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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-算筹的材料和形状" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#算筹的材料和形状"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>算筹的材料和形状</span> </div> </a> <ul id="toc-算筹的材料和形状-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-文獻記載" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#文獻記載"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>文獻記載</span> </div> </a> <ul id="toc-文獻記載-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-考古实物" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#考古实物"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>考古实物</span> </div> </a> <ul id="toc-考古实物-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">4</span> <span>計算方法</span> </div> </a> <ul id="toc-計算方法-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-参考文献" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#参考文献"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>参考文献</span> </div> </a> <ul id="toc-参考文献-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-参看" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#参看"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>参看</span> </div> </a> <ul id="toc-参看-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" title="目录" > <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">算筹</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。17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-17" 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">17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-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Numeraci%C3%B3_amb_varetes" title="Numeració amb varetes – 加泰罗尼亚语" lang="ca" hreflang="ca" data-title="Numeració amb varetes" 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-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Counting_rods" title="Counting rods – 英语" lang="en" hreflang="en" data-title="Counting rods" data-language-autonym="English" data-language-local-name="英语" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Numeraci%C3%B3n_con_varillas" title="Numeración con varillas – 西班牙语" lang="es" hreflang="es" data-title="Numeración con varillas" 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-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Puikkolaskenta" title="Puikkolaskenta – 芬兰语" lang="fi" hreflang="fi" data-title="Puikkolaskenta" data-language-autonym="Suomi" data-language-local-name="芬兰语" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Baguettes_%C3%A0_calculer" title="Baguettes à calculer – 法语" lang="fr" hreflang="fr" data-title="Baguettes à calculer" data-language-autonym="Français" data-language-local-name="法语" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%95%D7%98%D7%95%D7%AA_%D7%9E%D7%A0%D7%99%D7%99%D7%94" title="מוטות מנייה – 希伯来语" lang="he" hreflang="he" data-title="מוטות מנייה" 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-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Tongkat_penghitung" title="Tongkat penghitung – 印度尼西亚语" lang="id" hreflang="id" data-title="Tongkat penghitung" 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-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E7%AE%97%E6%9C%A8" title="算木 – 日语" lang="ja" hreflang="ja" data-title="算木" 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-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%82%B0%EA%B0%80%EC%A7%80" title="산가지 – 韩语" lang="ko" hreflang="ko" data-title="산가지" data-language-autonym="한국어" data-language-local-name="韩语" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Batang_pembilang" title="Batang pembilang – 马来语" lang="ms" hreflang="ms" data-title="Batang pembilang" 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-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Patyczki_liczbowe" title="Patyczki liczbowe – 波兰语" lang="pl" hreflang="pl" data-title="Patyczki liczbowe" 