平方根 - 维基百科，自由的百科全书

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href="/w/index.php?title=Special:%E5%88%9B%E5%BB%BA%E8%B4%A6%E6%88%B7&returnto=%E5%B9%B3%E6%96%B9%E6%A0%B9" title="我们推荐您创建账号并登录，但这不是强制性的"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>创建账号</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:%E7%94%A8%E6%88%B7%E7%99%BB%E5%BD%95&returnto=%E5%B9%B3%E6%96%B9%E6%A0%B9" 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-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">1</span> <span>历史</span> </div> </a> <ul id="toc-历史-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-正數" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#正數"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>正數</span> </div> </a> <ul id="toc-正數-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-负数与複數" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#负数与複數"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>负数与複數</span> </div> </a> <button aria-controls="toc-负数与複數-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>开关负数与複數子章节</span> </button> <ul id="toc-负数与複數-sublist" class="vector-toc-list"> <li id="toc-虚数的算术平方根" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#虚数的算术平方根"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>虚数的算术平方根</span> </div> </a> <ul id="toc-虚数的算术平方根-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-复数的算术平方根" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#复数的算术平方根"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>复数的算术平方根</span> </div> </a> <ul id="toc-复数的算术平方根-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-代数公式" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#代数公式"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>代数公式</span> </div> </a> <ul id="toc-代数公式-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-多项式" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#多项式"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>多项式</span> </div> </a> <ul id="toc-多项式-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-2的算术平方根" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#2的算术平方根"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>2的算术平方根</span> </div> </a> <ul id="toc-2的算术平方根-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-計算方法" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#計算方法"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>計算方法</span> </div> </a> <button aria-controls="toc-計算方法-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>开关計算方法子章节</span> </button> <ul id="toc-計算方法-sublist" class="vector-toc-list"> <li id="toc-因數計算" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#因數計算"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>因數計算</span> </div> </a> <ul id="toc-因數計算-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-中算开方" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#中算开方"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>中算开方</span> </div> </a> <ul id="toc-中算开方-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-長除式算法" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#長除式算法"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.3</span> <span>長除式算法</span> </div> </a> <ul id="toc-長除式算法-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-牛頓法" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#牛頓法"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.4</span> <span>牛頓法</span> </div> </a> <ul id="toc-牛頓法-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-連分數" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#連分數"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.5</span> <span>連分數</span> </div> </a> <ul id="toc-連分數-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-巴比倫方法" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#巴比倫方法"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.6</span> <span>巴比倫方法</span> </div> </a> <ul id="toc-巴比倫方法-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-重複的算術運算" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#重複的算術運算"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.7</span> <span>重複的算術運算</span> </div> </a> <ul id="toc-重複的算術運算-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-尺规作图" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#尺规作图"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.8</span> <span>尺规作图</span> </div> </a> <ul id="toc-尺规作图-sublist" class="vector-toc-list"> <li id="toc-問題" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#問題"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.8.1</span> <span>問題</span> </div> </a> <ul id="toc-問題-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-解法" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#解法"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.8.2</span> <span>解法</span> </div> </a> <ul id="toc-解法-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-證明" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#證明"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.8.3</span> <span>證明</span> </div> </a> <ul id="toc-證明-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-参见" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#参见"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>参见</span> </div> </a> <ul id="toc-参见-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-外部链接" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#外部链接"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>外部链接</span> </div> </a> <ul id="toc-外部链接-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-參考資料" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#參考資料"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>參考資料</span> </div> </a> <ul id="toc-參考資料-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">平方根</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。96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-96" 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">96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/Vierkantswortel" title="Vierkantswortel – 南非荷兰语" lang="af" hreflang="af" data-title="Vierkantswortel" data-language-autonym="Afrikaans" data-language-local-name="南非荷兰语" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AC%D8%B0%D8%B1_%D8%AA%D8%B1%D8%A8%D9%8A%D8%B9%D9%8A" title="جذر تربيعي – 阿拉伯语" lang="ar" hreflang="ar" data-title="جذر تربيعي" 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/Ra%C3%ADz_cuadrada" title="Raíz cuadrada – 阿斯图里亚斯语" lang="ast" hreflang="ast" data-title="Raíz cuadrada" 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/Kvadrat_k%C3%B6kl%C9%99r" title="Kvadrat köklər – 阿塞拜疆语" lang="az" hreflang="az" data-title="Kvadrat köklər" data-language-autonym="Azərbaycanca" data-language-local-name="阿塞拜疆语" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%B4%D1%80%D0%B0%D1%82_%D1%82%D0%B0%D0%BC%D1%8B%D1%80" title="Квадрат тамыр – 巴什基尔语" lang="ba" hreflang="ba" data-title="Квадрат тамыр" 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/Kwadradong_gamot" title="Kwadradong gamot – Central Bikol" lang="bcl" hreflang="bcl" data-title="Kwadradong gamot" 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%9A%D0%B2%D0%B0%D0%B4%D1%80%D0%B0%D1%82%D0%BD%D1%8B_%D0%BA%D0%BE%D1%80%D0%B0%D0%BD%D1%8C" title="Квадратны корань – 白俄罗斯语" lang="be" hreflang="be" data-title="Квадратны корань" 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%9A%D0%B2%D0%B0%D0%B4%D1%80%D0%B0%D1%82%D0%BD%D1%8B_%D0%BA%D0%BE%D1%80%D0%B0%D0%BD%D1%8C" title="Квадратны корань – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Квадратны корань" 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%9A%D0%B2%D0%B0%D0%B4%D1%80%D0%B0%D1%82%D0%B5%D0%BD_%D0%BA%D0%BE%D1%80%D0%B5%D0%BD" title="Квадратен корен – 保加利亚语" lang="bg" hreflang="bg" data-title="Квадратен корен" data-language-autonym="Български" data-language-local-name="保加利亚语" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AC%E0%A6%B0%E0%A7%8D%E0%A6%97%E0%A6%AE%E0%A7%82%E0%A6%B2" title="বর্গমূল – 孟加拉语" lang="bn" hreflang="bn" data-title="বর্গমূল" 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-br mw-list-item"><a href="https://br.wikipedia.org/wiki/Gwrizienn_garrez" title="Gwrizienn garrez – 布列塔尼语" lang="br" hreflang="br" data-title="Gwrizienn garrez" data-language-autonym="Brezhoneg" data-language-local-name="布列塔尼语" class="interlanguage-link-target"><span>Brezhoneg</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Kvadratni_korijen" title="Kvadratni korijen – 波斯尼亚语" lang="bs" hreflang="bs" data-title="Kvadratni korijen" data-language-autonym="Bosanski" data-language-local-name="波斯尼亚语" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Arrel_quadrada" title="Arrel quadrada – 加泰罗尼亚语" lang="ca" hreflang="ca" data-title="Arrel quadrada" 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%95%DB%95%DA%AF%DB%8C_%D8%AF%D9%88%D9%88%D8%AC%D8%A7" title="ڕەگی دووجا – 中库尔德语" lang="ckb" hreflang="ckb" data-title="ڕەگی دووجا" data-language-autonym="کوردی" data-language-local-name="中库尔德语" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Druh%C3%A1_odmocnina" title="Druhá odmocnina – 捷克语" lang="cs" hreflang="cs" data-title="Druhá odmocnina" 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%A2%C4%83%D0%B2%D0%B0%D1%82%D0%BA%D0%B0%D0%BB%D0%BB%D0%B0_%D1%82%D1%8B%D0%BC%D0%B0%D1%80" title="Тăваткалла тымар – 楚瓦什语" lang="cv" hreflang="cv" data-title="Тăваткалла тымар" 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/Ail_isradd" title="Ail isradd – 威尔士语" lang="cy" hreflang="cy" data-title="Ail isradd" 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/Kvadratrod" title="Kvadratrod – 丹麦语" lang="da" hreflang="da" data-title="Kvadratrod" 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/Quadratwurzel" title="Quadratwurzel – 德语" lang="de" hreflang="de" data-title="Quadratwurzel" data-language-autonym="Deutsch" data-language-local-name="德语" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A4%CE%B5%CF%84%CF%81%CE%B1%CE%B3%CF%89%CE%BD%CE%B9%CE%BA%CE%AE_%CF%81%CE%AF%CE%B6%CE%B1" title="Τετραγωνική ρίζα – 希腊语" lang="el" hreflang="el" data-title="Τετραγωνική ρίζα" data-language-autonym="Ελληνικά" data-language-local-name="希腊语" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Square_root" title="Square root – 英语" lang="en" hreflang="en" data-title="Square root" 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/Kvadrata_radiko" title="Kvadrata radiko – 世界语" lang="eo" hreflang="eo" data-title="Kvadrata radiko" 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/Ra%C3%ADz_cuadrada" title="Raíz cuadrada – 西班牙语" lang="es" hreflang="es" data-title="Raíz cuadrada" 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/Ruutjuur" title="Ruutjuur – 爱沙尼亚语" lang="et" hreflang="et" data-title="Ruutjuur" 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/Erro_karratu" title="Erro karratu – 巴斯克语" lang="eu" hreflang="eu" data-title="Erro karratu" 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%B1%DB%8C%D8%B4%D9%87_%D8%AF%D9%88%D9%85" title="ریشه دوم – 波斯语" lang="fa" hreflang="fa" data-title="ریشه دوم" 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/Neli%C3%B6juuri" title="Neliöjuuri – 芬兰语" lang="fi" hreflang="fi" data-title="Neliöjuuri" 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-fo mw-list-item"><a href="https://fo.wikipedia.org/wiki/Kvadratr%C3%B3t" title="Kvadratrót – 法罗语" lang="fo" hreflang="fo" data-title="Kvadratrót" 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/Racine_carr%C3%A9e" title="Racine carrée – 法语" lang="fr" hreflang="fr" data-title="Racine carrée" data-language-autonym="Français" data-language-local-name="法语" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-fur mw-list-item"><a href="https://fur.wikipedia.org/wiki/Lidr%C3%AEs_cuadrade" title="Lidrîs cuadrade – 弗留利语" lang="fur" hreflang="fur" data-title="Lidrîs cuadrade" 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-gan mw-list-item"><a href="https://gan.wikipedia.org/wiki/%E5%B9%B3%E6%96%B9%E6%A0%B9" title="平方根 – 赣语" lang="gan" hreflang="gan" data-title="平方根" 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/Rasin_di_roun_nonm" title="Rasin di roun nonm – Guianan Creole" lang="gcr" hreflang="gcr" data-title="Rasin di roun nonm" 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/Ra%C3%ADz_cadrada" title="Raíz cadrada – 加利西亚语" lang="gl" hreflang="gl" data-title="Raíz cadrada" 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-gu mw-list-item"><a href="https://gu.wikipedia.org/wiki/%E0%AA%B5%E0%AA%B0%E0%AB%8D%E0%AA%97%E0%AA%AE%E0%AB%82%E0%AA%B3" title="વર્ગમૂળ – 古吉拉特语" lang="gu" hreflang="gu" data-title="વર્ગમૂળ" data-language-autonym="ગુજરાતી" data-language-local-name="古吉拉特语" class="interlanguage-link-target"><span>ગુજરાતી</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A9%D7%95%D7%A8%D7%A9_%D7%A8%D7%99%D7%91%D7%95%D7%A2%D7%99" title="שורש ריבועי – 希伯来语" lang="he" hreflang="he" data-title="שורש ריבועי" 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%B0%E0%A5%8D%E0%A4%97%E0%A4%AE%E0%A5%82%E0%A4%B2" title="वर्गमूल – 印地语" lang="hi" hreflang="hi" data-title="वर्गमूल" 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/Kvadratni_korijen" title="Kvadratni korijen – 克罗地亚语" lang="hr" hreflang="hr" data-title="Kvadratni korijen" 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/N%C3%A9gyzetgy%C3%B6k" title="Négyzetgyök – 匈牙利语" lang="hu" hreflang="hu" data-title="Négyzetgyök" data-language-autonym="Magyar" data-language-local-name="匈牙利语" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%94%D5%A1%D5%BC%D5%A1%D5%AF%D5%B8%D6%82%D5%BD%D5%AB_%D5%A1%D6%80%D5%B4%D5%A1%D5%BF" title="Քառակուսի արմատ – 亚美尼亚语" lang="hy" hreflang="hy" data-title="Քառակուսի արմատ" data-language-autonym="Հայերեն" data-language-local-name="亚美尼亚语" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Akar_kuadrat" title="Akar kuadrat – 印度尼西亚语" lang="id" hreflang="id" data-title="Akar kuadrat" data-language-autonym="Bahasa Indonesia" data-language-local-name="印度尼西亚语" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Ferningsr%C3%B3t" title="Ferningsrót – 冰岛语" lang="is" hreflang="is" data-title="Ferningsrót" 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/Radice_quadrata" title="Radice quadrata – 意大利语" lang="it" hreflang="it" data-title="Radice quadrata" 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%B9%B3%E6%96%B9%E6%A0%B9" title="平方根 – 日语" lang="ja" hreflang="ja" data-title="平方根" 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/Skwier_ruut" title="Skwier ruut – 牙買加克里奧爾英文" lang="jam" hreflang="jam" data-title="Skwier ruut" 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-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%99%E1%83%95%E1%83%90%E1%83%93%E1%83%A0%E1%83%90%E1%83%A2%E1%83%A3%E1%83%9A%E1%83%98_%E1%83%A4%E1%83%94%E1%83%A1%E1%83%95%E1%83%98" title="კვადრატული ფესვი – 格鲁吉亚语" lang="ka" hreflang="ka" data-title="კვადრატული ფესვი" data-language-autonym="ქართული" data-language-local-name="格鲁吉亚语" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%B4%D1%80%D0%B0%D1%82%D1%82%D1%8B%D2%9B_%D1%82%D2%AF%D0%B1%D1%96%D1%80" title="Квадраттық түбір – 哈萨克语" lang="kk" hreflang="kk" data-title="Квадраттық түбір" 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-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%B5%E0%B2%B0%E0%B3%8D%E0%B2%97%E0%B2%AE%E0%B3%82%E0%B2%B2" title="ವರ್ಗಮೂಲ – 卡纳达语" lang="kn" hreflang="kn" data-title="ವರ್ಗಮೂಲ" 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%A0%9C%EA%B3%B1%EA%B7%BC" title="제곱근 – 韩语" lang="ko" hreflang="ko" data-title="제곱근" data-language-autonym="한국어" data-language-local-name="韩语" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%90%D1%80%D0%B8%D1%84%D0%BC%D0%B5%D1%82%D0%B8%D0%BA%D0%B0%D0%BB%D1%8B%D0%BA_%D0%BA%D0%B2%D0%B0%D0%B4%D1%80%D0%B0%D1%82_%D1%82%D0%B0%D0%BC%D1%8B%D1%80" title="Арифметикалык квадрат тамыр – 柯尔克孜语" lang="ky" hreflang="ky" data-title="Арифметикалык квадрат тамыр" 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/Radix_quadrata" title="Radix quadrata – 拉丁语" lang="la" hreflang="la" data-title="Radix quadrata" 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/Radis_cuadral" title="Radis cuadral – 新共同語言" lang="lfn" hreflang="lfn" data-title="Radis cuadral" 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-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Radis_quadrada" title="Radis quadrada – 倫巴底文" lang="lmo" hreflang="lmo" data-title="Radis quadrada" 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%AE%E0%BA%B2%E0%BA%81%E0%BA%AA%E0%BA%B5%E0%BB%88%E0%BA%AB%E0%BA%A5%E0%BB%88%E0%BA%BD%E0%BA%A1" title="ຮາກສີ່ຫລ່ຽມ – 老挝语" lang="lo" hreflang="lo" data-title="ຮາກສີ່ຫລ່ຽມ" 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/Kvadratin%C4%97_%C5%A1aknis" title="Kvadratinė šaknis – 立陶宛语" lang="lt" hreflang="lt" data-title="Kvadratinė šaknis" 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/Kvadr%C4%81tsakne" title="Kvadrātsakne – 拉脱维亚语" lang="lv" hreflang="lv" data-title="Kvadrātsakne" data-language-autonym="Latviešu" data-language-local-name="拉脱维亚语" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%B4%D1%80%D0%B0%D1%82%D0%B5%D0%BD_%D0%BA%D0%BE%D1%80%D0%B5%D0%BD" title="Квадратен корен – 马其顿语" lang="mk" hreflang="mk" data-title="Квадратен корен" 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%B5%BC%E0%B4%97%E0%B5%8D%E0%B4%97%E0%B4%AE%E0%B5%82%E0%B4%B2%E0%B4%82" title="വർഗ്ഗമൂലം – 马拉雅拉姆语" lang="ml" hreflang="ml" data-title="വർഗ്ഗമൂലം" 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%B0%E0%A5%8D%E0%A4%97%E0%A4%AE%E0%A5%82%E0%A4%B3" title="वर्गमूळ – 马拉地语" lang="mr" hreflang="mr" data-title="वर्गमूळ" 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/Punca_kuasa_dua" title="Punca kuasa dua – 马来语" lang="ms" hreflang="ms" data-title="Punca kuasa dua" data-language-autonym="Bahasa Melayu" data-language-local-name="马来语" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-mt mw-list-item"><a href="https://mt.wikipedia.org/wiki/G%C4%A7erq_kwadru" title="Għerq kwadru – 马耳他语" lang="mt" hreflang="mt" data-title="Għerq