%3Fid%3DGIR_SQAACAAJ%26dq%3D%25E7%25AE%2597%25E4%25B8%25A4%25E6%25AC%25A1%26hl%3Den%26sa%3DX%26ei%3Dhy3RUtGGHsejlQXT9oDQBg%26ved%3D0CCwQ6AEwAA&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text">類似地，有更一般的<a href="/wiki/%E6%AD%B8%E8%AC%AC%E6%B3%95" title="歸謬法">歸謬法</a>：若然假設該命題為真，在通过正确有效的推理后會得出邏輯上的矛盾、与某个事实或公理相悖的結論、或荒謬難以接受的結果，亦可以證明原来的命題為假。反證法是歸謬法的其中一種形式。</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><cite class="citation web"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20131228134723/http://www.lille.iufm.fr/IMG/pdf/Texte_Cambresy.pdf">Vinciane CAMBRÉSY-TANT，Dominique CAMBRÉSY，Stéphane CARPENTIER，''Autour du raisonnement par l'absurde''，IUFM Nord - Pas de Calais</a> <span style="font-size:85%;">(PDF)</span>. <span class="reference-accessdate"> [<span class="nowrap">2014-01-11</span>]</span>. （<a rel="nofollow" class="external text" href="http://www.lille.iufm.fr/IMG/pdf/Texte_Cambresy.pdf">原始内容</a> <span style="font-size:85%;">(PDF)</span>存档于2013-12-28）.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E6%95%B8%E5%AD%B8%E8%AD%89%E6%98%8E&rft.btitle=Vinciane+CAMBR%C3%89SY-TANT%EF%BC%8CDominique+CAMBR%C3%89SY%EF%BC%8CSt%C3%A9phane+CARPENTIER%EF%BC%8C%27%26%2339%3BAutour+du+raisonnement+par+l%27absurde%27%26%2339%3B%EF%BC%8CIUFM+Nord+-+Pas+de+Calais&rft.genre=unknown&rft_id=http%3A%2F%2Fwww.lille.iufm.fr%2FIMG%2Fpdf%2FTexte_Cambresy.pdf&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><cite class="citation web"><a rel="nofollow" class="external text" href="http://blms.oxfordjournals.org/cgi/reprint/26/6/593.pdf">THEODOR ESTERMANN，1975</a> <span style="font-size:85%;">(PDF)</span>. Blms.oxfordjournals.org. 2013-12-06 <span class="reference-accessdate"> [<span class="nowrap">2014-01-11</span>]</span>.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E6%95%B8%E5%AD%B8%E8%AD%89%E6%98%8E&rft.btitle=THEODOR+ESTERMANN%EF%BC%8C1975&rft.date=2013-12-06&rft.genre=unknown&rft.pub=Blms.oxfordjournals.org&rft_id=http%3A%2F%2Fblms.oxfordjournals.org%2Fcgi%2Freprint%2F26%2F6%2F593.pdf&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><cite class="citation book">苏淳. <a rel="nofollow" class="external text" href="http://books.google.com.tw/books?id=tww_SQAACAAJ&dq=%E6%BC%AB%E8%A9%B1%E6%95%B8%E5%AD%B8%E6%AD%B8%E7%B4%8D%E6%B3%95&hl=en&sa=X&ei=IS7RUpWSGYvHkwX6s4DgCA&redir_esc=y">《漫话数学归纳法》</a>. 中国科学技术大学出版社. 2009 <span class="reference-accessdate"> [<span class="nowrap">2014-01-11</span>]</span>. <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/7312024866" title="Special:网络书源/7312024866"><span title="国际标准书号">ISBN</span> 7312024866</a>. （原始内容<a rel="nofollow" class="external text" href="https://web.archive.org/web/20190503032307/https://books.google.com.tw/books?id=tww_SQAACAAJ&dq=%E6%BC%AB%E8%A9%B1%E6%95%B8%E5%AD%B8%E6%AD%B8%E7%B4%8D%E6%B3%95&hl=en&sa=X&ei=IS7RUpWSGYvHkwX6s4DgCA&redir_esc=y">存档</a>于2019-05-03）.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E6%95%B8%E5%AD%B8%E8%AD%89%E6%98%8E&rft.au=%E8%8B%8F%E6%B7%B3&rft.btitle=%E3%80%8A%E6%BC%AB%E8%AF%9D%E6%95%B0%E5%AD%A6%E5%BD%92%E7%BA%B3%E6%B3%95%E3%80%8B&rft.date=2009&rft.genre=book&rft.isbn=7312024866&rft.pub=%E4%B8%AD%E5%9B%BD%E7%A7%91%E5%AD%A6%E6%8A%80%E6%9C%AF%E5%A4%A7%E5%AD%A6%E5%87%BA%E7%89%88%E7%A4%BE&rft_id=http%3A%2F%2Fbooks.google.com.tw%2Fbooks%3Fid%3Dtww_SQAACAAJ%26dq%3D%25E6%25BC%25AB%25E8%25A9%25B1%25E6%2595%25B8%25E5%25AD%25B8%25E6%25AD%25B8%25E7%25B4%258D%25E6%25B3%2595%26hl%3Den%26sa%3DX%26ei%3DIS7RUpWSGYvHkwX6s4DgCA%26redir_esc%3Dy&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><cite class="citation web"><a rel="nofollow" class="external text" href="http://www.math.wustl.edu/~sk/eolss.pdf">The History and Concept of Mathematical Proof, Steven G. Krantz. 