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href="/w/index.php?title=Special:%E7%94%A8%E6%88%B7%E7%99%BB%E5%BD%95&returnto=%E5%AE%9E%E6%95%B0" 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-初等數學" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#初等數學"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>初等數學</span> </div> </a> <ul id="toc-初等數學-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-正数与负数" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#正数与负数"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>正数与负数</span> </div> </a> <ul id="toc-正数与负数-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-历史" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#历史"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>历史</span> </div> </a> <ul id="toc-历史-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-定义" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#定义"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>定义</span> </div> </a> <button aria-controls="toc-定义-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>开关定义子章节</span> </button> <ul id="toc-定义-sublist" class="vector-toc-list"> <li id="toc-從有理數构造實數" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#從有理數构造實數"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>從有理數构造實數</span> </div> </a> <ul id="toc-從有理數构造實數-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-公理化方法" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#公理化方法"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>公理化方法</span> </div> </a> <ul id="toc-公理化方法-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-例子" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#例子"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>例子</span> </div> </a> <ul id="toc-例子-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">6</span> <span>性质</span> </div> </a> <button aria-controls="toc-性质-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>开关性质子章节</span> </button> <ul id="toc-性质-sublist" class="vector-toc-list"> <li id="toc-基本运算" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#基本运算"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>基本运算</span> </div> </a> <ul id="toc-基本运算-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-连续性或完備性" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#连续性或完備性"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>连续性或完備性</span> </div> </a> <ul id="toc-连续性或完備性-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-完备的有序域（有序性）" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#完备的有序域（有序性）"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.3</span> <span>完备的有序域（有序性）</span> </div> </a> <ul id="toc-完备的有序域（有序性）-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-高级性质" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#高级性质"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.4</span> <span>高级性质</span> </div> </a> <ul id="toc-高级性质-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-拓撲性質" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#拓撲性質"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.5</span> <span>拓撲性質</span> </div> </a> <ul id="toc-拓撲性質-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-扩展与一般化" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#扩展与一般化"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>扩展与一般化</span> </div> </a> <ul id="toc-扩展与一般化-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-注释" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#注释"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>注释</span> </div> </a> <ul id="toc-注释-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-请参阅" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#请参阅"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>请参阅</span> </div> </a> <ul id="toc-请参阅-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">实数</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。118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-118" 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">118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/Re%C3%ABle_getal" title="Reële getal – 南非荷兰语" lang="af" hreflang="af" data-title="Reële 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-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Reelle_Zahl" title="Reelle Zahl – 瑞士德语" lang="gsw" hreflang="gsw" data-title="Reelle Zahl" data-language-autonym="Alemannisch" data-language-local-name="瑞士德语" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B9%D8%AF%D8%AF_%D8%AD%D9%82%D9%8A%D9%82%D9%8A" title="عدد حقيقي – 阿拉伯语" lang="ar" hreflang="ar" data-title="عدد حقيقي" 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-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/N%C3%BAmberu_real" title="Númberu real – 阿斯图里亚斯语" lang="ast" hreflang="ast" data-title="Númberu real" 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/H%C9%99qiqi_%C9%99d%C9%99dl%C9%99r" title="Həqiqi ədədlər – 阿塞拜疆语" lang="az" hreflang="az" data-title="Həqiqi 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-azb mw-list-item"><a href="https://azb.wikipedia.org/wiki/%D8%AD%D9%82%DB%8C%D9%82%DB%8C_%D8%B3%D8%A7%DB%8C%DB%8C%D9%84%D8%A7%D8%B1" title="حقیقی ساییلار – South Azerbaijani" lang="azb" hreflang="azb" data-title="حقیقی ساییلار" data-language-autonym="تۆرکجه" data-language-local-name="South Azerbaijani" class="interlanguage-link-target"><span>تۆرکجه</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%AB%D1%81%D1%8B%D0%BD_%D2%BB%D0%B0%D0%BD" title="Ысын һан – 巴什基尔语" lang="ba" hreflang="ba" data-title="Ысын 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-bcl mw-list-item"><a href="https://bcl.wikipedia.org/wiki/Tunay_na_bilang" title="Tunay na bilang – Central Bikol" lang="bcl" hreflang="bcl" data-title="Tunay na bilang" data-language-autonym="Bikol Central" data-language-local-name="Central Bikol" class="interlanguage-link-target"><span>Bikol Central</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%A0%D1%8D%D1%87%D0%B0%D1%96%D1%81%D0%BD%D1%8B_%D0%BB%D1%96%D0%BA" title="Рэчаісны лік – 白俄罗斯语" lang="be" hreflang="be" data-title="Рэчаісны 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-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%A0%D1%8D%D1%87%D0%B0%D1%96%D1%81%D0%BD%D1%8B_%D0%BB%D1%96%D0%BA" title="Рэчаісны лік – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Рэчаісны лік" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A0%D0%B5%D0%B0%D0%BB%D0%BD%D0%BE_%D1%87%D0%B8%D1%81%D0%BB%D0%BE" title="Реално число – 保加利亚语" lang="bg" hreflang="bg" data-title="Реално 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-bh mw-list-item"><a href="https://bh.wikipedia.org/wiki/%E0%A4%B5%E0%A4%BE%E0%A4%B8%E0%A5%8D%E0%A4%A4%E0%A4%B5%E0%A4%BF%E0%A4%95_%E0%A4%B8%E0%A4%82%E0%A4%96%E0%A5%8D%E0%A4%AF%E0%A4%BE" title="वास्तविक संख्या – Bhojpuri" lang="bh" hreflang="bh" data-title="वास्तविक संख्या" data-language-autonym="भोजपुरी" data-language-local-name="Bhojpuri" class="interlanguage-link-target"><span>भोजपुरी</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AC%E0%A6%BE%E0%A6%B8%E0%A7%8D%E0%A6%A4%E0%A6%AC_%E0%A6%B8%E0%A6%82%E0%A6%96%E0%A7%8D%E0%A6%AF%E0%A6%BE" title="বাস্তব সংখ্যা – 孟加拉语" lang="bn" hreflang="bn" data-title="বাস্তব 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/Realan_broj" title="Realan broj – 波斯尼亚语" lang="bs" hreflang="bs" data-title="Realan 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-bxr mw-list-item"><a href="https://bxr.wikipedia.org/wiki/%D0%91%D0%BE%D0%B4%D0%BE%D1%82%D0%BE_%D1%82%D0%BE%D0%BE" title="Бодото тоо – Russia Buriat" lang="bxr" hreflang="bxr" data-title="Бодото тоо" data-language-autonym="Буряад" data-language-local-name="Russia Buriat" class="interlanguage-link-target"><span>Буряад</span></a></li><li class="interlanguage-link interwiki-ca badge-Q17437798 badge-goodarticle mw-list-item" title="优良条目"><a href="https://ca.wikipedia.org/wiki/Nombre_real" title="Nombre real – 加泰罗尼亚语" lang="ca" hreflang="ca" data-title="Nombre real" 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_%DA%95%D8%A7%D8%B3%D8%AA%DB%95%D9%82%DB%8C%D9%86%DB%95" title="ژمارەی ڕاستەقینە – 中库尔德语" lang="ckb" hreflang="ckb" data-title="ژمارەی ڕاستەقینە" 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-crh mw-list-item"><a href="https://crh.wikipedia.org/wiki/Aqiqiy_say%C4%B1" title="Aqiqiy sayı – 克里米亚鞑靼语" lang="crh" hreflang="crh" data-title="Aqiqiy sayı" data-language-autonym="Qırımtatarca" data-language-local-name="克里米亚鞑靼语" class="interlanguage-link-target"><span>Qırımtatarca</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Re%C3%A1ln%C3%A9_%C4%8D%C3%ADslo" title="Reálné číslo – 捷克语" lang="cs" hreflang="cs" data-title="Reálné 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%A7%C4%83%D0%BD_%D1%85%D0%B8%D1%81%D0%B5%D0%BF" title="Чăн хисеп – 楚瓦什语" lang="cv" hreflang="cv" data-title="Чăн 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_real" title="Rhif real – 威尔士语" lang="cy" hreflang="cy" data-title="Rhif real" 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/Reelle_tal" title="Reelle tal – 丹麦语" lang="da" hreflang="da" data-title="Reelle 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/Reelle_Zahl" title="Reelle Zahl – 德语" lang="de" hreflang="de" data-title="Reelle 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-diq mw-list-item"><a href="https://diq.wikipedia.org/wiki/Amaro_reel" title="Amaro reel – Zazaki" lang="diq" hreflang="diq" data-title="Amaro reel" data-language-autonym="Zazaki" data-language-local-name="Zazaki" class="interlanguage-link-target"><span>Zazaki</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A0%CF%81%CE%B1%CE%B3%CE%BC%CE%B1%CF%84%CE%B9%CE%BA%CF%8C%CF%82_%CE%B1%CF%81%CE%B9%CE%B8%CE%BC%CF%8C%CF%82" title="Πραγματικός