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/Sistema_num%C3%A9rico_de_varas" title="Sistema numérico de varas – 葡萄牙语" lang="pt" hreflang="pt" data-title="Sistema numérico de varas" 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-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A1%D1%87%D1%91%D1%82%D0%BD%D1%8B%D0%B5_%D0%BF%D0%B0%D0%BB%D0%BE%D1%87%D0%BA%D0%B8" title="Счётные палочки – 俄语" lang="ru" hreflang="ru" data-title="Счётные палочки" data-language-autonym="Русский" data-language-local-name="俄语" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A0%D0%B0%D1%85%D1%83%D0%BD%D0%BA%D0%BE%D0%B2%D1%96_%D0%BF%D0%B0%D0%BB%D0%B8%D1%87%D0%BA%D0%B8" title="Рахункові палички – 乌克兰语" lang="uk" hreflang="uk" data-title="Рахункові палички" 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-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E7%AE%97%E7%AD%B9" title="算筹 – 吴语" lang="wuu" hreflang="wuu" data-title="算筹" 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-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E7%AE%97%E7%B1%8C" title="算籌 – 文言文" lang="lzh" hreflang="lzh" data-title="算籌" data-language-autonym="文言" data-language-local-name="文言文" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E7%AE%97%E7%B1%8C" title="算籌 – 粤语" lang="yue" hreflang="yue" data-title="算籌" data-language-autonym="粵語" data-language-local-name="粤语" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q583473#sitelinks-wikipedia" title="编辑跨语言链接" class="wbc-editpage">编辑链接</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="命名空间"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/%E7%AE%97%E7%AD%B9" title="浏览条目正文[c]" accesskey="c"><span>条目</span></a></li><li id="ca-talk" class="new vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Talk:%E7%AE%97%E7%AD%B9&action=edit&redlink=1" rel="discussion" class="new" title="关于此页面的讨论（页面不存在）[t]" accesskey="t"><span>讨论</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown " > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="更改语言变体" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">不转换</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-varlang-0" class="selected ca-variants-zh mw-list-item"><a href="/zh/%E7%AE%97%E7%AD%B9" lang="zh" hreflang="zh"><span>不转换</span></a></li><li id="ca-varlang-1" class="ca-variants-zh-Hans mw-list-item"><a href="/zh-hans/%E7%AE%97%E7%AD%B9" lang="zh-Hans" hreflang="zh-Hans"><span>简体</span></a></li><li id="ca-varlang-2" class="ca-variants-zh-Hant mw-list-item"><a href="/zh-hant/%E7%AE%97%E7%AD%B9" lang="zh-Hant" hreflang="zh-Hant"><span>繁體</span></a></li><li id="ca-varlang-3" class="ca-variants-zh-Hans-CN mw-list-item"><a href="/zh-cn/%E7%AE%97%E7%AD%B9" lang="zh-Hans-CN" hreflang="zh-Hans-CN"><span>大陆简体</span></a></li><li id="ca-varlang-4" class="ca-variants-zh-Hant-HK mw-list-item"><a href="/zh-hk/%E7%AE%97%E7%AD%B9" lang="zh-Hant-HK" hreflang="zh-Hant-HK"><span>香港繁體</span></a></li><li id="ca-varlang-5" class="ca-variants-zh-Hant-MO mw-list-item"><a href="/zh-mo/%E7%AE%97%E7%AD%B9" lang="zh-Hant-MO" hreflang="zh-Hant-MO"><span>澳門繁體</span></a></li><li id="ca-varlang-6" class="ca-variants-zh-Hans-MY mw-list-item"><a href="/zh-my/%E7%AE%97%E7%AD%B9" lang="zh-Hans-MY" hreflang="zh-Hans-MY"><span>大马简体</span></a></li><li id="ca-varlang-7" class="ca-variants-zh-Hans-SG mw-list-item"><a href="/zh-sg/%E7%AE%97%E7%AD%B9" lang="zh-Hans-SG" hreflang="zh-Hans-SG"><span>新加坡简体</span></a></li><li id="ca-varlang-8" class="ca-variants-zh-Hant-TW mw-list-item"><a href="/zh-tw/%E7%AE%97%E7%AD%B9" lang="zh-Hant-TW" hreflang="zh-Hant-TW"><span>臺灣正體</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="查看"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/%E7%AE%97%E7%AD%B9"><span>阅读</span></a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=%E7%AE%97%E7%AD%B9&action=edit" title="编辑该页面[e]" accesskey="e"><span>编辑</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=%E7%AE%97%E7%AD%B9&action=history" title="本页面的早前版本。