kwadru" data-language-autonym="Malti" data-language-local-name="马耳他语" class="interlanguage-link-target"><span>Malti</span></a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%94%E1%80%BE%E1%80%85%E1%80%BA%E1%80%91%E1%80%95%E1%80%BA%E1%80%80%E1%80%AD%E1%80%94%E1%80%BA%E1%80%B8%E1%80%9B%E1%80%84%E1%80%BA%E1%80%B8" title="နှစ်ထပ်ကိန်းရင်း – 缅甸语" lang="my" hreflang="my" data-title="နှစ်ထပ်ကိန်းရင်း" 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%B0%E0%A5%8D%E0%A4%97%E0%A4%AE%E0%A5%82%E0%A4%B2" title="वर्गमूल – 尼泊尔语" lang="ne" hreflang="ne" data-title="वर्गमूल" 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-new mw-list-item"><a href="https://new.wikipedia.org/wiki/%E0%A4%B5%E0%A4%B0%E0%A5%8D%E0%A4%97%E0%A4%AE%E0%A5%82%E0%A4%B2" title="वर्गमूल – 尼瓦尔语" lang="new" hreflang="new" data-title="वर्गमूल" data-language-autonym="नेपाल भाषा" data-language-local-name="尼瓦尔语" class="interlanguage-link-target"><span>नेपाल भाषा</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Vierkantswortel" title="Vierkantswortel – 荷兰语" lang="nl" hreflang="nl" data-title="Vierkantswortel" 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/Kvadratrot" title="Kvadratrot – 挪威尼诺斯克语" lang="nn" hreflang="nn" data-title="Kvadratrot" 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/Kvadratrot" title="Kvadratrot – 书面挪威语" lang="nb" hreflang="nb" data-title="Kvadratrot" 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/Rai%C3%A7_carrada" title="Raiç carrada – 奥克语" lang="oc" hreflang="oc" data-title="Raiç carrada" 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-om mw-list-item"><a href="https://om.wikipedia.org/wiki/Caaroo_Kaaslamee" title="Caaroo Kaaslamee – 奥罗莫语" lang="om" hreflang="om" data-title="Caaroo Kaaslamee" data-language-autonym="Oromoo" data-language-local-name="奥罗莫语" class="interlanguage-link-target"><span>Oromoo</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%B5%E0%A8%B0%E0%A8%97_%E0%A8%AE%E0%A9%82%E0%A8%B2" title="ਵਰਗ ਮੂਲ – 旁遮普语" lang="pa" hreflang="pa" data-title="ਵਰਗ ਮੂਲ" 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/Pierwiastek_kwadratowy" title="Pierwiastek kwadratowy – 波兰语" lang="pl" hreflang="pl" data-title="Pierwiastek kwadratowy" data-language-autonym="Polski" data-language-local-name="波兰语" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Raiz_quadrada" title="Raiz quadrada – 葡萄牙语" lang="pt" hreflang="pt" data-title="Raiz quadrada" 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/R%C4%83d%C4%83cin%C4%83_p%C4%83trat%C4%83" title="Rădăcină pătrată – 罗马尼亚语" lang="ro" hreflang="ro" data-title="Rădăcină pătrată" 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%9A%D0%B2%D0%B0%D0%B4%D1%80%D0%B0%D1%82%D0%BD%D1%8B%D0%B9_%D0%BA%D0%BE%D1%80%D0%B5%D0%BD%D1%8C" title="Квадратный корень – 俄语" lang="ru" hreflang="ru" data-title="Квадратный корень" data-language-autonym="Русский" data-language-local-name="俄语" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Radici_quatrata" title="Radici quatrata – 西西里语" lang="scn" hreflang="scn" data-title="Radici quatrata" 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/Kvadratni_koren" title="Kvadratni koren – 塞尔维亚-克罗地亚语" lang="sh" hreflang="sh" data-title="Kvadratni koren" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="塞尔维亚-克罗地亚语" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Square_root" title="Square root – Simple English" lang="en-simple" hreflang="en-simple" data-title="Square root" 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-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Kvadratni_koren" title="Kvadratni koren – 斯洛文尼亚语" lang="sl" hreflang="sl" data-title="Kvadratni koren" data-language-autonym="Slovenščina" data-language-local-name="斯洛文尼亚语" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Rr%C3%ABnja_katrore" title="Rrënja katrore – 阿尔巴尼亚语" lang="sq" hreflang="sq" data-title="Rrënja katrore" 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%9A%D0%B2%D0%B0%D0%B4%D1%80%D0%B0%D1%82%D0%BD%D0%B8_%D0%BA%D0%BE%D1%80%D0%B5%D0%BD" title="Квадратни корен – 塞尔维亚语" lang="sr" hreflang="sr" data-title="Квадратни корен" 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-su mw-list-item"><a href="https://su.wikipedia.org/wiki/Akar_kuadrat" title="Akar kuadrat – 巽他语" lang="su" hreflang="su" data-title="Akar kuadrat" data-language-autonym="Sunda" data-language-local-name="巽他语" class="interlanguage-link-target"><span>Sunda</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Kvadratrot" title="Kvadratrot – 瑞典语" lang="sv" hreflang="sv" data-title="Kvadratrot" 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/Kipeuo_cha_pili" title="Kipeuo cha pili – 斯瓦希里语" lang="sw" hreflang="sw" data-title="Kipeuo cha pili" data-language-autonym="Kiswahili" data-language-local-name="斯瓦希里语" class="interlanguage-link-target"><span>Kiswahili</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%B5%E0%AE%B0%E0%AF%8D%E0%AE%95%E0%AF%8D%E0%AE%95%E0%AE%AE%E0%AF%82%E0%AE%B2%E0%AE%AE%E0%AF%8D" title="வர்க்கமூலம் – 泰米尔语" lang="ta" hreflang="ta" data-title="வர்க்கமூலம்" data-language-autonym="தமிழ்" data-language-local-name="泰米尔语" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%B5%E0%B0%B0%E0%B1%8D%E0%B0%97%E0%B0%AE%E0%B1%82%E0%B0%B2%E0%B0%82" title="వర్గమూలం – 泰卢固语" lang="te" hreflang="te" data-title="వర్గమూలం" data-language-autonym="తెలుగు" data-language-local-name="泰卢固语" class="interlanguage-link-target"><span>తెలుగు</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Pariugat" title="Pariugat – 他加禄语" lang="tl" hreflang="tl" data-title="Pariugat" 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/Karek%C3%B6k" title="Karekök – 土耳其语" lang="tr" hreflang="tr" data-title="Karekök" 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%9A%D0%B2%D0%B0%D0%B4%D1%80%D0%B0%D1%82%D0%BD%D0%B8%D0%B9_%D0%BA%D0%BE%D1%80%D1%96%D0%BD%D1%8C" title="Квадратний корінь – 乌克兰语" lang="uk" hreflang="uk" data-title="Квадратний корінь" 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%AC%D8%B0%D8%B1_%D8%A7%D9%84%D9%85%D8%B1%D8%A8%D8%B9" title="جذر المربع – 乌尔都语" lang="ur" hreflang="ur" data-title="جذر المربع" 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/Kvadrat_ildiz" title="Kvadrat ildiz – 乌兹别克语" lang="uz" hreflang="uz" data-title="Kvadrat ildiz" 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/C%C4%83n_b%E1%BA%ADc_hai" title="Căn bậc hai – 越南语" lang="vi" hreflang="vi" data-title="Căn bậc hai" data-language-autonym="Tiếng Việt" data-language-local-name="越南语" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Kwadrado_nga_gamot" title="Kwadrado nga gamot – 瓦瑞语" lang="war" hreflang="war" data-title="Kwadrado nga gamot" data-language-autonym="Winaray" data-language-local-name="瓦瑞语" class="interlanguage-link-target"><span>Winaray</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E5%B9%B3%E6%96%B9%E6%A0%B9" title="平方根 – 吴语" lang="wuu" hreflang="wuu" data-title="平方根" data-language-autonym="吴语" data-language-local-name="吴语" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%A7%D7%95%D7%95%D7%90%D7%93%D7%A8%D7%90%D7%98_%D7%95%D7%95%D7%90%D7%A8%D7%A6%D7%9C" title="קוואדראט ווארצל – 意第绪语" lang="yi" hreflang="yi" data-title="קוואדראט ווארצל" 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 mw-list-item"><a href="https://yo.wikipedia.org/wiki/Gb%C3%B2ngb%C3%B2_al%C3%A1gb%C3%A1ram%C3%A9j%C3%AC" title="Gbòngbò alágbáraméjì – 约鲁巴语" lang="yo" hreflang="yo" data-title="Gbòngbò alágbáraméjì" data-language-autonym="Yorùbá" data-language-local-name="约鲁巴语" class="interlanguage-link-target"><span>Yorùbá</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/P%C3%AAng-hong-kin" title="Pêng-hong-kin – 闽南语" lang="nan" hreflang="nan" data-title="Pêng-hong-kin" 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/%E9%96%8B%E6%96%B9%E6%A0%B9" title="開方根 – 粤语" lang="yue" hreflang="yue" data-title="開方根" 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 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aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> <div id="mw-indicator-noteTA-8a12d924" class="mw-indicator"><div class="mw-parser-output"><span class="skin-invert" typeof="mw:File"><span title="本页使用了标题或全文手工转换"><img alt="本页使用了标题或全文手工转换" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Zh_conversion_icon_m.svg/35px-Zh_conversion_icon_m.svg.png" decoding="async" width="35" height="22" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Zh_conversion_icon_m.svg/53px-Zh_conversion_icon_m.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Zh_conversion_icon_m.svg/70px-Zh_conversion_icon_m.svg.png 2x" data-file-width="32" data-file-height="20" /></span></span></div></div> </div> <div id="siteSub" class="noprint">维基百科，自由的百科全书</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="zh" dir="ltr"><div id="noteTA-8a12d924" 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:数学对象;zh-tw:數學物件;"></div></div></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Nuvola_apps_edu_mathematics_blue-p.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/256px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="256" height="256" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/384px-Nuvola_apps_edu_mathematics_blue-p.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/512px-Nuvola_apps_edu_mathematics_blue-p.svg.png 2x" data-file-width="128" data-file-height="128" /></a><figcaption>算术平方根的數學表示式</figcaption></figure> <p>在<a href="/wiki/%E6%95%B8%E5%AD%B8" class="mw-redirect" title="數學">數學</a>中，一個數<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 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>的<b>平方根</b><span class="mwe-math-element"><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>指的是滿足<span class="mwe-math-element"><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^{2}=x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y^{2}=x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb1cfda1f3310a5c649d6847b1b2325968850889" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.643ex; height:3.009ex;" alt="{\displaystyle y^{2}=x}"></span>的數，即<a href="/wiki/%E5%B9%B3%E6%96%B9" title="平方">平方</a>結果等於<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 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>的數。例如，4和-4都是16的平方根，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 4^{2}=(-4)^{2}=16}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>4</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>16</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4^{2}=(-4)^{2}=16}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4fe90cd19335908e3320902af69ed20c7bc0396" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.573ex; height:3.176ex;" alt="{\displaystyle 4^{2}=(-4)^{2}=16}"></span>。 </p><p>任意非負<a href="/wiki/%E5%AF%A6%E6%95%B8" class="mw-redirect" title="實數">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 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>都有唯一的非負平方根，称为<b>算术平方根</b>或<b>主平方根</b>（英語：<span lang="en">principal square root</span>），記為<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 {\sqrt {x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>x</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d62b24be305beff66cba9bfbcc01a362ba390f44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.266ex; height:3.009ex;" alt="{\displaystyle {\sqrt {x}}}"></span>，其中的符号<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 {\sqrt {\quad }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mspace width="1em" /> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {\quad }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62a3244ffb6799c0ac7189a9acf3a4d7b2c55e8c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:4.258ex; height:3.009ex;" alt="{\displaystyle {\sqrt {\quad }}}"></span>称作<a href="/wiki/%E6%A0%B9%E5%8F%B7" title="根号">根号</a>。例如，9的算术平方根为3，记作 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {9}}=3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>9</mn> </msqrt> </mrow> <mo>=</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {9}}=3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44820d293d56e49db705b37ad756363767077121" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.359ex; height:2.843ex;" alt="{\displaystyle {\sqrt {9}}=3}"></span>，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 3^{2}=3\times 3=9}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>3</mn> <mo>×</mo> <mn>3</mn> <mo>=</mo> <mn>9</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3^{2}=3\times 3=9}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9de63fd232975081d29f13a5353a5f5bebe3785a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.741ex; height:2.676ex;" alt="{\displaystyle 3^{2}=3\times 3=9}"></span>并且3非负。被求平方根的数称作<b>被开方数</b>（英語：<span lang="en">radicand</span>），是根号下的数字或者表达式，即例子中的数字9。 </p><p><a href="/wiki/%E6%AD%A3%E6%95%B0" class="mw-redirect" title="正数">正数</a><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>有兩個互为<a href="/wiki/%E5%8A%A0%E6%B3%95%E9%80%86%E5%85%83" title="加法逆元">相反数</a>的平方根：正数<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 {\sqrt {x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>x</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d62b24be305beff66cba9bfbcc01a362ba390f44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.266ex; height:3.009ex;" alt="{\displaystyle {\sqrt {x}}}"></span>与负数<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 -{\sqrt {x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>x</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\sqrt {x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b2eaf3e2777b21ab1c891db88d1db286bd167d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.074ex; height:3.009ex;" alt="{\displaystyle -{\sqrt {x}}}"></span>，可以将两者一起记为<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 \pm {\sqrt {x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>x</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pm {\sqrt {x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/015b9bb6aff25be9b75393afd06d3976adeb11ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.074ex; height:3.009ex;" alt="{\displaystyle \pm {\sqrt {x}}}"></span>。 </p><p><a href="/wiki/%E8%B2%A0%E6%95%B8" class="mw-redirect" title="負數">負數</a>的平方根在<a href="/wiki/%E8%A4%87%E6%95%B8_(%E6%95%B8%E5%AD%B8)" class="mw-redirect" title="複數 (數學)">複數系</a>中有定義。而實際上，對任何定義了開平方運算的<a href="/wiki/%E6%95%B8%E5%AD%B8%E7%89%A9%E4%BB%B6" class="mw-redirect" title="數學物件">數學物件</a>都可考慮其“平方根”（例如<a href="/wiki/%E7%9F%A9%E9%98%B5%E7%9A%84%E5%B9%B3%E6%96%B9%E6%A0%B9" title="矩阵的平方根">矩陣的平方根</a>）。 </p> <ul><li>在MicroSoft的試算表軟體Excel與大部分程式語言中以 "sqrt()"表示求主平方根。</li></ul> <meta property="mw:PageProp/toc" /> <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%B9%B3%E6%96%B9%E6%A0%B9&action=edit&section=1" title="编辑章节：历史"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>耶鲁大学的巴比伦藏品<a href="/wiki/YBC_7289" title="YBC 7289">YBC 7289</a>是一块泥板，制作于<a href="/wiki/%E5%89%8D1800%E5%B9%B4" class="mw-redirect" title="前1800年">前1800年</a>到<a href="/wiki/%E5%89%8D1600%E5%B9%B4" class="mw-redirect" title="前1600年">前1600年</a>之间。泥板上是一个画了两条对角线正方形，标注了<span class="mwe-math-element"><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>的<a href="/wiki/%E5%85%AD%E5%8D%81%E9%80%B2%E5%88%B6" title="六十進制">六十进制</a>数字 1;24,51,10。<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>六十进制的 1;24,51,10 即十进制的 1.41421296，精确到了小数点后5位（1.41421356...）。 </p><p><a href="/wiki/%E8%8E%B1%E5%9B%A0%E5%BE%B7%E6%95%B0%E5%AD%A6%E7%BA%B8%E8%8D%89%E4%B9%A6" title="莱因德数学纸草书">莱因德数学纸草书</a>大约成书于<a href="/wiki/%E5%89%8D1650%E5%B9%B4" class="mw-redirect" title="前1650年">前1650年</a>，内容抄写自更早年代的教科书。书中展示了埃及人使用反比法求平方根的过程。<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p><a href="/wiki/%E5%8F%A4%E5%8D%B0%E5%BA%A6" class="mw-redirect" title="古印度">古印度</a>的《<a href="/wiki/%E7%BB%B3%E6%B3%95%E7%BB%8F" class="mw-redirect" title="绳法经">绳法经</a>》大约成书于<a href="/wiki/%E5%89%8D800%E5%B9%B4" class="mw-redirect" title="前800年">前800年</a>到<a href="/wiki/%E5%89%8D500%E5%B9%B4" title="前500年">前500年</a>之间，书中记载了将2的平方根的计算精确到小数点后5位的方法。 </p><p>古希腊的《<a href="/wiki/%E5%87%A0%E4%BD%95%E5%8E%9F%E6%9C%AC" title="几何原本">几何原本</a>》大约成书于<a href="/wiki/%E5%89%8D380%E5%B9%B4" title="前380年">前380年</a>，书中还阐述了如果<a href="/wiki/%E8%87%AA%E7%84%B6%E6%95%B0" title="自然数">正整数</a>不是<a href="/wiki/%E5%AE%8C%E5%85%A8%E5%B9%B3%E6%96%B9%E6%95%B0" class="mw-redirect" title="完全平方数">完全平方数</a>，那么它的平方根就一定是<a href="/wiki/%E6%97%A0%E7%90%86%E6%95%B0" class="mw-redirect" title="无理数">无理数</a>——一种无法以两个整数的<a href="/wiki/%E6%AF%94%E7%8E%87" class="mw-redirect" title="比率">比值</a>表示的数（无法写作<i>m/n</i>，其中<i>m</i>和<i>n</i>是整数）。<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>中国的《<a href="/wiki/%E4%B9%A6" class="mw-redirect" title="书">书</a>》成书于<a href="/wiki/%E6%B1%89%E6%9C%9D" title="汉朝">汉朝</a>（约<a href="/wiki/%E5%89%8D202%E5%B9%B4" title="前202年">前202年</a>到<a href="/wiki/%E5%89%8D186%E5%B9%B4" title="前186年">前186年</a>之间），书中介绍了使用<a href="/wiki/%E7%9B%88%E4%B8%8D%E8%B6%B3%E6%9C%AF" title="盈不足术">盈不足术</a>求平方根的方法。 </p><p>古代未有劃一的平方根符號時，人們通常使用他們語言內開方這個字的首個字母的變型作為開方號。 </p><p>中世紀時，<a href="/wiki/%E6%8B%89%E4%B8%81%E8%AA%9E" class="mw-redirect" title="拉丁語">拉丁語</a>中的<span lang="la" title="拉丁語文本">latus</span>（正方形邊）的首個字母“L”被不少歐洲人採用；<a href="/w/index.php?title=%E4%BA%A8%E5%88%A9%C2%B7%E5%B8%83%E9%87%8C%E6%A0%BC%E6%96%AF&action=edit&redlink=1" class="new" title="亨利·布里格斯（页面不存在）">亨利·布里格斯</a>在其著作《<span lang="la" title="拉丁語文本">Arithmetica Logarithmica</span>》中則用橫線當成<span lang="la" title="拉丁語文本">latus</span>的簡寫，在被開方的數下畫一線。 </p><p>最有影響的是拉丁語的<span lang="la" title="拉丁語文本">radix</span>（平方根），1220年Leconardo在《<i><span lang="la" title="拉丁語文本">Practica geometriae</span></i>》中使用℞（R右下角的有一斜劃，像P和x的合體）；⎷（沒有上面的橫劃）是由<a href="/w/index.php?title=%E5%85%8B%E9%87%8C%E6%96%AF%E5%A4%9A%E7%A6%8F%C2%B7%E9%AD%AF%E7%99%BB%E9%81%93%E5%A4%AB&action=edit&redlink=1" class="new" title="克里斯多福·魯登道夫（页面不存在）">克里斯多福·魯登道夫</a>在1525年的書<i>Coss</i>首次使用，據說是小寫r的變型；后来数学家<a href="/wiki/%E7%AC%9B%E5%8D%A1%E5%B0%94" class="mw-redirect" title="笛卡尔">笛卡尔</a>给其加上线括号，但与前面的方根符号是分开的（即“⎷‾”），因此在复杂的式子中它显得很乱。