αριθμός – 希腊语" lang="el" hreflang="el" data-title="Πραγματικός 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-eml mw-list-item"><a href="https://eml.wikipedia.org/wiki/N%C3%B3mmer_re%C3%A8l" title="Nómmer reèl – Emiliano-Romagnolo" lang="egl" hreflang="egl" data-title="Nómmer reèl" data-language-autonym="Emiliàn e rumagnòl" data-language-local-name="Emiliano-Romagnolo" class="interlanguage-link-target"><span>Emiliàn e rumagnòl</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Real_number" title="Real number – 英语" lang="en" hreflang="en" data-title="Real 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/Reelo" title="Reelo – 世界语" lang="eo" hreflang="eo" data-title="Reelo" 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_real" title="Número real – 西班牙语" lang="es" hreflang="es" data-title="Número real" 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/Reaalarv" title="Reaalarv – 爱沙尼亚语" lang="et" hreflang="et" data-title="Reaalarv" 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_erreal" title="Zenbaki erreal – 巴斯克语" lang="eu" hreflang="eu" data-title="Zenbaki erreal" 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_%D8%AD%D9%82%DB%8C%D9%82%DB%8C" title="عدد حقیقی – 波斯语" lang="fa" hreflang="fa" data-title="عدد حقیقی" 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/Reaaliluku" title="Reaaliluku – 芬兰语" lang="fi" hreflang="fi" data-title="Reaaliluku" 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/Reaalarv" title="Reaalarv – 佛羅文" lang="vro" hreflang="vro" data-title="Reaalarv" 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/Reelt_tal" title="Reelt tal – 法罗语" lang="fo" hreflang="fo" data-title="Reelt tal" 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 mw-list-item"><a href="https://fr.wikipedia.org/wiki/Nombre_r%C3%A9el" title="Nombre réel – 法语" lang="fr" hreflang="fr" data-title="Nombre réel" data-language-autonym="Français" data-language-local-name="法语" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Re%27el_taal" title="Re'el taal – 北弗里西亚语" lang="frr" hreflang="frr" data-title="Re'el taal" data-language-autonym="Nordfriisk" data-language-local-name="北弗里西亚语" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-fur mw-list-item"><a href="https://fur.wikipedia.org/wiki/Numars_re%C3%A2i" title="Numars reâi – 弗留利语" lang="fur" hreflang="fur" data-title="Numars reâi" data-language-autonym="Furlan" data-language-local-name="弗留利语" class="interlanguage-link-target"><span>Furlan</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/R%C3%A9aduimhir" title="Réaduimhir – 爱尔兰语" lang="ga" hreflang="ga" data-title="Réaduimhir" 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-gan mw-list-item"><a href="https://gan.wikipedia.org/wiki/%E5%AF%A6%E6%95%B8" title="實數 – 赣语" lang="gan" hreflang="gan" data-title="實數" data-language-autonym="贛語" data-language-local-name="赣语" class="interlanguage-link-target"><span>贛語</span></a></li><li class="interlanguage-link interwiki-gcr mw-list-item"><a href="https://gcr.wikipedia.org/wiki/Nonm_r%C3%A9y%C3%A8l" title="Nonm réyèl – Guianan Creole" lang="gcr" hreflang="gcr" data-title="Nonm réyèl" data-language-autonym="Kriyòl gwiyannen" data-language-local-name="Guianan Creole" class="interlanguage-link-target"><span>Kriyòl gwiyannen</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/N%C3%BAmero_real" title="Número real – 加利西亚语" lang="gl" hreflang="gl" data-title="Número real" 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-gv mw-list-item"><a href="https://gv.wikipedia.org/wiki/Feer_earroo" title="Feer earroo – 马恩语" lang="gv" hreflang="gv" data-title="Feer earroo" data-language-autonym="Gaelg" data-language-local-name="马恩语" class="interlanguage-link-target"><span>Gaelg</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%9E%D7%9E%D7%A9%D7%99" title="מספר ממשי – 希伯来语" lang="he" hreflang="he" data-title="מספר ממשי" 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%B5%E0%A4%BE%E0%A4%B8%E0%A5%8D%E0%A4%A4%E0%A4%B5%E0%A4%BF%E0%A4%95_%E0%A4%B8%E0%A4%82%E0%A4%96%E0%A5%8D%E0%A4%AF%E0%A4%BE" title="वास्तविक संख्या – 印地语" lang="hi" hreflang="hi" data-title="वास्तविक 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/Realni_broj" title="Realni broj – 克罗地亚语" lang="hr" hreflang="hr" data-title="Realni 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/Val%C3%B3s_sz%C3%A1mok" title="Valós számok – 匈牙利语" lang="hu" hreflang="hu" data-title="Valós 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%D6%80%D5%A1%D5%AF%D5%A1%D5%B6_%D5%A9%D5%AB%D5%BE" title="Իրական թիվ – 亚美尼亚语" lang="hy" hreflang="hy" data-title="Իրական 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-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Numero_real" title="Numero real – 国际语" lang="ia" hreflang="ia" data-title="Numero real" data-language-autonym="Interlingua" data-language-local-name="国际语" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-iba mw-list-item"><a href="https://iba.wikipedia.org/wiki/Lumur_bendar" title="Lumur bendar – 伊班语" lang="iba" hreflang="iba" data-title="Lumur bendar" data-language-autonym="Jaku Iban" data-language-local-name="伊班语" class="interlanguage-link-target"><span>Jaku Iban</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Bilangan_riil" title="Bilangan riil – 印度尼西亚语" lang="id" hreflang="id" data-title="Bilangan riil" 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/Reala_nombro" title="Reala nombro – 伊多语" lang="io" hreflang="io" data-title="Reala 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/Rauntala" title="Rauntala – 冰岛语" lang="is" hreflang="is" data-title="Rauntala" 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_reale" title="Numero reale – 意大利语" lang="it" hreflang="it" data-title="Numero reale" 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/%E5%AE%9F%E6%95%B0" title="実数 – 日语" lang="ja" hreflang="ja" data-title="実数" 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-jam mw-list-item"><a href="https://jam.wikipedia.org/wiki/Riil_nomba" title="Riil nomba – 牙買加克里奧爾英文" lang="jam" hreflang="jam" data-title="Riil nomba" data-language-autonym="Patois" data-language-local-name="牙買加克里奧爾英文" class="interlanguage-link-target"><span>Patois</span></a></li><li class="interlanguage-link interwiki-jbo mw-list-item"><a href="https://jbo.wikipedia.org/wiki/pavycimdyna%27u" title="pavycimdyna'u – 逻辑语" lang="jbo" hreflang="jbo" data-title="pavycimdyna'u" data-language-autonym="La .lojban." data-language-local-name="逻辑语" class="interlanguage-link-target"><span>La .lojban.</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%9C%E1%83%90%E1%83%9B%E1%83%93%E1%83%95%E1%83%98%E1%83%9A%E1%83%98_%E1%83%A0%E1%83%98%E1%83%AA%E1%83%AE%E1%83%95%E1%83%98" title="ნამდვილი რიცხვი – 格鲁吉亚语" lang="ka" hreflang="ka" data-title="ნამდვილი 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-kbp mw-list-item"><a href="https://kbp.wikipedia.org/wiki/Si%C5%8B%C5%8B_%C3%B1%CA%8A%C5%8B_(t%CA%8A%CA%8Az%CA%8A%CA%8A)" title="Siŋŋ ñʊŋ (tʊʊzʊʊ) – Kabiye" lang="kbp" hreflang="kbp" data-title="Siŋŋ ñʊŋ (tʊʊzʊʊ)" data-language-autonym="Kabɩyɛ" data-language-local-name="Kabiye" class="interlanguage-link-target"><span>Kabɩyɛ</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%9D%D0%B0%D2%9B%D1%82%D1%8B_%D1%81%D0%B0%D0%BD" title="Нақты сан – 哈萨克语" lang="kk" hreflang="kk" data-title="Нақты 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-km mw-list-item"><a href="https://km.wikipedia.org/wiki/%E1%9E%85%E1%9F%86%E1%9E%93%E1%9E%BD%E1%9E%93%E1%9E%96%E1%9E%B7%E1%9E%8F" title="ចំនួនពិត – 高棉语" lang="km" hreflang="km" data-title="ចំនួនពិត" data-language-autonym="ភាសាខ្មែរ" data-language-local-name="高棉语" class="interlanguage-link-target"><span>ភាសាខ្មែរ</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%A8%E0%B3%88%E0%B2%9C_%E0%B2%B8%E0%B2%82%E0%B2%96%E0%B3%8D%E0%B2%AF%E0%B3%86" title="ನೈಜ ಸಂಖ್ಯೆ – 卡纳达语" lang="kn" hreflang="kn" data-title="ನೈಜ ಸಂಖ್ಯೆ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="卡纳达语" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%8B%A4%EC%88%98" title="실수 – 韩语" lang="ko" hreflang="ko" data-title="실수" 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_rast%C3%AEn" title="Hejmarên rastîn – 库尔德语" lang="ku" hreflang="ku" data-title="Hejmarên rastîn" 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%90%D0%BD%D1%8B%D0%BA_%D1%81%D0%B0%D0%BD" title="Анык сан – 柯尔克孜语" lang="ky" hreflang="ky" data-title="Анык 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_realis" title="Numerus realis – 拉丁语" lang="la" hreflang="la" data-title="Numerus realis" 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-lfn mw-list-item"><a href="https://lfn.wikipedia.org/wiki/Numero_real" title="Numero real – 新共同語言" lang="lfn" hreflang="lfn" data-title="Numero real" data-language-autonym="Lingua Franca Nova" data-language-local-name="新共同語言" class="interlanguage-link-target"><span>Lingua Franca Nova</span></a></li><li class="interlanguage-link interwiki-li mw-list-item"><a href="https://li.wikipedia.org/wiki/Re%C3%ABel_getal" title="Reëel getal – 林堡语" lang="li" hreflang="li" data-title="Reëel getal" data-language-autonym="Limburgs" data-language-local-name="林堡语" class="interlanguage-link-target"><span>Limburgs</span></a></li><li class="interlanguage-link interwiki-lij mw-list-item"><a href="https://lij.wikipedia.org/wiki/Numeri_re%C3%A6" title="Numeri reæ – 利古里亚语" lang="lij" hreflang="lij" data-title="Numeri reæ" data-language-autonym="Ligure" data-language-local-name="利古里亚语" class="interlanguage-link-target"><span>Ligure</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Numer_real" title="Numer real – 倫巴底文" lang="lmo" hreflang="lmo" data-title="Numer real" 