[h]" accesskey="h"><span>查看历史</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="页面工具"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="工具" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">工具</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">工具</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">移至侧栏</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">隐藏</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="更多选项" > <div class="vector-menu-heading"> 操作 </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/%E7%AE%97%E7%AD%B9"><span>阅读</span></a></li><li id="ca-more-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=%E7%AE%97%E7%AD%B9&action=edit" title="编辑该页面[e]" accesskey="e"><span>编辑</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=%E7%AE%97%E7%AD%B9&action=history"><span>查看历史</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> 常规 </div> <div 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href="/w/index.php?title=Special:%E5%BC%95%E7%94%A8%E6%AD%A4%E9%A1%B5%E9%9D%A2&page=%E7%AE%97%E7%AD%B9&id=82736121&wpFormIdentifier=titleform" title="有关如何引用此页面的信息"><span>引用此页</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Special:URL%E7%BC%A9%E7%9F%AD%E7%A8%8B%E5%BA%8F&url=https%3A%2F%2Fzh.wikipedia.org%2Fwiki%2F%25E7%25AE%2597%25E7%25AD%25B9"><span>获取短链接</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Special:QrCode&url=https%3A%2F%2Fzh.wikipedia.org%2Fwiki%2F%25E7%25AE%2597%25E7%25AD%25B9"><span>下载二维码</span></a></li> </ul> </div> </div> <div id="p-electronpdfservice-sidebar-portlet-heading" class="vector-menu mw-portlet mw-portlet-electronpdfservice-sidebar-portlet-heading" > <div class="vector-menu-heading"> 打印/导出 </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="electron-print_pdf" class="mw-list-item"><a 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class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">隐藏</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">维基百科，自由的百科全书</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="zh" dir="ltr"><figure class="mw-halign-right" typeof="mw:File"><a href="/wiki/File:Yanghui_triangle.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ea/Yanghui_triangle.gif/200px-Yanghui_triangle.gif" decoding="async" width="200" height="311" class="mw-file-element" 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rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84265675" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84913072" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84265675" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84913072" /><table class="sidebar sidebar-collapse nomobile nowraplinks hlist"><tbody><tr><td class="sidebar-pretitle"><a href="/wiki/Category:%E6%95%B8%E5%88%B6" title="Category:數制">數制</a></td></tr><tr><th class="sidebar-title-with-pretitle"><a href="/wiki/%E8%AE%B0%E6%95%B0%E7%B3%BB%E7%BB%9F" title="记数系统">记数系统</a></th></tr><tr><td class="sidebar-content-with-subgroup"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title"><div class="sidebar-list-title-c"><a href="/wiki/%E8%BF%9B%E4%BD%8D%E5%88%B6" title="进位制">进位制</a></div></div><div class="sidebar-list-content mw-collapsible-content"><table class="sidebar-subgroup"><tbody><tr><td class="sidebar-content hlist"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title"><div class="sidebar-list-title-c"><a href="/wiki/%E5%8D%B0%E5%BA%A6-%E9%98%BF%E6%8B%89%E4%BC%AF%E6%95%B0%E5%AD%97%E7%B3%BB%E7%BB%9F" title="印度-阿拉伯数字系统">印阿體系</a></div></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/%E9%98%BF%E6%8B%89%E4%BC%AF%E6%95%B0%E5%AD%97" title="阿拉伯数字">西方阿拉伯數字</a></li> <li><a href="/wiki/%E9%98%BF%E6%8B%89%E4%BC%AF%E6%96%87%E6%95%B0%E5%AD%97" title="阿拉伯文数字">阿拉伯文数字</a></li></ul> <hr /> <ul><li><a href="/wiki/%E5%AD%9F%E5%8A%A0%E6%8B%89%E6%95%B0%E5%AD%97" title="孟加拉数字">孟加拉数字</a></li> <li><a href="/wiki/%E5%8D%B0%E5%BA%A6%E6%95%B8%E5%AD%97" title="印度數字">印度數字</a></li> <li><span class="ilh-all" data-orig-title="Gujarati