1. February 5, 2007</a> <span style="font-size:85%;">(PDF)</span>. <span class="reference-accessdate"> [<span class="nowrap">2014-01-11</span>]</span>. （原始内容<a rel="nofollow" class="external text" href="https://web.archive.org/web/20070221150523/http://www.math.wustl.edu/~sk/eolss.pdf">存档</a> <span style="font-size:85%;">(PDF)</span>于2007-02-21）.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E6%95%B8%E5%AD%B8%E8%AD%89%E6%98%8E&rft.btitle=The+History+and+Concept+of+Mathematical+Proof%2C+Steven+G.+Krantz.+1.+February+5%2C+2007&rft.genre=unknown&rft_id=http%3A%2F%2Fwww.math.wustl.edu%2F~sk%2Feolss.pdf&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><cite class="citation web"><a rel="nofollow" class="external text" href="http://www.kepu.net.cn/gb/basic/szsx/8/8_81/8_81_1007.htm">相信數學，或相信電腦</a>. 中國科普博覽. 2004-04-29 <span class="reference-accessdate"> [<span class="nowrap">2014-01-14</span>]</span>. （原始内容<a rel="nofollow" class="external text" href="https://web.archive.org/web/20170305011451/http://www.kepu.net.cn/gb/basic/szsx/8/8_81/8_81_1007.htm">存档</a>于2017-03-05）.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E6%95%B8%E5%AD%B8%E8%AD%89%E6%98%8E&rft.btitle=%E7%9B%B8%E4%BF%A1%E6%95%B8%E5%AD%B8%EF%BC%8C%E6%88%96%E7%9B%B8%E4%BF%A1%E9%9B%BB%E8%85%A6&rft.date=2004-04-29&rft.genre=unknown&rft.pub=%E4%B8%AD%E5%9C%8B%E7%A7%91%E6%99%AE%E5%8D%9A%E8%A6%BD&rft_id=http%3A%2F%2Fwww.kepu.net.cn%2Fgb%2Fbasic%2Fszsx%2F8%2F8_81%2F8_81_1007.htm&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> </ol> <div class="mw-heading mw-heading2"><h2 id="外部連結"><span id=".E5.A4.96.E9.83.A8.E9.80.A3.E7.B5.90"></span>外部連結</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E6%95%B8%E5%AD%B8%E8%AD%89%E6%98%8E&action=edit&section=20" title="编辑章节：外部連結"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://episte.math.ntu.edu.tw/articles/sm/sm_16_02_1/index.html">證明，曹亮吉</a> （<a rel="nofollow" class="external text" href="//web.archive.org/web/20200702201910/http://episte.math.ntu.edu.tw/articles/sm/sm_16_02_1/index.html">页面存档备份</a>，存于<a href="/wiki/%E4%BA%92%E8%81%94%E7%BD%91%E6%A1%A3%E6%A1%88%E9%A6%86" title="互联网档案馆">互联网档案馆</a>）</li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r84265675">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:" :"}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist-pipe dd::after,.mw-parser-output .hlist-pipe li::after{content:" | ";font-weight:normal}.mw-parser-output .hlist-hyphen dd::after,.mw-parser-output .hlist-hyphen li::after{content:" - ";font-weight:normal}.mw-parser-output .hlist-comma dd::after,.mw-parser-output .hlist-comma li::after{content:"、";font-weight:normal}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist li:last-child::after{content:none}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:"（"counter(listitem)"\a0 "}.mw-parser-output ul.cslist,.mw-parser-output ul.sslist{margin:0;padding:0;display:inline-block;list-style:none}.mw-parser-output .cslist li,.mw-parser-output .sslist li{margin:0;display:inline-block}.mw-parser-output .cslist li::after{content:"，"}.mw-parser-output .sslist li::after{content:"；"}.mw-parser-output .cslist li:last-child::after,.mw-parser-output .sslist li:last-child::after{content:none}</style><style data-mw-deduplicate="TemplateStyles:r84261037">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{text-align:center;padding-left:1em;padding-right:1em}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output .navbox-list{border-color:#fdfdfd}.mw-parser-output .navbox-list-with-group{text-align:left;border-left-width:2px;border-left-style:solid}.mw-parser-output tr+tr>.navbox-abovebelow,.mw-parser-output tr+tr>.navbox-group,.mw-parser-output tr+tr>.navbox-image,.mw-parser-output tr+tr>.navbox-list{border-top:2px