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%88%E0%BA%B4%E0%BA%87" title="ຈຳນວນຈິງ – 老挝语" lang="lo" hreflang="lo" data-title="ຈຳນວນຈິງ" 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/Realusis_skai%C4%8Dius" title="Realusis skaičius – 立陶宛语" lang="lt" hreflang="lt" data-title="Realusis 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/Re%C4%81ls_skaitlis" title="Reāls skaitlis – 拉脱维亚语" lang="lv" hreflang="lv" data-title="Reāls 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_voatsapa" title="Isa voatsapa – 马拉加斯语" lang="mg" hreflang="mg" data-title="Isa voatsapa" 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%A0%D0%B5%D0%B0%D0%BB%D0%B5%D0%BD_%D0%B1%D1%80%D0%BE%D1%98" title="Реален број – 马其顿语" lang="mk" hreflang="mk" data-title="Реален 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%B5%E0%B4%BE%E0%B4%B8%E0%B5%8D%E0%B4%A4%E0%B4%B5%E0%B4%BF%E0%B4%95%E0%B4%B8%E0%B4%82%E0%B4%96%E0%B5%8D%E0%B4%AF" title="വാസ്തവികസംഖ്യ – 马拉雅拉姆语" lang="ml" hreflang="ml" data-title="വാസ്തവികസംഖ്യ" data-language-autonym="മലയാളം" data-language-local-name="马拉雅拉姆语" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%B5%E0%A4%BE%E0%A4%B8%E0%A5%8D%E0%A4%A4%E0%A4%B5%E0%A4%BF%E0%A4%95_%E0%A4%B8%E0%A4%82%E0%A4%96%E0%A5%8D%E0%A4%AF%E0%A4%BE" title="वास्तविक संख्या – 马拉地语" lang="mr" hreflang="mr" data-title="वास्तविक 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_nyata" title="Nombor nyata – 马来语" lang="ms" hreflang="ms" data-title="Nombor nyata" 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-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%80%E1%80%AD%E1%80%94%E1%80%BA%E1%80%B8%E1%80%85%E1%80%85%E1%80%BA" title="ကိန်းစစ် – 缅甸语" lang="my" hreflang="my" data-title="ကိန်းစစ်" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="缅甸语" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-ne mw-list-item"><a href="https://ne.wikipedia.org/wiki/%E0%A4%B5%E0%A4%BE%E0%A4%B8%E0%A5%8D%E0%A4%A4%E0%A4%B5%E0%A4%BF%E0%A4%95_%E0%A4%B8%E0%A4%99%E0%A5%8D%E0%A4%96%E0%A5%8D%E0%A4%AF%E0%A4%BE" title="वास्तविक सङ्ख्या – 尼泊尔语" lang="ne" hreflang="ne" data-title="वास्तविक सङ्ख्या" data-language-autonym="नेपाली" data-language-local-name="尼泊尔语" class="interlanguage-link-target"><span>नेपाली</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Re%C3%ABel_getal" title="Reëel getal – 荷兰语" lang="nl" hreflang="nl" data-title="Reëel 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/Reelle_tal" title="Reelle tal – 挪威尼诺斯克语" lang="nn" hreflang="nn" data-title="Reelle 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/Reelt_tall" title="Reelt tall – 书面挪威语" lang="nb" hreflang="nb" data-title="Reelt 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-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Nombre_real" title="Nombre real – 奥克语" lang="oc" hreflang="oc" data-title="Nombre real" data-language-autonym="Occitan" data-language-local-name="奥克语" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-os mw-list-item"><a href="https://os.wikipedia.org/wiki/%D0%91%C3%A6%D0%BB%D0%B2%D1%8B%D1%80%D0%B4_%D0%BD%D1%8B%D0%BC%C3%A6%D1%86" title="Бæлвырд нымæц – 奥塞梯语" lang="os" hreflang="os" data-title="Бæлвырд нымæц" data-language-autonym="Ирон" data-language-local-name="奥塞梯语" class="interlanguage-link-target"><span>Ирон</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%B5%E0%A8%BE%E0%A8%B8%E0%A8%A4%E0%A8%B5%E0%A8%BF%E0%A8%95_%E0%A8%85%E0%A9%B0%E0%A8%95" title="ਵਾਸਤਵਿਕ ਅੰਕ – 旁遮普语" lang="pa" hreflang="pa" data-title="ਵਾਸਤਵਿਕ ਅੰਕ" 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_rzeczywiste" title="Liczby rzeczywiste – 波兰语" lang="pl" hreflang="pl" data-title="Liczby rzeczywiste" 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-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/N%C3%B9mer_real" title="Nùmer real – 皮埃蒙特文" lang="pms" hreflang="pms" data-title="Nùmer real" data-language-autonym="Piemontèis" data-language-local-name="皮埃蒙特文" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/N%C3%BAmero_real" title="Número real – 葡萄牙语" lang="pt" hreflang="pt" data-title="Número real" 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_real" title="Număr real – 罗马尼亚语" lang="ro" hreflang="ro" data-title="Număr real" 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%92%D0%B5%D1%89%D0%B5%D1%81%D1%82%D0%B2%D0%B5%D0%BD%D0%BD%D0%BE%D0%B5_%D1%87%D0%B8%D1%81%D0%BB%D0%BE" title="Вещественное число – 俄语" lang="ru" hreflang="ru" data-title="Вещественное 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-sah mw-list-item"><a href="https://sah.wikipedia.org/wiki/%D0%94%D1%8C%D0%B8%D2%A5%D0%BD%D1%8D%D1%8D%D1%85_%D1%87%D1%8B%D1%8B%D2%BB%D1%8B%D0%BB%D0%B0%D0%BB%D0%B0%D1%80" title="Дьиҥнээх чыыһылалар – 萨哈语" lang="sah" hreflang="sah" data-title="Дьиҥнээх чыыһылалар" data-language-autonym="Саха тыла" data-language-local-name="萨哈语" class="interlanguage-link-target"><span>Саха тыла</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/N%C3%B9mmuru_riali" title="Nùmmuru riali – 西西里语" lang="scn" hreflang="scn" data-title="Nùmmuru riali" 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/Realan_broj" title="Realan broj – 塞尔维亚-克罗地亚语" lang="sh" hreflang="sh" data-title="Realan 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-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B6%AD%E0%B7%8F%E0%B6%AD%E0%B7%8A%E0%B7%80%E0%B7%92%E0%B6%9A_%E0%B7%83%E0%B6%82%E0%B6%9B%E0%B7%8A%E2%80%8D%E0%B6%BA%E0%B7%8F" title="තාත්වික සංඛ්‍යා – 僧伽罗语" lang="si" hreflang="si" data-title="තාත්වික සංඛ්‍යා" data-language-autonym="සිංහල" data-language-local-name="僧伽罗语" class="interlanguage-link-target"><span>සිංහල</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Real_number" title="Real number – Simple English" lang="en-simple" hreflang="en-simple" data-title="Real 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/Re%C3%A1lne_%C4%8D%C3%ADslo" title="Reálne číslo – 斯洛伐克语" lang="sk" hreflang="sk" data-title="Reálne 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/Realno_%C5%A1tevilo" title="Realno število – 斯洛文尼亚语" lang="sl" hreflang="sl" data-title="Realno 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/Reaalloho" title="Reaalloho – 伊纳里萨米语" lang="smn" hreflang="smn" data-title="Reaalloho" 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_real%C3%AB" title="Numrat realë – 阿尔巴尼亚语" lang="sq" hreflang="sq" data-title="Numrat realë" 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%A0%D0%B5%D0%B0%D0%BB%D0%B0%D0%BD_%D0%B1%D1%80%D0%BE%D1%98" title="Реалан број – 塞尔维亚语" lang="sr" hreflang="sr" data-title="Реалан 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/Reella_tal" title="Reella tal – 瑞典语" lang="sv" hreflang="sv" data-title="Reella 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_halisi" title="Namba halisi – 斯瓦希里语" lang="sw" hreflang="sw" data-title="Namba halisi" 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%AE%E0%AF%86%E0%AE%AF%E0%AF%8D%E0%AE%AF%E0%AF%86%E0%AE%A3%E0%AF%8D" title="மெய்யெண் – 泰米尔语" lang="ta" hreflang="ta" data-title="மெய்யெண்" data-language-autonym="தமிழ்" data-language-local-name="泰米尔语" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%88%E0%B8%B3%E0%B8%99%E0%B8%A7%E0%B8%99%E0%B8%88%E0%B8%A3%E0%B8%B4%E0%B8%87" title="จำนวนจริง – 泰语" lang="th" hreflang="th" data-title="จำนวนจริง" data-language-autonym="ไทย" data-language-local-name="泰语" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Tunay_na_bilang" title="Tunay na bilang – 他加禄语" lang="tl" hreflang="tl" data-title="Tunay na bilang" data-language-autonym="Tagalog" data-language-local-name="他加禄语" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Reel_say%C4%B1lar" title="Reel sayılar – 土耳其语" lang="tr" hreflang="tr" data-title="Reel 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%94%D1%96%D0%B9%D1%81%D0%BD%D0%B5_%D1%87%D0%B8%D1%81%D0%BB%D0%BE" title="Дійсне число – 乌克兰语" lang="uk" hreflang="uk" data-title="Дійсне 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%AD%D9%82%DB%8C%D9%82%DB%8C_%D8%B9%D8%AF%D8%AF" title="حقیقی عدد – 乌尔都语" lang="ur" hreflang="ur" data-title="حقیقی 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/Haqiqiy_sonlar" title="Haqiqiy sonlar – 乌兹别克语" lang="uz" hreflang="uz" data-title="Haqiqiy 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_th%E1%BB%B1c" title="Số thực – 越南语" lang="vi" hreflang="vi" data-title="Số thực" 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-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E5%AE%9E%E6%95%B0" title="实数 – 吴语" lang="wuu" hreflang="wuu" data-title="实数" data-language-autonym="吴语" data-language-local-name="吴语" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-xal mw-list-item"><a href="https://xal.wikipedia.org/wiki/%D0%91%D3%99%D3%99%D0%BB%D2%BB%D0%B0%D0%BD_%D1%82%D0%BE%D0%B9%D0%B3" title="Бәәлһан тойг – 卡尔梅克语" lang="xal" hreflang="xal" data-title="Бәәлһан тойг" data-language-autonym="Хальмг" data-language-local-name="卡尔梅克语" class="interlanguage-link-target"><span>Хальмг</span></a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%A8%D7%A2%D7%90%D7%9C%D7%A2_%D7%A6%D7%90%D7%9C" title="רעאלע צאל – 意第绪语" lang="yi" hreflang="yi" data-title="רעאלע צאל" data-language-autonym="ייִדיש" data-language-local-name="意第绪语" class="interlanguage-link-target"><span>ייִדיש</span></a></li><li class="interlanguage-link interwiki-yo badge-Q17437798 badge-goodarticle mw-list-item" title="优良条目"><a href="https://yo.wikipedia.org/wiki/N%E1%BB%8D%CC%81mb%C3%A0_gidi" title="Nọ́mbà gidi – 约鲁巴语" lang="yo" hreflang="yo" data-title="Nọ́mbà gidi" 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-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E5%AF%A6%E6%95%B8" title="實數 – 文言文" lang="lzh" hreflang="lzh" data-title="實數" data-language-autonym="文言" data-language-local-name="文言文" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Si%CC%8Dt-s%C3%B2%CD%98" title="Si̍t-sò͘ – 闽南语" lang="nan" hreflang="nan" data-title="Si̍t-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/%E5%AF%A6%E6%95%B8" title="實數 – 粤语" lang="yue" hreflang="yue" data-title="實數" 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/Q12916#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 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.ambox{border-left-color:#36c!important}html.skin-theme-clientpref-os .mw-parser-output .ambox-speedy,html.skin-theme-clientpref-os .mw-parser-output .ambox-delete{border-left-color:#b32424!important}html.skin-theme-clientpref-os .mw-parser-output .ambox-speedy{background-color:#300!important}html.skin-theme-clientpref-os .mw-parser-output .ambox-content{border-left-color:#f28500!important}html.skin-theme-clientpref-os .mw-parser-output .ambox-style{border-left-color:#fc3!important}html.skin-theme-clientpref-os .mw-parser-output .ambox-move{border-left-color:#9932cc!important}html.skin-theme-clientpref-os .mw-parser-output .ambox-protection{border-left-color:#a2a9b1!important}}</style><table class="box-Unreferenced plainlinks metadata ambox ambox-content" role="presentation"><tbody><tr><td class="mbox-image"><div style="width:52px"><span typeof="mw:File"><a href="/wiki/File:Tango-nosources.