numerals" data-lang-code="en" data-lang-name="英语" data-foreign-title="Gujarati numerals"><span class="ilh-page"><a href="/w/index.php?title=Gujarati_numerals&action=edit&redlink=1" class="new" title="Gujarati numerals（页面不存在）">Gujarati</a></span><span class="noprint ilh-comment"><span class="ilh-paren">（</span><span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Gujarati_numerals" class="extiw" title="en:Gujarati numerals"><span lang="en" dir="auto">Gujarati numerals</span></a></span><span class="ilh-paren">）</span></span></span></li> <li><a href="/wiki/%E5%8F%A4%E6%9C%A8%E5%9F%BA%E6%96%87" title="古木基文">古木基文</a></li> <li><span class="ilh-all" data-orig-title="Odia numerals" data-lang-code="en" data-lang-name="英语" data-foreign-title="Odia numerals"><span class="ilh-page"><a href="/w/index.php?title=Odia_numerals&action=edit&redlink=1" class="new" title="Odia numerals（页面不存在）">Odia</a></span><span class="noprint ilh-comment"><span class="ilh-paren">（</span><span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Odia_numerals" class="extiw" title="en:Odia numerals"><span lang="en" dir="auto">Odia numerals</span></a></span><span class="ilh-paren">）</span></span></span></li> <li><span class="ilh-all" data-orig-title="Sinhala numerals" data-lang-code="en" data-lang-name="英语" data-foreign-title="Sinhala numerals"><span class="ilh-page"><a href="/w/index.php?title=Sinhala_numerals&action=edit&redlink=1" class="new" title="Sinhala numerals（页面不存在）">Sinhala</a></span><span class="noprint ilh-comment"><span class="ilh-paren">（</span><span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Sinhala_numerals" class="extiw" title="en:Sinhala numerals"><span lang="en" dir="auto">Sinhala numerals</span></a></span><span class="ilh-paren">）</span></span></span></li> <li><span class="ilh-all" data-orig-title="Tamil numerals" data-lang-code="en" data-lang-name="英语" data-foreign-title="Tamil numerals"><span class="ilh-page"><a href="/w/index.php?title=Tamil_numerals&action=edit&redlink=1" class="new" title="Tamil numerals（页面不存在）">Tamil</a></span><span class="noprint ilh-comment"><span class="ilh-paren">（</span><span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Tamil_numerals" class="extiw" title="en:Tamil numerals"><span lang="en" dir="auto">Tamil numerals</span></a></span><span class="ilh-paren">）</span></span></span></li> <li><span class="ilh-all" data-orig-title="Malayalam numerals" data-lang-code="en" data-lang-name="英语" data-foreign-title="Malayalam numerals"><span class="ilh-page"><a href="/w/index.php?title=Malayalam_numerals&action=edit&redlink=1" class="new" title="Malayalam numerals（页面不存在）">Malayalam</a></span><span class="noprint ilh-comment"><span class="ilh-paren">（</span><span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Malayalam_numerals" class="extiw" title="en:Malayalam numerals"><span lang="en" dir="auto">Malayalam numerals</span></a></span><span class="ilh-paren">）</span></span></span></li> <li><a href="/wiki/%E6%B3%B0%E5%8D%A2%E5%9B%BA%E6%96%87" title="泰卢固文">泰卢固文</a></li> <li><a href="/wiki/%E5%8D%A1%E7%B4%8D%E9%81%94%E6%96%87" title="卡納達文">卡納達文</a></li> <li><span class="ilh-all" data-orig-title="Dzongkha numerals" data-lang-code="en" data-lang-name="英语" data-foreign-title="Dzongkha numerals"><span class="ilh-page"><a href="/w/index.php?title=Dzongkha_numerals&action=edit&redlink=1" class="new" title="Dzongkha numerals（页面不存在）">Dzongkha</a></span><span class="noprint ilh-comment"><span class="ilh-paren">（</span><span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Dzongkha_numerals" class="extiw" title="en:Dzongkha numerals"><span lang="en" dir="auto">Dzongkha numerals</span></a></span><span class="ilh-paren">）</span></span></span></li></ul> <hr /> <ul><li><span class="ilh-all" data-orig-title="Tibetan numerals" data-lang-code="en" data-lang-name="英语" data-foreign-title="Tibetan numerals"><span class="ilh-page"><a href="/w/index.php?title=Tibetan_numerals&action=edit&redlink=1" class="new" title="Tibetan