無理數 - 维基百科，自由的百科全书

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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=%E7%84%A1%E7%90%86%E6%95%B8" 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-举例" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#举例"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>举例</span> </div> </a> <ul id="toc-举例-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-性质" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#性质"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>性质</span> </div> </a> <ul id="toc-性质-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-不知是否是無理數的數" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#不知是否是無理數的數"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>不知是否是無理數的數</span> </div> </a> <ul id="toc-不知是否是無理數的數-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-無理數集的特性" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#無理數集的特性"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>無理數集的特性</span> </div> </a> <ul id="toc-無理數集的特性-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-無理化作連分數的表達式" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#無理化作連分數的表達式"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>無理化作連分數的表達式</span> </div> </a> <ul id="toc-無理化作連分數的表達式-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-無理數之證" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#無理數之證"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>無理數之證</span> </div> </a> <button aria-controls="toc-無理數之證-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>开关無理數之證子章节</span> </button> <ul id="toc-無理數之證-sublist" class="vector-toc-list"> <li id="toc-證明'"`UNIQ--postMath-0000002A-QINU`"'是无理数" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#證明'"`UNIQ--postMath-0000002A-QINU`"'是无理数"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>證明'"`UNIQ--postMath-0000002A-QINU`"'是无理数</span> </div> </a> <ul id="toc-證明'"`UNIQ--postMath-0000002A-QINU`"'是无理数-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-證明'"`UNIQ--postMath-00000038-QINU`"'是无理数" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#證明'"`UNIQ--postMath-00000038-QINU`"'是无理数"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>證明'"`UNIQ--postMath-00000038-QINU`"'是无理数</span> </div> </a> <ul id="toc-證明'"`UNIQ--postMath-00000038-QINU`"'是无理数-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-證明'"`UNIQ--postMath-00000043-QINU`"'是无理数" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#證明'"`UNIQ--postMath-00000043-QINU`"'是无理数"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.3</span> <span>證明'"`UNIQ--postMath-00000043-QINU`"'是无理数</span> </div> </a> <ul id="toc-證明'"`UNIQ--postMath-00000043-QINU`"'是无理数-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-證明'"`UNIQ--postMath-00000059-QINU`"'是无理数" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#證明'"`UNIQ--postMath-00000059-QINU`"'是无理数"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.4</span> <span>證明'"`UNIQ--postMath-00000059-QINU`"'是无理数</span> </div> </a> <ul id="toc-證明'"`UNIQ--postMath-00000059-QINU`"'是无理数-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-参见" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#参见"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>参见</span> </div> </a> <ul id="toc-参见-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">8</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">無理數</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。89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-89" 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">89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/Irrasionale_getal" title="Irrasionale getal – 南非荷兰语" lang="af" hreflang="af" data-title="Irrasionale getal" 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-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B9%D8%AF%D8%AF_%D8%BA%D9%8A%D8%B1_%D9%83%D8%B3%D8%B1%D9%8A" title="عدد غير كسري – 阿拉伯语" lang="ar" hreflang="ar" data-title="عدد غير كسري" 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%85%E0%A6%AA%E0%A7%B0%E0%A6%BF%E0%A6%AE%E0%A7%87%E0%A6%AF%E0%A6%BC_%E0%A6%B8%E0%A6%82%E0%A6%96%E0%A7%8D%E0%A6%AF%E0%A6%BE" title="অপৰিমেয় সংখ্যা – 阿萨姆语" lang="as" hreflang="as" data-title="অপৰিমেয় সংখ্যা" 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/N%C3%BAmberu_irracional" title="Númberu irracional – 阿斯图里亚斯语" lang="ast" hreflang="ast" data-title="Númberu irracional" 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/%C4%B0rrasional_%C9%99d%C9%99dl%C9%99r" title="İrrasional ədədlər – 阿塞拜疆语" lang="az" hreflang="az" data-title="İrrasional ədədlər" 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%98%D1%80%D1%80%D0%B0%D1%86%D0%B8%D0%BE%D0%BD%D0%B0%D0%BB%D1%8C_%D2%BB%D0%B0%D0%BD" title="Иррациональ һан – 巴什基尔语" lang="ba" hreflang="ba" data-title="Иррациональ һан" 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-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%86%D1%80%D0%B0%D1%86%D1%8B%D1%8F%D0%BD%D0%B0%D0%BB%D1%8C%D0%BD%D1%8B_%D0%BB%D1%96%D0%BA" title="Ірацыянальны лік – 白俄罗斯语" lang="be" hreflang="be" data-title="Ірацыянальны лік" 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-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%98%D1%80%D0%B0%D1%86%D0%B8%D0%BE%D0%BD%D0%B0%D0%BB%D0%BD%D0%BE_%D1%87%D0%B8%D1%81%D0%BB%D0%BE" title="Ирационално число – 保加利亚语" lang="bg" hreflang="bg" data-title="Ирационално число" 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%85%E0%A6%AE%E0%A7%82%E0%A6%B2%E0%A6%A6_%E0%A6%B8%E0%A6%82%E0%A6%96%E0%A7%8D%E0%A6%AF%E0%A6%BE" title="অমূলদ সংখ্যা – 孟加拉语" lang="bn" hreflang="bn" data-title="অমূলদ সংখ্যা" 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/Iracionalan_broj" title="Iracionalan broj – 波斯尼亚语" lang="bs" hreflang="bs" data-title="Iracionalan broj" 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/Nombre_irracional" title="Nombre irracional – 加泰罗尼亚语" lang="ca" hreflang="ca" data-title="Nombre irracional" 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/%DA%98%D9%85%D8%A7%D8%B1%DB%95%DB%8C_%D9%86%D8%A7%DA%95%DB%8E%DA%98%DB%95%DB%8C%DB%8C" title="ژمارەی ناڕێژەیی – 中库尔德语" lang="ckb" hreflang="ckb" data-title="ژمارەی ناڕێژەیی" 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 mw-list-item"><a href="https://cs.wikipedia.org/wiki/Iracion%C3%A1ln%C3%AD_%C4%8D%C3%ADslo" title="Iracionální číslo – 捷克语" lang="cs" hreflang="cs" data-title="Iracionální číslo" 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/%D0%98%D1%80%D1%80%D0%B0%D1%86%D0%B8%D0%BE%D0%BD%D0%B0%D0%BB%D0%BB%C4%83_%D1%85%D0%B8%D1%81%D0%B5%D0%BF" title="Иррационаллă хисеп – 楚瓦什语" lang="cv" hreflang="cv" data-title="Иррационаллă хисеп" 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/Rhif_anghymarebol" title="Rhif anghymarebol – 威尔士语" lang="cy" hreflang="cy" data-title="Rhif anghymarebol" 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/Irrationale_tal" title="Irrationale tal – 丹麦语" lang="da" hreflang="da" data-title="Irrationale tal" 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/Irrationale_Zahl" title="Irrationale Zahl – 德语" lang="de" hreflang="de" data-title="Irrationale Zahl" 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%86%CF%81%CF%81%CE%B7%CF%84%CE%BF%CF%82_%CE%B1%CF%81%CE%B9%CE%B8%CE%BC%CF%8C%CF%82" title="Άρρητος αριθμός – 希腊语" lang="el" hreflang="el" data-title="Άρρητος αριθμός" 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/Irrational_number" title="Irrational number – 英语" lang="en" hreflang="en" data-title="Irrational number" 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 mw-list-item"><a href="https://eo.wikipedia.org/wiki/Neracionala_nombro" title="Neracionala nombro – 世界语" lang="eo" hreflang="eo" data-title="Neracionala nombro" 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/N%C3%BAmero_irracional" title="Número irracional – 西班牙语" lang="es" hreflang="es" data-title="Número irracional" 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/Irratsionaalarvud" title="Irratsionaalarvud – 爱沙尼亚语" lang="et" hreflang="et" data-title="Irratsionaalarvud" 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/Zenbaki_irrazional" title="Zenbaki irrazional – 巴斯克语" lang="eu" hreflang="eu" data-title="Zenbaki irrazional" 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%B9%D8%AF%D8%AF_%DA%AF%D9%86%DA%AF" title="عدد گنگ – 波斯语" lang="fa" hreflang="fa" data-title="عدد گنگ" 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/Irrationaaliluku" title="Irrationaaliluku – 芬兰语" lang="fi" hreflang="fi" data-title="Irrationaaliluku" 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-fiu-vro mw-list-item"><a href="https://fiu-vro.wikipedia.org/wiki/Irratsionaalarv" title="Irratsionaalarv – 佛羅文" lang="vro" hreflang="vro" data-title="Irratsionaalarv" data-language-autonym="Võro" data-language-local-name="佛羅文" class="interlanguage-link-target"><span>Võro</span></a></li><li class="interlanguage-link interwiki-fo mw-list-item"><a href="https://fo.wikipedia.org/wiki/Irrationell_t%C3%B8l" title="Irrationell tøl – 法罗语" lang="fo" hreflang="fo" data-title="Irrationell tøl" data-language-autonym="Føroyskt" data-language-local-name="法罗语" class="interlanguage-link-target"><span>Føroyskt</span></a></li><li class="interlanguage-link interwiki-fr badge-Q17437798 badge-goodarticle mw-list-item" title="优良条目"><a href="https://fr.wikipedia.org/wiki/Nombre_irrationnel" title="Nombre irrationnel – 法语" lang="fr" hreflang="fr" data-title="Nombre irrationnel" data-language-autonym="Français" data-language-local-name="法语" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Uimhir_%C3%A9ag%C3%B3imheasta" title="Uimhir éagóimheasta – 爱尔兰语" lang="ga" hreflang="ga" data-title="Uimhir éagóimheasta" data-language-autonym="Gaeilge" data-language-local-name="爱尔兰语" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/N%C3%BAmero_irracional" title="Número irracional – 加利西亚语" lang="gl" hreflang="gl" data-title="Número irracional" 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%9E%D7%A1%D7%A4%D7%A8_%D7%90%D7%99-%D7%A8%D7%A6%D7%99%D7%95%D7%A0%D7%9C%D7%99" title="מספר אי-רציונלי – 希伯来语" lang="he" hreflang="he" data-title="מספר אי-רציונלי" 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%85%E0%A4%AA%E0%A4%B0%E0%A4%BF%E0%A4%AE%E0%A5%87%E0%A4%AF_%E0%A4%B8%E0%A4%82%E0%A4%96%E0%A5%8D%E0%A4%AF%E0%A4%BE" title="अपरिमेय संख्या – 印地语" lang="hi" hreflang="hi" data-title="अपरिमेय संख्या" 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-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Iracionalni_broj" title="Iracionalni broj – 克罗地亚语" lang="hr" hreflang="hr" data-title="Iracionalni broj" 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/Irracion%C3%A1lis_sz%C3%A1mok" title="Irracionális számok – 匈牙利语" lang="hu" hreflang="hu" data-title="Irracionális számok" 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/%D4%BB%D5%BC%D5%A1%D6%81%D5%AB%D5%B8%D5%B6%D5%A1%D5%AC_%D5%A9%D5%AB%D5%BE" title="Իռացիոնալ թիվ – 亚美尼亚语" lang="hy" hreflang="hy" data-title="Իռացիոնալ թիվ" 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-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Bilangan_irasional" title="Bilangan irasional – 印度尼西亚语" lang="id" hreflang="id" data-title="Bilangan irasional" 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-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Neracionala_nombro" title="Neracionala nombro – 伊多语" lang="io" hreflang="io" data-title="Neracionala nombro" data-language-autonym="Ido" data-language-local-name="伊多语" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/%C3%93r%C3%A6%C3%B0ar_t%C3%B6lur" title="Óræðar tölur – 冰岛语" lang="is" hreflang="is" data-title="Óræðar tölur" 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/Numero_irrazionale" title="Numero irrazionale – 意大利语" lang="it" hreflang="it" data-title="Numero irrazionale" 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/%E7%84%A1%E7%90%86%E6%95%B0" title="無理数 – 日语" lang="ja" hreflang="ja" data-title="無理数" 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-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%98%E1%83%A0%E1%83%90%E1%83%AA%E1%83%98%E1%83%9D%E1%83%9C%E1%83%90%E1%83%9A%E1%83%A3%E1%83%A0%E1%83%98_%E1%83%A0%E1%83%98%E1%83%AA%E1%83%AE%E1%83%95%E1%83%98" title="ირაციონალური რიცხვი – 格鲁吉亚语" lang="ka" hreflang="ka" data-title="ირაციონალური რიცხვი" 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%A0%D0%B0%D0%B1%D0%B0%D0%B9%D1%81%D1%8B%D0%B7_%D1%81%D0%B0%D0%BD" title="Рабайсыз сан – 哈萨克语" lang="kk" hreflang="kk" data-title="Рабайсыз сан" 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/%EB%AC%B4%EB%A6%AC%EC%88%98" title="무리수 – 韩语" lang="ko" hreflang="ko" data-title="무리수" data-language-autonym="한국어" data-language-local-name="韩语" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-ku mw-list-item"><a href="https://ku.wikipedia.org/wiki/Hejmar%C3%AAn_na_aql%C3%AE" title="Hejmarên na aqlî – 库尔德语" lang="ku" hreflang="ku" data-title="Hejmarên na aqlî" data-language-autonym="Kurdî" data-language-local-name="库尔德语" class="interlanguage-link-target"><span>Kurdî</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%98%D1%80%D1%80%D0%B0%D1%86%D0%B8%D0%BE%D0%BD%D0%B0%D0%BB%D0%B4%D1%8B%D0%BA_%D1%81%D0%B0%D0%BD" title="Иррационалдык сан – 柯尔克孜语" lang="ky" hreflang="ky" data-title="Иррационалдык сан" data-language-autonym="Кыргызча" data-language-local-name="柯尔克孜语" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Numerus_irrationalis" title="Numerus irrationalis – 拉丁语" lang="la" hreflang="la" data-title="Numerus irrationalis" 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-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/N%C3%BCmar_irazziunaal" title="Nümar irazziunaal – 倫巴底文" lang="lmo" hreflang="lmo" data-title="Nümar irazziunaal" data-language-autonym="Lombard" data-language-local-name="倫巴底文" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-lo mw-list-item"><a href="https://lo.wikipedia.org/wiki/%E0%BA%88%E0%BA%B3%E0%BA%99%E0%BA%A7%E0%BA%99%E0%BA%AD%E0%BA%B0%E0%BA%9B%E0%BA%BB%E0%BA%81%E0%BA%81%E0%BA%B0%E0%BA%95%E0%BA%B4" title="ຈຳນວນອະປົກກະຕິ – 老挝语" lang="lo" hreflang="lo" data-title="ຈຳນວນອະປົກກະຕິ" data-language-autonym="ລາວ" data-language-local-name="老挝语" class="interlanguage-link-target"><span>ລາວ</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Iracionalusis_skai%C4%8Dius" title="Iracionalusis skaičius – 立陶宛语" lang="lt" hreflang="lt" data-title="Iracionalusis skaičius" 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/Iracion%C4%81ls_skaitlis" title="Iracionāls skaitlis – 拉脱维亚语" lang="lv" hreflang="lv" data-title="Iracionāls skaitlis" 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-mg mw-list-item"><a href="https://mg.wikipedia.org/wiki/Isa_tsivoasaina" title="Isa tsivoasaina – 马拉加斯语" lang="mg" hreflang="mg" data-title="Isa tsivoasaina" data-language-autonym="Malagasy" data-language-local-name="马拉加斯语" class="interlanguage-link-target"><span>Malagasy</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%98%D1%80%D0%B0%D1%86%D0%B8%D0%BE%D0%BD%D0%B0%D0%BB%D0%B5%D0%BD_%D0%B1%D1%80%D0%BE%D1%98" title="Ирационален број – 马其顿语" lang="mk" hreflang="mk" data-title="Ирационален број" 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%85%E0%B4%AD%E0%B4%BF%E0%B4%A8%E0%B5%8D%E0%B4%A8%E0%B4%95%E0%B4%B8%E0%B4%82%E0%B4%96%E0%B5%8D%E0%B4%AF" title="അഭിന്നകസംഖ്യ – 马拉雅拉姆语" lang="ml" hreflang="ml" data-title="അഭിന്നകസംഖ്യ" 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-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%98%D1%80%D1%80%D0%B0%D1%86%D0%B8%D0%BE%D0%BD%D0%B0%D0%BB_%D1%82%D0%BE%D0%BE" title="Иррационал тоо – 蒙古语" lang="mn" hreflang="mn" data-title="Иррационал тоо" data-language-autonym="Монгол" data-language-local-name="蒙古语" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%85%E0%A4%AA%E0%A4%B0%E0%A4%BF%E0%A4%AE%E0%A5%87%E0%A4%AF_%E0%A4%B8%E0%A4%82%E0%A4%96%E0%A5%8D%E0%A4%AF%E0%A4%BE" title="अपरिमेय संख्या – 马拉地语" lang="mr" hreflang="mr" data-title="अपरिमेय संख्या" 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/Nombor_bukan_nisbah" title="Nombor bukan nisbah – 马来语" lang="ms" hreflang="ms" data-title="Nombor bukan nisbah" 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-mt mw-list-item"><a href="https://mt.wikipedia.org/wiki/Numru_irrazzjonali" title="Numru irrazzjonali – 马耳他语" lang="mt" hreflang="mt" data-title="Numru irrazzjonali" data-language-autonym="Malti" data-language-local-name="马耳他语" class="interlanguage-link-target"><span>Malti</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Irrationaal_getal" title="Irrationaal getal – 荷兰语" lang="nl" hreflang="nl" data-title="Irrationaal getal" 