solid #fdfdfd}.mw-parser-output .navbox-title{background-color:#ccf;position:relative}.mw-parser-output .navbox-abovebelow,.mw-parser-output .navbox-group,.mw-parser-output .navbox-subgroup .navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em;width:auto;padding-left:0.2em;position:absolute;left:1em}.mw-parser-output .navbox .mw-collapsible-toggle{margin-left:0.5em;position:absolute;right:1em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="几何学术语" style="text-align: left;;padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="collapsible-title navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84265675"><style data-mw-deduplicate="TemplateStyles:r84244141">.mw-parser-output .navbar{display:inline;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:110%;margin:0 8em}.mw-parser-output .navbar-ct-mini{font-size:110%;margin:0 5em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:%E5%87%A0%E4%BD%95%E6%9C%AF%E8%AF%AD" title="Template:几何术语"><abbr title="查看该模板">查</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:%E5%87%A0%E4%BD%95%E6%9C%AF%E8%AF%AD" title="Template talk:几何术语"><abbr title="讨论该模板">论</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:%E7%BC%96%E8%BE%91%E9%A1%B5%E9%9D%A2/Template:%E5%87%A0%E4%BD%95%E6%9C%AF%E8%AF%AD" title="Special:编辑页面/Template:几何术语"><abbr title="编辑该模板">编</abbr></a></li></ul></div><div id="几何学术语" style="font-size:110%;margin:0 5em"><a href="/wiki/%E5%87%A0%E4%BD%95%E5%AD%A6" title="几何学">几何学</a>术语</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: right;"><a href="/wiki/%E7%82%B9" title="点">点</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E9%A0%82%E9%BB%9E_(%E5%B9%BE%E4%BD%95)" title="頂點 (幾何)">頂點</a></li> <li><a href="/wiki/%E4%BA%A4%E9%BB%9E" title="交點">交點</a></li> <li><a href="/wiki/%E4%B8%AD%E9%BB%9E" title="中點">中點</a></li> <li><a href="/wiki/%E8%A7%92" title="角">角</a> <ul><li><a href="/wiki/%E5%90%8C%E7%95%8C%E8%A7%92" title="同界角">同界角</a></li></ul></li> <li><a href="/wiki/%E6%9E%81%E5%80%BC%E7%82%B9" title="极值点">極值點</a></li> <li><a href="/w/index.php?title=%E6%9C%80%E5%80%BC%E9%BB%9E&action=edit&redlink=1" class="new" title="最值點（页面不存在）">最值點</a></li> <li><a href="/wiki/%E4%B8%B4%E7%95%8C%E7%82%B9_(%E6%95%B0%E5%AD%A6)" title="临界点 (数学)">臨界點</a></li> <li><a href="/wiki/%E9%A9%BB%E7%82%B9" title="驻点">驻点</a></li> <li><a href="/wiki/%E9%9E%8D%E9%BB%9E" title="鞍點">鞍點</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: right;"><a href="/wiki/%E7%9B%B4%E7%BA%BF" title="直线">直线</a>和<a href="/wiki/%E6%9B%B2%E7%BA%BF" title="曲线">曲线</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E7%BA%BF%E6%AE%B5" title="线段">线段</a></li> <li><a href="/wiki/%E5%B0%84%E7%BA%BF" title="射线">射线</a></li> <li><a href="/wiki/%E7%9B%B4%E7%BA%BF" title="直线">直线</a></li> <li><a href="/wiki/%E5%88%87%E7%BA%BF" title="切线">切线</a></li> <li><a href="/wiki/%E6%B3%95%E7%BA%BF" title="法线">（主）法线</a></li> <li><a href="/w/index.php?title=%E5%89%AF%E6%B3%95%E7%B7%9A&action=edit&redlink=1" class="new" title="副法線（页面不存在）">副法線</a></li> <li><a href="/wiki/%E6%9B%B2%E7%BA%BF" title="曲线">曲线</a></li> <li><a href="/wiki/%E5%9C%86%E9%94%A5%E6%9B%B2%E7%BA%BF" title="圆锥曲线">圆锥曲线</a></li> <li><a href="/wiki/%E5%8F%8C%E6%9B%B2%E7%BA%BF" title="双曲线">双曲线</a></li> <li><a href="/wiki/%E6%8A%9B%E7%89%A9%E7%BA%BF" title="抛物线">抛物线</a></li> <li><a href="/wiki/%E6%AD%A3%E5%BC%A6%E6%9B%B2%E7%B7%9A" title="正弦曲線">正弦曲線</a></li> <li><a href="/wiki/%E8%9A%8C%E7%BA%BF" title="蚌线">蚌线</a></li> <li><a href="/wiki/%E8%9C%97%E7%BA%BF" class="mw-redirect" title="蜗线">蜗线</a></li> <li><a href="/wiki/%E8%9E%BA%E7%BA%BF" title="螺线">螺线</a>（<a href="/wiki/%E9%98%BF%E5%9F%BA%E7%B1%B3%E5%BE%B7%E8%9E%BA%E7%BA%BF" title="阿基米德螺线">阿基米德螺线</a>、<a