numerals（页面不存在）">Tibetan</a></span><span class="noprint ilh-comment"><span class="ilh-paren">（</span><span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Tibetan_numerals" class="extiw" title="en:Tibetan numerals"><span lang="en" dir="auto">Tibetan numerals</span></a></span><span class="ilh-paren">）</span></span></span></li> <li><span class="ilh-all" data-orig-title="Balinese numerals" data-lang-code="en" data-lang-name="英语" data-foreign-title="Balinese numerals"><span class="ilh-page"><a href="/w/index.php?title=Balinese_numerals&action=edit&redlink=1" class="new" title="Balinese numerals（页面不存在）">Balinese</a></span><span class="noprint ilh-comment"><span class="ilh-paren">（</span><span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Balinese_numerals" class="extiw" title="en:Balinese numerals"><span lang="en" dir="auto">Balinese numerals</span></a></span><span class="ilh-paren">）</span></span></span></li> <li><a href="/wiki/%E7%B7%AC%E7%94%B8%E6%95%B8%E5%AD%97" title="緬甸數字">緬甸數字</a></li> <li><span class="ilh-all" data-orig-title="Javanese numerals" data-lang-code="en" data-lang-name="英语" data-foreign-title="Javanese numerals"><span class="ilh-page"><a href="/w/index.php?title=Javanese_numerals&action=edit&redlink=1" class="new" title="Javanese numerals（页面不存在）">Javanese</a></span><span class="noprint ilh-comment"><span class="ilh-paren">（</span><span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Javanese_numerals" class="extiw" title="en:Javanese numerals"><span lang="en" dir="auto">Javanese numerals</span></a></span><span class="ilh-paren">）</span></span></span></li> <li><a href="/wiki/%E9%AB%98%E6%A3%89%E6%95%B8%E5%AD%97" title="高棉數字">高棉數字</a></li> <li><a href="/wiki/%E8%80%81%E6%92%BE%E6%96%87" title="老撾文">老撾文</a></li> <li><span class="ilh-all" data-orig-title="Mongolian numerals" data-lang-code="en" data-lang-name="英语" data-foreign-title="Mongolian numerals"><span class="ilh-page"><a href="/w/index.php?title=Mongolian_numerals&action=edit&redlink=1" class="new" title="Mongolian numerals（页面不存在）">Mongolian</a></span><span class="noprint ilh-comment"><span class="ilh-paren">（</span><span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Mongolian_numerals" class="extiw" title="en:Mongolian numerals"><span lang="en" dir="auto">Mongolian numerals</span></a></span><span class="ilh-paren">）</span></span></span></li> <li><span class="ilh-all" data-orig-title="Sundanese numerals" data-lang-code="en" data-lang-name="英语" data-foreign-title="Sundanese numerals"><span class="ilh-page"><a href="/w/index.php?title=Sundanese_numerals&action=edit&redlink=1" class="new" title="Sundanese numerals（页面不存在）">Sundanese</a></span><span class="noprint ilh-comment"><span class="ilh-paren">（</span><span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Sundanese_numerals" class="extiw" title="en:Sundanese numerals"><span lang="en" dir="auto">Sundanese numerals</span></a></span><span class="ilh-paren">）</span></span></span></li> <li><a href="/wiki/%E6%B3%B0%E6%96%87%E6%95%B0%E5%AD%97" title="泰文数字">泰文数字</a></li></ul> <hr /></div></div></td> </tr><tr><td class="sidebar-content hlist"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title"><div class="sidebar-list-title-c">東亞體系</div></div><div class="sidebar-list-content mw-collapsible-content"> <dl><dt>當代</dt></dl> <ul><li><a href="/wiki/%E6%B1%89%E5%AD%97%E6%95%B0%E5%AD%97" title="汉字数字">中文数字</a> <ul><li><a href="/wiki/%E8%8B%8F%E5%B7%9E%E7%A0%81%E5%AD%90" title="苏州码子">苏州码子</a></li></ul></li> <li><span class="ilh-all" data-orig-title="閩南語數字" data-lang-code="en" data-lang-name="英语" data-foreign-title="Hokkien numerals"><span class="ilh-page"><a href="/w/index.php?title=%E9%96%A9%E5%8D%97%E8%AA%9E%E6%95%B8%E5%AD%97&action=edit&redlink=1" class="new" title="閩南語數字（页面不存在）">閩南語數字</a></span><span class="noprint ilh-comment"><span class="ilh-paren">（</span><span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Hokkien_numerals" class="extiw" title="en:Hokkien