直至18世纪中叶，数学家卢贝将前面的方根符号与线括号一笔写成，并将<a href="/w/index.php?title=%E6%A0%B9%E6%8C%87%E6%95%B0&action=edit&redlink=1" class="new" title="根指数（页面不存在）">根指数</a>写在<a href="/wiki/%E6%A0%B9%E5%8F%B7" title="根号">根号</a>的左上角，以表示高次方根（当根指数为2时，省略不写），从而形成了现在人們熟知的<a href="/wiki/%E9%96%8B%E6%96%B9" class="mw-redirect" title="開方">开方</a><a href="/wiki/%E8%BF%90%E7%AE%97" title="运算">运算</a>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 {\sqrt[{n}]{\,\,}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mrow> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{n}]{\,\,}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddcb8a480722f2e988fb8ebe5dfd905140a9a52a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:2.71ex; height:3.009ex;" alt="{\displaystyle {\sqrt[{n}]{\,\,}}}"></span>。 </p> <div class="mw-heading mw-heading2"><h2 id="正數"><span id=".E6.AD.A3.E6.95.B8"></span>正數</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%B9%B3%E6%96%B9%E6%A0%B9&action=edit&section=2" title="编辑章节：正數"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Square_root_0_25.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Square_root_0_25.svg/400px-Square_root_0_25.svg.png" decoding="async" width="400" height="208" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Square_root_0_25.svg/600px-Square_root_0_25.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Square_root_0_25.svg/800px-Square_root_0_25.svg.png 2x" data-file-width="535" data-file-height="278" /></a><figcaption>函數<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 f(x)={\sqrt {x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>x</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)={\sqrt {x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f4850536e7a37db22aacbc552b03f195a3eceaa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.782ex; height:3.009ex;" alt="{\displaystyle f(x)={\sqrt {x}}}"></span>圖，半<a href="/wiki/%E6%8B%8B%E7%89%A9%E7%B7%9A" class="mw-redirect" title="拋物線">拋物線</a>與垂直準線。</figcaption></figure> <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 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>的平方根亦可用<a href="/wiki/%E6%8C%87%E6%95%B0" class="mw-disambig" title="指数">指數</a>表示，如： </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{\frac {1}{2}}={\sqrt {x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>x</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{\frac {1}{2}}={\sqrt {x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc21e95c62d91b174b2a3d734ad06e046919e449" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.429ex; height:4.176ex;" alt="{\displaystyle x^{\frac {1}{2}}={\sqrt {x}}}"></span></dd></dl> <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 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>的<a href="/wiki/%E7%B5%95%E5%B0%8D%E5%80%BC" class="mw-redirect" title="絕對值">絕對值</a>可用<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 x^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf0bf28fd28f45d07e1ceb909ce333c18c558c93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.384ex; height:2.676ex;" alt="{\displaystyle x^{2}}"></span>的算數平方根表示： </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |x|={\sqrt {x^{2}}}\left(={\begin{cases}x&(x\geq 0)\\-x&(x<0)\end{cases}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mi>x</mi> </mtd> <mtd> <mo stretchy="false">(</mo> <mi>x</mi> <mo>≥</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mi>x</mi> </mtd> <mtd> <mo stretchy="false">(</mo> <mi>x</mi> <mo><</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |x|={\sqrt {x^{2}}}\left(={\begin{cases}x&(x\geq 0)\\-x&(x<0)\end{cases}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d60eb74e24a159278a3a3e8802d19ca94a174d7c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:32.046ex; height:6.176ex;" alt="{\displaystyle |x|={\sqrt {x^{2}}}\left(={\begin{cases}x&(x\geq 0)\\-x&(x<0)\end{cases}}\right)}"></span></dd></dl> <p>若正<a href="/wiki/%E6%95%B4%E6%95%B8" class="mw-redirect" title="整數">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 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>是<a href="/wiki/%E5%B9%B3%E6%96%B9%E6%95%B8" class="mw-redirect" title="平方數">平方數</a>，則其平方根是整數。若正整數<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 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>不是平方數，則其平方根是<a href="/wiki/%E7%84%A1%E7%90%86%E6%95%B8" title="無理數">無理數</a>。 </p><p>對於正數<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 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>、<span class="mwe-math-element"><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>，以下式成立： </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\sqrt {x}}{\sqrt {y}}&={\sqrt {xy}}\\{\frac {\sqrt {x}}{\sqrt {y}}}&={\sqrt {\frac {x}{y}}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>x</mi> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>y</mi> </msqrt> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>x</mi> <mi>y</mi> </msqrt> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mi>x</mi> </msqrt> <msqrt> <mi>y</mi> </msqrt> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mi>x</mi> <mi>y</mi> </mfrac> </msqrt> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\sqrt {x}}{\sqrt {y}}&={\sqrt {xy}}\\{\frac {\sqrt {x}}{\sqrt {y}}}&={\sqrt {\frac {x}{y}}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71d9792194237c76e1db95239136a148a22797a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.295ex; margin-bottom: -0.21ex; width:14.696ex; height:10.176ex;" alt="{\displaystyle {\begin{aligned}{\sqrt {x}}{\sqrt {y}}&={\sqrt {xy}}\\{\frac {\sqrt {x}}{\sqrt {y}}}&={\sqrt {\frac {x}{y}}}\end{aligned}}}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="负数与複數"><span id=".E8.B4.9F.E6.95.B0.E4.B8.8E.E8.A4.87.E6.95.B8"></span>负数与複數</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%B9%B3%E6%96%B9%E6%A0%B9&action=edit&section=3" title="编辑章节：负数与複數"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>正数和负数的平方都是正数，0的平方是0，因此负数没有<a href="/wiki/%E5%AE%9E%E6%95%B0" title="实数">实数</a>平方根。然而，我们可以把我们所使用的数字集合扩大，加入负数的平方根，这样的集合就是<a href="/wiki/%E8%A4%87%E6%95%B8_(%E6%95%B8%E5%AD%B8)" class="mw-redirect" title="複數 (數學)">複數</a>。首先需要引入一个实数集之外的新数字，记作<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span>（也可以记作<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f461e54f5c093e92a55547b9764291390f0b5d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}"></span>，比如<a href="/wiki/%E7%94%B5%E5%AD%A6" class="mw-redirect" title="电学">电学</a>场景中<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span>一般表示电流），称之为<a href="/wiki/%E8%99%9A%E6%95%B0%E5%8D%95%E4%BD%8D" class="mw-redirect" title="虚数单位">虚数单位</a>，定义即为<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 i^{2}=-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i^{2}=-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88e98a401d352e5037d5043028e2d7f449e83fa6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.926ex; height:2.843ex;" alt="{\displaystyle i^{2}=-1}"></span>，故<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span>是-1的平方根，而且<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 (-i)^{2}=i^{2}=-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mi>i</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-i)^{2}=i^{2}=-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f90874a1336cbf3e1de8a6534408f784bf81739" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.498ex; height:3.176ex;" alt="{\displaystyle (-i)^{2}=i^{2}=-1}"></span>，所以<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91fddb9f89a520937db3a8821575068cdcc76f60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.611ex; height:2.343ex;" alt="{\displaystyle -i}"></span>也是-1的平方根。通常称-1的算术平方根是<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 i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span>，如果<span class="mwe-math-element"><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>是任意非负实数，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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <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/ae55e66aeffc525917eed885b4b753ba5a7f8b3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.138ex; height:2.176ex;" alt="{\displaystyle -x}"></span>的算术平方根就是： </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {-x}}=i{\sqrt {x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mi>x</mi> </msqrt> </mrow> <mo>=</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>x</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {-x}}=i{\sqrt {x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/256fc4aeac40bb57320642c5564e5735199dd526" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.24ex; height:3.009ex;" alt="{\displaystyle {\sqrt {-x}}=i{\sqrt {x}}}"></span></dd></dl> <p>例如-5的平方根有两个，它们分别为<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 {5}}i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {5}}i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d588e7dbf7b30ee2ab7b5ad24373ce47d593da12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.901ex; height:2.843ex;" alt="{\displaystyle {\sqrt {5}}i}"></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 -{\sqrt {5}}i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\sqrt {5}}i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3eec4efefa52e663d17750e4dc9da16629856a05" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.709ex; height:2.843ex;" alt="{\displaystyle -{\sqrt {5}}i}"></span>。 </p><p>之所以等式右侧（包括其对应的负值）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 -x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <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/ae55e66aeffc525917eed885b4b753ba5a7f8b3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.138ex; height:2.176ex;" alt="{\displaystyle -x}"></span>的算术平方根，是因为： </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (i{\sqrt {x}})^{2}=i^{2}({\sqrt {x}})^{2}=(-1)x=-x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>x</mi> </msqrt> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>x</mi> </msqrt> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>x</mi> <mo>=</mo> <mo>−</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (i{\sqrt {x}})^{2}=i^{2}({\sqrt {x}})^{2}=(-1)x=-x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/158e5a5de426265a7c5a831cc5e391325c7a338b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:33.46ex; height:3.343ex;" alt="{\displaystyle (i{\sqrt {x}})^{2}=i^{2}({\sqrt {x}})^{2}=(-1)x=-x}"></span></dd></dl> <p>负数的兩個平方根为一对<a href="/wiki/%E5%85%B1%E8%BD%AD%E5%A4%8D%E6%95%B0" title="共轭复数">共轭</a>的<a href="/wiki/%E8%99%9A%E6%95%B0" title="虚数">纯虚数</a>。 </p><p>對於負數<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 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>、<span class="mwe-math-element"><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>，以下式成立： </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\sqrt {x}}{\sqrt {y}}&={\sqrt {-x}}\,i\times {\sqrt {-y}}\,i={\sqrt {xy}}\,i^{2}=-{\sqrt {xy}}\\{\frac {\sqrt {x}}{\sqrt {y}}}&={\frac {{\sqrt {-x}}i}{{\sqrt {-y}}i}}={\sqrt {\frac {-x}{-y}}}={\sqrt {\frac {x}{y}}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>x</mi> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>y</mi> </msqrt> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mi>x</mi> </msqrt> </mrow> <mspace width="thinmathspace" /> <mi>i</mi> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mi>y</mi> </msqrt> </mrow> <mspace width="thinmathspace" /> <mi>i</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>x</mi> <mi>y</mi> </msqrt> </mrow> <mspace width="thinmathspace" /> <msup> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>x</mi> <mi>y</mi> </msqrt> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mi>x</mi> </msqrt> <msqrt> <mi>y</mi> </msqrt> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mi>x</mi> </msqrt> </mrow> <mi>i</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mi>y</mi> </msqrt> </mrow> <mi>i</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mo>−</mo> <mi>x</mi> </mrow> <mrow> <mo>−</mo> <mi>y</mi> </mrow> </mfrac> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mi>x</mi> <mi>y</mi> </mfrac> </msqrt> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\sqrt {x}}{\sqrt {y}}&={\sqrt {-x}}\,i\times {\sqrt {-y}}\,i={\sqrt {xy}}\,i^{2}=-{\sqrt {xy}}\\{\frac {\sqrt {x}}{\sqrt {y}}}&={\frac {{\sqrt {-x}}i}{{\sqrt {-y}}i}}={\sqrt {\frac {-x}{-y}}}={\sqrt {\frac {x}{y}}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8251ec8a190d5d2889dc5cd025cf7681beeee20b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:44.878ex; height:10.509ex;" alt="{\displaystyle {\begin{aligned}{\sqrt {x}}{\sqrt {y}}&={\sqrt {-x}}\,i\times {\sqrt {-y}}\,i={\sqrt {xy}}\,i^{2}=-{\sqrt {xy}}\\{\frac {\sqrt {x}}{\sqrt {y}}}&={\frac {{\sqrt {-x}}i}{{\sqrt {-y}}i}}={\sqrt {\frac {-x}{-y}}}={\sqrt {\frac {x}{y}}}\end{aligned}}}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="虚数的算术平方根"><span id=".E8.99.9A.E6.95.B0.E7.9A.84.E7.AE.97.E6.9C.AF.E5.B9.B3.E6.96.B9.E6.A0.B9"></span>虚数的算术平方根</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%B9%B3%E6%96%B9%E6%A0%B9&action=edit&section=4" title="编辑章节：虚数的算术平方根"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Imaginary2Root.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/32/Imaginary2Root.svg/220px-Imaginary2Root.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/32/Imaginary2Root.svg/330px-Imaginary2Root.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/32/Imaginary2Root.svg/440px-Imaginary2Root.svg.png 2x" data-file-width="480" data-file-height="480" /></a><figcaption>复数平面中，<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span>的两个平方根</figcaption></figure> <p>虚数<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 i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span>的算术平方根可以根据以下公式计算： </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {i}}={\frac {\sqrt {2}}{2}}+i{\frac {\sqrt {2}}{2}}={\frac {\sqrt {2}}{2}}(1+i)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>i</mi> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>2</mn> </msqrt> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>2</mn> </msqrt> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>2</mn> </msqrt> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>i</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {i}}={\frac {\sqrt {2}}{2}}+i{\frac {\sqrt {2}}{2}}={\frac {\sqrt {2}}{2}}(1+i)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f385cc49a4503ce72ba10807dd4a98b57e6b195" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:30.996ex; height:5.843ex;" alt="{\displaystyle {\sqrt {i}}={\frac {\sqrt {2}}{2}}+i{\frac {\sqrt {2}}{2}}={\frac {\sqrt {2}}{2}}(1+i)}"></span></dd></dl> <p>这个公式可以通过用<a href="/wiki/%E4%BB%A3%E6%95%B0" title="代数">代数</a>方法推导，只需找到特定的实数<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 a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span>和<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span>，满足 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}i&=(a+bi)^{2}\\&=a^{2}+2abi-b^{2}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>i</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mi>i</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>a</mi> <mi>b</mi> <mi>i</mi> <mo>−</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}i&=(a+bi)^{2}\\&=a^{2}+2abi-b^{2}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a637db1a4010c34b92da34b725caeb554573760" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:18.861ex; height:6.509ex;" alt="{\displaystyle {\begin{aligned}i&=(a+bi)^{2}\\&=a^{2}+2abi-b^{2}\end{aligned}}}"></span></dd></dl> <p>就可以得到<a href="/wiki/%E6%96%B9%E7%A8%8B%E7%BB%84" title="方程组">方程组</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{cases}2ab=1\\a^{2}-b^{2}=0\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mn>2</mn> <mi>a</mi> <mi>b</mi> <mo>=</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>0</mn> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{cases}2ab=1\\a^{2}-b^{2}=0\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8988839e0f5c47007a7621e351f5d1658b59aaec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:13.932ex; height:6.176ex;" alt="{\displaystyle {\begin{cases}2ab=1\\a^{2}-b^{2}=0\end{cases}}}"></span></dd></dl> <p>的解： </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=b=\pm {\frac {\sqrt {2}}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mi>b</mi> <mo>=</mo> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>2</mn> </msqrt> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=b=\pm {\frac {\sqrt {2}}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f071ebbff8bb42ff4db288929a1a1d4da59673cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.167ex; height:5.843ex;" alt="{\displaystyle a=b=\pm {\frac {\sqrt {2}}{2}}}"></span></dd></dl> <p>其中，算术平方根即为 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=b={\frac {\sqrt {2}}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mi>b</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>2</mn> </msqrt> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=b={\frac {\sqrt {2}}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7d2742273cb39bd3cff598f13e52a764579171c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.359ex; height:5.843ex;" alt="{\displaystyle a=b={\frac {\sqrt {2}}{2}}}"></span></dd></dl> <p>这个公式还可以通过<a href="/wiki/%E6%A3%A3%E8%8E%AB%E5%BC%97%E5%85%AC%E5%BC%8F" title="棣莫弗公式">棣莫弗公式</a>得到，设 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i=\cos \left({\frac {\pi }{2}}\right)+i\sin \left({\frac {\pi }{2}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i=\cos \left({\frac {\pi }{2}}\right)+i\sin \left({\frac {\pi }{2}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9953f8002884039314461306d40b398751c2e0e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:23.785ex; height:4.843ex;" alt="{\displaystyle i=\cos \left({\frac {\pi }{2}}\right)+i\sin \left({\frac {\pi }{2}}\right)}"></span></dd></dl> <p>就可以推出 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\sqrt {i}}&=\left[\cos \left({\frac {\pi }{2}}\right)+i\sin \left({\frac {\pi }{2}}\right)\right]^{\frac {1}{2}}\\&=\cos \left({\frac {\pi }{4}}\right)+i\sin \left({\frac {\pi }{4}}\right)\\&={\frac {\sqrt {2}}{2}}+i{\frac {\sqrt {2}}{2}}\\&={\frac {\sqrt {2}}{2}}(1+i)\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>i</mi> </msqrt> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mrow> <mo>[</mo> <mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>2</mn> </msqrt> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>2</mn> </msqrt> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>2</mn> </msqrt> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>i</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\sqrt {i}}&=\left[\cos \left({\frac {\pi }{2}}\right)+i\sin \left({\frac {\pi }{2}}\right)\right]^{\frac {1}{2}}\\&=\cos \left({\frac {\pi }{4}}\right)+i\sin \left({\frac {\pi }{4}}\right)\\&={\frac {\sqrt {2}}{2}}+i{\frac {\sqrt {2}}{2}}\\&={\frac {\sqrt {2}}{2}}(1+i)\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f86b3b7203a50d96454199f6b6b0ca30d029303" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.838ex; width:30.403ex; height:22.843ex;" alt="{\displaystyle {\begin{aligned}{\sqrt {i}}&=\left[\cos \left({\frac {\pi }{2}}\right)+i\sin \left({\frac {\pi }{2}}\right)\right]^{\frac {1}{2}}\\&=\cos \left({\frac {\pi }{4}}\right)+i\sin \left({\frac {\pi }{4}}\right)\\&={\frac {\sqrt {2}}{2}}+i{\frac {\sqrt {2}}{2}}\\&={\frac {\sqrt {2}}{2}}(1+i)\end{aligned}}}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="复数的算术平方根"><span id=".E5.A4.8D.E6.95.B0.E7.9A.84.E7.AE.97.E6.9C.AF.E5.B9.B3.E6.96.B9.E6.A0.B9"></span>复数的算术平方根</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%B9%B3%E6%96%B9%E6%A0%B9&action=edit&section=5" title="编辑章节：复数的算术平方根"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Visualisation_complex_number_roots.