href="/wiki/%E7%AD%89%E8%A7%92%E8%9E%BA%E7%BA%BF" title="等角螺线">等角螺线</a>……）</li> <li><a href="/wiki/%E6%91%86%E7%BA%BF" title="摆线">摆线</a>（<a href="/wiki/%E6%9C%80%E9%80%9F%E9%99%8D%E7%B7%9A%E5%95%8F%E9%A1%8C" title="最速降線問題">最速降線問題</a>）</li> <li><a href="/wiki/%E6%82%AC%E9%93%BE%E7%BA%BF" title="悬链线">悬链线</a></li> <li><a href="/wiki/%E6%9B%B3%E7%89%A9%E7%BA%BF" title="曳物线">曳物线</a></li> <li><a href="/wiki/%E6%BC%B8%E4%BC%B8%E7%B7%9A" title="漸伸線">漸開線</a></li> <li><a href="/wiki/%E6%B8%90%E5%B1%88%E7%BA%BF" title="渐屈线">渐屈线</a></li> <li><a href="/wiki/%E6%B8%90%E8%BF%91%E7%BA%BF" title="渐近线">渐近线</a></li> <li><a href="/wiki/%E6%B5%8B%E5%9C%B0%E7%BA%BF" title="测地线">测地线</a></li> <li><a href="/wiki/%E9%82%8A_(%E5%B9%BE%E4%BD%95)" title="邊 (幾何)">邊</a></li> <li><a href="/wiki/%E5%91%A8%E9%95%BF" title="周长">周界</a></li> <li><a href="/wiki/%E5%BC%A6_(%E5%B9%BE%E4%BD%95)" title="弦 (幾何)">弦</a></li> <li><a href="/wiki/%E5%BC%A7" title="弧">弧</a></li> <li><a href="/wiki/%E7%9F%A2_(%E5%B9%BE%E4%BD%95)" title="矢 (幾何)">矢</a></li> <li><a href="/wiki/%E5%9E%82%E7%9B%B4%E5%B9%B3%E5%88%86%E7%B7%9A" title="垂直平分線">垂直平分線</a></li> <li><a href="/wiki/%E4%BB%A3%E6%95%B8%E6%9B%B2%E7%B7%9A" title="代數曲線">代數曲線</a></li> <li><a href="/wiki/%E6%A4%AD%E5%9C%86%E6%9B%B2%E7%BA%BF" title="椭圆曲线">椭圆曲线</a></li> <li><a href="/wiki/%E8%B6%85%E6%A9%A2%E5%9C%93" title="超橢圓">超橢圓</a></li> <li><a href="/wiki/%E6%98%9F%E5%BD%A2%E7%BA%BF" title="星形线">星形线</a></li> <li><a href="/wiki/%E4%B8%89%E5%B0%96%E7%93%A3%E7%BA%BF" title="三尖瓣线">三尖瓣线</a></li> <li><a href="/wiki/%E6%96%B9%E5%9C%93%E5%BD%A2" title="方圓形">方圓形</a></li> <li><a href="/wiki/%E5%8B%92%E6%B4%9B%E4%B8%89%E8%A7%92%E5%BD%A2" title="勒洛三角形">勒洛三角形</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: right;"><a href="/wiki/%E5%B9%B3%E9%9D%A2_(%E6%95%B0%E5%AD%A6)" title="平面 (数学)">平面</a>圖形</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E5%9C%86" title="圆">圆</a>（<a href="/wiki/%E5%B9%BF%E4%B9%89%E5%9C%86" title="广义圆">广义圆</a>）</li> <li><a href="/wiki/%E6%A4%AD%E5%9C%86" title="椭圆">椭圆</a></li> <li><a href="/wiki/%E6%89%87%E5%BD%A2" title="扇形">扇形</a></li> <li><a href="/wiki/%E5%BC%93%E5%BD%A2" title="弓形">弓形</a></li> <li><a href="/wiki/%E7%8E%AF%E5%BD%A2" title="环形">环形</a></li> <li><a href="/wiki/%E5%A4%9A%E8%BE%B9%E5%BD%A2" title="多边形">多边形</a></li> <li><a href="/wiki/%E4%B8%89%E8%A7%92%E5%BD%A2" title="三角形">三角形</a></li> <li><a href="/wiki/%E5%9B%9B%E9%82%8A%E5%BD%A2" title="四邊形">四邊形</a></li> <li><a href="/wiki/%E4%BA%94%E8%BE%B9%E5%BD%A2" title="五边形">五边形</a></li> <li><a href="/wiki/%E5%85%AD%E8%BE%B9%E5%BD%A2" title="六边形">六边形</a></li> <li><a href="/wiki/%E5%A4%9A%E8%BE%B9%E5%BD%A2" title="多边形">多边形</a></li> <li><a href="/wiki/%E6%AD%A3%E5%A4%9A%E8%BE%B9%E5%BD%A2" title="正多边形">正多边形</a></li> <li><a href="/wiki/%E6%A2%AF%E5%BD%A2" title="梯形">梯形</a></li> <li><a href="/wiki/%E5%B9%B3%E8%A1%8C%E5%9B%9B%E8%BE%B9%E5%BD%A2" title="平行四边形">平行四边形</a></li> <li><a href="/wiki/%E8%8F%B1%E5%BD%A2" title="菱形">菱形</a></li> <li><a href="/wiki/%E7%9F%A9%E5%BD%A2" title="矩形">矩形</a></li> <li><a href="/wiki/%E6%AD%A3%E6%96%B9%E5%BD%A2" title="正方形">正方形</a></li> <li><a href="/wiki/%E9%B7%82%E5%BD%A2" title="鷂形">鷂形</a></li> <li><a href="/wiki/%E5%8D%B5%E5%BD%A2%E7%BA%BF" title="卵形线">卵形线</a></li> <li><a href="/wiki/%E7%B4%A1%E9%8C%98%E5%BD%A2" title="紡錘形">梭形</a></li> <li><a href="/wiki/%E6%98%9F%E5%BD%A2%E5%A4%9A%E9%82%8A%E5%BD%A2" title="星形多邊形">星形</a></li> <li><a href="/wiki/%E4%BA%94%E8%A7%92%E6%98%9F" title="五角星">五角星</a></li> <li><a href="/wiki/%E5%85%AD%E8%A7%92%E6%98%9F" title="六角星">六角星</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: right;"><a href="/wiki/%E4%B8%89%E7%B6%AD%E7%A9%BA%E9%96%93" title="三維空間">立體</a>圖形</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E5%A4%9A%E9%9D%A2%E4%BD%93" title="多面体">多面体</a></li> <li><a