numerals"><span lang="en" dir="auto">Hokkien numerals</span></a></span><span class="ilh-paren">）</span></span></span></li> <li><a href="/wiki/%E6%97%A5%E8%AA%9E%E6%95%B8%E5%AD%97" title="日語數字">日語數字</a></li> <li><a href="/wiki/%E9%9F%93%E8%AA%9E%E6%95%B8%E5%AD%97" title="韓語數字">韓語數字</a></li> <li><a href="/wiki/%E8%B6%8A%E5%8D%97%E8%AF%AD%E6%95%B0%E5%AD%97" title="越南语数字">越南语数字</a></li></ul> <hr /> <dl><dt>古代</dt></dl> <ul><li><a class="mw-selflink selflink">算筹</a></li> <li><span class="ilh-all" data-orig-title="西夏數字" data-lang-code="en" data-lang-name="英语" data-foreign-title="Tangut numerals"><span class="ilh-page"><a href="/w/index.php?title=%E8%A5%BF%E5%A4%8F%E6%95%B8%E5%AD%97&action=edit&redlink=1" class="new" title="西夏數字（页面不存在）">西夏數字</a></span><span class="noprint ilh-comment"><span class="ilh-paren">（</span><span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Tangut_numerals" class="extiw" title="en:Tangut numerals"><span lang="en" dir="auto">Tangut numerals</span></a></span><span class="ilh-paren">）</span></span></span></li></ul></div></div></td> </tr><tr><td class="sidebar-content hlist"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title"><div class="sidebar-list-title-c">美洲體系</div></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/%E7%8E%9B%E9%9B%85%E6%95%B0%E5%AD%97" title="玛雅数字">玛雅数字</a></li> <li><span class="ilh-all" data-orig-title="穆伊斯卡數字" data-lang-code="en" data-lang-name="英语" data-foreign-title="Muisca numerals"><span class="ilh-page"><a href="/w/index.php?title=%E7%A9%86%E4%BC%8A%E6%96%AF%E5%8D%A1%E6%95%B8%E5%AD%97&action=edit&redlink=1" class="new" title="穆伊斯卡數字（页面不存在）">穆伊斯卡數字</a></span><span class="noprint ilh-comment"><span class="ilh-paren">（</span><span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Muisca_numerals" class="extiw" title="en:Muisca numerals"><span lang="en" dir="auto">Muisca numerals</span></a></span><span class="ilh-paren">）</span></span></span></li> <li><a href="/wiki/%E5%A5%87%E6%99%AE" title="奇普">奇普</a></li></ul> ...</div></div></td> </tr><tr><td class="sidebar-content hlist"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title"><div class="sidebar-list-title-c">其它體系</div></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><span class="ilh-all" data-orig-title="古數制史" data-lang-code="en" data-lang-name="英语" data-foreign-title="History of ancient numeral systems"><span class="ilh-page"><a href="/w/index.php?title=%E5%8F%A4%E6%95%B8%E5%88%B6%E5%8F%B2&action=edit&redlink=1" class="new" title="古數制史（页面不存在）">古數制史</a></span><span class="noprint ilh-comment"><span class="ilh-paren">（</span><span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/History_of_ancient_numeral_systems" class="extiw" title="en:History of ancient numeral systems"><span lang="en" dir="auto">History of ancient numeral systems</span></a></span><span class="ilh-paren">）</span></span></span></li></ul> <hr /> <dl><dt><a href="/wiki/%E5%8F%A4%E4%BB%A3%E5%8F%B2" title="古代史">古代史</a></dt></dl> <ul><li><a href="/wiki/%E5%B7%B4%E6%AF%94%E4%BC%A6%E6%95%B0%E5%AD%97" title="巴比伦数字">巴比伦数字</a></li></ul> <hr /> <dl><dt><a href="/wiki/%E5%90%8E%E5%8F%A4%E4%BB%A3%E5%8F%B2" class="mw-redirect" title="后古代史">後古典</a></dt></dl> <ul><li><span class="ilh-all" data-orig-title="西多會數字" data-lang-code="en" data-lang-name="英语" data-foreign-title="Cistercian numerals"><span class="ilh-page"><a href="/w/index.php?title=%E8%A5%BF%E5%A4%9A%E6%9C%83%E6%95%B8%E5%AD%97&action=edit&redlink=1" class="new" title="西多會數字（页面不存在）">西多會數字</a></span><span class="noprint ilh-comment"><span class="ilh-paren">（</span><span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Cistercian_numerals" class="extiw" title="en:Cistercian numerals"><span lang="en" dir="auto">Cistercian numerals</span></a></span><span class="ilh-paren">）</span></span></span></li> <li><a href="/wiki/%E7%8E%9B%E9%9B%85%E6%95%B0%E5%AD%97" title="玛雅数字">玛雅数字</a></li> <li><span