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/Irrasjonale_tal" title="Irrasjonale tal – 挪威尼诺斯克语" lang="nn" hreflang="nn" data-title="Irrasjonale tal" 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/Irrasjonalt_tall" title="Irrasjonalt tall – 书面挪威语" lang="nb" hreflang="nb" data-title="Irrasjonalt tall" 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-or mw-list-item"><a href="https://or.wikipedia.org/wiki/%E0%AC%85%E0%AC%AA%E0%AC%B0%E0%AC%BF%E0%AC%AE%E0%AD%87%E0%AD%9F_%E0%AC%B8%E0%AC%82%E0%AC%96%E0%AD%8D%E0%AD%9F%E0%AC%BE" title="ଅପରିମେୟ ସଂଖ୍ୟା – 奥里亚语" lang="or" hreflang="or" data-title="ଅପରିମେୟ ସଂଖ୍ୟା" data-language-autonym="ଓଡ଼ିଆ" data-language-local-name="奥里亚语" class="interlanguage-link-target"><span>ଓଡ଼ିଆ</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%97%E0%A8%BC%E0%A9%88%E0%A8%B0-%E0%A8%AC%E0%A8%9F%E0%A9%87%E0%A8%A8%E0%A9%81%E0%A8%AE%E0%A8%BE_%E0%A8%B8%E0%A9%B0%E0%A8%96%E0%A8%BF%E0%A8%86" title="ਗ਼ੈਰ-ਬਟੇਨੁਮਾ ਸੰਖਿਆ – 旁遮普语" lang="pa" hreflang="pa" data-title="ਗ਼ੈਰ-ਬਟੇਨੁਮਾ ਸੰਖਿਆ" 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/Liczby_niewymierne" title="Liczby niewymierne – 波兰语" lang="pl" hreflang="pl" data-title="Liczby niewymierne" 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-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/N%C3%BAmero_irracional" title="Número irracional – 葡萄牙语" lang="pt" hreflang="pt" data-title="Número irracional" 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/Num%C4%83r_ira%C8%9Bional" title="Număr irațional – 罗马尼亚语" lang="ro" hreflang="ro" data-title="Număr irațional" 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%98%D1%80%D1%80%D0%B0%D1%86%D0%B8%D0%BE%D0%BD%D0%B0%D0%BB%D1%8C%D0%BD%D0%BE%D0%B5_%D1%87%D0%B8%D1%81%D0%BB%D0%BE" title="Иррациональное число – 俄语" lang="ru" hreflang="ru" data-title="Иррациональное число" data-language-autonym="Русский" data-language-local-name="俄语" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/N%C3%B9mmuru_irrazziunali" title="Nùmmuru irrazziunali – 西西里语" lang="scn" hreflang="scn" data-title="Nùmmuru irrazziunali" 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/Iracionalni_broj" title="Iracionalni broj – 塞尔维亚-克罗地亚语" lang="sh" hreflang="sh" data-title="Iracionalni broj" 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-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Irrational_number" title="Irrational number – Simple English" lang="en-simple" hreflang="en-simple" data-title="Irrational number" 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 mw-list-item"><a href="https://sk.wikipedia.org/wiki/Iracion%C3%A1lne_%C4%8D%C3%ADslo" title="Iracionálne číslo – 斯洛伐克语" lang="sk" hreflang="sk" data-title="Iracionálne číslo" 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/Iracionalno_%C5%A1tevilo" title="Iracionalno število – 斯洛文尼亚语" lang="sl" hreflang="sl" data-title="Iracionalno število" 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-smn mw-list-item"><a href="https://smn.wikipedia.org/wiki/Irrationaalloho" title="Irrationaalloho – 伊纳里萨米语" lang="smn" hreflang="smn" data-title="Irrationaalloho" data-language-autonym="Anarâškielâ" data-language-local-name="伊纳里萨米语" class="interlanguage-link-target"><span>Anarâškielâ</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Numrat_irracional%C3%AB" title="Numrat irracionalë – 阿尔巴尼亚语" lang="sq" hreflang="sq" data-title="Numrat irracionalë" 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%98%D1%80%D0%B0%D1%86%D0%B8%D0%BE%D0%BD%D0%B0%D0%BB%D0%B0%D0%BD_%D0%B1%D1%80%D0%BE%D1%98" title="Ирационалан број – 塞尔维亚语" lang="sr" hreflang="sr" data-title="Ирационалан број" 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/Irrationella_tal" title="Irrationella tal – 瑞典语" lang="sv" hreflang="sv" data-title="Irrationella tal" 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/Namba_isiyowiana" title="Namba isiyowiana – 斯瓦希里语" lang="sw" hreflang="sw" data-title="Namba isiyowiana" 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%B5%E0%AE%BF%E0%AE%95%E0%AE%BF%E0%AE%A4%E0%AE%AE%E0%AF%81%E0%AE%B1%E0%AE%BE_%E0%AE%8E%E0%AE%A3%E0%AF%8D" title="விகிதமுறா எண் – 泰米尔语" lang="ta" hreflang="ta" data-title="விகிதமுறா எண்" 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-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%85%E0%B0%A8%E0%B0%BF%E0%B0%B7%E0%B1%8D%E0%B0%AA_%E0%B0%B8%E0%B0%82%E0%B0%96%E0%B1%8D%E0%B0%AF" title="అనిష్ప సంఖ్య – 泰卢固语" lang="te" hreflang="te" data-title="అనిష్ప సంఖ్య" data-language-autonym="తెలుగు" data-language-local-name="泰卢固语" class="interlanguage-link-target"><span>తెలుగు</span></a></li><li class="interlanguage-link interwiki-tg mw-list-item"><a href="https://tg.wikipedia.org/wiki/%D0%90%D0%B4%D0%B0%D0%B4%D2%B3%D0%BE%D0%B8_%D0%B8%D1%80%D1%80%D0%B0%D1%82%D1%81%D0%B8%D0%BE%D0%BD%D0%B0%D0%BB%D3%A3" title="Ададҳои ирратсионалӣ – 塔吉克语" lang="tg" hreflang="tg" data-title="Ададҳои ирратсионалӣ" data-language-autonym="Тоҷикӣ" data-language-local-name="塔吉克语" class="interlanguage-link-target"><span>Тоҷикӣ</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%88%E0%B8%B3%E0%B8%99%E0%B8%A7%E0%B8%99%E0%B8%AD%E0%B8%95%E0%B8%A3%E0%B8%A3%E0%B8%81%E0%B8%A2%E0%B8%B0" title="จำนวนอตรรกยะ – 泰语" lang="th" hreflang="th" data-title="จำนวนอตรรกยะ" data-language-autonym="ไทย" data-language-local-name="泰语" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/%C4%B0rrasyonel_say%C4%B1lar" title="İrrasyonel sayılar – 土耳其语" lang="tr" hreflang="tr" data-title="İrrasyonel sayılar" 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-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%86%D1%80%D1%80%D0%B0%D1%86%D1%96%D0%BE%D0%BD%D0%B0%D0%BB%D1%8C%D0%BD%D0%B5_%D1%87%D0%B8%D1%81%D0%BB%D0%BE" title="Ірраціональне число – 乌克兰语" lang="uk" hreflang="uk" data-title="Ірраціональне число" 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%BA%DB%8C%D8%B1%D9%86%D8%A7%D8%B7%D9%82_%D8%B9%D8%AF%D8%AF" title="غیرناطق عدد – 乌尔都语" lang="ur" hreflang="ur" data-title="غیرناطق عدد" 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-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Irratsional_sonlar" title="Irratsional sonlar – 乌兹别克语" lang="uz" hreflang="uz" data-title="Irratsional sonlar" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="乌兹别克语" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/S%E1%BB%91_v%C3%B4_t%E1%BB%89" title="Số vô tỉ – 越南语" lang="vi" hreflang="vi" data-title="Số vô tỉ" 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-vls mw-list-item"><a href="https://vls.wikipedia.org/wiki/Irrationoale_getalln" title="Irrationoale getalln – 西佛蘭德文" lang="vls" hreflang="vls" data-title="Irrationoale getalln" data-language-autonym="West-Vlams" data-language-local-name="西佛蘭德文" class="interlanguage-link-target"><span>West-Vlams</span></a></li><li class="interlanguage-link interwiki-yo mw-list-item"><a href="https://yo.wikipedia.org/wiki/N%E1%BB%8D%CC%81mb%C3%A0_al%C3%A1%C3%ACn%C3%AD%C3%ACp%C3%ADn" title="Nọ́mbà aláìníìpín – 约鲁巴语" lang="yo" hreflang="yo" data-title="Nọ́mbà aláìníìpín" data-language-autonym="Yorùbá" data-language-local-name="约鲁巴语" class="interlanguage-link-target"><span>Yorùbá</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/B%C3%BB-l%C3%AD-s%C3%B2%CD%98" title="Bû-lí-sò͘ – 闽南语" lang="nan" hreflang="nan" data-title="Bû-lí-sò͘" 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/%E7%84%A1%E7%90%86%E6%95%B8" title="無理數 – 粤语" lang="yue" hreflang="yue" data-title="無理數" 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/Q607728#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/%E7%84%A1%E7%90%86%E6%95%B8" 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:%E7%84%A1%E7%90%86%E6%95%B8" rel="discussion" title="关于此页面的讨论[t]" accesskey="t"><span>讨论</span></a></li> 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class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/60px-Text_document_with_red_question_mark.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/80px-Text_document_with_red_question_mark.svg.png 2x" data-file-width="48" data-file-height="48" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">此條目已<a href="/wiki/Wikipedia:%E5%88%97%E6%98%8E%E4%BE%86%E6%BA%90" class="mw-redirect" title="Wikipedia:列明來源">列出參考文獻</a>，但<b>因為沒有<a href="/wiki/Help:%E8%84%9A%E6%B3%A8" title="Help:脚注">文內引註</a>而使來源仍然不明</b>。<span class="hide-when-compact"></span> <small class="date-container"><i>(<span class="date">2021年3月26日</span>)</i></small><span class="hide-when-compact"><br /><small>请加上合适的文內引註来<a class="external text" href="https://zh.wikipedia.org/w/index.php?title=%E7%84%A1%E7%90%86%E6%95%B8&action=edit">改善这篇条目</a>。</small></span><span class="hide-when-compact"></span></div></td></tr></tbody></table> <table class="floatright toc" style="width:260px; margin: 0 0 1em 1em"> <tbody><tr style="background:#ccccff" align="center"> <td style="border-bottom: 2px solid #303060"><b>各种各样的<a href="/wiki/%E6%95%B0" title="数">数</a></b> </td></tr> <tr style="background:#ccccff" align="center"> <td><b>基本</b> </td></tr> <tr align="center"> <td> <p><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 \mathbb {N} \subseteq \mathbb {Z} \subseteq \mathbb {Q} \subseteq \mathbb {R} \subseteq \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo>⊆</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>⊆</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo>⊆</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>⊆</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} \subseteq \mathbb {Z} \subseteq \mathbb {Q} \subseteq \mathbb {R} \subseteq \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ba5a9ecbc18a9d9c1b0af89662b4452b7e9c0a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.787ex; height:2.509ex;" alt="{\displaystyle \mathbb {N} \subseteq \mathbb {Z} \subseteq \mathbb {Q} \subseteq \mathbb {R} \subseteq \mathbb {C} }"></span> <span class="skin-invert" typeof="mw:File"><a href="/wiki/File:NumberSetinC.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a0/NumberSetinC.svg/250px-NumberSetinC.svg.png" decoding="async" width="250" height="196" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a0/NumberSetinC.svg/375px-NumberSetinC.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a0/NumberSetinC.svg/500px-NumberSetinC.svg.png 2x" data-file-width="600" data-file-height="470" /></a></span> <style data-mw-deduplicate="TemplateStyles:r82553231">@media all and (max-width:720px){.mw-parser-output table.multicol>tr>td,.mw-parser-output table.multicol>tbody>tr>td{display:block!important;width:100%!important;padding:0!important}}.mw-parser-output table.multicol{border:0;border-collapse:collapse;background-color:transparent;color:inherit;padding:0}.mw-parser-output table.multicol>tr>td,.mw-parser-output table.multicol>tbody>tr>td{vertical-align:top!important}</style> </p> <div>   <table class="multicol" role="presentation" style="width:100%;"><tbody><tr> <td style="width: 50%;text-align: left;vertical-align: top;"> <p><a href="/wiki/%E6%AD%A3%E6%95%B8" title="正數">正數</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 \mathbb {R} ^{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{+}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97dc5e850d079061c24290bac160c8d3b62ee139" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.189ex; height:2.509ex;" alt="{\displaystyle \mathbb {R} ^{+}}"></span><br /> <a href="/wiki/%E8%87%AA%E7%84%B6%E6%95%B0" title="自然数">自然数</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 \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} }"></span><br /> <a href="/wiki/%E8%87%AA%E7%84%B6%E6%95%B0" title="自然数">正整數</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 \mathbb {Z} ^{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} ^{+}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/628778fcf14bd3629e9b9ebacffa172b0ad6ce41" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.061ex; height:2.509ex;" alt="{\displaystyle \mathbb {Z} ^{+}}"></span><br /> <a href="/wiki/%E5%B0%8F%E6%95%B0" title="小数">小数</a><br /> <a href="/wiki/%E6%9C%89%E9%99%90%E5%B0%8F%E6%95%B0" title="有限小数">有限小数</a><br /> <a href="/wiki/%E6%97%A0%E9%99%90%E5%B0%8F%E6%95%B0" title="无限小数">无限小数</a><br /> <a href="/wiki/%E5%BE%AA%E7%8E%AF%E5%B0%8F%E6%95%B0" title="循环小数">循环小数</a><br /> <a href="/wiki/%E6%9C%89%E7%90%86%E6%95%B0" title="有理数">有理数</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 \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }"></span><br /> <a href="/wiki/%E4%BB%A3%E6%95%B8%E6%95%B8" title="代數數">代數數</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 \mathbb {A} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {A} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3fb423c16a5f403edbaf66438b75e7a36e725af6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {A} }"></span><br /> <a href="/wiki/%E5%AE%9E%E6%95%B0" title="实数">实数</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 \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span><br /> <a href="/wiki/%E5%A4%8D%E6%95%B0_(%E6%95%B0%E5%AD%A6)" title="复数 (数学)">複數</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 \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }"></span><br /> <a href="/wiki/%E9%AB%98%E6%96%AF%E6%95%B4%E6%95%B8" title="高斯整數">高斯整數</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 \mathbb {Z} [i]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo stretchy="false">[</mo> <mi>i</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} [i]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ffa94e9e2e6d9e5e5373d5fafb954b902743fde" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.646ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} [i]}"></span><br />   </p> </td> <td style="width: 50%;text-align: left;vertical-align: top;"> <p><a href="/wiki/%E8%B4%9F%E6%95%B0" title="负数">负数</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 \mathbb {R} ^{-}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{-}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/158001a03e958f49f5885033776a420fc47b7267" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.189ex; height:2.509ex;" alt="{\displaystyle \mathbb {R} ^{-}}"></span><br /> <a href="/wiki/%E6%95%B4%E6%95%B0" title="整数">整数</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 \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></span><br /> <a href="/wiki/%E8%B2%A0%E6%95%B4%E6%95%B8" title="負整數">负整數</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 \mathbb {Z} ^{-}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} ^{-}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d265f6ad41c1623a6477b2cb4336208c7b6c1d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.061ex; height:2.509ex;" alt="{\displaystyle \mathbb {Z} ^{-}}"></span><br /> <a href="/wiki/%E5%88%86%E6%95%B8" title="分數">分數</a><br /> <a href="/wiki/%E5%96%AE%E4%BD%8D%E5%88%86%E6%95%B8" title="單位分數">單位分數</a><br /> <a href="/wiki/%E4%BA%8C%E8%BF%9B%E5%88%86%E6%95%B0" title="二进分数">二进分数</a><br /> <a href="/wiki/%E8%A6%8F%E7%9F%A9%E6%95%B8" title="規矩數">規矩數</a><br /> <a class="mw-selflink selflink">無理數</a><br /> <a href="/wiki/%E8%B6%85%E8%B6%8A%E6%95%B8" title="超越數">超越數</a><br /> <a href="/wiki/%E8%99%9A%E6%95%B0" title="虚数">虚数</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 \mathbb {I} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">I</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {I} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8205f06e0d279689ed04a1ac04a3d9c249c637df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.905ex; height:2.176ex;" alt="{\displaystyle \mathbb {I} }"></span><br /> <a href="/wiki/%E4%BA%8C%E6%AC%A1%E7%84%A1%E7%90%86%E6%95%B8" title="二次無理數">二次無理數</a><br /> <a href="/wiki/%E8%89%BE%E6%A3%AE%E6%96%AF%E5%9D%A6%E6%95%B4%E6%95%B0" title="艾森斯坦整数">艾森斯坦整数</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 \mathbb {Z} [\omega ]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo stretchy="false">[</mo> <mi>ω</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} [\omega ]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ae955a9a0d0f342fc73aaafe28af604d23267f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.29ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} [\omega ]}"></span><br />   </p> </td></tr></tbody></table></div> </td></tr> <tr style="background:#ccccff" align="center"> <td><b>延伸</b> </td></tr> <tr align="center"> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r82553231"> <div>   <table class="multicol" role="presentation" style="width:100%;"><tbody><tr> <td style="width: 50%;text-align: left;vertical-align: top;"> <p><a