href="/wiki/%E6%9F%8F%E6%8B%89%E5%9C%96%E7%AB%8B%E9%AB%94" title="柏拉圖立體">正多面體</a></li> <li><a href="/wiki/%E5%9B%9B%E9%9D%A2%E9%AB%94" title="四面體">四面體</a></li> <li><a href="/wiki/%E9%95%B7%E6%96%B9%E9%AB%94" title="長方體">長方體</a></li> <li><a href="/wiki/%E7%AB%8B%E6%96%B9%E9%AB%94" title="立方體">立方體</a></li> <li><a href="/wiki/%E5%B9%B3%E8%A1%8C%E5%85%AD%E9%9D%A2%E4%BD%93" title="平行六面体">平行六面体</a></li> <li><a href="/wiki/%E6%A3%B1%E6%9F%B1" title="棱柱">棱柱</a></li> <li><a href="/wiki/%E5%8F%8D%E6%A3%B1%E6%9F%B1" title="反棱柱">反棱柱</a></li> <li><a href="/wiki/%E6%A3%B1%E9%94%A5" title="棱锥">棱锥</a></li> <li><a href="/wiki/%E9%94%A5%E5%8F%B0" title="锥台">棱台</a></li> <li><a href="/wiki/%E5%9C%86%E6%9F%B1%E4%BD%93" title="圆柱体">圆柱体</a></li> <li><a href="/wiki/%E5%9C%86%E9%94%A5" title="圆锥">圆锥</a></li> <li><a href="/wiki/%E5%9C%86%E5%8F%B0" title="圆台">圆台</a></li> <li><a href="/wiki/%E6%A4%AD%E7%90%83" title="椭球">椭球</a>（<a href="/wiki/%E9%A1%9E%E7%90%83%E9%9D%A2" title="類球面">長球體</a>、<a href="/wiki/%E9%A1%9E%E7%90%83%E9%9D%A2" title="類球面">扁球體</a>）</li> <li><a href="/wiki/%E7%90%83_(%E6%95%B0%E5%AD%A6)" title="球 (数学)">球體</a></li> <li><a href="/wiki/%E7%90%83%E7%BC%BA" title="球缺">球缺</a></li> <li><a href="/wiki/%E7%90%83%E5%86%A0" title="球冠">球冠</a></li> <li><a href="/wiki/%E7%90%83%E5%8F%B0" title="球台">球台</a></li> <li><a href="/wiki/%E9%9B%A2%E5%BF%83%E7%8E%87" title="離心率">準線</a></li> <li><a href="/w/index.php?title=%E6%AF%8D%E7%BA%BF_(%E5%87%A0%E4%BD%95)&action=edit&redlink=1" class="new" title="母线 (几何)（页面不存在）">母線</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: right;"><a href="/wiki/%E6%9B%B2%E9%9D%A2" title="曲面">曲面</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E4%BA%8C%E6%AC%A1%E6%9B%B2%E9%9D%A2" title="二次曲面">二次曲面</a></li> <li><a href="/wiki/%E6%97%8B%E8%BD%89%E6%9B%B2%E9%9D%A2" title="旋轉曲面">旋轉曲面</a></li> <li><a href="/wiki/%E6%8A%9B%E7%89%A9%E9%9D%A2" title="抛物面">抛物面</a></li> <li><a href="/wiki/%E9%9B%99%E6%9B%B2%E9%9D%A2" title="雙曲面">雙曲面</a></li> <li><a href="/w/index.php?title=%E9%A9%AC%E9%9E%8D%E9%9D%A2&action=edit&redlink=1" class="new" title="马鞍面（页面不存在）">马鞍面</a></li> <li><a href="/wiki/%E7%90%83%E9%9D%A2" title="球面">球面</a></li> <li><a href="/wiki/%E6%A4%AD%E7%90%83" title="椭球">橢球面</a></li> <li><a href="/wiki/%E9%A1%9E%E7%90%83%E9%9D%A2" title="類球面">類球面</a></li> <li><a href="/wiki/%E7%8E%AF%E9%9D%A2" title="环面">环面</a></li> <li><a href="/wiki/%E8%8E%AB%E6%AF%94%E4%B9%8C%E6%96%AF%E5%B8%A6" title="莫比乌斯带">莫比乌斯带</a></li> <li><a href="/wiki/%E6%B5%81%E5%BD%A2" title="流形">流形</a></li> <li><a href="/wiki/%E9%BB%8E%E6%9B%BC%E6%9B%B2%E9%9D%A2" title="黎曼曲面">黎曼曲面</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: right;">高維空間</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E8%B6%85%E5%B9%B3%E9%9D%A2" title="超平面">超平面</a></li> <li><a href="/wiki/%E7%B6%AD%E9%9D%A2" title="維面">超面</a></li> <li><a href="/wiki/%E8%B6%85%E6%9B%B2%E9%9D%A2" title="超曲面">超曲面</a></li> <li><a href="/wiki/%E8%83%9E_(%E7%B5%90%E6%A7%8B)" title="胞 (結構)">胞</a></li> <li><a href="/wiki/%E5%A4%9A%E8%83%9E%E5%BD%A2" title="多胞形">多胞形</a></li> <li><a href="/wiki/%E8%B6%85%E7%90%83%E9%9D%A2" title="超球面">超球體</a></li> <li><a href="/wiki/%E8%B6%85%E6%96%B9%E5%BD%A2" title="超方形">超方形</a></li> <li><a href="/wiki/%E5%9B%9B%E7%B6%AD%E8%B6%85%E6%AD%A3%E6%96%B9%E9%AB%94" title="四維超正方體">超立方體</a></li> <li><a href="/wiki/%E5%85%8B%E8%8E%B1%E5%9B%A0%E7%93%B6" title="克莱因瓶">克莱因瓶</a></li> <li><a href="/wiki/%E5%9B%9B%E7%B6%AD%E6%9F%B1%E9%AB%94%E6%9F%B1" title="四維柱體柱">四維柱體柱</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: right;">圖形關係</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E7%9B%B8%E4%BC%BC_(%E5%B9%BE%E4%BD%95)" title="相似 (幾何)">相似</a></li> <li><a href="/wiki/%E5%85%A8%E7%AD%89" title="全等">全等</a></li> <li><a