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4e/Tango-nosources.svg/45px-Tango-nosources.svg.png" decoding="async" width="45" height="45" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4e/Tango-nosources.svg/68px-Tango-nosources.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4e/Tango-nosources.svg/90px-Tango-nosources.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<b>没有列出任何<a href="/wiki/Wikipedia:%E5%88%97%E6%98%8E%E6%9D%A5%E6%BA%90" title="Wikipedia:列明来源">参考或来源</a></b>。<span class="hide-when-compact"></span> <small class="date-container"><i>(<span class="date">2018年8月13日</span>)</i></small><span class="hide-when-compact"><br /><small>維基百科所有的內容都應該<a href="/wiki/Wikipedia:%E5%8F%AF%E4%BE%9B%E6%9F%A5%E8%AD%89" title="Wikipedia:可供查證">可供查證</a>。请协助補充<a href="/wiki/Wikipedia:%E5%8F%AF%E9%9D%A0%E6%9D%A5%E6%BA%90" title="Wikipedia:可靠来源">可靠来源</a>以<a class="external text" href="https://zh.wikipedia.org/w/index.php?title=%E5%AE%9E%E6%95%B0&action=edit">改善这篇条目</a>。无法查证的內容可能會因為異議提出而被移除。</small></span><span class="hide-when-compact"></span></div></td></tr></tbody></table> <div id="noteTA-fb278067" class="noteTA"><div class="noteTA-group"><div data-noteta-group-source="module" data-noteta-group="Math"></div></div><div class="noteTA-local"><div data-noteta-code="zh-cn:域; zh-tw:體;"></div></div></div> <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 class="mw-selflink selflink">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 href="/wiki/%E7%84%A1%E7%90%86%E6%95%B8" title="無理數">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> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Latex_real_numbers_square.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d0/Latex_real_numbers_square.svg/120px-Latex_real_numbers_square.svg.png" decoding="async" width="120" height="119" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d0/Latex_real_numbers_square.svg/180px-Latex_real_numbers_square.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d0/Latex_real_numbers_square.svg/240px-Latex_real_numbers_square.svg.png 2x" data-file-width="343" data-file-height="341" /></a><figcaption><b>實數</b>集的表示符號 (<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>)</figcaption></figure> <p>在<a href="/wiki/%E6%95%B8%E5%AD%B8" class="mw-redirect" title="數學">數學</a>中，<b>实数</b>（英語：<span lang="en">real number</span>）是<a href="/wiki/%E6%9C%89%E7%90%86%E6%95%B8" class="mw-redirect" title="有理數">有理數</a>和<a href="/wiki/%E7%84%A1%E7%90%86%E6%95%B8" title="無理數">無理數</a>的总称，前者如<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 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"></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 -4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -4}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/addc8519724f81b43a6883c5eb2c996f9fc2996f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle -4}"></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 {81}{7}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>81</mn> <mn>7</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {81}{7}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/772fe391e9a16ae2073fa6da4415099488acef71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:3.161ex; height:5.343ex;" alt="{\displaystyle {\frac {81}{7}}}"></span>；后者如<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 {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>、<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 }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }"></span>等。实数可以<a href="/wiki/%E7%9B%B4%E8%A7%80" class="mw-redirect" title="直觀">直观</a>地看作<a href="/wiki/%E5%B0%8F%E6%95%B8" class="mw-redirect" title="小數">小數</a>（<a href="/wiki/%E6%9C%89%E9%99%90%E5%B0%8F%E6%95%B0" title="有限小数">有限</a>或<a href="/wiki/%E6%97%A0%E9%99%90%E5%B0%8F%E6%95%B0" title="无限小数">無限</a>的），它們能把<a href="/wiki/%E6%95%B0%E8%BD%B4" class="mw-redirect" title="数轴">数轴</a>「填滿」。但僅僅以<a href="/wiki/%E6%9E%9A%E4%B8%BE" title="枚举">枚舉</a>的方式不能描述實數的全體。实数和<a href="/wiki/%E8%99%9A%E6%95%B0" title="虚数">虚数</a>共同构成<a href="/wiki/%E5%A4%8D%E6%95%B0_(%E6%95%B0%E5%AD%A6)" title="复数 (数学)">复数</a>。 </p><p>根据日常经验，<a href="/wiki/%E6%9C%89%E7%90%86%E6%95%B8%E9%9B%86" class="mw-redirect" title="有理數集">有理數集</a>在數軸上似乎是「<a href="/wiki/%E7%A8%A0%E5%AF%86" class="mw-redirect" title="稠密">稠密</a>」的，于是古人一直认为用有理數即能滿足<a href="/wiki/%E6%B8%AC%E9%87%8F" class="mw-disambig" title="測量">測量</a>上的實際需要。以邊長為<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 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span>公分的正方形為例，其對角線有多長？在規定的精度下（比如<a href="/wiki/%E8%AA%A4%E5%B7%AE" class="mw-redirect" title="誤差">誤差</a>小於<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 0.001}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0.001</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0.001}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3361411641bbe8caea51bf59e30f7caf87d0d7cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.297ex; height:2.176ex;" alt="{\displaystyle 0.001}"></span>公分），總可以用有理數來表示足夠精確的測量結果（比如<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 1.414}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1.414</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1.414}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60806674f032e76236bfdbd8d69fb325223726a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.297ex; height:2.176ex;" alt="{\displaystyle 1.414}"></span>公分）。但是，<a href="/wiki/%E5%8F%A4%E5%B8%8C%E8%87%98" class="mw-redirect" title="古希臘">古希臘</a><a href="/wiki/%E6%AF%95%E8%BE%BE%E5%93%A5%E6%8B%89%E6%96%AF%E5%AD%A6%E6%B4%BE" class="mw-redirect" title="毕达哥拉斯学派">畢達哥拉斯學派</a>的數學家發現，只使用有理數無法完全<a href="/wiki/%E7%B2%BE%E7%A2%BA" class="mw-redirect" title="精確">精確</a>地表示這條對角線的長度，這徹底地打擊了他們的數學理念；他們原以為： </p> <ul><li>任何兩條線段（的長度）的比，可以用<a href="/wiki/%E8%87%AA%E7%84%B6%E6%95%B8" class="mw-redirect" title="自然數">自然數</a>的比來表示。</li></ul> <p>正因如此，<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>本人甚至有「萬物皆數」的信念，這裡的數是指自然數（<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1,2,3,\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1,2,3,\ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee2ad0c84eec6102f704be18a9e709cec4029faf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.312ex; height:2.509ex;" alt="{\displaystyle 1,2,3,\ldots }"></span>），而由自然數的比就得到所有正有理數，而有理數集存在「縫隙」這一事實，對當時很多數學家來說可謂極大的打擊；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>從<a href="/wiki/%E5%8F%A4%E5%B8%8C%E8%87%98" class="mw-redirect" title="古希臘">古希臘</a>一直到17世紀，<a href="/wiki/%E6%95%B8%E5%AD%B8%E5%AE%B6" class="mw-redirect" title="數學家">數學家</a>們才慢慢接受無理數的存在，並把它和有理數平等地看作<a href="/wiki/%E6%95%B8" class="mw-redirect" title="數">數</a>；後來有<a href="/wiki/%E8%99%9A%E6%95%B0" title="虚数">虚数</a>概念的引入，為加以區別而稱作“實數”，意即“實在的數”。在當時，儘管虛數已經出現並廣為使用，實數的嚴格定義卻仍然是個難題，以至<a href="/wiki/%E5%87%BD%E6%95%B8" class="mw-redirect" title="函數">函數</a>、<a href="/wiki/%E6%9E%81%E9%99%90_(%E6%95%B0%E5%AD%A6)" title="极限 (数学)">極限</a>和<a href="/w/index.php?title=%E6%94%B6%E6%96%82%E6%80%A7&action=edit&redlink=1" class="new" title="收斂性（页面不存在）">收斂性</a>的概念都被定義清楚之後，才由十九世紀末的<a href="/wiki/%E6%88%B4%E5%BE%B7%E9%87%91" class="mw-redirect" title="戴德金">戴德金</a>、<a href="/wiki/%E5%BA%B7%E6%89%98%E5%B0%94" class="mw-redirect" title="康托尔">康托尔</a>等人對實數進行了嚴格處理。 </p><p>所有实数的集合則可稱為<b>实数系</b>（real number system）或<b>实数连续统</b>。任何一个完备的阿基米德有序域均可称为实数系。在保序同构意义下它是惟一的，常用<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>表示。由于<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \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>是定义了算数运算的运算系统，故有实数系这个名称。<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="初等數學"><span id=".E5.88.9D.E7.AD.89.E6.95.B8.E5.AD.B8"></span>初等數學</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%AE%9E%E6%95%B0&action=edit&section=1" title="编辑章节：初等數學"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>在目前的<a href="/wiki/%E5%88%9D%E7%AD%89%E6%95%B0%E5%AD%A6" title="初等数学">初等數學</a>中，没有對實數進行嚴格的定義，而一般把實數看作<a href="/wiki/%E5%B0%8F%E6%95%B8" class="mw-redirect" title="小數">小數</a>（有限或無限的）。实数的完备性可以利用幾何加以说明，即数轴上的點與實數一一對應；見<a href="/wiki/%E6%95%B0%E8%BD%B4" class="mw-redirect" title="数轴">数轴</a>。 </p><p>实数可以分为<a href="/wiki/%E6%9C%89%E7%90%86%E6%95%B0" title="有理数">有理数</a>（如<a href="/wiki/42" title="42"><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 42}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>42</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 42}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5abbb0f0316fecd2958f85a7d83b9f3df0ddd0e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle 42}"></span></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 -{\frac {23}{129}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>23</mn> <mn>129</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\frac {23}{129}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e491a3d46481a29e73e1d295021802eca734e4b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:6.132ex; height:5.176ex;" alt="{\displaystyle -{\frac {23}{129}}}"></span>）和<a href="/wiki/%E6%97%A0%E7%90%86%E6%95%B0" class="mw-redirect" title="无理数">无理数</a>（如<a href="/wiki/%E5%9C%93%E5%91%A8%E7%8E%87" title="圓周率"><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 }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }"></span></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 {\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>），或者<a href="/wiki/%E4%BB%A3%E6%95%B0%E6%95%B0" class="mw-redirect" title="代数数">代数数</a>和<a href="/wiki/%E8%B6%85%E8%B6%8A%E6%95%B0" class="mw-redirect" title="超越数">超越数</a>（有理數都是代數數）两类。实数<a href="/wiki/%E9%9B%86%E5%90%88_(%E6%95%B0%E5%AD%A6)" title="集合 (数学)">集合</a>通常用字母<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 R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span>或<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>表示。而<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} ^{n}}"> <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"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"></span>表示<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span><a href="/wiki/%E7%BB%B4" class="mw-redirect" title="维">维</a>实数空间。实数是不可数的。实数是<a href="/wiki/%E5%AE%9E%E5%88%86%E6%9E%90" class="mw-redirect" title="实分析">实分析</a>的核心研究对象。 </p><p>实数可以用来测量<a href="/wiki/%E8%BF%9E%E7%BB%AD" class="mw-redirect" title="连续">连续</a>變化的量。理论上，任何实数都可以用无限小数的方式表示，小数点的右边是一个无穷的<a href="/wiki/%E6%95%B0%E5%88%97" title="数列">数列</a>（可以是循环的，也可以是非循环的）。在实际运用中，实数经常被近似成一个有限小数（保留小数点后<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>位，<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>为正整数）。在计算机领域，由于计算机只能存储有限的小数位数，实数经常用<a href="/wiki/%E6%B5%AE%E7%82%B9%E6%95%B0" class="mw-redirect" title="浮点数">浮点数</a>来表示。 </p> <div class="mw-heading mw-heading2"><h2 id="正数与负数"><span id=".E6.AD.A3.E6.95.B0.E4.B8.8E.E8.B4.9F.E6.95.B0"></span>正数与负数</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%AE%9E%E6%95%B0&action=edit&section=2" title="编辑章节：正数与负数"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r85100532">.mw-parser-output .hatnote{font-size:small}.mw-parser-output div.hatnote{padding-left:2em;margin-bottom:0.8em;margin-top:0.8em}.mw-parser-output .hatnote-notice-img::after{content:"\202f \202f \202f \202f "}.mw-parser-output .hatnote-notice-img-small::after{content:"\202f \202f "}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}body.skin-minerva .mw-parser-output .hatnote-notice-img,body.skin-minerva .mw-parser-output .hatnote-notice-img-small{display:none}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">主条目：<a href="/wiki/%E6%AD%A3%E6%95%B0" class="mw-redirect" title="正数">正数</a>和<a href="/wiki/%E8%B4%9F%E6%95%B0" title="负数">负数</a></div> <p>实数是一个<a href="/wiki/%E9%9B%86%E5%90%88_(%E6%95%B0%E5%AD%A6)" title="集合 (数学)">集合</a>，通常可以分为<a href="/wiki/%E6%AD%A3%E6%95%B0" class="mw-redirect" title="正数">正数</a>、<a href="/wiki/%E8%B4%9F%E6%95%B0" title="负数">负数</a>和<a href="/wiki/0" title="0">零</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 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"></span>）三类。「正数」（符号：<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>）即<a href="/wiki/%E5%A4%A7%E4%BA%8E" class="mw-redirect" title="大于">大于</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 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"></span>的实数，而「负数」（符号：<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>）即<a href="/wiki/%E5%B0%8F%E4%BA%8E" class="mw-redirect" title="小于">小于</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 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"></span>的实数。与实数一样，两者都是<a href="/wiki/%E4%B8%8D%E5%8F%AF%E6%95%B8" class="mw-redirect" title="不可數">不可數</a>的<a href="/wiki/%E6%97%A0%E9%99%90%E9%9B%86%E5%90%88" title="无限集合">無限集合</a>。正数的<a href="/wiki/%E7%9B%B8%E5%8F%8D%E6%95%B0" class="mw-redirect" title="相反数">相反数</a>一定是负数，负数的相反数也一定是正数。除正數和負數外，通常将<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 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"></span>與正數统称为「非負數」（符号：<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} _{0}^{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} _{0}^{+}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d54edc82900153fe95d7604ca3418e80cff281e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.189ex; height:3.176ex;" alt="{\displaystyle \mathbb {R} _{0}^{+}}"></span>），而将<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 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"></span>與負數统称为「非正數」（符号：<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} _{0}^{-}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} _{0}^{-}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04a0ccd8b5b081f8906b167734243b7785e80dbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.189ex; height:3.176ex;" alt="{\displaystyle \mathbb {R} _{0}^{-}}"></span>）。这和<a href="/wiki/%E6%95%B4%E6%95%B0" title="整数">整数</a>可以分为<a href="/wiki/%E6%AD%A3%E6%95%B4%E6%95%B8" class="mw-redirect" title="正整數">正整數</a>、<a href="/wiki/%E8%B4%9F%E6%95%B4%E6%95%B0" class="mw-redirect" title="负整数">负整数</a>和零（<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 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"></span>），而<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 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"></span>與正整數通常统称为非負整數、<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 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"></span>與負整數则通常统称为非正整數非常相似。另外，只有实数可以分为正和负等，<a href="/wiki/%E8%99%9A%E6%95%B0" title="虚数">虚数</a>是没有正负之分的。 </p> <div class="mw-heading mw-heading2"><h2 id="历史"><span id=".E5.8E.86.E5.8F.B2"></span>历史</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%AE%9E%E6%95%B0&action=edit&section=3" title="编辑章节：历史"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>在公元前500年左右，以<a href="/wiki/%E6%AF%95%E8%BE%BE%E5%93%A5%E6%8B%89%E6%96%AF" title="毕达哥拉斯">毕达哥拉斯</a>为首的<a href="/wiki/%E5%B8%8C%E8%85%8A" title="希腊">希腊</a><a href="/wiki/%E6%95%B0%E5%AD%A6%E5%AE%B6" title="数学家">数学家</a>们認識到有理數在幾何上不能滿足需要，但<a href="/wiki/%E6%AF%95%E8%BE%BE%E5%93%A5%E6%8B%89%E6%96%AF" title="毕达哥拉斯">毕达哥拉斯</a>本身並不承認無理數的存在。直到17世纪，实数才在欧洲被广泛接受。18世纪，<a href="/wiki/%E5%BE%AE%E7%A7%AF%E5%88%86%E5%AD%A6" title="微积分学">微积分学</a>在实数的基础上发展起来。直到1871年，<a href="/wiki/%E5%BE%B7%E5%9B%BD" title="德国">德国</a><a href="/wiki/%E6%95%B0%E5%AD%A6%E5%AE%B6" title="数学家">数学家</a><a href="/wiki/%E5%BA%B7%E6%89%98%E5%B0%94" class="mw-redirect" title="康托尔">康托尔</a>第一次提出了实数的严格定义。 </p> <div class="mw-heading mw-heading2"><h2 id="定义"><span id=".E5.AE.9A.E4.B9.89"></span>定义</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%AE%9E%E6%95%B0&action=edit&section=4" title="编辑章节：定义"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="從有理數构造實數"><span id=".E5.BE.9E.E6.9C.89.E7.90.86.E6.95.B8.E6.9E.84.E9.80.A0.E5.AF.A6.E6.95.B8"></span>從有理數构造實數</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%AE%9E%E6%95%B0&action=edit&section=5" title="编辑章节：從有理數构造實數"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>实数可以用通过收敛于一个唯一实数的十进制或二进制展开如<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 {3,3.1,3.14,3.141,3.1415,3.14159...}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mo>,</mo> <mn>3.1</mn> <mo>,</mo> <mn>3.14</mn> <mo>,</mo> <mn>3.141</mn> <mo>,</mo> <mn>3.1415</mn> <mo>,</mo> <mn>3.14159...</mn> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {3,3.1,3.14,3.141,3.1415,3.14159...}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2441e33c4dc9cb3d5c7aeb75cc0f1d534436cc38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:34.756ex; height:2.509ex;" alt="{\displaystyle {3,3.1,3.14,3.141,3.1415,3.14159...}}"></span>所定义的序列的方式而构造为有理数的<a href="/wiki/%E5%AE%8C%E5%82%99%E5%8C%96" class="mw-redirect" title="完備化">完備化</a>。實數可以不同方式從<a href="/wiki/%E6%9C%89%E7%90%86%E6%95%B8" class="mw-redirect" title="有理數">有理數</a>构造出來。这里给出其中一种，其他方法请詳見<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>。 </p> <div class="mw-heading mw-heading3"><h3 id="公理化方法"><span id=".E5.85.AC.E7.90.86.E5.8C.96.E6.96.B9.E6.B3.95"></span>公理化方法</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%AE%9E%E6%95%B0&action=edit&section=6" title="编辑章节：公理化方法"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>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 \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> 是所有实数的<a href="/wiki/%E9%9B%86%E5%90%88_(%E6%95%B0%E5%AD%A6)" title="集合 (数学)">集合</a>，则： </p> <ul><li>集合 <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> 是一个<a href="/wiki/%E5%9F%9F_(%E6%95%B0%E5%AD%A6)" title="域 (数学)">域</a>：可以作<a href="/wiki/%E5%8A%A0%E6%B3%95" title="加法">加</a>、<a href="/wiki/%E5%87%8F%E6%B3%95" class="mw-redirect" title="减法">减</a>、<a href="/wiki/%E4%B9%98%E6%B3%95" title="乘法">乘</a>、<a href="/wiki/%E9%99%A4%E6%B3%95" title="除法">除</a>运算，且有如<a href="/wiki/%E4%BA%A4%E6%8D%A2%E5%BE%8B" class="mw-redirect" title="交换律">交换律</a>，<a href="/wiki/%E7%BB%93%E5%90%88%E5%BE%8B" title="结合律">结合律</a>等常见性质。