href="/wiki/%E4%BA%8C%E5%85%83%E6%95%B0" title="二元数">二元数</a><br /> <a href="/wiki/%E5%9B%9B%E5%85%83%E6%95%B8" title="四元數">四元數</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 \mathbb {H} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">H</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {H} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e050965453c42bcc6bd544546703c836bdafeac9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \mathbb {H} }"></span><br /> <a href="/wiki/%E5%85%AB%E5%85%83%E6%95%B0" title="八元数">八元数</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 \mathbb {O} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">O</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {O} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1ed2664a4fe515e6fbed25a7193ce663b82920c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \mathbb {O} }"></span><br /> <a href="/wiki/%E5%8D%81%E5%85%AD%E5%85%83%E6%95%B8" title="十六元數">十六元數</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 \mathbb {S} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">S</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {S} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f9d5874c5d7f68eba1cec9da9ccbe53903303bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.293ex; height:2.176ex;" alt="{\displaystyle \mathbb {S} }"></span><br /> <a href="/wiki/%E8%B6%85%E5%AE%9E%E6%95%B0_(%E9%9D%9E%E6%A0%87%E5%87%86%E5%88%86%E6%9E%90)" title="超实数 (非标准分析)">超實數</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 ^{*}\mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ^{*}\mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4df367c74a9138be68469102a92d2fa8cbc15f6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.343ex;" alt="{\displaystyle ^{*}\mathbb {R} }"></span><br /> <a href="/w/index.php?title=%E5%A4%A7%E5%AF%A6%E6%95%B8&action=edit&redlink=1" class="new" title="大實數（页面不存在）">大實數</a><br /> <a href="/w/index.php?title=%E4%B8%8A%E8%B6%85%E5%AF%A6%E6%95%B8&action=edit&redlink=1" class="new" title="上超實數（页面不存在）">上超實數</a><br />   </p> </td> <td style="width: 50%;text-align: left;vertical-align: top;"> <p><a href="/wiki/%E9%9B%99%E6%9B%B2%E8%A4%87%E6%95%B8" title="雙曲複數">雙曲複數</a><br /> <a href="/wiki/%E9%9B%99%E8%A4%87%E6%95%B8" title="雙複數">雙複數</a><br /> <a href="/wiki/%E8%A4%87%E5%9B%9B%E5%85%83%E6%95%B8" title="複四元數">複四元數</a><br /> <span class="ilh-all" data-orig-title="共四元數" data-lang-code="en" data-lang-name="英语" data-foreign-title="Dual quaternion"><span class="ilh-page"><a href="/w/index.php?title=%E5%85%B1%E5%9B%9B%E5%85%83%E6%95%B8&action=edit&redlink=1" class="new" title="共四元數（页面不存在）">共四元數</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/Dual_quaternion" class="extiw" title="en:Dual quaternion"><span lang="en" dir="auto">Dual quaternion</span></a></span>）</span></span><br /> <a href="/wiki/%E8%B6%85%E5%A4%8D%E6%95%B0" title="超复数">超复数</a><br /> <a href="/w/index.php?title=%E8%B6%85%E6%95%B8&action=edit&redlink=1" class="new" title="超數（页面不存在）">超數</a><br /> <a href="/wiki/%E8%B6%85%E7%8F%BE%E5%AF%A6%E6%95%B8" title="超現實數">超現實數</a><br />   </p> </td></tr></tbody></table></div> </td></tr> <tr style="background:#ccccff" align="center"> <td><b>其他</b> </td></tr> <tr align="center"> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r82553231"> <div>   <table class="multicol" role="presentation" style="width:100%;"><tbody><tr> <td style="width: 50%;text-align: left;vertical-align: top;"> <p><a href="/wiki/%E8%B4%A8%E6%95%B0" title="质数">質數</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 \mathbb {P} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {P} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1053af9e662ceaf56c4455f90e0f67273422eded" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.42ex; height:2.176ex;" alt="{\displaystyle \mathbb {P} }"></span><br /> <a href="/wiki/%E5%8F%AF%E8%A8%88%E7%AE%97%E6%95%B8" title="可計算數">可計算數</a><br /> <a href="/wiki/%E5%9F%BA%E6%95%B0_(%E6%95%B0%E5%AD%A6)" title="基数 (数学)">基數</a><br /> <a href="/wiki/%E9%98%BF%E5%88%97%E5%A4%AB%E6%95%B8" title="阿列夫數">阿列夫數</a><br /> <a href="/wiki/%E5%90%8C%E9%A4%98" title="同餘">同餘</a><br /> <a href="/wiki/%E6%95%B4%E6%95%B8%E6%95%B8%E5%88%97" title="整數數列">整數數列</a><br /> <a href="/w/index.php?title=%E5%85%AC%E7%A8%B1%E5%80%BC&action=edit&redlink=1" class="new" title="公稱值（页面不存在）">公稱值</a><br />   </p> </td> <td style="width: 50%;text-align: left;vertical-align: top;"> <p><a href="/wiki/%E8%A6%8F%E7%9F%A9%E6%95%B8" title="規矩數">規矩數</a><br /> <a href="/wiki/%E5%8F%AF%E5%AE%9A%E4%B9%89%E6%95%B0" title="可定义数">可定义数</a><br /> <a href="/wiki/%E5%BA%8F%E6%95%B0" title="序数">序数</a><br /> <a href="/wiki/%E8%B6%85%E9%99%90%E6%95%B0" title="超限数">超限数</a><br /> <a href="/wiki/P%E9%80%B2%E6%95%B8" title="P進數"><style data-mw-deduplicate="TemplateStyles:r58896141">'"`UNIQ--templatestyles-0000001B-QINU`"'</style><span class="serif"><span class="texhtml"><i>p</i></span></span>進數</a><br /> <a href="/wiki/%E6%95%B0%E5%AD%A6%E5%B8%B8%E6%95%B0" title="数学常数">数学常数</a><br />   </p> </td></tr></tbody></table></div> <p><a href="/wiki/%E5%9C%93%E5%91%A8%E7%8E%87" title="圓周率">圓周率</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 \pi =3.14159265}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo>=</mo> <mn>3.14159265</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi =3.14159265}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d789e69af1e86cd0404c764423ff0c104108f1c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.539ex; height:2.176ex;" alt="{\displaystyle \pi =3.14159265}"></span>…<br /> <a href="/wiki/E_(%E6%95%B0%E5%AD%A6%E5%B8%B8%E6%95%B0)" title="E (数学常数)">自然對數的底</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 e=2.718281828}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mo>=</mo> <mn>2.718281828</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e=2.718281828}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1855c2b5b2d2768a0b909ffbc16cf9a18bb11845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:16.453ex; height:2.176ex;" alt="{\displaystyle e=2.718281828}"></span>…<br /> <a href="/wiki/%E8%99%9B%E6%95%B8%E5%96%AE%E4%BD%8D" title="虛數單位">虛數單位</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 i={\sqrt {-{1}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i={\sqrt {-{1}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b13bc2b4f7e103e92342633692e46d585913f342" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.807ex; height:3.009ex;" alt="{\displaystyle i={\sqrt {-{1}}}}"></span><br /> <a href="/wiki/%E6%97%A0%E7%A9%B7" title="无穷">無限大</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 \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c26c105004f30c27aa7c2a9c601550a4183b1f21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.676ex;" alt="{\displaystyle \infty }"></span> </p> </td></tr> <tr> <td align="right"><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: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:%E6%95%B8" title="Template:數"><abbr title="查看该模板">查</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:%E6%95%B8" title="Template talk:數"><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:%E6%95%B8" title="Special:编辑页面/Template:數"><abbr title="编辑该模板">编</abbr></a></li></ul></div> </td></tr></tbody></table> <p><b>無理數</b>（irrational number）是指<a href="/wiki/%E6%9C%89%E7%90%86%E6%95%B0" title="有理数">有理数</a>以外的<a href="/wiki/%E5%AE%9E%E6%95%B0" title="实数">实数</a>，當中的「理」字来自于<a href="/wiki/%E6%8B%89%E4%B8%81%E8%AF%AD" title="拉丁语">拉丁语</a>的rationalis，意思是「理解」，实际是拉丁文对于logos「说明」的翻译，是指无法用两<a href="/wiki/%E6%95%B4%E6%95%B0" title="整数">整数</a>之比来说明的无理数。 </p><p>非<a href="/wiki/%E6%9C%89%E7%90%86%E6%95%B8" class="mw-redirect" title="有理數">有理數</a>之<a href="/wiki/%E5%AF%A6%E6%95%B8" class="mw-redirect" title="實數">實數</a>不能寫作兩整數之比。若將它寫成<a href="/wiki/%E5%B0%8F%E6%95%B8" class="mw-redirect" title="小數">小數</a>形式，小數點後有無限多<a href="/wiki/%E6%95%B0%E4%BD%8D_(%E6%95%B0%E5%AD%A6)" title="数位 (数学)">位</a>，並且不會循環，即无限不循环小数（任何有限或无限循环小数可表示成两整数的比）。常見無理數有大部分的<a href="/wiki/%E5%B9%B3%E6%96%B9%E6%A0%B9" title="平方根">平方根</a>、<a href="/wiki/%E5%9C%93%E5%91%A8%E7%8E%87" title="圓周率">π</a>和<a href="/wiki/E_(%E6%95%B0%E5%AD%A6%E5%B8%B8%E6%95%B0)" title="E (数学常数)">e</a>（後兩者同時為<a href="/wiki/%E8%B6%85%E8%B6%8A%E6%95%B8" title="超越數">超越數</a>）等。無理數另一特徵是無限的<a href="/wiki/%E9%80%A3%E5%88%86%E6%95%B8" class="mw-redirect" title="連分數">連分數</a><a href="/wiki/%E8%A1%A8%E9%81%94%E5%BC%8F" title="表達式">表達式</a>。 </p><p><a href="/wiki/%E5%82%B3%E8%AA%AA" class="mw-redirect" title="傳說">傳說</a>中，无理数最早由<a href="/wiki/%E7%95%A2%E9%81%94%E5%93%A5%E6%8B%89%E6%96%AF" class="mw-redirect" title="畢達哥拉斯">畢達哥拉斯</a>學派弟子<a href="/wiki/%E5%B8%8C%E4%BC%AF%E6%96%AF" class="mw-redirect" title="希伯斯">希伯斯</a>发现，他以幾何方法證明<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 {\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>無法用<a href="/wiki/%E6%95%B4%E6%95%B0" title="整数">整数</a>及<a href="/wiki/%E5%88%86%E6%95%B8" title="分數">分數</a>表示；而畢達哥拉斯深信任意数均可用整数及分数表示，不相信無理數存在，後來希伯斯触犯学派章程，将无理数透露给外人，因而被扔进海中处死，其罪名竟然等同于“渎神”。另見<a href="/wiki/%E7%AC%AC%E4%B8%80%E6%AC%A1%E6%95%B8%E5%AD%B8%E5%8D%B1%E6%A9%9F" title="第一次數學危機">第一次數學危機</a>。 </p><p>無理數可以通過有理數的<a href="/wiki/%E5%88%86%E5%88%92" class="mw-redirect mw-disambig" title="分划">分划</a>的概念來定義。 </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="举例"><span id=".E4.B8.BE.E4.BE.8B"></span>举例</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%84%A1%E7%90%86%E6%95%B8&action=edit&section=1" title="编辑章节：举例"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <ol><li><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 {3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b19c09494138b5082459afac7f9a8d99c546fcd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:2.843ex;" alt="{\displaystyle {\sqrt {3}}}"></span>＝1.73205080…</li> <li><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 \log _{10}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{10}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e1be2659542850a5a4def5da1ca2bb402fbb4e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.848ex; height:2.676ex;" alt="{\displaystyle \log _{10}}"></span>3＝0.47712125…</li> <li>e＝2.71828182845904523536…</li> <li>sin 45°＝<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="{\textstyle {\frac {\sqrt {2}}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>2</mn> </msqrt> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\frac {\sqrt {2}}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b4ed07b85acbae37e000a62a14bdb10f70d7d4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.027ex; height:4.176ex;" alt="{\textstyle {\frac {\sqrt {2}}{2}}}"></span>＝0.70710678…</li> <li>π＝3.141592653589793238462…</li></ol> <div class="mw-heading mw-heading2"><h2 id="性质"><span id=".E6.80.A7.E8.B4.A8"></span>性质</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%84%A1%E7%90%86%E6%95%B8&action=edit&section=2" title="编辑章节：性质"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>无理数加或减无理数不一定得无理数，如<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 \log _{10}2+\log _{10}5=\log _{10}10=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mn>2</mn> <mo>+</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mn>5</mn> <mo>=</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mn>10</mn> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{10}2+\log _{10}5=\log _{10}10=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec61b31e697d95cfa66203840e5886eb9e9051b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.555ex; height:2.676ex;" alt="{\displaystyle \log _{10}2+\log _{10}5=\log _{10}10=1}"></span>。</li> <li>无理数乘不等于0的有理数必得无理数。</li> <li>无理数的<a href="/wiki/%E5%B9%B3%E6%96%B9%E6%A0%B9" title="平方根">平方根</a>、<a href="/wiki/%E7%AB%8B%E6%96%B9%E6%A0%B9" title="立方根">立方根</a>等次方根必得无理数。</li></ul> <div class="mw-heading mw-heading2"><h2 id="不知是否是無理數的數"><span id=".E4.B8.8D.E7.9F.A5.E6.98.AF.E5.90.A6.E6.98.AF.E7.84.A1.E7.90.86.E6.95.B8.E7.9A.84.E6.95.B8"></span>不知是否是無理數的數</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%84%A1%E7%90%86%E6%95%B8&action=edit&section=3" title="编辑章节：不知是否是無理數的數"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>π＋e、π－e等，事实上，對于任何非零整數<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 m\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37129e832c2c81b9f146dd22228d409bd099b295" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.427ex; height:1.676ex;" alt="{\displaystyle m\,}"></span>及<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> <mspace width="thinmathspace" /> </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/205e33e6845813cc72ca346b896a7945f90ca373" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.782ex; height:1.676ex;" alt="{\displaystyle n\,}"></span>，不知道<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 m\pi +ne\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mi>π</mi> <mo>+</mo> <mi>n</mi> <mi>e</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\pi +ne\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f42ed77e562b47afeb45200d1b7c497a27f0fc86" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.078ex; height:2.176ex;" alt="{\displaystyle m\pi +ne\,}"></span>是否無理數。 </p><p>無理數與無理數的<a href="/wiki/%E5%9B%9B%E5%88%99%E8%BF%90%E7%AE%97" title="四则运算">四則運算</a>的結果往往不知道是否無理數，只有π－π＝0、<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 {3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}+{\sqrt {3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30e4e4dd36596db46670acf71aab9ca89616ae05" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.037ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}+{\sqrt {3}}}"></span>等除外。 </p><p>我們亦不知道<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 2^{e}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msup> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{e}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb9caabba16c459c44efd52343b238e89a30a086" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.548ex; height:2.343ex;" alt="{\displaystyle 2^{e}\,}"></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 \pi ^{e}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msup> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi ^{e}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98859147900026b613e608be505afe370c1fdbcb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.72ex; height:2.343ex;" alt="{\displaystyle \pi ^{e}\,}"></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 \pi ^{\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi ^{\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1bf856484c8277fb3aec5271829d51c32de27ca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.757ex; height:3.009ex;" alt="{\displaystyle \pi ^{\sqrt {2}}}"></span>、<a href="/wiki/%E6%AC%A7%E6%8B%89-%E9%A9%AC%E6%AD%87%E7%BD%97%E5%B0%BC%E5%B8%B8%E6%95%B0" class="mw-redirect" title="欧拉-马歇罗尼常数">欧拉－马歇罗尼常数</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 \gamma \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma \,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65da7961fee8269d576e5d06e838bf8695fc5179" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.649ex; height:2.176ex;" alt="{\displaystyle \gamma \,}"></span>、<a href="/wiki/%E5%8D%A1%E5%A1%94%E5%85%B0%E5%B8%B8%E6%95%B0" title="卡塔兰常数">卡塔兰常数</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 G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span>和<a href="/wiki/%E8%B2%BB%E6%A0%B9%E9%AE%91%E5%A7%86%E5%B8%B8%E6%95%B8" title="費根鮑姆常數">费根鲍姆常数</a>是否無理數。 </p> <div class="mw-heading mw-heading2"><h2 id="無理數集的特性"><span id=".E7.84.A1.E7.90.86.E6.95.B8.E9.9B.86.E7.9A.84.E7.89.B9.E6.80.A7"></span>無理數集的特性</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%84%A1%E7%90%86%E6%95%B8&action=edit&section=4" title="编辑章节：無理數集的特性"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>無理數集是<a href="/wiki/%E4%B8%8D%E5%8F%AF%E6%95%B8%E9%9B%86" title="不可數集">不可數集</a>（有理數集是<a href="/wiki/%E5%8F%AF%E6%95%B8%E9%9B%86" title="可數集">可數集</a>而實數集是不可數集）。無理數集是不<a href="/wiki/%E5%AE%8C%E5%A4%87" class="mw-redirect" title="完备">完備</a>的<a href="/wiki/%E6%8B%93%E6%89%91%E7%A9%BA%E9%97%B4" title="拓扑空间">拓撲空間</a>，它與所有<a href="/wiki/%E6%AD%A3%E6%95%B8" title="正數">正數</a><a href="/wiki/%E6%95%B0%E5%88%97" title="数列">數列</a>的集<a href="/w/index.php?title=%E6%8B%93%E6%92%B2%E5%90%8C%E6%A7%8B&action=edit&redlink=1" class="new" title="拓撲同構（页面不存在）">拓撲同構</a>，當中的同構<a href="/wiki/%E6%98%A0%E5%B0%84" title="映射">映射</a>是無理數的<a href="/wiki/%E9%80%A3%E5%88%86%E6%95%B8" class="mw-redirect" title="連分數">連分數</a>開展，因而<a href="/wiki/%E8%B4%9D%E5%B0%94%E7%BA%B2%E5%AE%9A%E7%90%86" title="贝尔纲定理">贝尔纲定理</a>可應用於無數間的拓撲空間。 </p> <div class="mw-heading mw-heading2"><h2 id="無理化作連分數的表達式"><span id=".E7.84.A1.E7.90.86.E5.8C.96.E4.BD.9C.E9.80.A3.E5.88.86.E6.95.B8.E7.9A.84.E8.A1.A8.E9.81.94.E5.BC.8F"></span>無理化作連分數的表達式</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%84%A1%E7%90%86%E6%95%B8&action=edit&section=5" title="编辑章节：無理化作連分數的表達式"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <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 x^{2}=c\qquad (c>0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>c</mi> <mspace width="2em" /> <mo stretchy="false">(</mo> <mi>c</mi> <mo>></mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}=c\qquad (c>0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2619afbad2db923fdb169cd09655962520b5d7ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.211ex; height:3.176ex;" alt="{\displaystyle x^{2}=c\qquad (c>0)}"></span>，</dd></dl> <p>選取正實數<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 \rho \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho \,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1d651c28959a0f15127c097ff4488b123d9e708" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.589ex; height:2.176ex;" alt="{\displaystyle \rho \,}"></span>使 </p> <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 \rho ^{2}<c\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo><</mo> <mi>c</mi> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho ^{2}<c\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92efe14527bc1876558d322d340aba56dff32fdd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.38ex; width:6.355ex; height:3.176ex;" alt="{\displaystyle \rho ^{2}<c\!}"></span>。</dd></dl> <p>經由遞迴處理 </p> <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 {\begin{aligned}x^{2}\ -\!\rho ^{2}&=c\ -\!\rho ^{2}\\(x\ -\!\rho )(x\ +\!\rho )&=c\ -\!\rho ^{2}\\x\ -\!\rho &={\frac {c\ -\!\rho ^{2}}{\rho \ +\!x}}\\x&=\rho \ +\!{\frac {c\ -\!\rho ^{2}}{\rho \ +\!x}}\\&=\rho \ +\!{\cfrac {c\ -\!\rho ^{2}}{\rho \ +\!\left(\rho \ +\!{\cfrac {c\ -\!\rho ^{2}}{\rho \ +\!x}}\right)}}\\&=\rho \ +\!{\cfrac {c\ -\!\rho ^{2}}{2\rho \ +\!{\cfrac {c\ -\!\rho ^{2}}{2\rho \ +\!{\cfrac {c\ -\!\rho ^{2}}{\ddots \,}}}}}}={\sqrt {c}}\,\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mtext> </mtext> <mo>−</mo> <mspace width="negativethinmathspace" /> <msup> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>c</mi> <mtext> </mtext> <mo>−</mo> <mspace width="negativethinmathspace" /> <msup> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mo stretchy="false">(</mo> <mi>x</mi> <mtext> </mtext> <mo>−</mo> <mspace width="negativethinmathspace" /> <mi>ρ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mtext> </mtext> <mo>+</mo> <mspace width="negativethinmathspace" /> <mi>ρ</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>c</mi> <mtext> </mtext> <mo>−</mo> <mspace width="negativethinmathspace" /> <msup> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mi>x</mi> <mtext> </mtext> <mo>−</mo> <mspace width="negativethinmathspace" /> <mi>ρ</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>c</mi> <mtext> </mtext> <mo>−</mo> <mspace width="negativethinmathspace" /> <msup> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mi>ρ</mi> <mtext> </mtext> <mo>+</mo> <mspace width="negativethinmathspace" /> <mi>x</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>x</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>ρ</mi> <mtext> </mtext> <mo>+</mo> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>c</mi> <mtext> </mtext> <mo>−</mo> <mspace width="negativethinmathspace" /> <msup> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mi>ρ</mi> <mtext> </mtext> <mo>+</mo> <mspace width="negativethinmathspace" /> <mi>x</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>ρ</mi> <mtext> </mtext> <mo>+</mo> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mtext> </mtext> <mo>−</mo> <mspace width="negativethinmathspace" /> <msup> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>ρ</mi> <mtext> </mtext> <mo>+</mo> <mspace width="negativethinmathspace" /> <mrow> <mo>(</mo> <mrow> <mi>ρ</mi> <mtext> </mtext> <mo>+</mo> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mtext> </mtext> <mo>−</mo> <mspace width="negativethinmathspace" /> <msup> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>ρ</mi> <mtext> </mtext> <mo>+</mo> <mspace width="negativethinmathspace" /> <mi>x</mi> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>ρ</mi> <mtext> </mtext> <mo>+</mo> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mtext> </mtext> <mo>−</mo> <mspace width="negativethinmathspace" /> <msup> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>ρ</mi> <mtext> </mtext> <mo>+</mo> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mtext> </mtext> <mo>−</mo> <mspace width="negativethinmathspace" /> <msup> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>ρ</mi> <mtext> </mtext> <mo>+</mo> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mtext> </mtext> <mo>−</mo> <mspace width="negativethinmathspace" /> <msup> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo>⋱</mo> <mspace width="thinmathspace" /> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>c</mi> </msqrt> </mrow> <mspace width="thinmathspace" /> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}x^{2}\ -\!\rho ^{2}&=c\ -\!\rho ^{2}\\(x\ -\!\rho )(x\ +\!\rho )&=c\ -\!\rho ^{2}\\x\ -\!\rho &={\frac {c\ -\!\rho ^{2}}{\rho \ +\!x}}\\x&=\rho \ +\!{\frac {c\ -\!\rho ^{2}}{\rho \ +\!x}}\\&=\rho \ +\!{\cfrac {c\ -\!\rho ^{2}}{\rho \ +\!\left(\rho \ +\!{\cfrac {c\ -\!\rho ^{2}}{\rho \ +\!x}}\right)}}\\&=\rho \ +\!{\cfrac {c\ -\!\rho ^{2}}{2\rho \ +\!{\cfrac {c\ -\!\rho ^{2}}{2\rho \ +\!{\cfrac {c\ -\!\rho ^{2}}{\ddots \,}}}}}}={\sqrt {c}}\,\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02d71cedbdafaf3072ea9ae6f6d99d121e55968d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -22.649ex; margin-bottom: -0.189ex; width:48.866ex; height:46.843ex;" alt="{\displaystyle {\begin{aligned}x^{2}\ -\!\rho ^{2}&=c\ -\!\rho ^{2}\\(x\ -\!\rho )(x\ +\!\rho )&=c\ -\!\rho ^{2}\\x\ -\!\rho &={\frac {c\ -\!\rho ^{2}}{\rho \ +\!x}}\\x&=\rho \ +\!{\frac {c\ -\!\rho ^{2}}{\rho \ +\!x}}\\&=\rho \ +\!{\cfrac {c\ -\!\rho ^{2}}{\rho \ +\!\left(\rho \ +\!{\cfrac {c\ -\!\rho ^{2}}{\rho \ +\!x}}\right)}}\\&=\rho \ +\!{\cfrac {c\ -\!\rho ^{2}}{2\rho \ +\!{\cfrac {c\ -\!\rho ^{2}}{2\rho \ +\!{\cfrac {c\ -\!\rho ^{2}}{\ddots \,}}}}}}={\sqrt {c}}\,\end{aligned}}}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="無理數之證"><span id=".E7.84.A1.E7.90.86.E6.95.B8.E4.B9.8B.E8.AD.89"></span>無理數之證</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%84%A1%E7%90%86%E6%95%B8&action=edit&section=6" title="编辑章节：無理數之證"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="證明'"`UNIQ--postMath-0000002A-QINU`"'是无理数"><span id=".E8.AD.89.E6.98.8E.7F.27.22.60UNIQ--postMath-0000002A-QINU.60.22.27.7F.E6.98.AF.E6.97.A0.E7.90.86.E6.95.B0"></span>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 {\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>是无理数</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%84%A1%E7%90%86%E6%95%B8&action=edit&section=7" title="编辑章节：證明'"`UNIQ--postMath-0000002A-QINU`"'是无理数"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>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，且<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}}={\frac {p}{q}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>p</mi> <mi>q</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}={\frac {p}{q}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1e7be04f0750138ae80b3822554a69f9f32c925" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:8.202ex; height:5.343ex;" alt="{\displaystyle {\sqrt {2}}={\frac {p}{q}}}"></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 {\frac {p}{q}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>p</mi> <mi>q</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {p}{q}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9903bc1de26879e5fc4c7f78b54b952bcbb800f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:2.006ex; height:5.343ex;" alt="{\displaystyle {\frac {p}{q}}}"></span>是最简分数。 </p><p>两边平方，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 2={\frac {p^{2}}{q^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2={\frac {p^{2}}{q^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c475d692e2822f2180219a815aca7faea86d73d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:7.321ex; height:6.343ex;" alt="{\displaystyle 2={\frac {p^{2}}{q^{2}}}}"></span>。将此式改写为<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 2q^{2}=p^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2q^{2}=p^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd9d3cd7669a54a0b759112dbacadaa7816bea4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.618ex; height:3.009ex;" alt="{\displaystyle 2q^{2}=p^{2}}"></span>，可见<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^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef685027b97072ee63a8c738f395cd40f63767e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.313ex; height:3.009ex;" alt="{\displaystyle p^{2}}"></span>为偶数。 </p><p>因为平方运算保持奇偶性，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 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>只能为偶数。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 p=2p_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mn>2</mn> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=2p_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d2821629cb5a459e6e20308bae03b43194f56a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:7.743ex; height:2.509ex;" alt="{\displaystyle p=2p_{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 p_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9b58f22283ca46dd5da309cc34303b06a797783" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.313ex; height:2.009ex;" alt="{\displaystyle p_{1}}"></span>为整数。 </p><p>代入可得<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 q^{2}=2p_{1}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>2</mn> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q^{2}=2p_{1}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d8ef176b208755eaa761dbed2f634317ae72345" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.618ex; height:3.343ex;" alt="{\displaystyle q^{2}=2p_{1}^{2}}"></span>。同理可得<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 q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span>亦为偶数。 </p><p>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 {\frac {p}{q}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>p</mi> <mi>q</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {p}{q}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9903bc1de26879e5fc4c7f78b54b952bcbb800f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:2.006ex; height:5.343ex;" alt="{\displaystyle {\frac {p}{q}}}"></span>为最简分数的假设矛盾，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 {\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>是有理数的假设不成立。 </p> <div class="mw-heading mw-heading3"><h3 id="證明'"`UNIQ--postMath-00000038-QINU`"'是无理数"><span id=".E8.AD.89.E6.98.8E.7F.27.22.60UNIQ--postMath-00000038-QINU.60.22.27.7F.E6.98.AF.E6.97.A0.E7.90.86.E6.95.B0"></span>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 {\sqrt {2}}+{\sqrt {3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}+{\sqrt {3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30e4e4dd36596db46670acf71aab9ca89616ae05" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.037ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}+{\sqrt {3}}}"></span>是无理数</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%84%A1%E7%90%86%E6%95%B8&action=edit&section=8" title="编辑章节：證明'"`UNIQ--postMath-00000038-QINU`"'是无理数"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>假設<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 {3}}=p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> <mo>=</mo> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}+{\sqrt {3}}=p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ee160bbb00cdf19d973cb36f0b78ee16372eab7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.305ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}+{\sqrt {3}}=p}"></span>是有理數，兩邊平方得 </p><p><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 5+2{\sqrt {6}}=p^{2}\Rightarrow {\sqrt {6}}={\frac {p^{2}-5}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> <mo>+</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>6</mn> </msqrt> </mrow> <mo>=</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">⇒</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>6</mn> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>5</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5+2{\sqrt {6}}=p^{2}\Rightarrow {\sqrt {6}}={\frac {p^{2}-5}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/024e6b834977209a28b3128f5dd5819a760bc139" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:30.459ex; height:5.676ex;" alt="{\displaystyle 5+2{\sqrt {6}}=p^{2}\Rightarrow {\sqrt {6}}={\frac {p^{2}-5}{2}}}"></span> </p><p>其中因為<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>是有理數，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 {\frac {p^{2}-5}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>5</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {p^{2}-5}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f278da72a036e726e98ac92d6e4a63229a85eb12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.063ex; height:5.676ex;" alt="{\displaystyle {\frac {p^{2}-5}{2}}}"></span>也是有理數。 </p><p>透過證明<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 {\sqrt {a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>a</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afccc332c876539296df1a980127d86173e59ef0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.166ex; height:3.009ex;" alt="{\displaystyle {\sqrt {a}}}"></span>為無理數的方法，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"> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </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/ecaee8e0957dc3d0a7a3c7da3b54def4bcd27062" 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>為一非完全平方数 </p><p>可以證明<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 {\sqrt {6}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>6</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {6}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a857de6bca2591cfad08e4378634825b6be66a01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:2.843ex;" alt="{\displaystyle {\sqrt {6}}}"></span>是無理數 </p><p>同樣也推出<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 {p^{2}-5}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>5</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {p^{2}-5}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f278da72a036e726e98ac92d6e4a63229a85eb12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.063ex; height:5.676ex;" alt="{\displaystyle {\frac {p^{2}-5}{2}}}"></span>是無理數 </p><p>但這又和<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 {\frac {p^{2}-5}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>5</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {p^{2}-5}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f278da72a036e726e98ac92d6e4a63229a85eb12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.063ex; height:5.676ex;" alt="{\displaystyle {\frac {p^{2}-5}{2}}}"></span>是有理數互相矛盾 </p><p>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 {\sqrt {2}}+{\sqrt {3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}+{\sqrt {3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30e4e4dd36596db46670acf71aab9ca89616ae05" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.037ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}+{\sqrt {3}}}"></span>是一無理數 </p> <div class="mw-heading mw-heading3"><h3 id="證明'"`UNIQ--postMath-00000043-QINU`"'是无理数"><span id=".E8.AD.89.E6.98.8E.7F.27.22.60UNIQ--postMath-00000043-QINU.60.22.27.7F.E6.98.AF.E6.97.A0.E7.90.86.E6.95.B0"></span>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 {\sqrt {2}}+{\sqrt {3}}+{\sqrt {5}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}+{\sqrt {3}}+{\sqrt {5}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce20fffa5b955ea2e19a35afb06e091c0f726ef9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.976ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}+{\sqrt {3}}+{\sqrt {5}}}"></span>是无理数</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%84%A1%E7%90%86%E6%95%B8&action=edit&section=9" title="编辑章节：證明'"`UNIQ--postMath-00000043-QINU`"'是无理数"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>證一</b> </p><p>同樣，假設<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 {3}}+{\sqrt {5}}=p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> <mo>=</mo> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}+{\sqrt {3}}+{\sqrt {5}}=p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c8562eb1a5fabe1d1dc3fed51fc585170de6e3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.244ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}+{\sqrt {3}}+{\sqrt {5}}=p}"></span>是有理數，兩邊平方得 </p><p><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 10+2{\sqrt {6}}+2{\sqrt {10}}+2{\sqrt {15}}=p^{2}\Rightarrow {\sqrt {6}}+{\sqrt {10}}+{\sqrt {15}}={\frac {p^{2}-10}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>10</mn> <mo>+</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>6</mn> </msqrt> </mrow> <mo>+</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>10</mn> </msqrt> </mrow> <mo>+</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>15</mn> </msqrt> </mrow> <mo>=</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">⇒</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>6</mn> </msqrt> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>10</mn> </msqrt> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>15</mn> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>10</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 10+2{\sqrt {6}}+2{\sqrt {10}}+2{\sqrt {15}}=p^{2}\Rightarrow {\sqrt {6}}+{\sqrt {10}}+{\sqrt {15}}={\frac {p^{2}-10}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c46a91b444c584d0bda7b1cff973b39c5b7a2c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:63.514ex; height:5.676ex;" alt="{\displaystyle 10+2{\sqrt {6}}+2{\sqrt {10}}+2{\sqrt {15}}=p^{2}\Rightarrow {\sqrt {6}}+{\sqrt {10}}+{\sqrt {15}}={\frac {p^{2}-10}{2}}}"></span>， </p><p>於是<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 {6}}+{\sqrt {10}}+{\sqrt {15}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>6</mn> </msqrt> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>10</mn> </msqrt> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>15</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {6}}+{\sqrt {10}}+{\sqrt {15}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abeb8a9dc209d8904dabaeb358c092fa3b0c63bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.301ex; height:2.843ex;" alt="{\displaystyle {\sqrt {6}}+{\sqrt {10}}+{\sqrt {15}}}"></span>是有理數。兩邊再次平方，得： </p><p><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 31+10{\sqrt {6}}+6{\sqrt {10}}+4{\sqrt {15}}={\frac {(p^{2}-10)^{2}}{4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>31</mn> <mo>+</mo> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>6</mn> </msqrt> </mrow> <mo>+</mo> <mn>6</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>10</mn> </msqrt> </mrow> <mo>+</mo> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>15</mn> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>10</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mn>4</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 31+10{\sqrt {6}}+6{\sqrt {10}}+4{\sqrt {15}}={\frac {(p^{2}-10)^{2}}{4}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79063e2b86cc9d558e3dd51be6638a0fe0c6d937" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:41.303ex; height:5.843ex;" alt="{\displaystyle 31+10{\sqrt {6}}+6{\sqrt {10}}+4{\sqrt {15}}={\frac {(p^{2}-10)^{2}}{4}}}"></span>， </p><p>於是<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 5{\sqrt {6}}+3{\sqrt {10}}+2{\sqrt {15}}={\frac {{\frac {(p^{2}-10)^{2}}{8}}-31}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>6</mn> </msqrt> </mrow> <mo>+</mo> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>10</mn> </msqrt> </mrow> <mo>+</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>15</mn> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>10</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mn>8</mn> </mfrac> </mrow> <mo>−</mo> <mn>31</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5{\sqrt {6}}+3{\sqrt {10}}+2{\sqrt {15}}={\frac {{\frac {(p^{2}-10)^{2}}{8}}-31}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c77b51a97e3c08996a99040294f0a2e91fe43ce4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:37.416ex; height:7.343ex;" alt="{\displaystyle 5{\sqrt {6}}+3{\sqrt {10}}+2{\sqrt {15}}={\frac {{\frac {(p^{2}-10)^{2}}{8}}-31}{2}}}"></span> </p><p>由於<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 {6}}+{\sqrt {10}}+{\sqrt {15}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>6</mn> </msqrt> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>10</mn> </msqrt> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>15</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {6}}+{\sqrt {10}}+{\sqrt {15}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abeb8a9dc209d8904dabaeb358c092fa3b0c63bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.301ex; height:2.843ex;" alt="{\displaystyle {\sqrt {6}}+{\sqrt {10}}+{\sqrt {15}}}"></span>是有理數，所以 </p><p><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 3{\sqrt {6}}+{\sqrt {10}}+2({\sqrt {6}}+{\sqrt {10}}+{\sqrt {15}})={\frac {{\frac {(p^{2}-10)^{2}}{4}}-31}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>6</mn> </msqrt> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>10</mn> </msqrt> </mrow> <mo>+</mo> <mn>2</mn> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>6</mn> </msqrt> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>10</mn> </msqrt> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>15</mn> </msqrt> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>10</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mn>4</mn> </mfrac> </mrow> <mo>−</mo> <mn>31</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3{\sqrt {6}}+{\sqrt {10}}+2({\sqrt {6}}+{\sqrt {10}}+{\sqrt {15}})={\frac {{\frac {(p^{2}-10)^{2}}{4}}-31}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ced647731dc91df691d42ae11a75ba58dfd82866" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:51.103ex; height:7.343ex;" alt="{\displaystyle 3{\sqrt {6}}+{\sqrt {10}}+2({\sqrt {6}}+{\sqrt {10}}+{\sqrt {15}})={\frac {{\frac {(p^{2}-10)^{2}}{4}}-31}{2}}}"></span> </p><p><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 \Rightarrow 3{\sqrt {6}}+{\sqrt {10}}={\frac {{\frac {(p^{2}-10)^{2}}{4}}-31}{2}}-2({\sqrt {6}}+{\sqrt {10}}+{\sqrt {15}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇒</mo> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>6</mn> </msqrt> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>10</mn> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>10</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mn>4</mn> </mfrac> </mrow> <mo>−</mo> <mn>31</mn> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mn>2</mn> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>6</mn> </msqrt> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>10</mn> </msqrt> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>15</mn> </msqrt> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Rightarrow 3{\sqrt {6}}+{\sqrt {10}}={\frac {{\frac {(p^{2}-10)^{2}}{4}}-31}{2}}-2({\sqrt {6}}+{\sqrt {10}}+{\sqrt {15}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8821e306621549e9f37a552cc4dd62b79029758" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:54.072ex; height:7.343ex;" alt="{\displaystyle \Rightarrow 3{\sqrt {6}}+{\sqrt {10}}={\frac {{\frac {(p^{2}-10)^{2}}{4}}-31}{2}}-2({\sqrt {6}}+{\sqrt {10}}+{\sqrt {15}})}"></span> </p><p>透過證明形如<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 {a}}+{\sqrt {b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>a</mi> </msqrt> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>b</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {a}}+{\sqrt {b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13b3f0e835e36b51c9c14af39c2eadd416af8f55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.94ex; height:3.343ex;" alt="{\displaystyle {\sqrt {a}}+{\sqrt {b}}}"></span>的數是無理數的方法，得出<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 3{\sqrt {6}}+{\sqrt {10}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>6</mn> </msqrt> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>10</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3{\sqrt {6}}+{\sqrt {10}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7171f02c3cf0442042c51652394e8fa53eb2107" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.362ex; height:2.843ex;" alt="{\displaystyle 3{\sqrt {6}}+{\sqrt {10}}}"></span>也是一無理數 </p><p>但這結果明顯和<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 {\frac {{\frac {(p^{2}-10)^{2}}{8}}-31}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>10</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mn>8</mn> </mfrac> </mrow> <mo>−</mo> <mn>31</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {{\frac {(p^{2}-10)^{2}}{8}}-31}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb9cce07c895595e386fac59f8bdef4ef5ad6a8b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.529ex; height:7.343ex;" alt="{\displaystyle {\frac {{\frac {(p^{2}-10)^{2}}{8}}-31}{2}}}"></span>與<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 2({\sqrt {6}}+{\sqrt {10}}+{\sqrt {15}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>6</mn> </msqrt> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>10</mn> </msqrt> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>15</mn> </msqrt> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2({\sqrt {6}}+{\sqrt {10}}+{\sqrt {15}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/096e3aef1c90e949d281cd1082914b6322468f5b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.272ex; height:3.009ex;" alt="{\displaystyle 2({\sqrt {6}}+{\sqrt {10}}+{\sqrt {15}})}"></span>皆為有理數出現矛盾，故<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 {3}}+{\sqrt {5}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}+{\sqrt {3}}+{\sqrt {5}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce20fffa5b955ea2e19a35afb06e091c0f726ef9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.976ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}+{\sqrt {3}}+{\sqrt {5}}}"></span>為無理數 </p><p><b>證二</b> </p><p>同樣假設<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 {3}}+{\sqrt {5}}=p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> <mo>=</mo> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}+{\sqrt {3}}+{\sqrt {5}}=p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c8562eb1a5fabe1d1dc3fed51fc585170de6e3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.244ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}+{\sqrt {3}}+{\sqrt {5}}=p}"></span>是有理數， </p><p><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 {3}}+{\sqrt {5}}=p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> <mo>=</mo> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}+{\sqrt {3}}+{\sqrt {5}}=p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c8562eb1a5fabe1d1dc3fed51fc585170de6e3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.244ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}+{\sqrt {3}}+{\sqrt {5}}=p}"></span> </p><p><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 \Rightarrow {\sqrt {2}}+{\sqrt {3}}=p-{\sqrt {5}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇒</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> <mo>=</mo> <mi>p</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Rightarrow {\sqrt {2}}+{\sqrt {3}}=p-{\sqrt {5}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d92a2351b570d079a7d31408c89d169ae769ac80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.213ex; height:3.009ex;" alt="{\displaystyle \Rightarrow {\sqrt {2}}+{\sqrt {3}}=p-{\sqrt {5}}}"></span>，兩邊平方： </p><p><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 \Rightarrow ({\sqrt {2}}+{\sqrt {3}})^{2}=(p-{\sqrt {5}})^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇒</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Rightarrow ({\sqrt {2}}+{\sqrt {3}})^{2}=(p-{\sqrt {5}})^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d891983d3ec8777b401493b676af8ec37e6f9e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.94ex; height:3.176ex;" alt="{\displaystyle \Rightarrow ({\sqrt {2}}+{\sqrt {3}})^{2}=(p-{\sqrt {5}})^{2}}"></span> </p><p><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 \Rightarrow 5+2{\sqrt {6}}=p^{2}+5-2p{\sqrt {5}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇒</mo> <mn>5</mn> <mo>+</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>6</mn> </msqrt> </mrow> <mo>=</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>5</mn> <mo>−</mo> <mn>2</mn> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Rightarrow 5+2{\sqrt {6}}=p^{2}+5-2p{\sqrt {5}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21b6bb8571e5f9de1fd91713a7e3b03ef67b4671" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:28.828ex; height:3.009ex;" alt="{\displaystyle \Rightarrow 5+2{\sqrt {6}}=p^{2}+5-2p{\sqrt {5}}}"></span> </p><p><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 \Rightarrow 2({\sqrt {6}}+p{\sqrt {5}})=p^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇒</mo> <mn>2</mn> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>6</mn> </msqrt> </mrow> <mo>+</mo> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Rightarrow 2({\sqrt {6}}+p{\sqrt {5}})=p^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4112fc23a421c45335f63101011b1608c7185eae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.469ex; height:3.176ex;" alt="{\displaystyle \Rightarrow 2({\sqrt {6}}+p{\sqrt {5}})=p^{2}}"></span> </p><p>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 {\sqrt {a}}+{\sqrt {b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>a</mi> </msqrt> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>b</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {a}}+{\sqrt {b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13b3f0e835e36b51c9c14af39c2eadd416af8f55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.94ex; height:3.343ex;" alt="{\displaystyle {\sqrt {a}}+{\sqrt {b}}}"></span>形式的數是無理數的方法，得出<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 {6}}+p{\sqrt {5}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>6</mn> </msqrt> </mrow> <mo>+</mo> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {6}}+p{\sqrt {5}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a2eff76b67c5c97d5399fa272e1d10e54400cca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.206ex; height:2.843ex;" alt="{\displaystyle {\sqrt {6}}+p{\sqrt {5}}}"></span>是無理數 </p><p>也是矛盾的。 </p> <div class="mw-heading mw-heading3"><h3 id="證明'"`UNIQ--postMath-00000059-QINU`"'是无理数"><span id=".E8.AD.89.E6.98.8E.7F.27.22.60UNIQ--postMath-00000059-QINU.60.22.27.7F.E6.98.AF.E6.97.A0.E7.90.86.E6.95.B0"></span>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 {\sqrt {2}}+{\sqrt {3}}+{\sqrt {5}}+{\sqrt {7}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>7</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}+{\sqrt {3}}+{\sqrt {5}}+{\sqrt {7}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01703b5b0518428b8c412cdc55735510f5d4313c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.915ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}+{\sqrt {3}}+{\sqrt {5}}+{\sqrt {7}}}"></span>是无理数</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%84%A1%E7%90%86%E6%95%B8&action=edit&section=10" title="编辑章节：證明'"`UNIQ--postMath-00000059-QINU`"'是无理数"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><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 {3}}+{\sqrt {5}}+{\sqrt {7}}=p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>7</mn> </msqrt> </mrow> <mo>=</mo> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}+{\sqrt {3}}+{\sqrt {5}}+{\sqrt {7}}=p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce65795fe7018c9004e409f0b438e7d43cc3bf74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:25.182ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}+{\sqrt {3}}+{\sqrt {5}}+{\sqrt {7}}=p}"></span> </p><p><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 \Rightarrow {\sqrt {2}}+{\sqrt {3}}+{\sqrt {5}}=p-{\sqrt {7}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇒</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> <mo>=</mo> <mi>p</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>7</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Rightarrow {\sqrt {2}}+{\sqrt {3}}+{\sqrt {5}}=p-{\sqrt {7}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06aacb113b1b6a146f477f5ab3bf8d837c9a0402" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:28.151ex; height:3.009ex;" alt="{\displaystyle \Rightarrow {\sqrt {2}}+{\sqrt {3}}+{\sqrt {5}}=p-{\sqrt {7}}}"></span>，兩邊平方得 </p><p><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 \Rightarrow 10+2{\sqrt {6}}+2{\sqrt {10}}+2{\sqrt {15}}=p^{2}+7-2p{\sqrt {7}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇒</mo> <mn>10</mn> <mo>+</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>6</mn> </msqrt> </mrow> <mo>+</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>10</mn> </msqrt> </mrow> <mo>+</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>15</mn> </msqrt> </mrow> <mo>=</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>7</mn> <mo>−</mo> <mn>2</mn> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>7</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Rightarrow 10+2{\sqrt {6}}+2{\sqrt {10}}+2{\sqrt {15}}=p^{2}+7-2p{\sqrt {7}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc3d8b69b0bf1c35cc659ce29c60878fe315033f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:46.518ex; height:3.009ex;" alt="{\displaystyle \Rightarrow 10+2{\sqrt {6}}+2{\sqrt {10}}+2{\sqrt {15}}=p^{2}+7-2p{\sqrt {7}}}"></span> </p><p><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 \Rightarrow {\sqrt {6}}+{\sqrt {10}}+{\sqrt {15}}+p{\sqrt {7}}={\frac {p^{2}}{2}}-{\frac {3}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇒</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>6</mn> </msqrt> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>10</mn> </msqrt> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>15</mn> </msqrt> </mrow> <mo>+</mo> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>7</mn> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Rightarrow {\sqrt {6}}+{\sqrt {10}}+{\sqrt {15}}+p{\sqrt {7}}={\frac {p^{2}}{2}}-{\frac {3}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/005e1614be1056be6362af067074375c89a539b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:38.375ex; height:5.676ex;" alt="{\displaystyle \Rightarrow {\sqrt {6}}+{\sqrt {10}}+{\sqrt {15}}+p{\sqrt {7}}={\frac {p^{2}}{2}}-{\frac {3}{2}}}"></span>，得到<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 {6}}+{\sqrt {10}}+{\sqrt {15}}+p{\sqrt {7}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>6</mn> </msqrt> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>10</mn> </msqrt> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>15</mn> </msqrt> </mrow> <mo>+</mo> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>7</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {6}}+{\sqrt {10}}+{\sqrt {15}}+p{\sqrt {7}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c586c4cb0813f8ca3b1094ec2426b1c6710e8dd4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:24.409ex; height:3.009ex;" alt="{\displaystyle {\sqrt {6}}+{\sqrt {10}}+{\sqrt {15}}+p{\sqrt {7}}}"></span>為一有理數 </p><p><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 \Rightarrow {\sqrt {6}}+{\sqrt {10}}+{\sqrt {15}}={\frac {p^{2}}{2}}-{\frac {3}{2}}-p{\sqrt {7}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇒</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>6</mn> </msqrt> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>10</mn> </msqrt> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>15</mn> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>7</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Rightarrow {\sqrt {6}}+{\sqrt {10}}+{\sqrt {15}}={\frac {p^{2}}{2}}-{\frac {3}{2}}-p{\sqrt {7}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c359f9bf9fdaa0639f52328f2b3d6e1dd2b773f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:38.375ex; height:5.676ex;" alt="{\displaystyle \Rightarrow {\sqrt {6}}+{\sqrt {10}}+{\sqrt {15}}={\frac {p^{2}}{2}}-{\frac {3}{2}}-p{\sqrt {7}}}"></span>，兩邊繼續平方： </p><p><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 \Rightarrow \left({\sqrt {6}}+{\sqrt {10}}+{\sqrt {15}}\right)^{2}=\left(p^{2}-{\frac {3}{2}}-p{\sqrt {7}}\right)^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇒</mo> <msup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>6</mn> </msqrt> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>10</mn> </msqrt> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>15</mn> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>7</mn> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Rightarrow \left({\sqrt {6}}+{\sqrt {10}}+{\sqrt {15}}\right)^{2}=\left(p^{2}-{\frac {3}{2}}-p{\sqrt {7}}\right)^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1503ff8c20fc6ff4f5ec46378bb67fc6abf213ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:45.198ex; height:6.509ex;" alt="{\displaystyle \Rightarrow \left({\sqrt {6}}+{\sqrt {10}}+{\sqrt {15}}\right)^{2}=\left(p^{2}-{\frac {3}{2}}-p{\sqrt {7}}\right)^{2}}"></span> </p><p><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 \Rightarrow \left({\sqrt {6}}+{\sqrt {10}}+{\sqrt {15}}\right)^{2}=\left[\left(p^{2}-{\frac {3}{2}}\right)-p{\sqrt {7}}\right]^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇒</mo> <msup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>6</mn> </msqrt> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>10</mn> </msqrt> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>15</mn> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mrow> <mo>[</mo> <mrow> <mrow> <mo>(</mo> <mrow> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>7</mn> </msqrt> </mrow> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Rightarrow \left({\sqrt {6}}+{\sqrt {10}}+{\sqrt {15}}\right)^{2}=\left[\left(p^{2}-{\frac {3}{2}}\right)-p{\sqrt {7}}\right]^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/908ddcae4876aa0183c6675eee3658f9106185db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:47.653ex; height:6.509ex;" alt="{\displaystyle \Rightarrow \left({\sqrt {6}}+{\sqrt {10}}+{\sqrt {15}}\right)^{2}=\left[\left(p^{2}-{\frac {3}{2}}\right)-p{\sqrt {7}}\right]^{2}}"></span> </p><p><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 \Rightarrow 31+2{\sqrt {60}}+2{\sqrt {90}}+2{\sqrt {150}}=\left(p^{2}-{\frac {3}{2}}\right)^{2}+(-p{\sqrt {7}})^{2}-2\times {p}{\sqrt {7}}\times \left(p^{2}-{\frac {3}{2}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇒</mo> <mn>31</mn> <mo>+</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>60</mn> </msqrt> </mrow> <mo>+</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>90</mn> </msqrt> </mrow> <mo>+</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>150</mn> </msqrt> </mrow> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>7</mn> </msqrt> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>7</mn> </msqrt> </mrow> <mo>×</mo> <mrow> <mo>(</mo> <mrow> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Rightarrow 31+2{\sqrt {60}}+2{\sqrt {90}}+2{\sqrt {150}}=\left(p^{2}-{\frac {3}{2}}\right)^{2}+(-p{\sqrt {7}})^{2}-2\times {p}{\sqrt {7}}\times \left(p^{2}-{\frac {3}{2}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1b2a5105232f12428a5cb553595c2c2264041c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:82.099ex; height:6.509ex;" alt="{\displaystyle \Rightarrow 31+2{\sqrt {60}}+2{\sqrt {90}}+2{\sqrt {150}}=\left(p^{2}-{\frac {3}{2}}\right)^{2}+(-p{\sqrt {7}})^{2}-2\times {p}{\sqrt {7}}\times \left(p^{2}-{\frac {3}{2}}\right)}"></span> </p><p><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 \Rightarrow 31+4{\sqrt {15}}+6{\sqrt {10}}+10{\sqrt {6}}=\left(p^{2}-{\frac {3}{2}}\right)^{2}+7p^{2}-p(2p^{2}-3){\sqrt {7}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇒</mo> <mn>31</mn> <mo>+</mo> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>15</mn> </msqrt> </mrow> <mo>+</mo> <mn>6</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>10</mn> </msqrt> </mrow> <mo>+</mo> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>6</mn> </msqrt> </mrow> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>7</mn> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>p</mi> <mo stretchy="false">(</mo> <mn>2</mn> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>7</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Rightarrow 31+4{\sqrt {15}}+6{\sqrt {10}}+10{\sqrt {6}}=\left(p^{2}-{\frac {3}{2}}\right)^{2}+7p^{2}-p(2p^{2}-3){\sqrt {7}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3cd9f559f069e39cacb018bcb269aa93d750fb7f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:67.254ex; height:6.509ex;" alt="{\displaystyle \Rightarrow 31+4{\sqrt {15}}+6{\sqrt {10}}+10{\sqrt {6}}=\left(p^{2}-{\frac {3}{2}}\right)^{2}+7p^{2}-p(2p^{2}-3){\sqrt {7}}}"></span> </p><p><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 \Rightarrow 2{\sqrt {10}}+6{\sqrt {6}}+p(2p^{2}-3){\sqrt {7}}=\left(p^{2}-{\frac {3}{2}}\right)^{2}+7p^{2}-4\left({\sqrt {6}}+{\sqrt {10}}+{\sqrt {15}}+p{\sqrt {7}}\right)-31}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇒</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>10</mn> </msqrt> </mrow> <mo>+</mo> <mn>6</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>6</mn> </msqrt> </mrow> <mo>+</mo> <mi>p</mi> <mo stretchy="false">(</mo> <mn>2</mn> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>7</mn> </msqrt> </mrow> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>7</mn> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>4</mn> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>6</mn> </msqrt> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>10</mn> </msqrt> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>15</mn> </msqrt> </mrow> <mo>+</mo> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>7</mn> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mn>31</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Rightarrow 2{\sqrt {10}}+6{\sqrt {6}}+p(2p^{2}-3){\sqrt {7}}=\left(p^{2}-{\frac {3}{2}}\right)^{2}+7p^{2}-4\left({\sqrt {6}}+{\sqrt {10}}+{\sqrt {15}}+p{\sqrt {7}}\right)-31}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8987de98c0f01893cac67ad09439ab2d1ae59513" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:88.757ex; height:6.509ex;" alt="{\displaystyle \Rightarrow 2{\sqrt {10}}+6{\sqrt {6}}+p(2p^{2}-3){\sqrt {7}}=\left(p^{2}-{\frac {3}{2}}\right)^{2}+7p^{2}-4\left({\sqrt {6}}+{\sqrt {10}}+{\sqrt {15}}+p{\sqrt {7}}\right)-31}"></span> </p><p>由於<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 {6}}+{\sqrt {10}}+{\sqrt {15}}+p{\sqrt {7}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>6</mn> </msqrt> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>10</mn> </msqrt> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>15</mn> </msqrt> </mrow> <mo>+</mo> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>7</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {6}}+{\sqrt {10}}+{\sqrt {15}}+p{\sqrt {7}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c586c4cb0813f8ca3b1094ec2426b1c6710e8dd4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:24.409ex; height:3.009ex;" alt="{\displaystyle {\sqrt {6}}+{\sqrt {10}}+{\sqrt {15}}+p{\sqrt {7}}}"></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 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>皆為有理數 </p><p>設<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 {10}}+6{\sqrt {6}}+p(2p^{2}-3){\sqrt {7}}=q=\left(p^{2}-{\frac {3}{2}}\right)^{2}+7p^{2}-4\left({\sqrt {6}}+{\sqrt {10}}+{\sqrt {15}}+p{\sqrt {7}}\right)-31}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>10</mn> </msqrt> </mrow> <mo>+</mo> <mn>6</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>6</mn> </msqrt> </mrow> <mo>+</mo> <mi>p</mi> <mo stretchy="false">(</mo> <mn>2</mn> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>7</mn> </msqrt> </mrow> <mo>=</mo> <mi>q</mi> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>7</mn> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>4</mn> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>6</mn> </msqrt> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>10</mn> </msqrt> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>15</mn> </msqrt> </mrow> <mo>+</mo> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>7</mn> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mn>31</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {10}}+6{\sqrt {6}}+p(2p^{2}-3){\sqrt {7}}=q=\left(p^{2}-{\frac {3}{2}}\right)^{2}+7p^{2}-4\left({\sqrt {6}}+{\sqrt {10}}+{\sqrt {15}}+p{\sqrt {7}}\right)-31}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/597af4b0eba73292f7e48dbb143d4777d52b4de4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:88.793ex; height:6.509ex;" alt="{\displaystyle {\sqrt {10}}+6{\sqrt {6}}+p(2p^{2}-3){\sqrt {7}}=q=\left(p^{2}-{\frac {3}{2}}\right)^{2}+7p^{2}-4\left({\sqrt {6}}+{\sqrt {10}}+{\sqrt {15}}+p{\sqrt {7}}\right)-31}"></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 q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span>亦為有理數 </p><p>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 {\sqrt {a}}+{\sqrt {b}}+{\sqrt {c}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>a</mi> </msqrt> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>b</mi> </msqrt> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>c</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {a}}+{\sqrt {b}}+{\sqrt {c}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bcf2b7a02b4ae28cd6f3217e5b424afb60fadbaf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.723ex; height:3.343ex;" alt="{\displaystyle {\sqrt {a}}+{\sqrt {b}}+{\sqrt {c}}}"></span>形式的數是無理數的方法可知<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 {10}}+6{\sqrt {6}}+p(2p^{2}-3){\sqrt {7}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>10</mn> </msqrt> </mrow> <mo>+</mo> <mn>6</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>6</mn> </msqrt> </mrow> <mo>+</mo> <mi>p</mi> <mo stretchy="false">(</mo> <mn>2</mn> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>7</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {10}}+6{\sqrt {6}}+p(2p^{2}-3){\sqrt {7}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a13230ebb7c931bb4470826ea621b64504d22ba7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.668ex; height:3.176ex;" alt="{\displaystyle {\sqrt {10}}+6{\sqrt {6}}+p(2p^{2}-3){\sqrt {7}}}"></span>為無理數 </p><p>這和<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 q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span>是有理數衝突 </p><p>所以得證<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 {3}}+{\sqrt {5}}+{\sqrt {7}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>7</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}+{\sqrt {3}}+{\sqrt {5}}+{\sqrt {7}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01703b5b0518428b8c412cdc55735510f5d4313c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.915ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}+{\sqrt {3}}+{\sqrt {5}}+{\sqrt {7}}}"></span>為無理數 </p> <div class="mw-heading mw-heading2"><h2 id="参见"><span id=".E5.8F.82.E8.A7.81"></span>参见</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%84%A1%E7%90%86%E6%95%B8&action=edit&section=11" title="编辑章节：参见"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/%E5%88%86%E5%88%92" class="mw-redirect mw-disambig" title="分划">分划</a></li> <li><a href="/wiki/%E4%B8%A2%E7%95%AA%E5%9B%BE%E9%80%BC%E8%BF%91" class="mw-redirect" title="丢番图逼近">丢番图逼近</a></li> <li><a href="/wiki/3%E7%9A%84%E7%AE%97%E8%A1%93%E5%B9%B3%E6%96%B9%E6%A0%B9" title="3的算術平方根">3的算术平方根</a></li> <li><a href="/wiki/5%E7%9A%84%E7%AE%97%E6%9C%AF%E5%B9%B3%E6%96%B9%E6%A0%B9" class="mw-redirect" title="5的算术平方根">5的算术平方根</a></li> <li><a href="/wiki/%E8%B6%85%E8%B6%8A%E6%95%B0" class="mw-redirect" title="超越数">超越数</a></li> <li><a href="/wiki/%E7%AC%AC%E4%B8%80%E6%AC%A1%E6%95%B8%E5%AD%B8%E5%8D%B1%E6%A9%9F" title="第一次數學危機">第一次數學危機</a></li> <li><a href="/wiki/%E6%AD%A3%E8%A6%8F%E6%95%B8" class="mw-redirect" title="正規數">正規數</a></li></ul> <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=%E7%84%A1%E7%90%86%E6%95%B8&action=edit&section=12" 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_26_02_1/page4.html">從畢氏學派到歐氏幾何的誕生，蔡聰明</a> （<a rel="nofollow" class="external text" href="//web.archive.org/web/20210402044906/http://episte.math.ntu.edu.tw/articles/sm/sm_26_02_1/page4.