href="/wiki/%E5%B0%8D%E7%A8%B1" title="對稱">對稱</a></li> <li><a href="/wiki/%E5%B9%B3%E8%A1%8C" title="平行">平行</a></li> <li><a href="/wiki/%E5%9E%82%E7%9B%B4" title="垂直">垂直</a></li> <li><a href="/wiki/%E7%9B%B8%E4%BA%A4" title="相交">相交</a></li> <li><a href="/wiki/%E5%88%87%E7%BA%BF" title="切线">相切</a></li> <li><a href="/w/index.php?title=%E7%9B%B8%E9%9B%A2&action=edit&redlink=1" class="new" title="相離（页面不存在）">相離</a></li> <li><a href="/wiki/%E9%95%9C%E5%83%8F_(%E5%87%A0%E4%BD%95)" title="镜像 (几何)">镜像</a></li> <li><a href="/wiki/%E6%97%8B%E8%BD%AC" title="旋转">旋转</a></li> <li><a href="/wiki/%E5%8F%8D%E6%BC%94" title="反演">反演</a></li> <li><a href="/wiki/%E6%88%AA%E9%9D%A2_(%E5%B9%BE%E4%BD%95)" title="截面 (幾何)">截面</a></li> <li><a href="/wiki/%E7%BC%A9%E6%94%BE" title="缩放">缩放</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: right;"><a href="/wiki/%E4%B8%89%E8%A7%92%E5%BD%A2" title="三角形">三角形</a>關係</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E7%9B%B8%E4%BC%BC%E4%B8%89%E8%A7%92%E5%BD%A2" title="相似三角形">相似三角形</a></li> <li><a href="/wiki/%E5%85%A8%E7%AD%89%E4%B8%89%E8%A7%92%E5%BD%A2" title="全等三角形">全等三角形</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: right;"><a href="/wiki/%E9%87%8F_(%E6%95%B0%E5%AD%A6)" title="量 (数学)">量</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E8%B7%9D%E7%A6%BB" title="距离">距离</a></li> <li><a href="/wiki/%E9%95%BF%E5%BA%A6" title="长度">长度</a></li> <li><a href="/wiki/%E5%91%A8%E9%95%BF" title="周长">周长</a></li> <li><a href="/wiki/%E5%BC%A7%E9%95%BF" title="弧长">弧长</a></li> <li><a href="/wiki/%E9%AB%98%E5%BA%A6" title="高度">高度</a></li> <li><a href="/wiki/%E9%9D%A2%E7%A7%AF" title="面积">面积</a></li> <li><a href="/wiki/%E8%A1%A8%E9%9D%A2%E7%A9%8D" title="表面積">表面積</a></li> <li><a href="/wiki/%E4%BD%93%E7%A7%AF" title="体积">体积</a></li> <li><a href="/wiki/%E5%AE%B9%E7%A9%8D" class="mw-redirect" title="容積">容積</a></li> <li><a href="/wiki/%E5%BA%A6_(%E8%A7%92)" title="度 (角)">角度</a></li> <li><a href="/wiki/%E6%9B%B2%E7%8E%87" title="曲率">曲率</a></li> <li><a href="/wiki/%E6%9B%B2%E7%BA%BF%E7%9A%84%E6%8C%A0%E7%8E%87" title="曲线的挠率">撓率</a></li> <li><a href="/wiki/%E9%9B%A2%E5%BF%83%E7%8E%87" title="離心率">離心率</a></li> <li><a href="/wiki/%E5%87%B9%E5%87%B8%E6%80%A7_(%E5%B9%BE%E4%BD%95)" title="凹凸性 (幾何)">凹凸性</a></li> <li><a href="/w/index.php?title=%E6%9C%89%E5%90%91%E6%9B%B2%E9%9D%A2&action=edit&redlink=1" class="new" title="有向曲面（页面不存在）">有向曲面</a></li> <li><a href="/wiki/%E5%8F%AF%E5%B1%95%E6%9B%B2%E9%9D%A2" title="可展曲面">可展曲面</a></li> <li><a href="/wiki/%E7%9B%B4%E7%B4%8B%E6%9B%B2%E9%9D%A2" title="直紋曲面">直紋曲面</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: right;">作圖</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E5%B0%BA" title="尺">尺</a> <ul><li><a href="/wiki/%E7%9B%B4%E5%B0%BA" title="直尺">直尺</a></li> <li><a href="/wiki/%E4%B8%89%E8%A7%92%E5%B0%BA" title="三角尺">三角尺</a></li></ul></li> <li><a href="/wiki/%E5%9C%86%E8%A7%84" title="圆规">圆规</a></li> <li><a href="/wiki/%E5%B0%BA%E8%A7%84%E4%BD%9C%E5%9B%BE" title="尺规作图">尺规作图</a></li> <li><a href="/wiki/%E4%BA%8C%E5%88%BB%E5%B0%BA%E4%BD%9C%E5%9C%96" title="二刻尺作圖">二刻尺作圖</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: right;">分支</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E6%AC%A7%E5%87%A0%E9%87%8C%E5%BE%97%E5%87%A0%E4%BD%95" title="欧几里得几何">平面幾何</a></li> <li><a href="/wiki/%E7%AB%8B%E4%BD%93%E5%87%A0%E4%BD%95" title="立体几何">立体几何</a></li> <li><a href="/wiki/%E4%B8%89%E8%A7%92%E5%AD%A6" title="三角学">三角学</a></li> <li><a href="/wiki/%E8%A7%A3%E6%9E%90%E5%87%A0%E4%BD%95" title="解析几何">解析几何</a></li> <li><a href="/wiki/%E5%BE%AE%E5%88%86%E5%87%A0%E4%BD%95" title="微分几何">微分几何</a></li> <li><a href="/wiki/%E6%8B%93%E6%89%91%E5%AD%A6" title="拓扑学">拓扑学</a></li> <li><a href="/wiki/%E5%9B%BE%E8%AE%BA" title="图论">图论</a></li> <li><a