class="ilh-all" data-orig-title="穆伊斯卡數字" data-lang-code="en" data-lang-name="英语" data-foreign-title="Muisca numerals"><span class="ilh-page"><a href="/w/index.php?title=%E7%A9%86%E4%BC%8A%E6%96%AF%E5%8D%A1%E6%95%B8%E5%AD%97&action=edit&redlink=1" class="new" title="穆伊斯卡數字（页面不存在）">穆伊斯卡數字</a></span><span class="noprint ilh-comment"><span class="ilh-paren">（</span><span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Muisca_numerals" class="extiw" title="en:Muisca numerals"><span lang="en" dir="auto">Muisca numerals</span></a></span><span class="ilh-paren">）</span></span></span></li> <li><span class="ilh-all" data-orig-title="納維亞五元數字" data-lang-code="en" data-lang-name="英语" data-foreign-title="Pentadic numerals"><span class="ilh-page"><a href="/w/index.php?title=%E7%B4%8D%E7%B6%AD%E4%BA%9E%E4%BA%94%E5%85%83%E6%95%B8%E5%AD%97&action=edit&redlink=1" class="new" title="納維亞五元數字（页面不存在）">納維亞五元數字</a></span><span class="noprint ilh-comment"><span class="ilh-paren">（</span><span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Pentadic_numerals" class="extiw" title="en:Pentadic numerals"><span lang="en" dir="auto">Pentadic numerals</span></a></span><span class="ilh-paren">）</span></span></span></li> <li><a href="/wiki/%E5%A5%87%E6%99%AE" title="奇普">奇普</a></li> <li><span class="ilh-all" data-orig-title="魯米數字符號" data-lang-code="en" data-lang-name="英语" data-foreign-title="Rumi Numeral Symbols"><span class="ilh-page"><a href="/w/index.php?title=%E9%AD%AF%E7%B1%B3%E6%95%B8%E5%AD%97%E7%AC%A6%E8%99%9F&action=edit&redlink=1" class="new" title="魯米數字符號（页面不存在）">魯米數字符號</a></span><span class="noprint ilh-comment"><span class="ilh-paren">（</span><span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Rumi_Numeral_Symbols" class="extiw" title="en:Rumi Numeral Symbols"><span lang="en" dir="auto">Rumi Numeral Symbols</span></a></span><span class="ilh-paren">）</span></span></span></li></ul> <hr /> <dl><dt>當代</dt></dl> <ul><li><a href="/wiki/%E5%88%87%E7%BD%97%E5%9F%BA%E9%9F%B3%E8%8A%82%E6%96%87%E5%AD%97" title="切罗基音节文字">切罗基音节文字</a></li> <li><a href="/wiki/%E5%8D%A1%E5%85%8B%E6%89%98%E7%BB%B4%E5%85%8B%E6%95%B0%E5%AD%97" title="卡克托维克数字">卡克托维克数字</a>（Iñupiaq）</li></ul></div></div></td> </tr><tr><td class="sidebar-content hlist"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title"><div class="sidebar-list-title-c">各<a href="/wiki/%E5%BA%95%E6%95%B0_(%E8%BF%9B%E5%88%B6)" title="底数 (进制)">底數</a>的<a href="/wiki/%E9%80%B2%E4%BD%8D%E5%88%B6" class="mw-redirect" title="進位制">進位制</a></div></div><div class="sidebar-list-content mw-collapsible-content"> <dl><dt>一般進位</dt></dl> <ul><li><a href="/wiki/%E4%BA%8C%E8%BF%9B%E5%88%B6" title="二进制">二进制</a></li> <li><a href="/wiki/%E4%B8%89%E8%BF%9B%E5%88%B6" title="三进制">三进制</a></li> <li><a href="/wiki/%E5%9B%9B%E9%80%B2%E5%88%B6" title="四進制">四進制</a></li> <li><a href="/wiki/%E4%BA%94%E9%80%B2%E5%88%B6" title="五進制">五進制</a></li> <li><a href="/wiki/%E5%85%AD%E8%BF%9B%E5%88%B6" title="六进制">六进制</a></li> <li><a href="/wiki/%E5%85%AB%E8%BF%9B%E5%88%B6" title="八进制">八进制</a></li> <li><a href="/wiki/%E4%B9%9D%E9%80%B2%E4%BD%8D" title="九進位">九進位</a></li> <li><a href="/wiki/%E5%8D%81%E8%BF%9B%E5%88%B6" title="十进制">十进制</a></li> <li><a href="/wiki/%E5%8D%81%E4%B8%80%E8%BF%9B%E5%88%B6" title="十一进制">十一進制</a></li> <li><a href="/wiki/%E5%8D%81%E4%BA%8C%E8%BF%9B%E5%88%B6" title="十二进制">十二进制</a></li> <li><a href="/wiki/%E5%8D%81%E5%85%AD%E8%BF%9B%E5%88%B6" title="十六进制">十六进制</a></li> <li><a href="/wiki/%E4%BA%8C%E5%8D%81%E8%BF%9B%E5%88%B6" title="二十进制">二十进制</a></li> <li><a href="/wiki/%E4%B8%89%E5%8D%81%E5%85%AD%E8%BF%9B%E5%88%B6" title="三十六进制">三十六進制</a></li> <li><a href="/wiki/%E5%85%AD%E5%8D%81%E9%80%B2%E5%88%B6" title="六十進制">六十進制</a></li> <li><a href="/wiki/%E9%80%B2%E5%88%B6%E8%A1%A8" title="進制表">進制表</a></li></ul> <hr /> <dl><dt><a href="/wiki/%E5%B9%BF%E4%B9%89%E7%9A%84%E8%BF%9B%E5%88%B6%E7%B3%BB%E7%BB%9F" title="广义的进制系统">广义的进制系统</a></dt></dl> <ul><li><a