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>），有畢氏弄石法的證明</li> <li><a rel="nofollow" class="external text" href="http://www.math.ust.hk/excalibur/v3_n1.pdf"><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>是無理數的六個證明，香港大學數學系蕭文強</a> （<a rel="nofollow" class="external text" href="//web.archive.org/web/20201120151352/http://www.math.ust.hk/excalibur/v3_n1.pdf">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>）（Mathematical Excalibur Vol.3 No.1 Page 2）</li> <li><a rel="nofollow" class="external text" href="http://www.math.sinica.edu.tw/math_media/pdf.php?m_file=ZDMwNC8zMDQwNA==">舊題新解—根號2是無理數，張海潮張鎮華</a><sup class="noprint Inline-Template"><span style="white-space: nowrap;">[<a href="/wiki/Wikipedia:%E5%A4%B1%E6%95%88%E9%93%BE%E6%8E%A5" title="Wikipedia:失效链接"><span title="自2018年1月失效">永久失效連結</span></a>]</span></sup>（數學傳播第30卷第4期）</li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84265675"><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><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r58896141"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r58896141"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r58896141"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r58896141"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r58896141"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r58896141"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r58896141"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r58896141"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r58896141"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r58896141"></div><div role="navigation" class="navbox" aria-labelledby="無理數" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="collapsible-title navbox-title" colspan="3"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84265675"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84244141"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:%E7%84%A1%E7%90%86%E6%95%B8%E5%B0%8E%E8%88%AA" title="Template:無理數導航"><abbr title="查看该模板">查</abbr></a></li><li class="nv-talk"><a href="/w/index.php?title=Template_talk:%E7%84%A1%E7%90%86%E6%95%B8%E5%B0%8E%E8%88%AA&action=edit&redlink=1" class="new" title="Template talk:無理數導航（页面不存在）"><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:%E7%84%A1%E7%90%86%E6%95%B8%E5%B0%8E%E8%88%AA" title="Special:编辑页面/Template:無理數導航"><abbr title="编辑该模板">编</abbr></a></li></ul></div><div id="無理數" style="font-size:110%;margin:0 5em"><a class="mw-selflink selflink">無理數</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E6%9F%B4%E5%BB%B7%E5%B8%B8%E6%95%B8" title="柴廷常數">柴廷常數</a> (<span class="serif"><span class="texhtml">Ω</span></span>)</li> <li><a href="/wiki/%E5%88%98%E7%BB%B4%E5%B0%94%E6%95%B0" title="刘维尔数">刘维尔数</a></li> <li><span class="ilh-all" data-orig-title="質常數" data-lang-code="en" data-lang-name="英语" data-foreign-title="Prime constant"><span class="ilh-page"><a href="/w/index.php?title=%E8%B3%AA%E5%B8%B8%E6%95%B8&action=edit&redlink=1" class="new" title="質常數（页面不存在）">質常數</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/Prime_constant" class="extiw" title="en:Prime constant"><span lang="en" dir="auto">Prime constant</span></a></span>）</span></span> (<span class="serif"><span class="texhtml mvar" style="font-style:italic;margin-left:2px;margin-right:2px;">ρ</span></span>)</li> <li><a href="/wiki/%E6%AC%A7%E7%B1%B3%E5%8A%A0%E5%B8%B8%E6%95%B0" title="欧米加常数">欧米加常数</a></li> <li><a href="/wiki/%E5%8D%A1%E6%BC%A2%E5%B8%B8%E6%95%B8" title="卡漢常數">卡漢常數</a></li> <li><a href="/wiki/2%E7%9A%84%E8%87%AA%E7%84%B6%E5%AF%B9%E6%95%B0" title="2的自然对数">2的自然对数</a></li> <li><a href="/wiki/%E9%AB%98%E6%96%AF%E5%B8%B8%E6%95%B8" title="高斯常數">高斯常數</a> (<span class="serif"><span class="texhtml mvar" style="font-style:italic;margin-left:2px;margin-right:2px;">G</span></span>)</li> <li><a href="/wiki/2%E7%9A%8412%E6%AC%A1%E6%96%B9%E6%A0%B9" title="2的12次方根">2的12次方根</a></li> <li><a href="/wiki/%E9%98%BF%E5%9F%B9%E9%87%8C%E5%B8%B8%E6%95%B0" title="阿培里常数">阿培里常数</a> (<span class="serif"><span class="texhtml"><i>ζ</i>(3)</span></span>)</li> <li><a href="/wiki/%E5%A1%91%E8%86%A0%E6%95%B8" title="塑膠數">塑膠數</a> (<span class="serif"><span class="texhtml mvar" style="font-style:italic;margin-left:2px;margin-right:2px;">ρ</span></span>)</li> <li><a href="/wiki/2%E7%9A%84%E7%AE%97%E8%A1%93%E5%B9%B3%E6%96%B9%E6%A0%B9" title="2的算術平方根">2的算術平方根</a></li></ul> <ul><li><a href="/wiki/%E8%B6%85%E9%BB%83%E9%87%91%E6%AF%94%E4%BE%8B" title="超黃金比例">超黃金比例</a> (<span class="serif"><span class="texhtml mvar" style="font-style:italic;margin-left:2px;margin-right:2px;">ψ</span></span>)</li> <li><a href="/wiki/%E5%9F%83%E5%B0%94%E5%BE%B7%E4%BB%80-%E6%B3%A2%E6%B8%A9%E5%B8%B8%E6%95%B0" title="埃尔德什-波温常数">埃尔德什-波温常数</a> (<span class="serif"><span class="texhtml mvar" style="font-style:italic;margin-left:2px;margin-right:2px;">E</span></span>)</li> <li><a href="/wiki/%E9%BB%84%E9%87%91%E5%88%86%E5%89%B2%E7%8E%87" title="黄金分割率">黄金分割率</a> (<span class="serif"><span class="texhtml mvar" style="font-style:italic;margin-left:2px;margin-right:2px;">φ</span></span>)</li> <li><a href="/wiki/3%E7%9A%84%E7%AE%97%E8%A1%93%E5%B9%B3%E6%96%B9%E6%A0%B9" title="3的算術平方根">3的算術平方根</a></li> <li><a href="/wiki/5%E7%9A%84%E7%AE%97%E8%A1%93%E5%B9%B3%E6%96%B9%E6%A0%B9" title="5的算術平方根">5的算術平方根</a></li> <li><a href="/wiki/%E7%99%BD%E9%8A%80%E6%AF%94%E4%BE%8B" title="白銀比例">白銀比例</a> (<span class="serif"><span class="texhtml"><i>δ</i><sub><i>S</i></sub></span></span>)</li> <li><span class="ilh-all" data-orig-title="6的算術平方根" data-lang-code="en" data-lang-name="英语" data-foreign-title="Square root of 6"><span class="ilh-page"><a href="/w/index.php?title=6%E7%9A%84%E7%AE%97%E8%A1%93%E5%B9%B3%E6%96%B9%E6%A0%B9&action=edit&redlink=1" class="new" title="6的算術平方根（页面不存在）">6的算術平方根</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/Square_root_of_6" class="extiw" title="en:Square root of 6"><span lang="en" dir="auto">Square root of 6</span></a></span>）</span></span></li> <li><span class="ilh-all" data-orig-title="7的算術平方根" data-lang-code="en" data-lang-name="英语" data-foreign-title="Square root of 7"><span class="ilh-page"><a href="/w/index.php?title=7%E7%9A%84%E7%AE%97%E8%A1%93%E5%B9%B3%E6%96%B9%E6%A0%B9&action=edit&redlink=1" class="new" title="7的算術平方根（页面不存在）">7的算術平方根</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/Square_root_of_7" class="extiw" title="en:Square root of 7"><span lang="en" dir="auto">Square root of 7</span></a></span>）</span></span></li> <li><a href="/wiki/E_(%E6%95%B0%E5%AD%A6%E5%B8%B8%E6%95%B0)" title="E (数学常数)">自然常數/尤拉數</a> (<span class="serif"><span class="texhtml mvar" style="font-style:italic;margin-left:2px;margin-right:2px;">e</span></span>)</li> <li><a href="/wiki/%E5%9C%93%E5%91%A8%E7%8E%87" title="圓周率">圓周率</a> (<span class="texhtml">π</span>)</li> <li><a href="/wiki/%E9%9D%92%E9%8A%85%E6%AF%94%E4%BE%8B" title="青銅比例">青銅比例</a></li></ul> </div></td><td class="noviewer navbox-image" rowspan="2" style="width:1px;padding:0px 0px 0px 2px"><div><span typeof="mw:File"><a href="/wiki/File:Gold,_square_root_of_2,_and_square_root_of_3_rectangles.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/34/Gold%2C_square_root_of_2%2C_and_square_root_of_3_rectangles.png/50px-Gold%2C_square_root_of_2%2C_and_square_root_of_3_rectangles.png" decoding="async" width="50" height="89" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/34/Gold%2C_square_root_of_2%2C_and_square_root_of_3_rectangles.png/75px-Gold%2C_square_root_of_2%2C_and_square_root_of_3_rectangles.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/34/Gold%2C_square_root_of_2%2C_and_square_root_of_3_rectangles.png/100px-Gold%2C_square_root_of_2%2C_and_square_root_of_3_rectangles.png 2x" data-file-width="1047" data-file-height="1873" /></a></span></div></td></tr><tr><td colspan="2" class="navbox-list navbox-even hlist" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><span class="ilh-all" data-orig-title="分裂數" data-lang-code="en" data-lang-name="英语" data-foreign-title="Schizophrenic number"><span class="ilh-page"><a href="/w/index.php?title=%E5%88%86%E8%A3%82%E6%95%B8&action=edit&redlink=1" class="new" title="分裂數（页面不存在）">分裂數</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/Schizophrenic_number" class="extiw" title="en:Schizophrenic number"><span lang="en" dir="auto">Schizophrenic number</span></a></span>）</span></span></li> <li><a href="/wiki/%E8%B6%85%E8%B6%8A%E6%95%B8" title="超越數">超越數</a></li> <li><a href="/wiki/%E4%B8%89%E8%A7%92%E5%87%BD%E6%95%B8%E6%95%B8" class="mw-redirect" title="三角函數數">三角函數數</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84265675"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84261037"></div><div role="navigation" class="navbox" aria-labelledby="实数" style="padding:3px"><table class="nowraplinks 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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84244141"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:%E5%AF%A6%E6%95%B8" title="Template:實數"><abbr title="查看该模板">查</abbr></a></li><li class="nv-talk"><a href="/w/index.php?title=Template_talk:%E5%AF%A6%E6%95%B8&action=edit&redlink=1" class="new" title="Template talk:實數（页面不存在）"><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%AF%A6%E6%95%B8" title="Special:编辑页面/Template:實數"><abbr title="编辑该模板">编</abbr></a></li></ul></div><div id="实数" style="font-size:110%;margin:0 5em"><a href="/wiki/%E5%AE%9E%E6%95%B0" title="实数">实数</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/0.999%E2%80%A6" title="0.999…">0.999…</a></li> <li><span class="ilh-all" data-orig-title="絕對差量" data-lang-code="en" data-lang-name="英语" data-foreign-title="Absolute difference"><span class="ilh-page"><a href="/w/index.php?title=%E7%B5%95%E5%B0%8D%E5%B7%AE%E9%87%8F&action=edit&redlink=1" class="new" title="絕對差量（页面不存在）">絕對差量</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/Absolute_difference" class="extiw" title="en:Absolute difference"><span lang="en" dir="auto">Absolute difference</span></a></span>）</span></span></li> <li><a href="/wiki/%E5%BA%B7%E6%89%98%E5%B0%94%E9%9B%86" title="康托尔集">康托尔集</a></li> <li><span class="ilh-all" data-orig-title="康托爾–戴德金公理" data-lang-code="en" data-lang-name="英语" data-foreign-title="Cantor–Dedekind axiom"><span class="ilh-page"><a href="/w/index.php?title=%E5%BA%B7%E6%89%98%E7%88%BE%E2%80%93%E6%88%B4%E5%BE%B7%E9%87%91%E5%85%AC%E7%90%86&action=edit&redlink=1" class="new" title="康托爾–戴德金公理（页面不存在）">康托爾–戴德金公理</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/Cantor%E2%80%93Dedekind_axiom" class="extiw" title="en:Cantor–Dedekind axiom"><span lang="en" dir="auto">Cantor–Dedekind axiom</span></a></span>）</span></span></li> <li><a href="/wiki/%E5%AE%9E%E6%95%B0%E5%AE%8C%E5%A4%87%E6%80%A7" title="实数完备性">实数完备性</a></li> <li><a href="/wiki/%E5%AF%A6%E6%95%B8%E7%9A%84%E6%A7%8B%E9%80%A0" title="實數的構造">實數的構造</a></li> <li><span class="ilh-all" data-orig-title="實數的一階理論可決定性" data-lang-code="en" data-lang-name="英语" data-foreign-title="Decidability of first-order theories of the real numbers"><span class="ilh-page"><a href="/w/index.php?title=%E5%AF%A6%E6%95%B8%E7%9A%84%E4%B8%80%E9%9A%8E%E7%90%86%E8%AB%96%E5%8F%AF%E6%B1%BA%E5%AE%9A%E6%80%A7&action=edit&redlink=1" class="new" title="實數的一階理論可決定性（页面不存在）">實數的一階理論可決定性</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/Decidability_of_first-order_theories_of_the_real_numbers" class="extiw" title="en:Decidability of first-order theories of the real numbers"><span lang="en" dir="auto">Decidability of first-order theories of the real numbers</span></a></span>）</span></span></li> <li><a href="/wiki/%E6%93%B4%E5%B1%95%E5%AF%A6%E6%95%B8%E7%B7%9A" title="擴展實數線">擴展實數線</a></li> <li><span class="ilh-all" data-orig-title="格雷果里數" data-lang-code="en" data-lang-name="英语" data-foreign-title="Gregory number"><span class="ilh-page"><a href="/w/index.php?title=%E6%A0%BC%E9%9B%B7%E6%9E%9C%E9%87%8C%E6%95%B8&action=edit&redlink=1" class="new" title="格雷果里數（页面不存在）">格雷果里數</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/Gregory_number" class="extiw" title="en:Gregory number"><span lang="en" dir="auto">Gregory number</span></a></span>）</span></span></li> <li><a class="mw-selflink selflink">無理數</a></li> <li><a href="/wiki/%E6%AD%A3%E8%A7%84%E6%95%B0" title="正规数">正规数</a></li> <li><a href="/wiki/%E6%9C%89%E7%90%86%E6%95%B0" title="有理数">有理数</a></li> <li><span class="ilh-all" data-orig-title="有理ζ级数" data-lang-code="en" data-lang-name="英语" data-foreign-title="Rational zeta series"><span class="ilh-page"><a href="/w/index.php?title=%E6%9C%89%E7%90%86%CE%B6%E7%BA%A7%E6%95%B0&action=edit&redlink=1" class="new" title="有理ζ级数（页面不存在）">有理ζ级数</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/Rational_zeta_series" class="extiw" title="en:Rational zeta series"><span lang="en" dir="auto">Rational zeta series</span></a></span>）</span></span></li> <li><span class="ilh-all" data-orig-title="實坐標空間" data-lang-code="en" data-lang-name="英语" data-foreign-title="Real coordinate space"><span class="ilh-page"><a href="/w/index.php?title=%E5%AF%A6%E5%9D%90%E6%A8%99%E7%A9%BA%E9%96%93&action=edit&redlink=1" class="new" title="實坐標空間（页面不存在）">實坐標空間</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/Real_coordinate_space" class="extiw" title="en:Real coordinate space"><span lang="en" dir="auto">Real coordinate space</span></a></span>）</span></span></li> <li><a href="/wiki/%E5%AF%A6%E6%95%B8%E7%B7%9A" title="實數線">實數線</a></li> <li><span class="ilh-all" data-orig-title="塔尔斯基的实数公理化" data-lang-code="en" data-lang-name="英语" data-foreign-title="Tarski's axiomatization of the reals"><span class="ilh-page"><a href="/w/index.php?title=%E5%A1%94%E5%B0%94%E6%96%AF%E5%9F%BA%E7%9A%84%E5%AE%9E%E6%95%B0%E5%85%AC%E7%90%86%E5%8C%96&action=edit&redlink=1" class="new" title="塔尔斯基的实数公理化（页面不存在）">塔尔斯基的实数公理化</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/Tarski%27s_axiomatization_of_the_reals" class="extiw" title="en:Tarski's axiomatization of the reals"><span lang="en" dir="auto">Tarski's axiomatization of the reals</span></a></span>）</span></span></li> <li><a href="/wiki/%E7%BB%B4%E5%A1%94%E5%88%A9%E9%9B%86%E5%90%88" title="维塔利集合">维塔利集合</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84265675"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84261037"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r58896141"></div><div role="navigation" class="navbox" aria-labelledby="数的系統" style="padding:3px"><table class="nowraplinks 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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84244141"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:%E6%95%B8%E7%9A%84%E7%B3%BB%E7%B5%B1" title="Template:數的系統"><abbr title="查看该模板">查</abbr></a></li><li class="nv-talk"><a href="/w/index.php?title=Template_talk:%E6%95%B8%E7%9A%84%E7%B3%BB%E7%B5%B1&action=edit&redlink=1" class="new" title="Template talk:數的系統（页面不存在）"><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:%E6%95%B8%E7%9A%84%E7%B3%BB%E7%B5%B1" title="Special:编辑页面/Template:數的系統"><abbr title="编辑该模板">编</abbr></a></li></ul></div><div id="数的系統" style="font-size:110%;margin:0 5em"><a href="/wiki/%E6%95%B0" title="数">数</a>的系統</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%E5%8F%AF%E6%95%B8%E9%9B%86" title="可數集">可數集</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E8%87%AA%E7%84%B6%E6%95%B0" title="自然数">自然数</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 \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} }"></span>)</li> <li><a href="/wiki/%E6%95%B4%E6%95%B0" title="整数">整数</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 \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></span>)</li> <li><a href="/wiki/%E6%9C%89%E7%90%86%E6%95%B0" title="有理数">有理数</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 \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }"></span>)</li> <li><a href="/wiki/%E8%A6%8F%E7%9F%A9%E6%95%B8" title="規矩數">規矩數</a></li> <li><a href="/wiki/%E4%BB%A3%E6%95%B8%E6%95%B8" title="代數數">代數數</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 \mathbb {A} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {A} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3fb423c16a5f403edbaf66438b75e7a36e725af6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {A} }"></span>)</li> <li><span class="ilh-all" data-orig-title="周期 (代數幾何)" data-lang-code="en" data-lang-name="英语" data-foreign-title="Period (algebraic geometry)"><span class="ilh-page"><a href="/w/index.php?title=%E5%91%A8%E6%9C%9F_(%E4%BB%A3%E6%95%B8%E5%B9%BE%E4%BD%95)&action=edit&redlink=1" class="new" title="周期 (代數幾何)（页面不存在）">周期</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/Period_(algebraic_geometry)" class="extiw" title="en:Period (algebraic geometry)"><span lang="en" dir="auto">Period (algebraic geometry)</span></a></span>）</span></span></li> <li><a href="/wiki/%E5%8F%AF%E8%A8%88%E7%AE%97%E6%95%B8" title="可計算數">可計算數</a></li> <li><a href="/wiki/%E5%8F%AF%E5%AE%9A%E4%B9%89%E6%95%B0" title="可定义数">可定义数</a></li> <li><a