</li> <li>域 <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> 是个<a href="/wiki/%E6%9C%89%E5%BA%8F%E5%9F%9F" title="有序域">有序域</a>，即存在<a href="/wiki/%E5%85%A8%E5%BA%8F%E5%85%B3%E7%B3%BB" title="全序关系">全序关系</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 \geq }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≥</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \geq }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bcef7c0e95bb77a35fd1a874ca91f425215f3c26" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \geq }"></span> ，对所有实数<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 x,y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ea0abffd33a692ded22accc104515a032851dff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.519ex; height:2.009ex;" alt="{\displaystyle x,y}"></span>和<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"></span>： <ul><li>若<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\geq y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>≥</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\geq y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aded94d634c48071188bf96a76a4d3b7dfb28470" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.584ex; height:2.343ex;" alt="{\displaystyle x\geq y}"></span>则<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+z\geq y+z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>+</mo> <mi>z</mi> <mo>≥</mo> <mi>y</mi> <mo>+</mo> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x+z\geq y+z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84694c0444e70f202529044131df4020dbc5605e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.441ex; height:2.343ex;" alt="{\displaystyle x+z\geq y+z}"></span>·</li> <li>若<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\geq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2608e2b392b079f5b763f27bf52883dbee3b64a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.591ex; height:2.343ex;" alt="{\displaystyle x\geq 0}"></span>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 y\geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y\geq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/130e8795bc869a5b823133c5a0972693605c00bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.416ex; height:2.509ex;" alt="{\displaystyle y\geq 0}"></span>则<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 xy\geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mi>y</mi> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle xy\geq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7d706d1523aed20dc910962ade635dd7ccabe5b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.746ex; height:2.509ex;" alt="{\displaystyle xy\geq 0}"></span></li></ul></li> <li>集合 <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> 满足<a href="/wiki/%E6%88%B4%E5%BE%B7%E9%87%91%E5%AE%8C%E5%A4%87%E6%80%A7" class="mw-redirect" title="戴德金完备性">戴德金完备性</a>，即任意 <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> 的非空子集<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 S(S\subseteq \mathbb {R} ,S\neq \varnothing )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo>⊆</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>,</mo> <mi>S</mi> <mo>≠</mo> <mi class="MJX-variant">∅</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S(S\subseteq \mathbb {R} ,S\neq \varnothing )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d252b534421a991cdaa615b74797382a16aa015a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.024ex; height:2.843ex;" alt="{\displaystyle S(S\subseteq \mathbb {R} ,S\neq \varnothing )}"></span>，若<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span>在 <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> 内有<a href="/wiki/%E4%B8%8A%E7%95%8C" class="mw-redirect" title="上界">上界</a>，那么<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 S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span>在 <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> 内有<a href="/wiki/%E4%B8%8A%E7%A1%AE%E7%95%8C" class="mw-redirect" title="上确界">上确界</a>。</li></ul> <p>最后一条是区分实数和有理数的关键。例如所有<a href="/wiki/%E5%B9%B3%E6%96%B9" title="平方">平方</a>小于<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/901fc910c19990d0dbaaefe4726ceb1a4e217a0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 2}"></span>的有理数的集合存在有理数上界，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 1.5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1.5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1.5}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de4cf334519dcf06a7941d7bb9411c202f77c4ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.972ex; height:2.176ex;" alt="{\displaystyle 1.5}"></span>；但是不存在有理数上确界（因为<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 {\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4afc1e27d418021bf10898eb44a7f5f315735ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}}"></span>不是有理数）。 </p><p>實數通过上述性质唯一确定。更准确的说，给定任意两个戴德金完备的有序域 <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} _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} _{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/880c089509beeddaae996a6985f29fb00a7f45e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.732ex; height:2.509ex;" alt="{\displaystyle \mathbb {R} _{1}}"></span>和 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} _{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/139c792da4e0dfb9cca58316eb540fb919dce79d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.732ex; height:2.509ex;" alt="{\displaystyle \mathbb {R} _{2}}"></span>，存在从 <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} _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} _{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/880c089509beeddaae996a6985f29fb00a7f45e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.732ex; height:2.509ex;" alt="{\displaystyle \mathbb {R} _{1}}"></span>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 \mathbb {R} _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} _{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/139c792da4e0dfb9cca58316eb540fb919dce79d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.732ex; height:2.509ex;" alt="{\displaystyle \mathbb {R} _{2}}"></span>的唯一的域<a href="/wiki/%E5%90%8C%E6%A7%8B" class="mw-redirect" title="同構">同構</a>，即代數學上兩者可看作是相同。 </p> <div class="mw-heading mw-heading2"><h2 id="例子"><span id=".E4.BE.8B.E5.AD.90"></span>例子</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%AE%9E%E6%95%B0&action=edit&section=7" title="编辑章节：例子"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><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 15}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>15</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 15}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea331af19ed2ccc36bb864409b6c305e18cff30f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle 15}"></span>（整数）</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 2.121}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2.121</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2.121}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa5a0a23ef3446d8b9b64e926dd303050552ecb2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.297ex; height:2.176ex;" alt="{\displaystyle 2.121}"></span>（有限小数）</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 -1.3333333\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>1.3333333</mn> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -1.3333333\ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c77e7f37d6a58ac51d95f7542a033e31750f1608" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:14.865ex; height:2.343ex;" alt="{\displaystyle -1.3333333\ldots }"></span>（无限循环小数）</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 \pi =3.1415926\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo>=</mo> <mn>3.1415926</mn> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi =3.1415926\ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f97bc1f3178a625a35786af4c2e8e2c287581e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:17.487ex; height:2.176ex;" alt="{\displaystyle \pi =3.1415926\ldots }"></span> （无理数）</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 {\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>（无理数）</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 {\frac {1}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7093420fb1b77a06432a4e0d9eba91705cef6d02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:1.999ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{3}}}"></span> （分数）</li></ul> <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=%E5%AE%9E%E6%95%B0&action=edit&section=8" title="编辑章节：性质"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="基本运算"><span id=".E5.9F.BA.E6.9C.AC.E8.BF.90.E7.AE.97"></span>基本运算</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%AE%9E%E6%95%B0&action=edit&section=9" title="编辑章节：基本运算"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>在实数域内，可实现的基本<a href="/wiki/%E8%BF%90%E7%AE%97" title="运算">运算</a>有<a href="/wiki/%E5%8A%A0" class="mw-redirect" title="加">加</a>、<a href="/wiki/%E5%87%8F" class="mw-redirect" title="减">减</a>、<a href="/wiki/%E4%B9%98" class="mw-redirect" title="乘">乘</a>、<a href="/wiki/%E9%99%A4" class="mw-redirect" title="除">除</a>、<a href="/wiki/%E4%B9%98%E6%96%B9" class="mw-redirect" title="乘方">乘方</a>等，对非负数还可以进行<a href="/wiki/%E9%96%8B%E6%96%B9" class="mw-redirect" title="開方">开方</a>运算。实数加、减、乘、除（除数不为零）、平方后结果还是实数。任何实数都可以开奇次方，结果仍是实数；只有非负实数才能开偶次方，其结果还是实数。 </p> <div class="mw-heading mw-heading3"><h3 id="连续性或完備性"><span id=".E8.BF.9E.E7.BB.AD.E6.80.A7.E6.88.96.E5.AE.8C.E5.82.99.E6.80.A7"></span>连续性或完備性</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%AE%9E%E6%95%B0&action=edit&section=10" title="编辑章节：连续性或完備性"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>作为<a href="/wiki/%E5%BA%A6%E9%87%8F%E7%A9%BA%E9%97%B4" title="度量空间">度量空間</a>或<a href="/wiki/%E4%B8%80%E8%87%B4%E7%A9%BA%E9%96%93" class="mw-redirect" title="一致空間">一致空間</a>，實數集合是一个<a href="/wiki/%E5%AE%8C%E5%A4%87%E7%A9%BA%E9%97%B4" title="完备空间">完备空间</a>，它有以下性质： </p> <dl><dd>所有實數的<a href="/wiki/%E6%9F%AF%E8%A5%BF%E5%BA%8F%E5%88%97" title="柯西序列">柯西序列</a>都有一個實數<a href="/wiki/%E6%9E%81%E9%99%90_(%E5%BA%8F%E5%88%97)" class="mw-redirect" title="极限 (序列)">極限</a>。</dd></dl> <p>有理數集合就不是完备空间。例如，<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 (1,1.4,1.41,1.414,1.4142,1.41421,\ldots )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>1.4</mn> <mo>,</mo> <mn>1.41</mn> <mo>,</mo> <mn>1.414</mn> <mo>,</mo> <mn>1.4142</mn> <mo>,</mo> <mn>1.41421</mn> <mo>,</mo> <mo>…</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1,1.4,1.41,1.414,1.4142,1.41421,\ldots )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f5a6b9c901d3d4f7c77a8c3046b982be3d80d4c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.382ex; height:2.843ex;" alt="{\displaystyle (1,1.4,1.41,1.414,1.4142,1.41421,\ldots )}"></span>是有理數的柯西序列，但沒有有理數極限。实际上，它有個實數極限<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>。實數是有理數的<a href="/wiki/%E5%AE%8C%E5%82%99%E5%8C%96" class="mw-redirect" title="完備化">完备化</a>：這亦是构造實數集合的一种方法。 </p> <div class="mw-heading mw-heading3"><h3 id="完备的有序域（有序性）"><span id=".E5.AE.8C.E5.A4.87.E7.9A.84.E6.9C.89.E5.BA.8F.E5.9F.9F.EF.BC.88.E6.9C.89.E5.BA.8F.E6.80.A7.EF.BC.89"></span>完备的有序域（有序性）</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%AE%9E%E6%95%B0&action=edit&section=11" title="编辑章节：完备的有序域（有序性）"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>实数集合通常被描述为“完备的有序域”，这可以几种解释。 </p> <ul><li>首先，有序域可以是<a href="/wiki/%E5%AE%8C%E5%A4%87%E6%A0%BC" class="mw-redirect" title="完备格">完备格</a>。然而，很容易发现没有有序域会是完备格。这是由于有序域没有最大元素（对任意元素<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 z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"></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 z+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bd946396897f1d63172e5a7d20c83ec160071e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.091ex; height:2.343ex;" alt="{\displaystyle z+1}"></span>将更大）。所以，这裡的“完备”不是完备格的意思。</li> <li>另外，有序域满足<a href="/wiki/%E6%88%B4%E5%BE%B7%E9%87%91%E5%AE%8C%E5%A4%87%E6%80%A7" class="mw-redirect" title="戴德金完备性">戴德金完备性</a>，这在上述「公理」中已经定义。上述的唯一性也说明了这裡的“完备”是指戴德金完备性的意思。这个完备性的意思非常接近采用<a href="/wiki/%E6%88%B4%E5%BE%B7%E9%87%91%E5%88%86%E5%89%B2" title="戴德金分割">戴德金分割</a>来构造实数的方法，即从（有理数）有序域出发，通过标准的方法建立戴德金完备性。</li> <li>这两个完备性的概念都忽略了域的结构。然而，有序<a href="/wiki/%E7%BE%A4" title="群">群</a>（域是种特殊的群）可以定义一致空间，而一致空间又有<a href="/wiki/%E5%AE%8C%E5%A4%87%E7%A9%BA%E9%97%B4" title="完备空间">完备空间</a>的概念。上述「完备性」中所述的只是一个特例。（这里采用一致空间中的完备性概念，而不是相关的人们熟知的度量空间的完备性，这是由于度量空间的定义依赖于实数的性质。）当然，<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>并不是<i>唯一的</i>一致完备的有序域，但它是唯一的一致完备的「<a href="/w/index.php?title=%E9%98%BF%E5%9F%BA%E7%B1%B3%E5%BE%B7%E5%9F%9F&action=edit&redlink=1" class="new" title="阿基米德域（页面不存在）">阿基米德域</a>」。实际上，“完备的阿基米德域”比“完备的有序域”更常见。可以证明，任意一致完备的阿基米德域必然是戴德金完备的（当然反之亦然）。这个完备性的意思非常接近采用柯西序列来构造实数的方法，即从（有理数）阿基米德域出发，通过标准的方法建立一致完备性。</li> <li>“完备的阿基米德域”最早是由<a href="/wiki/%E5%B8%8C%E5%B0%94%E4%BC%AF%E7%89%B9" class="mw-redirect" title="希尔伯特">希尔伯特</a>提出来的，他还想表达一些不同于上述的意思。他认为，实数构成了「最大的」阿基米德域，即所有其他的阿基米德域都是<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 \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>的子域。这样<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>是“完备的”是指，在其中加入任何元素都将使它不再是阿基米德域。这个完备性的意思非常接近用<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)">超实数</a>来构造实数的方法，即从某个包含所有（超实数）有序域的纯类出发，从其子域中找出最大的阿基米德域。</li></ul> <div class="mw-heading mw-heading3"><h3 id="高级性质"><span id=".E9.AB.98.E7.BA.A7.E6.80.A7.E8.B4.A8"></span>高级性质</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%AE%9E%E6%95%B0&action=edit&section=12" title="编辑章节：高级性质"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>实数集是<a href="/wiki/%E4%B8%8D%E5%8F%AF%E6%95%B8" class="mw-redirect" title="不可數">不可数</a>的，也就是说，实数的个数严格多于<a href="/wiki/%E8%87%AA%E7%84%B6%E6%95%B0" title="自然数">自然数</a>的个数（尽管两者都是<a href="/wiki/%E6%97%A0%E7%A9%B7%E5%A4%A7" class="mw-redirect" title="无穷大">无穷大</a>）。这一点，可以通过<a href="/wiki/%E5%BA%B7%E6%89%98%E7%88%BE%E5%B0%8D%E8%A7%92%E7%B7%9A%E6%96%B9%E6%B3%95" class="mw-redirect" title="康托爾對角線方法">康托尔对角线方法</a>证明。实际上，实数集的<a href="/wiki/%E5%8A%BF_(%E6%95%B0%E5%AD%A6)" title="势 (数学)">势</a>为2<sup><i>ω</i></sup>（请参见<a href="/wiki/%E8%BF%9E%E7%BB%AD%E7%BB%9F%E7%9A%84%E5%8A%BF" title="连续统的势">连续统的势</a>），即<a href="/wiki/%E8%87%AA%E7%84%B6%E6%95%B0" title="自然数">自然数</a>集的<a href="/wiki/%E5%86%AA%E9%9B%86" title="冪集">幂集</a>的势。由于实数集中只有<a href="/wiki/%E5%8F%AF%E6%95%B0%E9%9B%86" class="mw-redirect" title="可数集">可数集</a>个数的元素可能是<a href="/wiki/%E4%BB%A3%E6%95%B0%E6%95%B0" class="mw-redirect" title="代数数">代数数</a>，<a href="/w/index.php?title=%E7%BB%9D%E5%A4%A7%E5%A4%9A%E6%95%B0&action=edit&redlink=1" class="new" title="绝大多数（页面不存在）">绝大多数</a>实数是<a href="/wiki/%E8%B6%85%E8%B6%8A%E6%95%B0" class="mw-redirect" title="超越数">超越数</a>。实数集的子集中，不存在其势严格大于自然数集的势且严格小于实数集的势的集合，这就是<a href="/wiki/%E8%BF%9E%E7%BB%AD%E7%BB%9F%E5%81%87%E8%AE%BE" title="连续统假设">连续统假设</a>。该假设不能被证明是否正确，这是因为它和<a href="/wiki/%E9%9B%86%E5%90%88%E8%AE%BA" title="集合论">集合论</a>的<a href="/wiki/%E7%AD%96%E6%A2%85%E6%B4%9B-%E5%BC%97%E5%85%B0%E5%85%8B%E5%B0%94%E9%9B%86%E5%90%88%E8%AE%BA" title="策梅洛-弗兰克尔集合论">ZF(ZFC)公理系统</a>相互独立。</li> <li>所有非负实数的<a href="/wiki/%E5%B9%B3%E6%96%B9%E6%A0%B9" title="平方根">平方根</a>属于<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span>，但这对负数不成立。这表明<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 R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span>上的序是由其代数结构确定的。而且，所有奇数次多项式至少有一个根属于<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span>。这两个性质使<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 R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span>成为<a href="/wiki/%E5%AF%A6%E5%B0%81%E9%96%89%E5%9F%9F" class="mw-redirect" title="實封閉域">实封闭域</a>的最主要的实例。证明这一点就是对<a href="/wiki/%E4%BB%A3%E6%95%B0%E5%9F%BA%E6%9C%AC%E5%AE%9A%E7%90%86" title="代数基本定理">代数基本定理</a>的证明的前半部分。</li> <li>实数集拥有一个规范的<a href="/wiki/%E6%B5%8B%E5%BA%A6" title="测度">测度</a>，即<a href="/wiki/%E5%8B%92%E8%B4%9D%E6%A0%BC%E6%B5%8B%E5%BA%A6" title="勒贝格测度">勒贝格测度</a>。</li> <li>实数集的上确界公理用到了实数集的子集，这是一种二阶逻辑的陈述。不可能只采用<a href="/wiki/%E4%B8%80%E9%98%B6%E9%80%BB%E8%BE%91" title="一阶逻辑">一阶逻辑</a>来刻画实数集：1. <a href="/wiki/L%C3%B6wenheim-Skolem%E5%AE%9A%E7%90%86" class="mw-redirect" title="Löwenheim-Skolem定理">Löwenheim-Skolem定理</a>说明，存在一个实数集的可数稠密子集，它在一阶逻辑中正好满足和实数集自身完全相同的命题；2. <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)">超实数</a>的集合远远大于<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span>，但也同样满足和<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 R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span>一样的一阶逻辑命题。满足和<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 R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span>一样的一阶逻辑命题的有序域称为<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 R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span>的<a href="/wiki/%E9%9D%9E%E6%A0%87%E5%87%86%E6%A8%A1%E5%9E%8B" title="非标准模型">非标准模型</a>。这就是<a href="/wiki/%E9%9D%9E%E6%A0%87%E5%87%86%E5%88%86%E6%9E%90" title="非标准分析">非标准分析</a>的研究内容，在非标准模型中证明一阶逻辑命题（可能比在<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 R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span>中证明要简单一些），从而确定这些命题在<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 R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span>中也成立。</li></ul> <div class="mw-heading mw-heading3"><h3 id="拓撲性質"><span id=".E6.8B.93.E6.92.B2.E6.80.A7.E8.B3.AA"></span>拓撲性質</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%AE%9E%E6%95%B0&action=edit&section=13" title="编辑章节：拓撲性質"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>實數集構成一個<a href="/wiki/%E5%BA%A6%E9%87%8F%E7%A9%BA%E9%97%B4" title="度量空间">度量空間</a>：<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>和<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span>間的距離定為<a href="/wiki/%E7%B5%95%E5%B0%8D%E5%80%BC" class="mw-redirect" title="絕對值">絕對值</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |x-y|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |x-y|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9eca622011e2b10520e69a7fa83f3bd75159ab66" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.619ex; height:2.843ex;" alt="{\displaystyle |x-y|}"></span>。作為一個<a href="/wiki/%E5%85%A8%E5%BA%8F%E9%9B%86" class="mw-redirect" title="全序集">全序集</a>，它也具有<a href="/wiki/%E5%BA%8F%E6%8B%93%E6%92%B2" title="序拓撲">序拓撲</a>。這裡，從度量和序關係得到的拓撲相同。實數集又是一<a href="/wiki/%E7%B6%AD" class="mw-redirect" title="維">維</a>的<a href="/wiki/%E5%8F%AF%E7%BC%A9%E7%A9%BA%E9%97%B4" class="mw-redirect" title="可缩空间">可縮空間</a>（所以也是<a href="/wiki/%E9%80%A3%E9%80%9A%E7%A9%BA%E9%96%93" class="mw-redirect" title="連通空間">連通空間</a>）、<a href="/wiki/%E5%B1%80%E9%83%A8%E7%B4%A7%E8%87%B4%E7%A9%BA%E9%97%B4" class="mw-redirect" title="局部紧致空间">局部緊緻空間</a>、<a href="/wiki/%E5%8F%AF%E5%88%86%E7%A9%BA%E9%97%B4" title="可分空间">可分空間</a>、<span class="ilh-all" data-orig-title="貝爾空間" data-lang-code="en" data-lang-name="英语" data-foreign-title="Baire space"><span class="ilh-page"><a href="/w/index.php?title=%E8%B2%9D%E7%88%BE%E7%A9%BA%E9%96%93&action=edit&redlink=1" class="new" title="貝爾空間（页面不存在）">貝爾空間</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/Baire_space" class="extiw" title="en:Baire space"><span lang="en" dir="auto">Baire space</span></a></span>）</span></span>。但實數集不是<a href="/wiki/%E7%B4%A7%E8%87%B4%E7%A9%BA%E9%97%B4" class="mw-redirect" title="紧致空间">緊緻空間</a>。