href="/wiki/%E6%91%BA%E7%B4%99%E6%95%B8%E5%AD%B8" title="摺紙數學">摺紙數學</a></li> <li><a href="/wiki/%E6%AC%A7%E5%87%A0%E9%87%8C%E5%BE%97%E5%87%A0%E4%BD%95" title="欧几里得几何">欧几里得几何</a></li> <li><a href="/wiki/%E9%9D%9E%E6%AC%A7%E5%87%A0%E9%87%8C%E5%BE%97%E5%87%A0%E4%BD%95" title="非欧几里得几何">非欧几里得几何</a>（<a href="/wiki/%E5%8F%8C%E6%9B%B2%E5%87%A0%E4%BD%95" title="双曲几何">双曲几何</a>、<a href="/wiki/%E7%90%83%E9%9D%A2%E5%B9%BE%E4%BD%95%E5%AD%B8" title="球面幾何學">球面幾何</a>……）</li> <li><a href="/wiki/%E5%88%86%E5%BD%A2" title="分形">分形</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: right;">理論</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E5%AE%9A%E7%90%86" title="定理">定理</a></li> <li><a href="/wiki/%E5%85%AC%E7%90%86" title="公理">公理</a></li> <li><a href="/wiki/%E5%AE%9A%E4%B9%89" title="定义">定义</a></li> <li><a class="mw-selflink selflink">數學證明</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><span typeof="mw:File"><span title="分类"><img alt="分类" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span><b><a href="/wiki/Category:%E5%87%A0%E4%BD%95%E5%AD%A6" title="Category:几何学">分类</a></b></li> <li><span typeof="mw:File"><span title="主题"><img alt="主题" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e2/Symbol_portal_class.svg/16px-Symbol_portal_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e2/Symbol_portal_class.svg/23px-Symbol_portal_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e2/Symbol_portal_class.svg/31px-Symbol_portal_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span><b><a href="/wiki/Portal:%E5%87%A0%E4%BD%95%E5%AD%A6" title="Portal:几何学">主题</a></b></li> <li><span typeof="mw:File"><span title="共享资源页面"><img alt="共享资源页面" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/12px-Commons-logo.svg.png" decoding="async" width="12" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/18px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/24px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span><b><a href="https://commons.wikimedia.org/wiki/Category:Geometry" class="extiw" title="commons:Category:Geometry">共享资源</a></b></li> <li><span typeof="mw:File"><span title="专题"><img alt="专题" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/37/People_icon.svg/16px-People_icon.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/37/People_icon.svg/24px-People_icon.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/37/People_icon.svg/32px-People_icon.svg.png 2x" data-file-width="100" data-file-height="100" /></span></span><b><a href="/wiki/WikiProject:%E5%87%A0%E4%BD%95%E5%AD%A6" title="WikiProject:几何学">专题</a></b></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">检索自“<a dir="ltr" href="https://zh.wikipedia.org/w/index.php?title=數學證明&oldid=79648586">https://zh.wikipedia.org/w/index.php?title=數學證明&oldid=79648586</a>”</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Special:%E9%A1%B5%E9%9D%A2%E5%88%86%E7%B1%BB" title="Special:页面分类">分类</a>：<ul><li><a href="/wiki/Category:%E5%87%A0%E4%BD%95%E6%9C%AF%E8%AF%AD" title="Category:几何术语">几何术语</a></li><li><a href="/wiki/Category:%E6%95%B8%E5%AD%B8%E6%8E%A8%E7%90%86" title="Category:數學推理">數學推理</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">隐藏分类：<ul><li><a href="/wiki/Category:%E6%9C%89%E8%93%9D%E9%93%BE%E5%8D%B4%E6%9C%AA%E7%A7%BB%E9%99%A4%E5%86%85%E9%83%A8%E9%93%BE%E6%8E%A5%E5%8A%A9%E6%89%8B%E6%A8%A1%E6%9D%BF%E7%9A%84%E9%A1%B5%E9%9D%A2" title="Category:有蓝链却未移除内部链接助手模板的页面">有蓝链却未移除内部链接助手模板的页面</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> 本页面最后修订于2023年11月6日 (星期一) 08:38。</li> <li id="footer-info-copyright">本站的全部文字在<a rel="nofollow" class="external text" href="//creativecommons.org/licenses/by-sa/4.0/deed.zh">知识共享署名-相同方式共享 4.0协议</a>之条款下提供，附加条款亦可能应用。