href="/wiki/%E9%AB%98%E6%96%AF%E6%95%B4%E6%95%B8" title="高斯整數">高斯整數</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 \mathbb {Z} [i]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo stretchy="false">[</mo> <mi>i</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} [i]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ffa94e9e2e6d9e5e5373d5fafb954b902743fde" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.646ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} [i]}"></span>)</li> <li><a href="/wiki/%E8%89%BE%E6%A3%AE%E6%96%AF%E5%9D%A6%E6%95%B4%E6%95%B0" title="艾森斯坦整数">艾森斯坦整数</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><span class="ilh-all" data-orig-title="合成代數" data-lang-code="en" data-lang-name="英语" data-foreign-title="Composition algebra"><span class="ilh-page"><a href="/w/index.php?title=%E5%90%88%E6%88%90%E4%BB%A3%E6%95%B8&action=edit&redlink=1" class="new" title="合成代數（页面不存在）">合成代數</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/Composition_algebra" class="extiw" title="en:Composition algebra"><span lang="en" dir="auto">Composition algebra</span></a></span>）</span></span></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><span class="ilh-all" data-orig-title="可除代數" data-lang-code="en" data-lang-name="英语" data-foreign-title="Division algebra"><span class="ilh-page"><a href="/w/index.php?title=%E5%8F%AF%E9%99%A4%E4%BB%A3%E6%95%B8&action=edit&redlink=1" class="new" title="可除代數（页面不存在）">可除代數</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/Division_algebra" class="extiw" title="en:Division algebra"><span lang="en" dir="auto">Division algebra</span></a></span>）</span></span>：<a href="/wiki/%E5%AE%9E%E6%95%B0" title="实数">实数</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 \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span>)</li> <li><a href="/wiki/%E5%A4%8D%E6%95%B0_(%E6%95%B0%E5%AD%A6)" title="复数 (数学)">複數</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 \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }"></span>)</li> <li><a href="/wiki/%E5%9B%9B%E5%85%83%E6%95%B8" title="四元數">四元數</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 \mathbb {H} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">H</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {H} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e050965453c42bcc6bd544546703c836bdafeac9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \mathbb {H} }"></span>)</li> <li><a href="/wiki/%E5%85%AB%E5%85%83%E6%95%B0" title="八元数">八元数</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 \mathbb {O} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">O</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {O} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1ed2664a4fe515e6fbed25a7193ce663b82920c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \mathbb {O} }"></span>)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%E5%87%AF%E8%8E%B1-%E8%BF%AA%E5%85%8B%E6%A3%AE%E7%BB%93%E6%9E%84" title="凯莱-迪克森结构">凯莱－迪克森结构</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E5%AE%9E%E6%95%B0" title="实数">实数</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 \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span>)</li> <li><a href="/wiki/%E5%A4%8D%E6%95%B0_(%E6%95%B0%E5%AD%A6)" title="复数 (数学)">複數</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 \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }"></span>)</li> <li><a href="/wiki/%E5%9B%9B%E5%85%83%E6%95%B8" title="四元數">四元數</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 \mathbb {H} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">H</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {H} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e050965453c42bcc6bd544546703c836bdafeac9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \mathbb {H} }"></span>)</li> <li><a href="/wiki/%E5%85%AB%E5%85%83%E6%95%B0" title="八元数">八元数</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 \mathbb {O} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">O</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {O} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1ed2664a4fe515e6fbed25a7193ce663b82920c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \mathbb {O} }"></span>)</li> <li><a href="/wiki/%E5%8D%81%E5%85%AD%E5%85%83%E6%95%B8" title="十六元數">十六元數</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 \mathbb {S} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">S</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {S} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f9d5874c5d7f68eba1cec9da9ccbe53903303bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.293ex; height:2.176ex;" alt="{\displaystyle \mathbb {S} }"></span>)</li> <li><a href="/wiki/%E4%B8%89%E5%8D%81%E4%BA%8C%E5%85%83%E6%95%B8" title="三十二元數">三十二元數</a></li> <li><a href="/wiki/%E5%85%AD%E5%8D%81%E5%9B%9B%E5%85%83%E6%95%B8" class="mw-redirect" title="六十四元數">六十四元數</a></li> <li><a href="/wiki/%E4%B8%80%E7%99%BE%E4%BA%8C%E5%8D%81%E5%85%AB%E5%85%83%E6%95%B8" class="mw-redirect" title="一百二十八元數">一百二十八元數</a></li> <li>二百五十六元數……</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">分裂<br />形式</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li>於<a href="/wiki/%E5%AE%9E%E6%95%B0" title="实数"><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 \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span></a>：<a href="/wiki/%E9%9B%99%E6%9B%B2%E8%A4%87%E6%95%B8" title="雙曲複數">雙曲複數</a></li> <li><a href="/wiki/%E5%88%86%E8%A3%82%E5%9B%9B%E5%85%83%E6%95%B0" title="分裂四元数">分裂四元数</a></li> <li><span class="ilh-all" data-orig-title="分裂複四元數" data-lang-code="en" data-lang-name="英语" data-foreign-title="Split-biquaternion"><span class="ilh-page"><a href="/w/index.php?title=%E5%88%86%E8%A3%82%E8%A4%87%E5%9B%9B%E5%85%83%E6%95%B8&action=edit&redlink=1" class="new" title="分裂複四元數（页面不存在）">分裂複四元數</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/Split-biquaternion" class="extiw" title="en:Split-biquaternion"><span lang="en" dir="auto">Split-biquaternion</span></a></span>）</span></span></li> <li><span class="ilh-all" data-orig-title="分裂八元數" data-lang-code="en" data-lang-name="英语" data-foreign-title="Split-octonion"><span class="ilh-page"><a href="/w/index.php?title=%E5%88%86%E8%A3%82%E5%85%AB%E5%85%83%E6%95%B8&action=edit&redlink=1" class="new" title="分裂八元數（页面不存在）">分裂八元數</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/Split-octonion" class="extiw" title="en:Split-octonion"><span lang="en" dir="auto">Split-octonion</span></a></span>）</span></span><br /> 於<a href="/wiki/%E5%A4%8D%E6%95%B0_(%E6%95%B0%E5%AD%A6)" title="复数 (数学)"><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 \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }"></span></a>：<a href="/wiki/%E9%9B%99%E8%A4%87%E6%95%B8" title="雙複數">雙複數</a></li> <li><a href="/wiki/%E8%A4%87%E5%9B%9B%E5%85%83%E6%95%B8" title="複四元數">複四元數</a></li> <li><span class="ilh-all" data-orig-title="複八元數" data-lang-code="en" data-lang-name="英语" data-foreign-title="Bioctonion"><span class="ilh-page"><a href="/w/index.php?title=%E8%A4%87%E5%85%AB%E5%85%83%E6%95%B8&action=edit&redlink=1" class="new" title="複八元數（页面不存在）">複八元數</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/Bioctonion" class="extiw" title="en:Bioctonion"><span lang="en" dir="auto">Bioctonion</span></a></span>）</span></span></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">其他<a href="/wiki/%E8%B6%85%E5%A4%8D%E6%95%B0" title="超复数">超複數</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E4%BA%8C%E5%85%83%E6%95%B0" title="二元数">二元数</a></li> <li><span class="ilh-all" data-orig-title="二元四元數" data-lang-code="en" data-lang-name="英语" data-foreign-title="Dual quaternion"><span class="ilh-page"><a href="/w/index.php?title=%E4%BA%8C%E5%85%83%E5%9B%9B%E5%85%83%E6%95%B8&action=edit&redlink=1" class="new" title="二元四元數（页面不存在）">二元四元數</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/Dual_quaternion" class="extiw" title="en:Dual quaternion"><span lang="en" dir="auto">Dual quaternion</span></a></span>）</span></span></li> <li><span class="ilh-all" data-orig-title="二元複數" data-lang-code="en" data-lang-name="英语" data-foreign-title="Applications of dual quaternions to 2D geometry"><span class="ilh-page"><a href="/w/index.php?title=%E4%BA%8C%E5%85%83%E8%A4%87%E6%95%B8&action=edit&redlink=1" class="new" title="二元複數（页面不存在）">二元複數</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/Applications_of_dual_quaternions_to_2D_geometry" class="extiw" title="en:Applications of dual quaternions to 2D geometry"><span lang="en" dir="auto">Applications of dual quaternions to 2D geometry</span></a></span>）</span></span></li> <li><span class="ilh-all" data-orig-title="雙曲四元數" data-lang-code="en" data-lang-name="英语" data-foreign-title="Hyperbolic quaternion"><span class="ilh-page"><a href="/w/index.php?title=%E9%9B%99%E6%9B%B2%E5%9B%9B%E5%85%83%E6%95%B8&action=edit&redlink=1" class="new" title="雙曲四元數（页面不存在）">雙曲四元數</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/Hyperbolic_quaternion" class="extiw" title="en:Hyperbolic quaternion"><span lang="en" dir="auto">Hyperbolic quaternion</span></a></span>）</span></span></li> <li><a href="/wiki/%E5%8D%81%E5%85%AD%E5%85%83%E6%95%B8" title="十六元數">十六元數</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 \mathbb {S} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">S</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {S} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f9d5874c5d7f68eba1cec9da9ccbe53903303bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.293ex; height:2.176ex;" alt="{\displaystyle \mathbb {S} }"></span>)</li> <li><a href="/wiki/%E4%B8%89%E5%8D%81%E4%BA%8C%E5%85%83%E6%95%B8" title="三十二元數">三十二元數</a></li> <li><span class="ilh-all" data-orig-title="分裂複四元數" data-lang-code="en" data-lang-name="英语" data-foreign-title="Split-biquaternion"><span class="ilh-page"><a href="/w/index.php?title=%E5%88%86%E8%A3%82%E8%A4%87%E5%9B%9B%E5%85%83%E6%95%B8&action=edit&redlink=1" class="new" title="分裂複四元數（页面不存在）">分裂複四元數</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/Split-biquaternion" class="extiw" title="en:Split-biquaternion"><span lang="en" dir="auto">Split-biquaternion</span></a></span>）</span></span></li> <li><a href="/wiki/%E5%A4%9A%E9%87%8D%E5%A4%8D%E6%95%B0" title="多重复数">多重複數</a></li> <li><span class="ilh-all" data-orig-title="幾何代數" data-lang-code="en" data-lang-name="英语" data-foreign-title="Geometric algebra"><span class="ilh-page"><a href="/wiki/%E5%87%A0%E4%BD%95%E4%BB%A3%E6%95%B0" title="几何代数">幾何代數</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/Geometric_algebra" class="extiw" title="en:Geometric algebra"><span lang="en" dir="auto">Geometric algebra</span></a></span>）</span></span> <ul><li><span class="ilh-all" data-orig-title="物理空間代數" data-lang-code="en" data-lang-name="英语" data-foreign-title="Algebra of physical space"><span class="ilh-page"><a href="/wiki/%E7%89%A9%E7%90%86%E7%A9%BA%E9%97%B4%E4%BB%A3%E6%95%B0" title="物理空间代数">物理空間代數</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/Algebra_of_physical_space" class="extiw" title="en:Algebra of physical space"><span lang="en" dir="auto">Algebra of physical space</span></a></span>）</span></span></li> <li><span class="ilh-all" data-orig-title="時空代數" data-lang-code="en" data-lang-name="英语" data-foreign-title="Spacetime algebra"><span class="ilh-page"><a href="/wiki/%E6%97%B6%E7%A9%BA%E4%BB%A3%E6%95%B0" title="时空代数">時空代數</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/Spacetime_algebra" class="extiw" title="en:Spacetime algebra"><span lang="en" dir="auto">Spacetime algebra</span></a></span>）</span></span></li></ul></li> <li><a href="/wiki/%E4%B8%89%E5%85%83%E6%95%B8" title="三元數">三元數</a> <ul><li>無法良好構建</li></ul></li> <li><a href="/wiki/%E5%85%AD%E5%85%83%E6%95%B8" class="mw-redirect" title="六元數">六元數</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">其他系統</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E5%9F%BA%E6%95%B0_(%E6%95%B0%E5%AD%A6)" title="基数 (数学)">基數</a></li> <li><span class="ilh-all" data-orig-title="擴展自然數" data-lang-code="en" data-lang-name="英语" data-foreign-title="Extended natural numbers"><span class="ilh-page"><a href="/w/index.php?title=%E6%93%B4%E5%B1%95%E8%87%AA%E7%84%B6%E6%95%B8&action=edit&redlink=1" class="new" title="擴展自然數（页面不存在）">擴展自然數</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/Extended_natural_numbers" class="extiw" title="en:Extended natural numbers"><span lang="en" dir="auto">Extended natural numbers</span></a></span>）</span></span></li> <li><a class="mw-selflink selflink">無理數</a></li> <li><span class="ilh-all" data-orig-title="模糊數" data-lang-code="en" data-lang-name="英语" data-foreign-title="Fuzzy number"><span class="ilh-page"><a href="/w/index.php?title=%E6%A8%A1%E7%B3%8A%E6%95%B8&action=edit&redlink=1" class="new" title="模糊數（页面不存在）">模糊數</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/Fuzzy_number" class="extiw" title="en:Fuzzy number"><span lang="en" dir="auto">Fuzzy number</span></a></span>）</span></span></li> <li><a href="/wiki/%E8%B6%85%E5%AE%9E%E6%95%B0_(%E9%9D%9E%E6%A0%87%E5%87%86%E5%88%86%E6%9E%90)" title="超实数 (非标准分析)">超實數</a></li> <li><span class="ilh-all" data-orig-title="列維-奇維塔域" data-lang-code="en" data-lang-name="英语" data-foreign-title="Levi-Civita field"><span class="ilh-page"><a href="/w/index.php?title=%E5%88%97%E7%B6%AD-%E5%A5%87%E7%B6%AD%E5%A1%94%E5%9F%9F&action=edit&redlink=1" class="new" title="列維-奇維塔域（页面不存在）">列維-奇維塔域</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/Levi-Civita_field" class="extiw" title="en:Levi-Civita field"><span lang="en" dir="auto">Levi-Civita field</span></a></span>）</span></span></li> <li><a href="/wiki/%E8%B6%85%E7%8F%BE%E5%AF%A6%E6%95%B8" title="超現實數">超現實數</a></li> <li><a href="/wiki/%E8%B6%85%E8%B6%8A%E6%95%B8" title="超越數">超越數</a></li> <li><a href="/wiki/%E5%BA%8F%E6%95%B0" title="序数">序数</a></li> <li><a href="/wiki/P%E9%80%B2%E6%95%B8" title="P進數"><span class="serif"><span class="texhtml"><i>p</i></span></span>進數</a></li> <li><span class="ilh-all" data-orig-title="超自然數" data-lang-code="en" data-lang-name="英语" data-foreign-title="Supernatural number"><span class="ilh-page"><a href="/w/index.php?title=%E8%B6%85%E8%87%AA%E7%84%B6%E6%95%B8&action=edit&redlink=1" class="new" title="超自然數（页面不存在）">超自然數</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/Supernatural_number" class="extiw" title="en:Supernatural number"><span lang="en" dir="auto">Supernatural number</span></a></span>）</span></span></li> <li><span class="ilh-all" data-orig-title="上超實數" data-lang-code="en" data-lang-name="英语" data-foreign-title="Superreal number"><span class="ilh-page"><a href="/w/index.php?title=%E4%B8%8A%E8%B6%85%E5%AF%A6%E6%95%B8&action=edit&redlink=1" class="new" title="上超實數（页面不存在）">上超實數</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/Superreal_number" class="extiw" title="en:Superreal number"><span lang="en" dir="auto">Superreal number</span></a></span>）</span></span></li></ul> </div></td></tr><tr><td class="navbox-abovebelow hlist" colspan="2"><div> <ul><li><a href="/wiki/%E6%95%B0#數的類別" title="数">分類</a></li> <li><span typeof="mw:File"><span title="列表级条目"><img alt="列表级条目" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/db/Symbol_list_class.svg/16px-Symbol_list_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/db/Symbol_list_class.svg/23px-Symbol_list_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/db/Symbol_list_class.svg/31px-Symbol_list_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <span class="ilh-all" data-orig-title="數的類別列表" data-lang-code="en" data-lang-name="英语" data-foreign-title="List of types of numbers"><span class="ilh-page"><a href="/w/index.php?title=%E6%95%B8%E7%9A%84%E9%A1%9E%E5%88%A5%E5%88%97%E8%A1%A8&action=edit&redlink=1" class="new" title="數的類別列表（页面不存在）">列表</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/List_of_types_of_numbers" class="extiw" title="en:List of types of numbers"><span lang="en" dir="auto">List of types of numbers</span></a></span>）</span></span></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84265675"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84261037"><style data-mw-deduplicate="TemplateStyles:r79005747">.mw-parser-output .tooltip-dotted{border-bottom:1px dotted;cursor:help}</style></div><div role="navigation" class="navbox authority-control" aria-label="Navbox" style="padding:3px"><table class="nowraplinks hlist navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="row" class="navbox-group" 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