這些可以通過特定的性質來確定，例如，無限連續可分的<a href="/wiki/%E5%BA%8F%E6%8B%93%E6%92%B2" title="序拓撲">序拓撲</a>必須和實數集<a href="/wiki/%E5%90%8C%E8%83%9A" title="同胚">同胚</a>。以下是實數的拓撲性質總覽： </p> <ul><li>令<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mspace width="thickmathspace" /> </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/ac43f811d675e91eb98db33b8d59ed11866220a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.875ex; height:1.676ex;" alt="{\displaystyle a\;}"></span>為一實數。<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 a\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mspace width="thickmathspace" /> </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/ac43f811d675e91eb98db33b8d59ed11866220a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.875ex; height:1.676ex;" alt="{\displaystyle a\;}"></span>的<a href="/wiki/%E9%84%B0%E5%9F%9F" class="mw-redirect" title="鄰域">鄰域</a>是實數集中一個包括一段含有<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mspace width="thickmathspace" /> </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/ac43f811d675e91eb98db33b8d59ed11866220a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.875ex; height:1.676ex;" alt="{\displaystyle a\;}"></span>的線段的子集。</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 \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>是<a href="/wiki/%E5%8F%AF%E5%88%86%E7%A9%BA%E9%97%B4" title="可分空间">可分空間</a>。</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 \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>在<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>中處處稠密。</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 \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>的<a href="/wiki/%E5%BC%80%E9%9B%86" title="开集">開集</a>是開區間的聯集。</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 \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>的緊子集等价于有界閉集。特別是：所有含端點的有限線段都是緊子集。</li> <li>每個<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>中的有界序列都有收斂子序列。</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 \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>是連通且<a href="/wiki/%E5%96%AE%E9%80%A3%E9%80%9A" title="單連通">單連通</a>的。</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 \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>中的連通子集是線段、射線與<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 \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>本身。由此性質可迅速導出<a href="/wiki/%E4%B8%AD%E9%96%93%E5%80%BC%E5%AE%9A%E7%90%86" class="mw-redirect" title="中間值定理">中間值定理</a>。</li> <li>區間套定理：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 (F_{n})_{n\in \mathbb {N} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (F_{n})_{n\in \mathbb {N} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d217766d14654c8751bbcedfd13baefcd874ef1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.024ex; height:2.843ex;" alt="{\displaystyle (F_{n})_{n\in \mathbb {N} }}"></span>為一個有界閉集的序列，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 F_{n}\supset F_{n+1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>⊃</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}\supset F_{n+1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae596e63c84e4b8e0964d14901e93921bc4f5809" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.625ex; height:2.509ex;" alt="{\displaystyle F_{n}\supset F_{n+1}}"></span>，則其交集非空。嚴格表法如下：</li></ul> <dl><dd><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 \forall n\in \mathbb {N} \;\forall m>n\quad F_{m}\subset F_{n}\quad \Rightarrow \quad \bigcap _{n\in \mathbb {N} }F_{n}\;\neq \varnothing \;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mspace width="thickmathspace" /> <mi mathvariant="normal">∀</mi> <mi>m</mi> <mo>></mo> <mi>n</mi> <mspace width="1em" /> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>⊂</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mspace width="1em" /> <mo stretchy="false">⇒</mo> <mspace width="1em" /> <munder> <mo>⋂</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </munder> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mspace width="thickmathspace" /> <mo>≠</mo> <mi class="MJX-variant">∅</mi> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall n\in \mathbb {N} \;\forall m>n\quad F_{m}\subset F_{n}\quad \Rightarrow \quad \bigcap _{n\in \mathbb {N} }F_{n}\;\neq \varnothing \;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b663e65999ae96842327f07d675f7e07b70aba78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:47.806ex; height:5.676ex;" alt="{\displaystyle \forall n\in \mathbb {N} \;\forall m>n\quad F_{m}\subset F_{n}\quad \Rightarrow \quad \bigcap _{n\in \mathbb {N} }F_{n}\;\neq \varnothing \;}"></span>.</dd></dl></dd></dl> <div class="mw-heading mw-heading2"><h2 id="扩展与一般化"><span id=".E6.89.A9.E5.B1.95.E4.B8.8E.E4.B8.80.E8.88.AC.E5.8C.96"></span>扩展与一般化</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%AE%9E%E6%95%B0&action=edit&section=14" title="编辑章节：扩展与一般化"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>实数集可以在几种不同的方面进行扩展和一般化： </p> <ul><li>最自然的扩展可能就是<a href="/wiki/%E5%A4%8D%E6%95%B0_(%E6%95%B0%E5%AD%A6)" title="复数 (数学)">复数</a>了。复数集是包含了所有<a href="/wiki/%E5%A4%9A%E9%A1%B9%E5%BC%8F" class="mw-redirect" title="多项式">多项式</a>的根的<a href="/wiki/%E4%BB%A3%E6%95%B8%E9%96%89%E5%9F%9F" title="代數閉域">代数闭域</a>。但是，复数集不是一个<a href="/wiki/%E6%9C%89%E5%BA%8F%E5%9F%9F" title="有序域">有序域</a>。</li></ul> <ul><li>实数集扩展的有序域是<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)">超实数</a>的集合，包含<a href="/wiki/%E6%97%A0%E7%A9%B7%E5%B0%8F" class="mw-redirect" title="无穷小">无穷小</a>和<a href="/wiki/%E6%97%A0%E7%A9%B7%E5%A4%A7" class="mw-redirect" title="无穷大">无穷大</a>。它不是一个<a href="/w/index.php?title=%E9%98%BF%E5%9F%BA%E7%B1%B3%E5%BE%B7%E5%9F%9F&action=edit&redlink=1" class="new" title="阿基米德域（页面不存在）">阿基米德域</a>。</li></ul> <ul><li>有时候，形式元素 +∞和 -∞加入实数集，构成<a href="/wiki/%E6%89%A9%E5%B1%95%E7%9A%84%E5%AE%9E%E6%95%B0%E8%BD%B4" class="mw-redirect" title="扩展的实数轴">扩展的实数轴</a>。它是一个紧致空间，而不是一个域，但它保留了许多实数的性质。</li></ul> <ul><li><a href="/wiki/%E5%B8%8C%E5%B0%94%E4%BC%AF%E7%89%B9%E7%A9%BA%E9%97%B4" title="希尔伯特空间">希尔伯特空间</a>的<a href="/wiki/%E8%87%AA%E4%BC%B4%E9%9A%8F%E7%AE%97%E5%AD%90" class="mw-redirect" title="自伴随算子">自伴随算子</a>在许多方面一般化实数集：它们可以是有序的（尽管不一定全序）、完备的；它们所有的<a href="/wiki/%E7%89%B9%E5%BE%81%E5%80%BC" class="mw-redirect" title="特征值">特征值</a>都是实数；它们构成一个实<a href="/wiki/%E7%B5%90%E5%90%88%E4%BB%A3%E6%95%B8" title="結合代數">结合代数</a>。</li></ul> <div class="mw-heading mw-heading2"><h2 id="注释"><span id=".E6.B3.A8.E9.87.8A"></span>注释</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%AE%9E%E6%95%B0&action=edit&section=15" title="编辑章节：注释"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="reflist" style="list-style-type: decimal;"> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">《数学辞海（第一卷）》山西教育出版社中国科学技术出版社东南大学出版社</span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="请参阅"><span id=".E8.AF.B7.E5.8F.82.E9.98.85"></span>请参阅</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%AE%9E%E6%95%B0&action=edit&section=16" title="编辑章节：请参阅"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/%E6%9C%89%E7%90%86%E6%95%B0" title="有理数">有理数</a></li> <li><a href="/wiki/%E6%97%A0%E7%90%86%E6%95%B0" class="mw-redirect" title="无理数">无理数</a></li> <li><a href="/wiki/%E8%99%9A%E6%95%B0" title="虚数">虚数</a></li> <li><a href="/wiki/%E5%A4%8D%E6%95%B0_(%E6%95%B0%E5%AD%A6)" title="复数 (数学)">复数</a></li> <li><a href="/w/index.php?title=%E5%AE%9E%E6%95%B0%E7%B3%BB%E7%9A%84%E8%BF%9E%E7%BB%AD%E6%80%A7&action=edit&redlink=1" class="new" title="实数系的连续性（页面不存在）">实数系的连续性</a></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></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 class="mw-selflink selflink">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 href="/wiki/%E7%84%A1%E7%90%86%E6%95%B8" title="無理數">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"></div><div role="navigation" class="navbox" aria-labelledby="复数" 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:%E8%A4%87%E6%95%B8" title="Template:複數"><abbr title="查看该模板">查</abbr></a></li><li class="nv-talk"><a href="/w/index.php?title=Template_talk:%E8%A4%87%E6%95%B8&action=edit&redlink=1" class="new" title="Template talk:複數（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:%E8%A4%87%E6%95%B8" title="Special:编辑页面/Template:複數"><abbr title="编辑该模板">编</abbr></a></li></ul></div><div id="复数" style="font-size:110%;margin:0 5em"><a href="/wiki/%E5%A4%8D%E6%95%B0_(%E6%95%B0%E5%AD%A6)" title="复数 (数学)">复数</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/%E5%85%B1%E8%BD%AD%E5%A4%8D%E6%95%B0" title="共轭复数">共轭复数</a></li> <li><a href="/wiki/%E5%A4%8D%E5%B9%B3%E9%9D%A2" title="复平面">复平面</a></li> <li><a href="/wiki/%E8%99%9A%E6%95%B0" title="虚数">虚数</a></li> <li><a class="mw-selflink selflink">实数</a></li> <li><a href="/wiki/%E5%9C%93%E7%BE%A4" title="圓群">單位複數</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"><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 class="mw-selflink selflink">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 class="mw-selflink selflink">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 class="mw-selflink selflink"><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 href="/wiki/%E7%84%A1%E7%90%86%E6%95%B8" title="無理數">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><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">检索自“<a dir="ltr" href="https://zh.wikipedia.org/w/index.php?title=实数&oldid=84567670">https://zh.wikipedia.org/w/index.php?title=实数&oldid=84567670</a>”</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Special:%E9%A1%B5%E9%9D%A2%E5%88%86%E7%B1%BB" 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