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Visualisation_complex_number_roots.svg/220px-Visualisation_complex_number_roots.svg.png" decoding="async" width="220" height="293" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Visualisation_complex_number_roots.svg/330px-Visualisation_complex_number_roots.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Visualisation_complex_number_roots.svg/440px-Visualisation_complex_number_roots.svg.png 2x" data-file-width="512" data-file-height="683" /></a><figcaption>极坐标下，复数<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 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>的几个方根</figcaption></figure> <p>对于任何一个非零的复数<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 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>都存在两个複数<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 w}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>w</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88b1e0c8e1be5ebe69d18a8010676fa42d7961e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.664ex; height:1.676ex;" alt="{\displaystyle w}"></span>使得<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w^{2}=z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w^{2}=z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5a953292d49e530fb8ab87db8adff44bae46a2b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.905ex; height:2.676ex;" alt="{\displaystyle w^{2}=z}"></span>。 </p><p>首先，我们将复数<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+iy}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>+</mo> <mi>i</mi> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x+iy}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ceb1c6ce62a20dbfe9cb3d82dca889577b469703" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.128ex; height:2.509ex;" alt="{\displaystyle x+iy}"></span> 看作是平面上的点，即<a href="/wiki/%E7%AC%9B%E5%8D%A1%E5%B0%94%E5%9D%90%E6%A0%87%E7%B3%BB" title="笛卡尔坐标系">笛卡尔坐标系</a>中的<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </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/41cf50e4a314ca8e2c30964baa8d26e5be7a9386" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.328ex; height:2.843ex;" alt="{\displaystyle (x,y)}"></span>点。这个点也可以写作<a href="/wiki/%E6%9E%81%E5%9D%90%E6%A0%87" class="mw-redirect" title="极坐标">极坐标</a>的<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (r,\varphi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>r</mi> <mo>,</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (r,\varphi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/deec3051ba5a9c7e5b676df779673dfb5e37a0a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.412ex; height:2.843ex;" alt="{\displaystyle (r,\varphi )}"></span>，其中<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r\geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r\geq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa96c19954fcda2695f988938ccf091d2bc2bbae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.31ex; height:2.343ex;" alt="{\displaystyle r\geq 0}"></span>，是该点到坐标原点的距离，<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"></span>则是从原点到该点的直线与实数坐标轴（<span class="mwe-math-element"><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>轴）的夹角。<a href="/wiki/%E5%A4%8D%E5%88%86%E6%9E%90" class="mw-redirect" title="复分析">复分析</a>中，通常把该点记作<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle re^{i\varphi }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>φ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle re^{i\varphi }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a3baa9456e8a581a66e0427dd9fcd05b345904c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.007ex; height:2.676ex;" alt="{\displaystyle re^{i\varphi }}"></span>。如果 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=re^{i\varphi },-\pi <\varphi \leq \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mi>r</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>φ</mi> </mrow> </msup> <mo>,</mo> <mo>−</mo> <mi>π</mi> <mo><</mo> <mi>φ</mi> <mo>≤</mo> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=re^{i\varphi },-\pi <\varphi \leq \pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1b503fb9a34cfca1198fca478ee130982ac4c12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.416ex; height:3.176ex;" alt="{\displaystyle z=re^{i\varphi },-\pi <\varphi \leq \pi }"></span></dd></dl> <p>那么我们将<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 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>的算术平方根定义为： </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {z}}={\sqrt {r}}e^{\frac {i\varphi }{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>z</mi> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>r</mi> </msqrt> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>i</mi> <mi>φ</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {z}}={\sqrt {r}}e^{\frac {i\varphi }{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abbc8e02d775ee11153d1b017a4fff17312570af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.592ex; height:4.509ex;" alt="{\displaystyle {\sqrt {z}}={\sqrt {r}}e^{\frac {i\varphi }{2}}}"></span></dd></dl> <p>因此，平方根函数除了在非正实数轴上以外是处处<a href="/wiki/%E5%85%A8%E7%BA%AF%E5%87%BD%E6%95%B0" title="全纯函数">全纯</a>的。<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {1+x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>+</mo> <mi>x</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {1+x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8b52f282420b9d2b6625eb1889455838334976f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.268ex; height:3.009ex;" alt="{\displaystyle {\sqrt {1+x}}}"></span> 的泰勒级数也适用于复数<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(\left\vert x\right\vert <1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">(</mo> <mrow> <mo>|</mo> <mi>x</mi> <mo>|</mo> </mrow> <mo><</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x(\left\vert x\right\vert <1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5fbe9b64d54849fab34ed889c15060ca15261c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.023ex; height:2.843ex;" alt="{\displaystyle x(\left\vert x\right\vert <1)}"></span>。 </p><p>上面的公式还可以用<a href="/wiki/%E4%B8%89%E8%A7%92%E5%87%BD%E6%95%B0" title="三角函数">三角函数</a>的形式表达： </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {r\left(\cos \varphi +i\sin \varphi \right)}}={\sqrt {r}}\left(\cos {\frac {\varphi }{2}}+i\sin {\frac {\varphi }{2}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>r</mi> <mrow> <mo>(</mo> <mrow> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <mo>+</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> </mrow> <mo>)</mo> </mrow> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>r</mi> </msqrt> </mrow> <mrow> <mo>(</mo> <mrow> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>φ</mi> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>φ</mi> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {r\left(\cos \varphi +i\sin \varphi \right)}}={\sqrt {r}}\left(\cos {\frac {\varphi }{2}}+i\sin {\frac {\varphi }{2}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9654238ad2d7347c9ccb86c5b0d670b31c1efd12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:44.109ex; height:4.843ex;" alt="{\displaystyle {\sqrt {r\left(\cos \varphi +i\sin \varphi \right)}}={\sqrt {r}}\left(\cos {\frac {\varphi }{2}}+i\sin {\frac {\varphi }{2}}\right)}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="代数公式"><span id=".E4.BB.A3.E6.95.B0.E5.85.AC.E5.BC.8F"></span>代数公式</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%B9%B3%E6%96%B9%E6%A0%B9&action=edit&section=6" title="编辑章节：代数公式"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>如果使用笛卡尔坐标的形式表达复数 <i>z</i>，其算术平方根可以使用如下公式：<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {z}}={\sqrt {\frac {|z|+\Re (z)}{2}}}\pm i{\sqrt {\frac {|z|-\Re (z)}{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>z</mi> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>+</mo> <mi mathvariant="normal">ℜ</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </msqrt> </mrow> <mo>±</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>−</mo> <mi mathvariant="normal">ℜ</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {z}}={\sqrt {\frac {|z|+\Re (z)}{2}}}\pm i{\sqrt {\frac {|z|-\Re (z)}{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2b1c4487834a52c5d0440965acd4fd6f9c67ac0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:36.173ex; height:7.676ex;" alt="{\displaystyle {\sqrt {z}}={\sqrt {\frac {|z|+\Re (z)}{2}}}\pm i{\sqrt {\frac {|z|-\Re (z)}{2}}}}"></span></dd></dl> <p>其中，方根虚部的<a href="/wiki/%E7%AC%A6%E5%8F%B7%E5%87%BD%E6%95%B0" title="符号函数">符号</a>与被开方数虚部的符号相同（为0时取正）；<span class="ilh-all" data-orig-title="主值" data-lang-code="en" data-lang-name="英语" data-foreign-title="Principal value"><span class="ilh-page"><a href="/w/index.php?title=%E4%B8%BB%E5%80%BC&action=edit&redlink=1" class="new" title="主值（页面不存在）">主值</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/Principal_value" class="extiw" title="en:Principal value"><span lang="en" dir="auto">Principal value</span></a></span>）</span></span>实部永远非负。 </p><p>在虛數裡，平方根函數的值不是連續的，以下等式不一定成立： </p> <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 {\sqrt {zw}}={\sqrt {z}}{\sqrt {w}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>z</mi> <mi>w</mi> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>z</mi> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>w</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {zw}}={\sqrt {z}}{\sqrt {w}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69736e0fca5366225d71b03cbe0ae84ef8d7d9bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.411ex; height:3.009ex;" alt="{\displaystyle {\sqrt {zw}}={\sqrt {z}}{\sqrt {w}}}"></span></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 {\sqrt {w}}{\sqrt {z}}}={\sqrt {\frac {w}{z}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mi>w</mi> </msqrt> <msqrt> <mi>z</mi> </msqrt> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mi>w</mi> <mi>z</mi> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\sqrt {w}}{\sqrt {z}}}={\sqrt {\frac {w}{z}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32905bc5f10ca0ce6afeea72186f2ac85d9d91ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:12.359ex; height:6.843ex;" alt="{\displaystyle {\frac {\sqrt {w}}{\sqrt {z}}}={\sqrt {\frac {w}{z}}}}"></span></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 {z^{*}}}=\left({\sqrt {z}}\right)^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </msqrt> </mrow> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>z</mi> </msqrt> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {z^{*}}}=\left({\sqrt {z}}\right)^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7de0bb8f7c38b7810de19ebe31b599569de31b15" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.066ex; height:3.343ex;" alt="{\displaystyle {\sqrt {z^{*}}}=\left({\sqrt {z}}\right)^{*}}"></span></li></ul> <p>所以這是錯誤的： </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -1=i\cdot i={\sqrt {-1}}\cdot {\sqrt {-1}}={\sqrt {(-1)\cdot (-1)}}={\sqrt {1}}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>1</mn> <mo>=</mo> <mi>i</mi> <mo>⋅</mo> <mi>i</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> </msqrt> </mrow> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -1=i\cdot i={\sqrt {-1}}\cdot {\sqrt {-1}}={\sqrt {(-1)\cdot (-1)}}={\sqrt {1}}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/208ed50475bd6661822dc3df5288d2d27ca87044" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:51.062ex; height:4.843ex;" alt="{\displaystyle -1=i\cdot i={\sqrt {-1}}\cdot {\sqrt {-1}}={\sqrt {(-1)\cdot (-1)}}={\sqrt {1}}=1}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="多项式"><span id=".E5.A4.9A.E9.A1.B9.E5.BC.8F"></span>多项式</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%B9%B3%E6%96%B9%E6%A0%B9&action=edit&section=7" title="编辑章节：多项式"><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">参见：<a href="/wiki/%E5%9B%A0%E5%BC%8F%E5%88%86%E8%A7%A3" title="因式分解">因式分解</a></div> <p>例：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\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9c6d458566aec47a7259762034790c8981aefab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.848ex; height:2.176ex;" alt="{\displaystyle x\in \mathbb {R} }"></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 {\sqrt {x^{4}+2x^{2}+1}}={\sqrt {(x^{2}+1)^{2}}}=|x^{2}+1|=x^{2}+1\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {x^{4}+2x^{2}+1}}={\sqrt {(x^{2}+1)^{2}}}=|x^{2}+1|=x^{2}+1\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6cc6016d5f0a6bb5e05c7c26c2531128979a72b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; margin-right: -0.387ex; width:50.421ex; height:4.843ex;" alt="{\displaystyle {\sqrt {x^{4}+2x^{2}+1}}={\sqrt {(x^{2}+1)^{2}}}=|x^{2}+1|=x^{2}+1\,\!}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="2的算术平方根"><span id="2.E7.9A.84.E7.AE.97.E6.9C.AF.E5.B9.B3.E6.96.B9.E6.A0.B9"></span>2的算术平方根</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%B9%B3%E6%96%B9%E6%A0%B9&action=edit&section=8" title="编辑章节：2的算术平方根"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>數學史中，最重要的平方根可以說是<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {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>，它代表邊長為1的<a href="/wiki/%E6%AD%A3%E6%96%B9%E5%BD%A2" title="正方形">正方形</a>的<a href="/wiki/%E5%B0%8D%E8%A7%92%E7%B7%9A" title="對角線">對角線</a>長，是第一個公認的<a href="/wiki/%E7%84%A1%E7%90%86%E6%95%B8" title="無理數">無理數</a>，也叫<a href="/wiki/2%E7%9A%84%E7%AE%97%E8%A1%93%E5%B9%B3%E6%96%B9%E6%A0%B9" title="2的算術平方根">毕达哥拉斯常数</a>，其值到小數點14位約為1.4142135623731。 </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4afc1e27d418021bf10898eb44a7f5f315735ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}}"></span>是無理數，可由<a href="/wiki/%E6%AD%B8%E8%AC%AC%E6%B3%95" title="歸謬法">歸謬法</a>證明： </p> <ol><li>設<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2}}}"> <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>為<a href="/wiki/%E6%9C%89%E7%90%86%E6%95%B8" class="mw-redirect" title="有理數">有理數</a>，可表示為<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 {\frac {p}{q}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>p</mi> <mi>q</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {p}{q}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9903bc1de26879e5fc4c7f78b54b952bcbb800f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:2.006ex; height:5.343ex;" alt="{\displaystyle {\frac {p}{q}}}"></span>，其中<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span>、<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span>為<a href="/wiki/%E4%BA%92%E8%B3%AA" title="互質">互質</a>之正整數。</li> <li>因為<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left({\sqrt {2}}\right)^{2}={\frac {p^{2}}{q^{2}}}=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\sqrt {2}}\right)^{2}={\frac {p^{2}}{q^{2}}}=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02c123f765e90c034a2ae7f48a614d15a86619be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.702ex; height:6.343ex;" alt="{\displaystyle \left({\sqrt {2}}\right)^{2}={\frac {p^{2}}{q^{2}}}=2}"></span>，故<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef685027b97072ee63a8c738f395cd40f63767e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.313ex; height:3.009ex;" alt="{\displaystyle p^{2}}"></span>是2的倍數，<i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span></i>也是2的倍數，記為<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 2k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab358eb7defb4d2b0fc1f9e8a4e2d189fe600eb6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.374ex; height:2.176ex;" alt="{\displaystyle 2k}"></span>，其中<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span>為正整數。