（请参阅<a class="external text" href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Terms_of_Use">使用条款</a>）<br /> Wikipedia®和维基百科标志是<a rel="nofollow" class="external text" href="https://wikimediafoundation.org/zh">维基媒体基金会</a>的注册商标；维基™是维基媒体基金会的商标。<br /> 维基媒体基金会是按美国国內稅收法501(c)(3)登记的<a class="external text" href="https://donate.wikimedia.org/wiki/Special:MyLanguage/Tax_deductibility">非营利慈善机构</a>。<br /></li> </ul> <ul id="footer-places"> <li id="footer-places-privacy"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy">隐私政策</a></li> <li id="footer-places-about"><a href="/wiki/Wikipedia:%E5%85%B3%E4%BA%8E">关于维基百科</a></li> <li id="footer-places-disclaimers"><a href="/wiki/Wikipedia:%E5%85%8D%E8%B4%A3%E5%A3%B0%E6%98%8E">免责声明</a></li> <li id="footer-places-wm-codeofconduct"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Universal_Code_of_Conduct">行为准则</a></li> <li id="footer-places-developers"><a href="https://developer.wikimedia.org">开发者</a></li> <li id="footer-places-statslink"><a href="https://stats.wikimedia.org/#/zh.wikipedia.org">统计</a></li> <li id="footer-places-cookiestatement"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Cookie_statement">Cookie声明</a></li> <li id="footer-places-mobileview"><a href="//zh.m.wikipedia.org/w/index.php?title=%E6%95%B8%E5%AD%B8%E8%AD%89%E6%98%8E&mobileaction=toggle_view_mobile" class="noprint stopMobileRedirectToggle">手机版视图</a></li> </ul> <ul id="footer-icons" class="noprint"> <li id="footer-copyrightico"><a href="https://wikimediafoundation.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/static/images/footer/wikimedia-button.svg" width="84" height="29" alt="Wikimedia Foundation" loading="lazy"></a></li> <li id="footer-poweredbyico"><a href="https://www.mediawiki.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/w/resources/assets/poweredby_mediawiki.svg" alt="Powered by MediaWiki" width="88" height="31" loading="lazy"></a></li> </ul> </footer> </div> </div> </div> <div class="vector-settings" id="p-dock-bottom"> <ul></ul> </div><script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.main-6b7f745dd4-zrk7h","wgBackendResponseTime":163,"wgPageParseReport":{"limitreport":{"cputime":"0.368","walltime":"0.614","ppvisitednodes":{"value":1313,"limit":1000000},"postexpandincludesize":{"value":86969,"limit":2097152},"templateargumentsize":{"value":426,"limit":2097152},"expansiondepth":{"value":7,"limit":100},"expensivefunctioncount":{"value":7,"limit":500},"unstrip-depth":{"value":0,"limit":20},"unstrip-size":{"value":29814,"limit":5000000},"entityaccesscount":{"value":0,"limit":400},"timingprofile":["100.00% 276.596 1 -total"," 33.21% 91.855 1 Template:几何术语"," 32.53% 89.986 1 Template:Navbox"," 16.60% 45.902 5 Template:Cite_web"," 16.19% 44.781 1 Template:NoteTA"," 7.95% 21.995 5 Template:Cite_book"," 7.32% 20.256 3 Template:Main"," 6.51% 17.996 2 Template:Tsl"," 2.60% 7.186 4 Template:Icon"," 1.88% 5.197 2 Template:Link-en"]},"scribunto":{"limitreport-timeusage":{"value":"0.175","limit":"10.000"},"limitreport-memusage":{"value":3983837,"limit":52428800}},"cachereport":{"origin":"mw-api-int.codfw.main-849f99967d-6w6cz","timestamp":"20241124083719","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"\u6578\u5b78\u8b49\u660e","url":"https:\/\/zh.wikipedia.org\/wiki\/%E6%95%B8%E5%AD%B8%E8%AD%89%E6%98%8E","sameAs":"http:\/\/www.wikidata.org\/entity\/Q11538","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q11538","author":{"@type":"Organization","name":"\u7ef4\u57fa\u5a92\u4f53\u9879\u76ee\u8d21\u732e\u8005"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2004-12-25T02:08:17Z","dateModified":"2023-11-06T08:38:47Z","image":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/8\/8d\/P._Oxy._I_29.jpg"}</script> </body> </html>