</li> <li>但是<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 2q^{2}=p^{2}=4k^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>4</mn> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2q^{2}=p^{2}=4k^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e6aa204f2ea84ceff2b966f3e23659556f0a072" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.145ex; height:3.009ex;" alt="{\displaystyle 2q^{2}=p^{2}=4k^{2}}"></span>，故<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q^{2}=2k^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>2</mn> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q^{2}=2k^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c70a613fd80255a1f094bc317dc6374ed7dddf6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.66ex; height:3.009ex;" alt="{\displaystyle q^{2}=2k^{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 q^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/024d4dbdf3feb09055609f33baa8a7ae23aef1d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.134ex; height:3.009ex;" alt="{\displaystyle q^{2}}"></span>是2的倍數，<i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span></i>也是2的倍數。</li> <li>依上兩式，<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span>、<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span>都是2的倍數，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 p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span>、<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span>為互質之正整數的前題矛盾。依歸謬法，得證<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2}}}"> <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>不是有理數，即<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4afc1e27d418021bf10898eb44a7f5f315735ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}}"></span>是無理數。</li></ol> <div class="mw-heading mw-heading2"><h2 id="計算方法"><span id=".E8.A8.88.E7.AE.97.E6.96.B9.E6.B3.95"></span>計算方法</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%B9%B3%E6%96%B9%E6%A0%B9&action=edit&section=9" title="编辑章节：計算方法"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="因數計算"><span id=".E5.9B.A0.E6.95.B8.E8.A8.88.E7.AE.97"></span>因數計算</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%B9%B3%E6%96%B9%E6%A0%B9&action=edit&section=10" title="编辑章节：因數計算"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {24}}={\sqrt {2^{2}\cdot 6}}={\sqrt {2^{2}}}{\sqrt {6}}=2{\sqrt {6}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>24</mn> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⋅</mo> <mn>6</mn> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>6</mn> </msqrt> </mrow> <mo>=</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>6</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {24}}={\sqrt {2^{2}\cdot 6}}={\sqrt {2^{2}}}{\sqrt {6}}=2{\sqrt {6}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d0cbdf10067134f4a2303474ef6e5fc310bc996" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:32.838ex; height:3.509ex;" alt="{\displaystyle {\sqrt {24}}={\sqrt {2^{2}\cdot 6}}={\sqrt {2^{2}}}{\sqrt {6}}=2{\sqrt {6}}}"></span>。 </p><p><br /> </p><p>注意，6 的质因数分解为 2 × 3，不能写成某个数的平方，因此 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2{\sqrt {6}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>6</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2{\sqrt {6}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/975f782dae96fcca952fe4e47fc26d8846e04a2c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.261ex; height:2.843ex;" alt="{\displaystyle 2{\sqrt {6}}}"></span> 就是最简结果。 </p> <div class="mw-heading mw-heading3"><h3 id="中算开方"><span id=".E4.B8.AD.E7.AE.97.E5.BC.80.E6.96.B9"></span>中算开方</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%B9%B3%E6%96%B9%E6%A0%B9&action=edit&section=11" title="编辑章节：中算开方"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:JIA_XIAN_SQRT2.GIF" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/JIA_XIAN_SQRT2.GIF/300px-JIA_XIAN_SQRT2.GIF" decoding="async" width="300" height="206" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/JIA_XIAN_SQRT2.GIF/450px-JIA_XIAN_SQRT2.GIF 1.5x, //upload.wikimedia.org/wikipedia/commons/2/20/JIA_XIAN_SQRT2.GIF 2x" data-file-width="458" data-file-height="314" /></a><figcaption>北宋贾宪增乘开平方法</figcaption></figure> <p>《<a href="/wiki/%E4%B9%9D%E7%AB%A0%E7%AE%97%E6%9C%AF" title="九章算术">九章算术</a>》和《<a href="/wiki/%E5%AD%99%E5%AD%90%E7%AE%97%E7%BB%8F" title="孙子算经">孙子算经</a>》都有<a href="/wiki/%E7%AD%B9%E7%AE%97" title="筹算">筹算</a>的开方法。<a href="/wiki/%E5%AE%8B%E4%BB%A3" class="mw-redirect" title="宋代">宋代</a>数学家<a href="/wiki/%E8%B4%BE%E5%AE%AA" title="贾宪">贾宪</a>发明<a href="/w/index.php?title=%E9%87%8A%E9%94%81%E5%BC%80%E5%B9%B3%E6%96%B9%E6%B3%95&action=edit&redlink=1" class="new" title="释锁开平方法（页面不存在）">释锁开平方法</a>、<a href="/wiki/%E5%A2%9E%E4%B9%98%E5%BC%80%E5%B9%B3%E6%96%B9%E6%B3%95" title="增乘开平方法">增乘开平方法</a>；<a href="/wiki/%E6%98%8E%E4%BB%A3" class="mw-redirect" title="明代">明代</a>数学家<a href="/w/index.php?title=%E7%8E%8B%E7%B4%A0%E6%96%87&action=edit&redlink=1" class="new" title="王素文（页面不存在）">王素文</a>，<a href="/wiki/%E7%A8%8B%E5%A4%A7%E4%BD%8D" title="程大位">程大位</a>发明<a href="/w/index.php?title=%E7%8F%A0%E7%AE%97%E5%BC%80%E5%B9%B3%E6%96%B9%E6%B3%95&action=edit&redlink=1" class="new" title="珠算开平方法（页面不存在）">珠算开平方法</a>，而<a href="/wiki/%E6%9C%B1%E8%BD%BD%E5%A0%89" class="mw-redirect" title="朱载堉">朱载堉</a>《<a href="/wiki/%E7%AE%97%E5%AD%A6%E6%96%B0%E8%AF%B4" title="算学新说">算学新说</a>》首创用81位<a href="/wiki/%E7%AE%97%E7%9B%98" title="算盘">算盘</a>开方，精确到25位数字<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup>。 </p> <div class="mw-heading mw-heading3"><h3 id="長除式算法"><span id=".E9.95.B7.E9.99.A4.E5.BC.8F.E7.AE.97.E6.B3.95"></span>長除式算法</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%B9%B3%E6%96%B9%E6%A0%B9&action=edit&section=12" title="编辑章节：長除式算法"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>長除式算平方根的方式也稱為直式開方法，原理是<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 .(a+b)^{2}=a^{2}+2ab+b^{2}=a^{2}+(2a+b)b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>.</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>a</mi> <mi>b</mi> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle .(a+b)^{2}=a^{2}+2ab+b^{2}=a^{2}+(2a+b)b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8d6f7cbea09b87a9282db236cf0b6149c2f57c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:42.73ex; height:3.176ex;" alt="{\displaystyle .(a+b)^{2}=a^{2}+2ab+b^{2}=a^{2}+(2a+b)b}"></span>。 </p> <ol><li>首先將要開平方根的數從小數點分別向右及向左每兩個位一組分開，如98765.432內小數點前的65是一組，87是一組，9是一組，小數點後的43是一組，之後是單獨一個2，要補一個0而得20是一組。如1 04.85 73得四組，順序為1' 04. 85' 73'。</li> <li>將最左的一組的數減去最接近又少於它的平方數，並將該平方數的開方（應該是個位數）記下。</li> <li>將上一步所得之差乘100，和下一組數加起來。</li> <li>將記下的數乘20，然後將它加上某個個位數，再乘以該個個位數，令這個積不大於但最接近上一步所得之差，並將該個個位數記下，且將上一步所得之差減去所得之積。</li> <li>記下的數一次隔兩位記下。</li> <li>重覆第3步，直到找到答案。</li> <li>可以在數字的最右補上多組的00'以求得理想的精確度為止。</li></ol> <p>下面以<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {200}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>200</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {200}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ebf7c3e7850fc8a49ee456107b45fc64027b8b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.423ex; height:2.843ex;" alt="{\displaystyle {\sqrt {200}}}"></span>為例子： </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{ll}\quad {\color {Red}1}~~{\color {Green}4}.~~{\color {Blue}1}~~{\color {Purple}4}~~{\color {Orange}2}\\{\sqrt {2|00.00|00|00}}\\\quad {\underline {1\quad ~}}&\quad {\color {Red}1}\times {\color {Red}1}\leq 2\\\quad 1~00&a={\color {Red}1}0,b={\color {Green}4}\\\quad {\underline {~~\,96\quad ~}}&\quad \Rightarrow (2a+b)b=2{\color {Green}4}\times {\color {Green}4}=96\leq 100\\\qquad ~4~00&a={\color {Red}1}{\color {Green}4}0,b={\color {Blue}1}\\\qquad ~{\underline {2~81\quad ~}}&\quad \Rightarrow (2a+b)b=28{\color {Blue}1}\times {\color {Blue}1}=281\leq 400\\\qquad ~1~19~00&a={\color {Red}1}{\color {Green}4}{\color {Blue}1}0,b={\color {Purple}4}\\\qquad ~{\underline {1~12~96\quad ~}}&\quad \Rightarrow (2a+b)b=282{\color {Purple}4}\times {\color {Purple}4}=11296\leq 11900\\\qquad \quad ~~6~04~00&a={\color {Red}1}{\color {Green}4}{\color {Blue}1}{\color {Purple}4}0,b={\color {Orange}2}\\\qquad \quad ~~{\underline {5~65~64}}&\quad \Rightarrow (2a+b)b=2828{\color {Orange}2}\times {\color {Orange}2}=56564\leq 60400\\\qquad \quad \quad ~\,38~36\\\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="left left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#ED1B23"> <mn>1</mn> </mstyle> </mrow> <mtext> </mtext> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#55ff55"> <mn>4</mn> </mstyle> </mrow> <mo>.</mo> <mtext> </mtext> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#2D2F92"> <mn>1</mn> </mstyle> </mrow> <mtext> </mtext> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#99479B"> <mn>4</mn> </mstyle> </mrow> <mtext> </mtext> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#F58137"> <mn>2</mn> </mstyle> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>00.00</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>00</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>00</mn> </msqrt> </mrow> </mtd> </mtr> <mtr> <mtd> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <munder> <mrow> <mn>1</mn> <mspace width="1em" /> <mtext> </mtext> </mrow> <mo>_</mo> </munder> </mrow> </mtd> <mtd> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#ED1B23"> <mn>1</mn> </mstyle> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#ED1B23"> <mn>1</mn> </mstyle> </mrow> <mo>≤</mo> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mspace width="1em" /> <mn>1</mn> <mtext> </mtext> <mn>00</mn> </mtd> <mtd> <mi>a</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#ED1B23"> <mn>1</mn> </mstyle> </mrow> <mn>0</mn> <mo>,</mo> <mi>b</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#55ff55"> <mn>4</mn> </mstyle> </mrow> </mtd> </mtr> <mtr> <mtd> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <munder> <mrow> <mtext> </mtext> <mtext> </mtext> <mspace width="thinmathspace" /> <mn>96</mn> <mspace width="1em" /> <mtext> </mtext> </mrow> <mo>_</mo> </munder> </mrow> </mtd> <mtd> <mspace width="1em" /> <mo stretchy="false">⇒</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mi>b</mi> <mo>=</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#55ff55"> <mn>4</mn> </mstyle> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#55ff55"> <mn>4</mn> </mstyle> </mrow> <mo>=</mo> <mn>96</mn> <mo>≤</mo> <mn>100</mn> </mtd> </mtr> <mtr> <mtd> <mspace width="2em" /> <mtext> </mtext> <mn>4</mn> <mtext> </mtext> <mn>00</mn> </mtd> <mtd> <mi>a</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#ED1B23"> <mn>1</mn> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#55ff55"> <mn>4</mn> </mstyle> </mrow> <mn>0</mn> <mo>,</mo> <mi>b</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#2D2F92"> <mn>1</mn> </mstyle> </mrow> </mtd> </mtr> <mtr> <mtd> <mspace width="2em" /> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <munder> <mrow> <mn>2</mn> <mtext> </mtext> <mn>81</mn> <mspace width="1em" /> <mtext> </mtext> </mrow> <mo>_</mo> </munder> </mrow> </mtd> <mtd> <mspace width="1em" /> <mo stretchy="false">⇒</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mi>b</mi> <mo>=</mo> <mn>28</mn> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#2D2F92"> <mn>1</mn> </mstyle> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#2D2F92"> <mn>1</mn> </mstyle> </mrow> <mo>=</mo> <mn>281</mn> <mo>≤</mo> <mn>400</mn> </mtd> </mtr> <mtr> <mtd> <mspace width="2em" /> <mtext> </mtext> <mn>1</mn> <mtext> </mtext> <mn>19</mn> <mtext> </mtext> <mn>00</mn> </mtd> <mtd> <mi>a</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#ED1B23"> <mn>1</mn> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#55ff55"> <mn>4</mn> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#2D2F92"> <mn>1</mn> </mstyle> </mrow> <mn>0</mn> <mo>,</mo> <mi>b</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#99479B"> <mn>4</mn> </mstyle> </mrow> </mtd> </mtr> <mtr> <mtd> <mspace width="2em" /> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <munder> <mrow> <mn>1</mn> <mtext> </mtext> <mn>12</mn> <mtext> </mtext> <mn>96</mn> <mspace width="1em" /> <mtext> </mtext> </mrow> <mo>_</mo> </munder> </mrow> </mtd> <mtd> <mspace width="1em" /> <mo stretchy="false">⇒</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mi>b</mi> <mo>=</mo> <mn>282</mn> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#99479B"> <mn>4</mn> </mstyle> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#99479B"> <mn>4</mn> </mstyle> </mrow> <mo>=</mo> <mn>11296</mn> <mo>≤</mo> <mn>11900</mn> </mtd> </mtr> <mtr> <mtd> <mspace width="2em" /> <mspace width="1em" /> <mtext> </mtext> <mtext> </mtext> <mn>6</mn> <mtext> </mtext> <mn>04</mn> <mtext> </mtext> <mn>00</mn> </mtd> <mtd> <mi>a</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#ED1B23"> <mn>1</mn> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#55ff55"> <mn>4</mn> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#2D2F92"> <mn>1</mn> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#99479B"> <mn>4</mn> </mstyle> </mrow> <mn>0</mn> <mo>,</mo> <mi>b</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#F58137"> <mn>2</mn> </mstyle> </mrow> </mtd> </mtr> <mtr> <mtd> <mspace width="2em" /> <mspace width="1em" /> <mtext> </mtext> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <munder> <mrow> <mn>5</mn> <mtext> </mtext> <mn>65</mn> <mtext> </mtext> <mn>64</mn> </mrow> <mo>_</mo> </munder> </mrow> </mtd> <mtd> <mspace width="1em" /> <mo stretchy="false">⇒</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mi>b</mi> <mo>=</mo> <mn>2828</mn> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#F58137"> <mn>2</mn> </mstyle> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#F58137"> <mn>2</mn> </mstyle> </mrow> <mo>=</mo> <mn>56564</mn> <mo>≤</mo> <mn>60400</mn> </mtd> </mtr> <mtr> <mtd> <mspace width="2em" /> <mspace width="1em" /> <mspace width="1em" /> <mtext> </mtext> <mspace width="thinmathspace" /> <mn>38</mn> <mtext> </mtext> <mn>36</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{ll}\quad {\color {Red}1}~~{\color {Green}4}.~~{\color {Blue}1}~~{\color {Purple}4}~~{\color {Orange}2}\\{\sqrt {2|00.00|00|00}}\\\quad {\underline {1\quad ~}}&\quad {\color {Red}1}\times {\color {Red}1}\leq 2\\\quad 1~00&a={\color {Red}1}0,b={\color {Green}4}\\\quad {\underline {~~\,96\quad ~}}&\quad \Rightarrow (2a+b)b=2{\color {Green}4}\times {\color {Green}4}=96\leq 100\\\qquad ~4~00&a={\color {Red}1}{\color {Green}4}0,b={\color {Blue}1}\\\qquad ~{\underline {2~81\quad ~}}&\quad \Rightarrow (2a+b)b=28{\color {Blue}1}\times {\color {Blue}1}=281\leq 400\\\qquad ~1~19~00&a={\color {Red}1}{\color {Green}4}{\color {Blue}1}0,b={\color {Purple}4}\\\qquad ~{\underline {1~12~96\quad ~}}&\quad \Rightarrow (2a+b)b=282{\color {Purple}4}\times {\color {Purple}4}=11296\leq 11900\\\qquad \quad ~~6~04~00&a={\color {Red}1}{\color {Green}4}{\color {Blue}1}{\color {Purple}4}0,b={\color {Orange}2}\\\qquad \quad ~~{\underline {5~65~64}}&\quad \Rightarrow (2a+b)b=2828{\color {Orange}2}\times {\color {Orange}2}=56564\leq 60400\\\qquad \quad \quad ~\,38~36\\\end{array}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d54a63deea064813349d04a97ce6be458f4fa90" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -19.338ex; width:63.626ex; height:39.843ex;" alt="{\displaystyle {\begin{array}{ll}\quad {\color {Red}1}~~{\color {Green}4}.~~{\color {Blue}1}~~{\color {Purple}4}~~{\color {Orange}2}\\{\sqrt {2|00.00|00|00}}\\\quad {\underline {1\quad ~}}&\quad {\color {Red}1}\times {\color {Red}1}\leq 2\\\quad 1~00&a={\color {Red}1}0,b={\color {Green}4}\\\quad {\underline {~~\,96\quad ~}}&\quad \Rightarrow (2a+b)b=2{\color {Green}4}\times {\color {Green}4}=96\leq 100\\\qquad ~4~00&a={\color {Red}1}{\color {Green}4}0,b={\color {Blue}1}\\\qquad ~{\underline {2~81\quad ~}}&\quad \Rightarrow (2a+b)b=28{\color {Blue}1}\times {\color {Blue}1}=281\leq 400\\\qquad ~1~19~00&a={\color {Red}1}{\color {Green}4}{\color {Blue}1}0,b={\color {Purple}4}\\\qquad ~{\underline {1~12~96\quad ~}}&\quad \Rightarrow (2a+b)b=282{\color {Purple}4}\times {\color {Purple}4}=11296\leq 11900\\\qquad \quad ~~6~04~00&a={\color {Red}1}{\color {Green}4}{\color {Blue}1}{\color {Purple}4}0,b={\color {Orange}2}\\\qquad \quad ~~{\underline {5~65~64}}&\quad \Rightarrow (2a+b)b=2828{\color {Orange}2}\times {\color {Orange}2}=56564\leq 60400\\\qquad \quad \quad ~\,38~36\\\end{array}}}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {200}}\approx 14.14213562373095048801668872421}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>200</mn> </msqrt> </mrow> <mo>≈</mo> <mn>14.14213562373095048801668872421</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {200}}\approx 14.14213562373095048801668872421}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/353409596f12a8b943a90b8ef443cf9cd8bca51d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:45.205ex; height:2.843ex;" alt="{\displaystyle {\sqrt {200}}\approx 14.14213562373095048801668872421}"></span> </p><p><a href="/wiki/%E5%9B%9B%E6%8D%A8%E4%BA%94%E5%85%A5" class="mw-redirect" title="四捨五入">四捨五入</a>得答案為14.14。 </p><p>事實上，將算法稍作改動，可以開任何次方的根，詳見<a href="/w/index.php?title=N%E6%AC%A1%E6%96%B9%E7%AE%97%E6%B3%95&action=edit&redlink=1" class="new" title="N次方算法（页面不存在）">n次方算法</a>。 </p><p>利用高精度长式除法可以计算出1至20的平方根如下： </p> <table> <tbody><tr> <td align="right" style="padding-bottom:5px;"><span class="mwe-math-element"><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 {1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7bf2189fbb14a70abb4d7e3f2aedec1f3e787e6" 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 {1}}}"></span></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2219e60aaf3efae72388ea5a38538ff64f0ce00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.195ex; height:1.343ex;" alt="{\displaystyle =\,}"></span></td> <td>1 </td></tr> <tr> <td align="right" style="padding-bottom:5px;"><span class="mwe-math-element"><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></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \approx }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≈</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \approx }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f58f4c2b73283ce8a5ad28fb3746f2a8c998789" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.125ex; margin-bottom: -0.297ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \approx }"></span></td> <td>1.4142135623 7309504880 1688724209 6980785696 7187537694 8073176679 7379907324 78462 </td></tr> <tr> <td align="right" style="padding-bottom:5px;"><span class="mwe-math-element"><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></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \approx }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≈</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \approx }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f58f4c2b73283ce8a5ad28fb3746f2a8c998789" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.125ex; margin-bottom: -0.297ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \approx }"></span></td> <td>1.7320508075 6887729352 7446341505 8723669428 0525381038 0628055806 9794519330 16909 </td></tr> <tr> <td align="right" style="padding-bottom:5px;"><span class="mwe-math-element"><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 {4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>4</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {4}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ddc524346bd183be9f2c63f906d73910844ef1d" 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 {4}}}"></span></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2219e60aaf3efae72388ea5a38538ff64f0ce00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.195ex; height:1.343ex;" alt="{\displaystyle =\,}"></span></td> <td>2 </td></tr> <tr> <td align="right" style="padding-bottom:5px;"><span class="mwe-math-element"><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 {5}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {5}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b78ccdb7e18e02d4fc567c66aac99bf524acb5f" 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 {5}}}"></span></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \approx }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≈</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \approx }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f58f4c2b73283ce8a5ad28fb3746f2a8c998789" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.125ex; margin-bottom: -0.297ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \approx }"></span></td> <td>2.2360679774 9978969640 9173668731 2762354406 1835961152 5724270897 2454105209 25638 </td></tr> <tr> <td align="right" style="padding-bottom:5px;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {6}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>6</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {6}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a857de6bca2591cfad08e4378634825b6be66a01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:2.843ex;" alt="{\displaystyle {\sqrt {6}}}"></span></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \approx }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≈</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \approx }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f58f4c2b73283ce8a5ad28fb3746f2a8c998789" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.125ex; margin-bottom: -0.297ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \approx }"></span></td> <td>2.4494897427 8317809819 7284074705 8913919659 4748065667 0128432692 5672509603 77457 </td></tr> <tr> <td align="right" style="padding-bottom:5px;"><span class="mwe-math-element"><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 {7}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>7</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {7}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ca16e62fc47de4252d87457029895a954d91a42" 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 {7}}}"></span></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \approx }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≈</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \approx }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f58f4c2b73283ce8a5ad28fb3746f2a8c998789" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.125ex; margin-bottom: -0.297ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \approx }"></span></td> <td>2.6457513110 6459059050 1615753639 2604257102 5918308245 0180368334 4592010688 23230 </td></tr> <tr> <td align="right" style="padding-bottom:5px;"><span class="mwe-math-element"><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 {8}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>8</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {8}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74655655dfdd370266c9238e7ba06ff9cc9d43f9" 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 {8}}}"></span></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \approx }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≈</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \approx }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f58f4c2b73283ce8a5ad28fb3746f2a8c998789" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.125ex; margin-bottom: -0.297ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \approx }"></span></td> <td>2.8284271247 4619009760 3377448419 3961571393 4375075389 6146353359 4759814649 56924 </td></tr> <tr> <td align="right" style="padding-bottom:5px;"><span class="mwe-math-element"><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 {9}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>9</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {9}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac20efa072b0e9082e65122a4104a02f4a746986" 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 {9}}}"></span></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2219e60aaf3efae72388ea5a38538ff64f0ce00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.195ex; height:1.343ex;" alt="{\displaystyle =\,}"></span></td> <td>3 </td></tr> <tr> <td align="right" style="padding-bottom:5px;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {10}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>10</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {10}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd7409b0ddbc1f90280265e7bc95dd20626ebf1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.261ex; height:2.843ex;" alt="{\displaystyle {\sqrt {10}}}"></span></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \approx }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≈</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \approx }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f58f4c2b73283ce8a5ad28fb3746f2a8c998789" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.125ex; margin-bottom: -0.297ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \approx }"></span></td> <td>3.1622776601 6837933199 8893544432 7185337195 5513932521 6826857504 8527925944 38639 </td></tr> <tr> <td align="right" style="padding-bottom:5px;"><span class="mwe-math-element"><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 {11}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>11</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {11}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b713509221c99940e3bfc0eeb7ddafe6ec870ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.261ex; height:3.009ex;" alt="{\displaystyle {\sqrt {11}}}"></span></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \approx }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≈</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \approx }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f58f4c2b73283ce8a5ad28fb3746f2a8c998789" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.125ex; margin-bottom: -0.297ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \approx }"></span></td> <td>3.3166247903 5539984911 4932736670 6866839270 8854558935 3597058682 1461164846 42609 </td></tr> <tr> <td align="right" style="padding-bottom:5px;"><span class="mwe-math-element"><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 {12}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>12</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {12}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c878efdf227cf70df28aa7d43cea0069e6f515e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.261ex; height:3.009ex;" alt="{\displaystyle {\sqrt {12}}}"></span></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \approx }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≈</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \approx }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f58f4c2b73283ce8a5ad28fb3746f2a8c998789" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.125ex; margin-bottom: -0.297ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \approx }"></span></td> <td>3.4641016151 3775458705 4892683011 7447338856 1050762076 1256111613 9589038660 33818 </td></tr> <tr> <td align="right" style="padding-bottom:5px;"><span class="mwe-math-element"><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 {13}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>13</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {13}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a95228048246821171e1789114839cbd00978027" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.261ex; height:2.843ex;" alt="{\displaystyle {\sqrt {13}}}"></span></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \approx }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≈</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \approx }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f58f4c2b73283ce8a5ad28fb3746f2a8c998789" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.125ex; margin-bottom: -0.297ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \approx }"></span></td> <td>3.6055512754 6398929311 9221267470 4959462512 9657384524 6212710453 0562271669 48293 </td></tr> <tr> <td align="right" style="padding-bottom:5px;"><span class="mwe-math-element"><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 {14}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>14</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {14}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19763bff915d190ddb1eb7b8dd9beb7ede194bff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.261ex; height:3.009ex;" alt="{\displaystyle {\sqrt {14}}}"></span></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \approx }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≈</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \approx }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f58f4c2b73283ce8a5ad28fb3746f2a8c998789" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.125ex; margin-bottom: -0.297ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \approx }"></span></td> <td>3.7416573867 7394138558 3748732316 5493017560 1980777872 6946303745 4673200351 56307 </td></tr> <tr> <td align="right" style="padding-bottom:5px;"><span class="mwe-math-element"><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 {15}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>15</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {15}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e343ba1fba1d0222e6d6b02e264aec5717548f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.261ex; height:2.843ex;" alt="{\displaystyle {\sqrt {15}}}"></span></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \approx }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≈</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \approx }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f58f4c2b73283ce8a5ad28fb3746f2a8c998789" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.125ex; margin-bottom: -0.297ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \approx }"></span></td> <td>3.8729833462 0741688517 9265399782 3996108329 2170529159 0826587573 7661134830 91937 </td></tr> <tr> <td align="right" style="padding-bottom:5px;"><span class="mwe-math-element"><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 {16}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>16</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {16}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/513d1ff9b2ee38ab35caf25b67c82f17a6f99a9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.261ex; height:2.843ex;" alt="{\displaystyle {\sqrt {16}}}"></span></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2219e60aaf3efae72388ea5a38538ff64f0ce00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.195ex; height:1.343ex;" alt="{\displaystyle =\,}"></span></td> <td>4 </td></tr> <tr> <td align="right" style="padding-bottom:5px;"><span class="mwe-math-element"><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 {17}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>17</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {17}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d090c5c91c92d2926ceeece2133403c09bdf4dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.261ex; height:3.009ex;" alt="{\displaystyle {\sqrt {17}}}"></span></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \approx }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≈</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \approx }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f58f4c2b73283ce8a5ad28fb3746f2a8c998789" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.125ex; margin-bottom: -0.297ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \approx }"></span></td> <td>4.1231056256 1766054982 1409855974 0770251471 9922537362 0434398633 5730949543 46338 </td></tr> <tr> <td align="right" style="padding-bottom:5px;"><span class="mwe-math-element"><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 {18}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>18</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {18}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a72a94abb379143c58c86b6db1eddbba27a9b5b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.261ex; height:2.843ex;" alt="{\displaystyle {\sqrt {18}}}"></span></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \approx }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≈</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \approx }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f58f4c2b73283ce8a5ad28fb3746f2a8c998789" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.125ex; margin-bottom: -0.297ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \approx }"></span></td> <td>4.2426406871 1928514640 5066172629 0942357090 1562613084 4219530039 2139721974 35386 </td></tr> <tr> <td align="right" style="padding-bottom:5px;"><span class="mwe-math-element"><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 {19}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>19</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {19}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16a05218b63c0a1f5fb64f9bdd886e04ec9d1f3c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.261ex; height:2.843ex;" alt="{\displaystyle {\sqrt {19}}}"></span></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \approx }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≈</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \approx }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f58f4c2b73283ce8a5ad28fb3746f2a8c998789" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.125ex; margin-bottom: -0.297ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \approx }"></span></td> <td>4.3588989435 4067355223 6981983859 6156591370 0392523244 4936890344 1381595573 28203 </td></tr> <tr> <td align="right" style="padding-bottom:5px;"><span class="mwe-math-element"><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 {20}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>20</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {20}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9d32ce4e3e5160ec9e305bedcdd1fbbdd775c9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.261ex; height:2.843ex;" alt="{\displaystyle {\sqrt {20}}}"></span></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \approx }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≈</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \approx }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f58f4c2b73283ce8a5ad28fb3746f2a8c998789" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.125ex; margin-bottom: -0.297ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \approx }"></span></td> <td>4.4721359549 9957939281 8347337462 5524708812 3671922305 1448541794 4908210418 51276 </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="牛頓法"><span id=".E7.89.9B.E9.A0.93.E6.B3.95"></span>牛頓法</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%B9%B3%E6%96%B9%E6%A0%B9&action=edit&section=13" title="编辑章节：牛頓法"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>如果要求<span class="mwe-math-element"><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>1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <mi>S</mi> <mo>></mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S\,(S>1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6087ad4ea0346f41caab2b8fd102d34d4fbea47d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.456ex; height:2.843ex;" alt="{\displaystyle S\,(S>1)}"></span>的平方根，選取<span class="mwe-math-element"><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\,<\,x_{0}\,<\,S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mspace width="thinmathspace" /> <mo><</mo> <mspace width="thinmathspace" /> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mo><</mo> <mspace width="thinmathspace" /> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\,<\,x_{0}\,<\,S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ece7a438e6bbc834c9da5aca94cc3fa8d52c839b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.791ex; height:2.509ex;" alt="{\displaystyle 1\,<\,x_{0}\,<\,S}"></span> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{n+1}={\frac {1}{2}}\left(x_{n}+{\frac {S}{x_{n}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>S</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{n+1}={\frac {1}{2}}\left(x_{n}+{\frac {S}{x_{n}}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/856e2cc57765274ae3f7e80b2a0c0c274cae2003" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:22.327ex; height:6.176ex;" alt="{\displaystyle x_{n+1}={\frac {1}{2}}\left(x_{n}+{\frac {S}{x_{n}}}\right)}"></span></dd></dl> <p>例子：求<span class="mwe-math-element"><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 {125348}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>125348</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {125348}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3dc86773159d6b46cca89597b449800da9889b61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.911ex; height:3.009ex;" alt="{\displaystyle {\sqrt {125348}}}"></span>至6位<a href="/wiki/%E6%9C%89%E6%95%88%E6%95%B0%E5%AD%97" title="有效数字">有效數字</a>。 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}=3^{6}=729.000\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mo>=</mo> <mn>729.000</mn> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}=3^{6}=729.000\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f7e729729f59e7043e69945f0ea77b39ba1c0bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.387ex; width:18.806ex; height:3.009ex;" alt="{\displaystyle x_{0}=3^{6}=729.000\,\!}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1}={\frac {1}{2}}\left(x_{0}+{\frac {S}{x_{0}}}\right)={\frac {1}{2}}\left(729.000+{\frac {125348}{729.000}}\right)=450.472}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>S</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mn>729.000</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>125348</mn> <mn>729.000</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>450.472</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1}={\frac {1}{2}}\left(x_{0}+{\frac {S}{x_{0}}}\right)={\frac {1}{2}}\left(729.000+{\frac {125348}{729.000}}\right)=450.472}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/494afedc8a66d2d7d37ecac41bef3a124f0d6907" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:58.279ex; height:6.176ex;" alt="{\displaystyle x_{1}={\frac {1}{2}}\left(x_{0}+{\frac {S}{x_{0}}}\right)={\frac {1}{2}}\left(729.000+{\frac {125348}{729.000}}\right)=450.472}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{2}={\frac {1}{2}}\left(x_{1}+{\frac {S}{x_{1}}}\right)={\frac {1}{2}}\left(450.472+{\frac {125348}{450.472}}\right)=364.365}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>S</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mn>450.472</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>125348</mn> <mn>450.472</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>364.365</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{2}={\frac {1}{2}}\left(x_{1}+{\frac {S}{x_{1}}}\right)={\frac {1}{2}}\left(450.472+{\frac {125348}{450.472}}\right)=364.365}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3de23adf43a880d5e083e404b80bec3fed8e34bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:58.279ex; height:6.176ex;" alt="{\displaystyle x_{2}={\frac {1}{2}}\left(x_{1}+{\frac {S}{x_{1}}}\right)={\frac {1}{2}}\left(450.472+{\frac {125348}{450.472}}\right)=364.365}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{3}={\frac {1}{2}}\left(x_{2}+{\frac {S}{x_{2}}}\right)={\frac {1}{2}}\left(364.365+{\frac {125348}{364.365}}\right)=354.191}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>S</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mn>364.365</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>125348</mn> <mn>364.365</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>354.191</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{3}={\frac {1}{2}}\left(x_{2}+{\frac {S}{x_{2}}}\right)={\frac {1}{2}}\left(364.365+{\frac {125348}{364.365}}\right)=354.191}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d7304d3c0026dc9755ac41f62282f7535ad1633" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:58.279ex; height:6.176ex;" alt="{\displaystyle x_{3}={\frac {1}{2}}\left(x_{2}+{\frac {S}{x_{2}}}\right)={\frac {1}{2}}\left(364.365+{\frac {125348}{364.365}}\right)=354.191}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{4}={\frac {1}{2}}\left(x_{3}+{\frac {S}{x_{3}}}\right)={\frac {1}{2}}\left(354.191+{\frac {125348}{354.191}}\right)=354.045}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>S</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mn>354.191</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>125348</mn> <mn>354.191</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>354.045</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{4}={\frac {1}{2}}\left(x_{3}+{\frac {S}{x_{3}}}\right)={\frac {1}{2}}\left(354.191+{\frac {125348}{354.191}}\right)=354.045}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bba61b4866b035c837260aaa79c2aa85c9b098ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:58.279ex; height:6.176ex;" alt="{\displaystyle x_{4}={\frac {1}{2}}\left(x_{3}+{\frac {S}{x_{3}}}\right)={\frac {1}{2}}\left(354.191+{\frac {125348}{354.191}}\right)=354.045}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{5}={\frac {1}{2}}\left(x_{4}+{\frac {S}{x_{4}}}\right)={\frac {1}{2}}\left(354.045+{\frac {125348}{354.045}}\right)=354.045}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>S</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mn>354.045</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>125348</mn> <mn>354.045</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>354.045</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{5}={\frac {1}{2}}\left(x_{4}+{\frac {S}{x_{4}}}\right)={\frac {1}{2}}\left(354.045+{\frac {125348}{354.045}}\right)=354.045}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eec508225a99d1c2f03dfafcd3b3c6aec86bca3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:58.279ex; height:6.176ex;" alt="{\displaystyle x_{5}={\frac {1}{2}}\left(x_{4}+{\frac {S}{x_{4}}}\right)={\frac {1}{2}}\left(354.045+{\frac {125348}{354.045}}\right)=354.045}"></span></dd></dl> <p>因此<span class="mwe-math-element"><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 {125348}}\approx 354.045}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>125348</mn> </msqrt> </mrow> <mo>≈</mo> <mn>354.045</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {125348}}\approx 354.045}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/911a2fcae60110f45600cde15413db3f55b6a8d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.631ex; height:3.009ex;" alt="{\displaystyle {\sqrt {125348}}\approx 354.045}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="連分數"><span id=".E9.80.A3.E5.88.86.E6.95.B8"></span><a href="/wiki/%E9%80%A3%E5%88%86%E6%95%B8" class="mw-redirect" title="連分數">連分數</a></h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%B9%B3%E6%96%B9%E6%A0%B9&action=edit&section=14" title="编辑章节：連分數"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>平方根可以简便地用连分数的形式表示，关于连分数请见<a href="/wiki/%E8%BF%9E%E5%88%86%E6%95%B0" title="连分数">连分数</a>，其中1至20的算术平方根分别可用连分数表示为：<br /> <span class="mwe-math-element"><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 {1}}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> </msqrt> </mrow> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {1}}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5509eac62c3da5cefd034b3ca1f2b2f5f2e9c3ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.359ex; height:3.009ex;" alt="{\displaystyle {\sqrt {1}}=1}"></span><br /> <span class="mwe-math-element"><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}}=[1;2,2,2,2...]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>=</mo> <mo stretchy="false">[</mo> <mn>1</mn> <mo>;</mo> <mn>2</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>2...</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}=[1;2,2,2,2...]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5fc9b02798ffa8c81bf9bdcbc2f09e1e0aee4422" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.379ex; height:3.176ex;" alt="{\displaystyle {\sqrt {2}}=[1;2,2,2,2...]}"></span><br /> <span class="mwe-math-element"><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}}=[1;1,2,1,2...]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> <mo>=</mo> <mo stretchy="false">[</mo> <mn>1</mn> <mo>;</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2...</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {3}}=[1;1,2,1,2...]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df0c6eb26d9a8ea8fc7536e85d506c3970d25d3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.379ex; height:3.009ex;" alt="{\displaystyle {\sqrt {3}}=[1;1,2,1,2...]}"></span><br /> <span class="mwe-math-element"><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 {4}}=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>4</mn> </msqrt> </mrow> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {4}}=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d82f5e32bfe3dfb14b13ed6b632b48b063acd77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.359ex; height:3.009ex;" alt="{\displaystyle {\sqrt {4}}=2}"></span><br /> <span class="mwe-math-element"><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 {5}}=[2;4,4,4,4...]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> <mo>=</mo> <mo stretchy="false">[</mo> <mn>2</mn> <mo>;</mo> <mn>4</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>4...</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {5}}=[2;4,4,4,4...]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3db0b9a3b70d460c26600c2a48038f4044f30bf2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.379ex; height:3.009ex;" alt="{\displaystyle {\sqrt {5}}=[2;4,4,4,4...]}"></span><br /> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {6}}=[2;2,4,2,4...]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>6</mn> </msqrt> </mrow> <mo>=</mo> <mo stretchy="false">[</mo> <mn>2</mn> <mo>;</mo> <mn>2</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>4...</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {6}}=[2;2,4,2,4...]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a77c30771fc26826e8d2c5e7bed3f3a41850f47" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.379ex; height:3.009ex;" alt="{\displaystyle {\sqrt {6}}=[2;2,4,2,4...]}"></span><br /> <span class="mwe-math-element"><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 {7}}=[2;1,1,1,4,1,1,1,4...]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>7</mn> </msqrt> </mrow> <mo>=</mo> <mo stretchy="false">[</mo> <mn>2</mn> <mo>;</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>4...</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {7}}=[2;1,1,1,4,1,1,1,4...]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07f8ee5f5ca1b5e7fef75103bacd0ffbc18ab597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.165ex; height:3.176ex;" alt="{\displaystyle {\sqrt {7}}=[2;1,1,1,4,1,1,1,4...]}"></span><br /> <span class="mwe-math-element"><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 {8}}=[2;1,4,1,4...]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>8</mn> </msqrt> </mrow> <mo>=</mo> <mo stretchy="false">[</mo> <mn>2</mn> <mo>;</mo> <mn>1</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>4...</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {8}}=[2;1,4,1,4...]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f94114159ad39146dbee0db04f66f5fc6c5e15ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.379ex; height:3.009ex;" alt="{\displaystyle {\sqrt {8}}=[2;1,4,1,4...]}"></span><br /> <span class="mwe-math-element"><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 {9}}=3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>9</mn> </msqrt> </mrow> <mo>=</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {9}}=3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44820d293d56e49db705b37ad756363767077121" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.359ex; height:2.843ex;" alt="{\displaystyle {\sqrt {9}}=3}"></span><br /> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {10}}=[3;6,6,6,6...]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>10</mn> </msqrt> </mrow> <mo>=</mo> <mo stretchy="false">[</mo> <mn>3</mn> <mo>;</mo> <mn>6</mn> <mo>,</mo> <mn>6</mn> <mo>,</mo> <mn>6</mn> <mo>,</mo> <mn>6...</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {10}}=[3;6,6,6,6...]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/849b4bef93de145e137f648999d6d3c3fab2db5e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.541ex; height:3.009ex;" alt="{\displaystyle {\sqrt {10}}=[3;6,6,6,6...]}"></span><br /> <span class="mwe-math-element"><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 {11}}=[3;3,6,3,6...]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>11</mn> </msqrt> </mrow> <mo>=</mo> <mo stretchy="false">[</mo> <mn>3</mn> <mo>;</mo> <mn>3</mn> <mo>,</mo> <mn>6</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>6...</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {11}}=[3;3,6,3,6...]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf1bddf4ae2a7b8fa8a47e8996cd6f109ddeb0ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.541ex; height:3.176ex;" alt="{\displaystyle {\sqrt {11}}=[3;3,6,3,6...]}"></span><br /> <span class="mwe-math-element"><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 {12}}=[3;2,6,2,6...]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>12</mn> </msqrt> </mrow> <mo>=</mo> <mo stretchy="false">[</mo> <mn>3</mn> <mo>;</mo> <mn>2</mn> <mo>,</mo> <mn>6</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>6...</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {12}}=[3;2,6,2,6...]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7f86b2454a2b20534e09dbe888b7228db1d8a44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.541ex; height:3.176ex;" alt="{\displaystyle {\sqrt {12}}=[3;2,6,2,6...]}"></span><br /> <span class="mwe-math-element"><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 {13}}=[3;1,1,1,1,6,1,1,1,1,6...]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>13</mn> </msqrt> </mrow> <mo>=</mo> <mo stretchy="false">[</mo> <mn>3</mn> <mo>;</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>6</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>6...</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {13}}=[3;1,1,1,1,6,1,1,1,1,6...]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1270c1d403e3d9059ab4cf2c146ecb36c3dc08b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.72ex; height:3.009ex;" alt="{\displaystyle {\sqrt {13}}=[3;1,1,1,1,6,1,1,1,1,6...]}"></span><br /> <span class="mwe-math-element"><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 {14}}=[3;1,2,1,6,1,2,1,6...]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>14</mn> </msqrt> </mrow> <mo>=</mo> <mo stretchy="false">[</mo> <mn>3</mn> <mo>;</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>6</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>6...</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {14}}=[3;1,2,1,6,1,2,1,6...]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d9a7e23c50b6c14119333421f27b7381250c7fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.327ex; height:3.176ex;" alt="{\displaystyle {\sqrt {14}}=[3;1,2,1,6,1,2,1,6...]}"></span><br /> <span class="mwe-math-element"><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 {15}}=[3;1,6,1,6...]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>15</mn> </msqrt> </mrow> <mo>=</mo> <mo stretchy="false">[</mo> <mn>3</mn> <mo>;</mo> <mn>1</mn> <mo>,</mo> <mn>6</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>6...</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {15}}=[3;1,6,1,6...]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1883e99932565fc6b8e5a86a3934b130d3cb09be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.541ex; height:3.009ex;" alt="{\displaystyle {\sqrt {15}}=[3;1,6,1,6...]}"></span><br /> <span class="mwe-math-element"><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 {16}}=4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>16</mn> </msqrt> </mrow> <mo>=</mo> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {16}}=4}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c0d55c700c3a4b7cc23296f9a6c913fec320464" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.522ex; height:2.843ex;" alt="{\displaystyle {\sqrt {16}}=4}"></span><br /> <span class="mwe-math-element"><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 {17}}=[4;8,8,8,8...]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>17</mn> </msqrt> </mrow> <mo>=</mo> <mo stretchy="false">[</mo> <mn>4</mn> <mo>;</mo> <mn>8</mn> <mo>,</mo> <mn>8</mn> <mo>,</mo> <mn>8</mn> <mo>,</mo> <mn>8...</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {17}}=[4;8,8,8,8...]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/099316b44578e9592e1425571402bb24e76664fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.541ex; height:3.176ex;" alt="{\displaystyle {\sqrt {17}}=[4;8,8,8,8...]}"></span><br /> <span class="mwe-math-element"><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 {18}}=[4;4,8,4,8...]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>18</mn> </msqrt> </mrow> <mo>=</mo> <mo stretchy="false">[</mo> <mn>4</mn> <mo>;</mo> <mn>4</mn> <mo>,</mo> <mn>8</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>8...</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {18}}=[4;4,8,4,8...]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4a789b740adf75a585e1a4e3bb5d85a2fc23056" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.541ex; height:3.009ex;" alt="{\displaystyle {\sqrt {18}}=[4;4,8,4,8...]}"></span><br /> <span class="mwe-math-element"><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 {19}}=[4;2,1,3,1,2,8,2,1,3,1,2,8...]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>19</mn> </msqrt> </mrow> <mo>=</mo> <mo stretchy="false">[</mo> <mn>4</mn> <mo>;</mo> <mn>2</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>8</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>8...</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {19}}=[4;2,1,3,1,2,8,2,1,3,1,2,8...]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7cd3c48024a8f78a4b7ce606d442893d273e665" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.113ex; height:3.009ex;" alt="{\displaystyle {\sqrt {19}}=[4;2,1,3,1,2,8,2,1,3,1,2,8...]}"></span><br /> <span class="mwe-math-element"><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 {20}}=[4;2,8,2,8...]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>20</mn> </msqrt> </mrow> <mo>=</mo> <mo stretchy="false">[</mo> <mn>4</mn> <mo>;</mo> <mn>2</mn> <mo>,</mo> <mn>8</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>8...</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {20}}=[4;2,8,2,8...]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8ab7596515455a8476ccbcd6827283d0eab5ed0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.541ex; height:3.009ex;" alt="{\displaystyle {\sqrt {20}}=[4;2,8,2,8...]}"></span><br /> </p><p>连分数部分均循环，省略号前为2或4个循环节。 </p> <div class="mw-heading mw-heading3"><h3 id="巴比倫方法"><span id=".E5.B7.B4.E6.AF.94.E5.80.AB.E6.96.B9.E6.B3.95"></span>巴比倫方法</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%B9%B3%E6%96%B9%E6%A0%B9&action=edit&section=15" title="编辑章节：巴比倫方法"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r85100532"><div role="note" class="hatnote navigation-not-searchable">主条目：<span class="ilh-all" data-orig-title="巴比倫方法" data-lang-code="en" data-lang-name="英语" data-foreign-title="Babylonian method"><span class="ilh-page"><a href="/w/index.php?title=%E5%B7%B4%E6%AF%94%E5%80%AB%E6%96%B9%E6%B3%95&action=edit&redlink=1" class="new" title="巴比倫方法（页面不存在）">巴比倫方法</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Babylonian_method" class="extiw" title="en:Babylonian method"><span lang="en" dir="auto">Babylonian method</span></a></span>）</span></span></div> <p>巴比伦求平方根的算法实际上很简单：（假设要求一个数N的平方根） </p> <ol><li>预测一个平方根<span class="mwe-math-element"><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>，初始另一个值<span class="mwe-math-element"><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>，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 xy=N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mi>y</mi> <mo>=</mo> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle xy=N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffe65ea55977f9d8d0669eeef51fd37df5213c2d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.647ex; height:2.509ex;" alt="{\displaystyle xy=N}"></span></li> <li>求预测值与初始值的均值：<span class="mwe-math-element"><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={\frac {x+y}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mo>+</mo> <mi>y</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x={\frac {x+y}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26ee8a5e890f115ebca72754ee3c4e418b8a67f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.59ex; height:5.176ex;" alt="{\displaystyle x={\frac {x+y}{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 y={\frac {N}{x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>N</mi> <mi>x</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y={\frac {N}{x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/acc9957500cd702722de4d5a76789794114a345e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.154ex; height:5.176ex;" alt="{\displaystyle y={\frac {N}{x}}}"></span></li> <li>比较<span class="mwe-math-element"><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>的差值是否达到精度，如果无，继续步骤</li></ol> <div class="mw-heading mw-heading3"><h3 id="重複的算術運算"><span id=".E9.87.8D.E8.A4.87.E7.9A.84.E7.AE.97.E8.A1.93.E9.81.8B.E7.AE.97"></span>重複的算術運算</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%B9%B3%E6%96%B9%E6%A0%B9&action=edit&section=16" title="编辑章节：重複的算術運算"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>這個方法是從<a href="/wiki/%E4%BD%A9%E5%B0%94%E6%96%B9%E7%A8%8B" title="佩尔方程">佩爾方程</a>演變過來的，它通過不斷減去奇數來求得答案。 </p> <div class="mw-heading mw-heading3"><h3 id="尺规作图"><span id=".E5.B0.BA.E8.A7.84.E4.BD.9C.E5.9B.BE"></span><a href="/wiki/%E5%B0%BA%E8%A7%84%E4%BD%9C%E5%9B%BE" title="尺规作图">尺规作图</a></h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%B9%B3%E6%96%B9%E6%A0%B9&action=edit&section=17" title="编辑章节：尺规作图"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading4"><h4 id="問題"><span id=".E5.95.8F.E9.A1.8C"></span>問題</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%B9%B3%E6%96%B9%E6%A0%B9&action=edit&section=18" title="编辑章节：問題"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>給定線段<i>AB</i>和1，求一條長為<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 {\sqrt {AB}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>A</mi> <mi>B</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {AB}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f473632977440b95128dfdb62ae0d997d837ba7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.443ex; height:3.009ex;" alt="{\displaystyle {\sqrt {AB}}}"></span>的線段。 </p> <div class="mw-heading mw-heading4"><h4 id="解法"><span id=".E8.A7.A3.E6.B3.95"></span>解法</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%B9%B3%E6%96%B9%E6%A0%B9&action=edit&section=19" title="编辑章节：解法"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File"><a href="/wiki/File:Rcsquare_root.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/f/f4/Rcsquare_root.png" decoding="async" width="290" height="161" class="mw-file-element" data-file-width="290" data-file-height="161" /></a><figcaption></figcaption></figure> <ol><li>畫線<i>AB</i>，延長<i>BA</i>至<i>C</i>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 AC=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>C</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle AC=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbe81790332736420b0c659bb1294aaab8ce747f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.77ex; height:2.176ex;" alt="{\displaystyle AC=1}"></span></li> <li>以<i>BC</i>的中點為圓心，<i>OC</i>為半徑畫圓</li> <li>過<i>A</i>畫<i>BC</i>的垂直線，垂直線和圓弧交於<i>D</i>，<i>AD</i>即為所求之長度</li></ol> <div class="mw-heading mw-heading4"><h4 id="證明"><span id=".E8.AD.89.E6.98.8E"></span>證明</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%B9%B3%E6%96%B9%E6%A0%B9&action=edit&section=20" title="编辑章节：證明"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>將整個過程搬到<a href="/wiki/%E7%AC%9B%E5%8D%A1%E5%84%BF%E5%9D%90%E6%A0%87%E7%B3%BB" class="mw-redirect" title="笛卡儿坐标系">直角座標</a>上，已知<i>AC</i>=1，設 </p> <ul><li><i>O</i>=<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0,0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d630d3e781a53b0a3559ae7e5b45f9479a3141a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.168ex; height:2.843ex;" alt="{\displaystyle (0,0)}"></span></li> <li><i>AB</i>=<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span></li></ul> <ol><li>直徑為<i>BC</i>的圓就是<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 x^{2}+y^{2}=\left({\frac {n+1}{2}}\right)^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}+y^{2}=\left({\frac {n+1}{2}}\right)^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91a8b0a0e57ce5b19cde1672263fd47f566d5038" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:21.247ex; height:6.509ex;" alt="{\displaystyle x^{2}+y^{2}=\left({\frac {n+1}{2}}\right)^{2}}"></span>（圓的方程式：<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}+y^{2}=r^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}+y^{2}=r^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37dd4f282df84a83620f71dc52345122e0e3a514" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.64ex; height:3.009ex;" alt="{\displaystyle x^{2}+y^{2}=r^{2}}"></span>）（其中<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span>表示半径。）</li> <li>將<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left({\frac {n+1}{2}}-1\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\frac {n+1}{2}}-1\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1bcfb152675798373a6aebf9267e300d76d17ea6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:13.658ex; height:6.176ex;" alt="{\displaystyle \left({\frac {n+1}{2}}-1\right)}"></span>（<i>A</i>,<i>D</i>所在的<i>x</i>座標）代入上面的方程式</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 \left({\frac {n+1}{2}}-1\right)^{2}+y^{2}=\left({\frac {n+1}{2}}\right)^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\frac {n+1}{2}}-1\right)^{2}+y^{2}=\left({\frac {n+1}{2}}\right)^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7e860f9d7377d9c67676907b15d275a4b3793ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:33.575ex; height:6.509ex;" alt="{\displaystyle \left({\frac {n+1}{2}}-1\right)^{2}+y^{2}=\left({\frac {n+1}{2}}\right)^{2}}"></span></li> <li>解方程，得<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y={\sqrt {n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>n</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y={\sqrt {n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51fe35bc0d3d5cfadf39e4c9bc08ff19650e9f14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.585ex; height:3.009ex;" alt="{\displaystyle y={\sqrt {n}}}"></span>。</li></ol> <p>另也可参见<a href="/wiki/%E5%B0%84%E5%BD%B1%E5%AE%9A%E7%90%86" title="射影定理">射影定理</a>。 </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:%E5%B0%84%E5%BD%B1%E5%AE%9A%E7%90%86_2.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c6/%E5%B0%84%E5%BD%B1%E5%AE%9A%E7%90%86_2.jpg/220px-%E5%B0%84%E5%BD%B1%E5%AE%9A%E7%90%86_2.jpg" decoding="async" width="220" height="103" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c6/%E5%B0%84%E5%BD%B1%E5%AE%9A%E7%90%86_2.jpg/330px-%E5%B0%84%E5%BD%B1%E5%AE%9A%E7%90%86_2.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/c/c6/%E5%B0%84%E5%BD%B1%E5%AE%9A%E7%90%86_2.jpg 2x" data-file-width="388" data-file-height="182" /></a><figcaption>射影定理（图）</figcaption></figure> <div class="mw-heading mw-heading2"><h2 id="参见"><span id=".E5.8F.82.E8.A7.81"></span>参见</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%B9%B3%E6%96%B9%E6%A0%B9&action=edit&section=21" title="编辑章节：参见"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/%E6%96%B9%E6%A0%B9" title="方根">方根</a></li> <li><a href="/wiki/%E5%A2%9E%E4%B9%98%E5%BC%80%E5%B9%B3%E6%96%B9%E6%B3%95" title="增乘开平方法">增乘开平方法</a></li> <li><a href="/wiki/%E4%BA%8C%E9%A1%B9%E5%BC%8F%E5%AE%9A%E7%90%86" title="二项式定理">二项式定理</a></li> <li><a href="/wiki/%E7%89%9B%E9%A1%BF%E6%B3%95" title="牛顿法">牛顿法</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="外部链接"><span id=".E5.A4.96.E9.83.A8.E9.93.BE.E6.8E.A5"></span>外部链接</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%B9%B3%E6%96%B9%E6%A0%B9&action=edit&section=22" title="编辑章节：外部链接"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="https://web.archive.org/web/19990220034927/http://members.aol.com/jeff570/operation.html">Earliest Uses of Symbols of Operation</a><span style="font-family: sans-serif; cursor: default; color:var(--color-subtle, #54595d); font-size: 0.8em; bottom: 0.1em; font-weight: bold;" title="英語">（英文）</span></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20060821061824/http://www.roma.unisa.edu.au/07305/symbols.htm#Radical#Radical">The History of Mathematical Symbols: The radical symbol</a></li> <li><a rel="nofollow" class="external text" href="http://www.docin.com/p-133945458.html">开方公式的推导</a> （<a rel="nofollow" class="external text" href="//web.archive.org/web/20120622071420/http://www.docin.com/p-133945458.html">页面存档备份</a>，存于<a href="/wiki/%E4%BA%92%E8%81%94%E7%BD%91%E6%A1%A3%E6%A1%88%E9%A6%86" title="互联网档案馆">互联网档案馆</a>）</li> <li><a rel="nofollow" class="external text" href="http://www.calculatorsquareroot.com">平方根計算器</a> （<a rel="nofollow" class="external text" href="//web.archive.org/web/20200803211832/http://www.calculatorsquareroot.com/">页面存档备份</a>，存于<a href="/wiki/%E4%BA%92%E8%81%94%E7%BD%91%E6%A1%A3%E6%A1%88%E9%A6%86" title="互联网档案馆">互联网档案馆</a>）<span style="font-family: sans-serif; cursor: default; color:var(--color-subtle, #54595d); font-size: 0.8em; bottom: 0.1em; font-weight: bold;" title="英語">（英文）</span></li></ul> <div class="mw-heading mw-heading2"><h2 id="參考資料"><span id=".E5.8F.83.E8.80.83.E8.B3.87.E6.96.99"></span>參考資料</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%B9%B3%E6%96%B9%E6%A0%B9&action=edit&section=23" title="编辑章节：參考資料"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <div role="navigation" aria-label="Portals" class="noprint portal plainlist tright" style="margin:0.5em 0 0.5em 1em;border:solid #aaa 1px"> <ul style="display:table;box-sizing:border-box;padding:0.1em;max-width:175px;background:var(--background-color-base,#f9f9f9);font-size:85%;line-height:110%;font-weight:bold"> <li style="display:table-row"><span style="display:table-cell;padding:0.2em;vertical-align:middle;text-align:center"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Nuvola_apps_edu_mathematics_blue-p.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/28px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="28" height="28" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/42px-Nuvola_apps_edu_mathematics_blue-p.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/56px-Nuvola_apps_edu_mathematics_blue-p.svg.png 2x" data-file-width="128" data-file-height="128" /></a></span></span><span style="display:table-cell;padding:0.2em 0.2em 0.2em 0.3em;vertical-align:middle"><a href="/wiki/Portal:%E6%95%B0%E5%AD%A6" title="Portal:数学">数学主题</a></span></li></ul></div> <div class="references-small"> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><cite class="citation web"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20200312005425/http://www.math.ubc.ca/~cass/Euclid/ybc/analysis.html">Analysis of YBC 7289</a>. ubc.ca. <span class="reference-accessdate"> [<span class="nowrap">19 January</span> 2015]</span>. （<a rel="nofollow" class="external text" href="http://www.math.ubc.ca/~cass/Euclid/ybc/analysis.html">原始内容</a>存档于2020-03-12）.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%B9%B3%E6%96%B9%E6%A0%B9&rft.atitle=Analysis+of+YBC+7289&rft.genre=unknown&rft.jtitle=ubc.ca&rft_id=http%3A%2F%2Fwww.math.ubc.ca%2F~cass%2FEuclid%2Fybc%2Fanalysis.html&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">Anglin, W.S. (1994). <i>Mathematics: A Concise History and Philosophy</i>. New York: Springer-Verlag.</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite class="citation book">Heath, Sir Thomas L. <a rel="nofollow" class="external text" href="https://archive.org/stream/thirteenbookseu03heibgoog#page/n14/mode/1up">The Thirteen Books of The Elements, Vol. 3</a>. Cambridge University Press. 1908: 3.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%B9%B3%E6%96%B9%E6%A0%B9&rft.aufirst=Sir+Thomas+L.&rft.aulast=Heath&rft.btitle=The+Thirteen+Books+of+The+Elements%2C+Vol.+3&rft.date=1908&rft.genre=book&rft.pages=3&rft.pub=Cambridge+University+Press&rft_id=https%3A%2F%2Farchive.org%2Fstream%2Fthirteenbookseu03heibgoog%23page%2Fn14%2Fmode%2F1up&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><cite class="citation book">Abramowitz, Milton; Stegun, Irene A. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=MtU8uP7XMvoC">Handbook of mathematical functions with formulas, graphs, and mathematical tables</a>. Courier Dover Publications. 1964: 17. <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/0-486-61272-4" title="Special:网络书源/0-486-61272-4"><span title="国际标准书号">ISBN</span> 0-486-61272-4</a>. （原始内容<a rel="nofollow" class="external text" href="https://web.archive.org/web/20160423180235/https://books.google.com/books?id=MtU8uP7XMvoC">存档</a>于2016-04-23）.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%B9%B3%E6%96%B9%E6%A0%B9&rft.au=Stegun%2C+Irene+A.&rft.aufirst=Milton&rft.aulast=Abramowitz&rft.btitle=Handbook+of+mathematical+functions+with+formulas%2C+graphs%2C+and+mathematical+tables&rft.date=1964&rft.genre=book&rft.isbn=0-486-61272-4&rft.pages=17&rft.pub=Courier+Dover+Publications&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DMtU8uP7XMvoC&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span>, <a rel="nofollow" class="external text" href="http://www.math.sfu.ca/~cbm/aands/page_17.htm">Section 3.7.27, p. 17</a> <a href="/wiki/Wayback_Machine" class="mw-redirect" title="Wayback Machine">互联网档案馆</a>的<a rel="nofollow" class="external text" href="https://web.archive.org/web/20090910094533/http://www.math.sfu.ca/~cbm/aands/page_17.htm">存檔</a>，存档日期2009-09-10.</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><cite class="citation book">Cooke, Roger. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=lUcTsYopfhkC&pg=PA59">Classical algebra: its nature, origins, and uses</a>. John Wiley and Sons. 2008: 59. <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/0-470-25952-3" title="Special:网络书源/0-470-25952-3"><span title="国际标准书号">ISBN</span> 0-470-25952-3</a>. （原始内容<a rel="nofollow" class="external text" href="https://web.archive.org/web/20160423183239/https://books.google.com/books?id=lUcTsYopfhkC&pg=PA59">存档</a>于2016-04-23）.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%B9%B3%E6%96%B9%E6%A0%B9&rft.aufirst=Roger&rft.aulast=Cooke&rft.btitle=Classical+algebra%3A+its+nature%2C+origins%2C+and+uses&rft.date=2008&rft.genre=book&rft.isbn=0-470-25952-3&rft.pages=59&rft.pub=John+Wiley+and+Sons&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DlUcTsYopfhkC%26pg%3DPA59&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text">劳汉生《珠算与实用算术》<a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/7537518912" class="internal mw-magiclink-isbn">ISBN 7-5375-1891-2</a>/O</span> </li> </ol> </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=平方根&oldid=85110363">https://zh.wikipedia.org/w/index.php?title=平方根&oldid=85110363</a>”</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Special:%E9%A1%B5%E9%9D%A2%E5%88%86%E7%B1%BB" title="Special:页面分类">分类</a>：<ul><li><a href="/wiki/Category:%E5%88%9D%E7%AD%89%E4%BB%A3%E6%95%B0" title="Category:初等代数">初等代数</a></li><li><a href="/wiki/Category:%E7%AE%97%E6%9C%AF" title="Category:算术">算术</a></li><li><a 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