href="/wiki/%E9%9B%99%E5%B0%84%E8%A8%98%E6%95%B8" title="雙射記數">雙射記數</a><span class="nowrap"> </span>(<a href="/wiki/%E4%B8%80%E8%BF%9B%E5%88%B6" title="一进制">一进制</a>)</li> <li><span class="ilh-all" data-orig-title="有符號記數" data-lang-code="en" data-lang-name="英语" data-foreign-title="Signed-digit representation"><span class="ilh-page"><a href="/w/index.php?title=%E6%9C%89%E7%AC%A6%E8%99%9F%E8%A8%98%E6%95%B8&action=edit&redlink=1" class="new" title="有符號記數（页面不存在）">有符號記數</a></span><span class="noprint ilh-comment"><span class="ilh-paren">（</span><span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Signed-digit_representation" class="extiw" title="en:Signed-digit representation"><span lang="en" dir="auto">Signed-digit representation</span></a></span><span class="ilh-paren">）</span></span></span><span class="nowrap"> </span>(<a href="/wiki/%E5%B9%B3%E8%A1%A1%E4%B8%89%E9%80%B2%E4%BD%8D" title="平衡三進位">平衡三進位</a>)</li> <li><span class="ilh-all" data-orig-title="混合進制" data-lang-code="en" data-lang-name="英语" data-foreign-title="Mixed radix"><span class="ilh-page"><a href="/w/index.php?title=%E6%B7%B7%E5%90%88%E9%80%B2%E5%88%B6&action=edit&redlink=1" class="new" title="混合進制（页面不存在）">混合進制</a></span><span class="noprint ilh-comment"><span class="ilh-paren">（</span><span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Mixed_radix" class="extiw" title="en:Mixed radix"><span lang="en" dir="auto">Mixed radix</span></a></span><span class="ilh-paren">）</span></span></span><span class="nowrap"> </span>(<a href="/wiki/%E9%98%B6%E4%B9%98%E8%BF%9B%E5%88%B6" title="阶乘进制">阶乘进制</a>)</li> <li><span class="ilh-all" data-orig-title="負進制" data-lang-code="en" data-lang-name="英语" data-foreign-title="Negative base"><span class="ilh-page"><a href="/w/index.php?title=%E8%B2%A0%E9%80%B2%E5%88%B6&action=edit&redlink=1" class="new" title="負進制（页面不存在）">負進制</a></span><span class="noprint ilh-comment"><span class="ilh-paren">（</span><span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Negative_base" class="extiw" title="en:Negative base"><span lang="en" dir="auto">Negative base</span></a></span><span class="ilh-paren">）</span></span></span></li> <li><a href="/wiki/%E5%A4%8D%E5%BA%95%E6%95%B0%E8%BF%9B%E5%88%B6" title="复底数进制">复底数进制</a><span class="nowrap"> </span>(<a href="/wiki/2i%E9%80%B2%E5%88%B6" title="2i進制">2i進制</a>)</li> <li><a href="/wiki/%E9%9D%9E%E6%95%B4%E6%95%B0%E8%BF%9B%E4%BD%8D%E5%88%B6" title="非整数进位制">非整数进位制</a><span class="nowrap"> </span>(<a href="/wiki/%E9%BB%84%E9%87%91%E8%BF%9B%E5%88%B6" title="黄金进制">黄金进制</a>)</li> <li><span class="ilh-all" data-orig-title="非對稱進制" data-lang-code="en" data-lang-name="英语" data-foreign-title="Asymmetric numeral systems"><span class="ilh-page"><a href="/w/index.php?title=%E9%9D%9E%E5%B0%8D%E7%A8%B1%E9%80%B2%E5%88%B6&action=edit&redlink=1" class="new" title="非對稱進制（页面不存在）">非對稱進制</a></span><span class="noprint ilh-comment"><span class="ilh-paren">（</span><span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Asymmetric_numeral_systems" class="extiw" title="en:Asymmetric numeral systems"><span lang="en" dir="auto">Asymmetric numeral systems</span></a></span><span class="ilh-paren">）</span></span></span></li></ul></div></div></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title"><div class="sidebar-list-title-c"><a href="/wiki/%E7%AC%A6%E5%80%BC%E7%9B%B8%E5%8A%A0%E8%A8%98%E6%95%B8%E6%B3%95" title="符值相加記數法">符值相加記數法</a></div></div><div class="sidebar-list-content mw-collapsible-content"> <dl><dt>非字母</dt></dl> <ul><li><a href="/wiki/%E6%84%9B%E7%90%B4%E6%95%B8%E5%AD%97" title="愛琴數字">愛琴數字</a></li> <li><a href="/wiki/%E9%98%BF%E6%8F%90%E5%8D%A1%E6%95%B8%E5%AD%97" title="阿提卡數字">阿提卡數字</a></li> <li><a href="/wiki/%E9%98%BF%E8%8C%B2%E7%89%B9%E5%85%8B%E6%96%87%E5%AD%97" title="阿茲特克文字">阿茲特克文字</a></li> <li><span class="ilh-all" data-orig-title="婆羅米數字" data-lang-code="en" data-lang-name="英语" data-foreign-title="Brahmi numerals"><span class="ilh-page"><a href="/w/index.php?title=%E5%A9%86%E7%BE%85%E7%B1%B3%E6%95%B8%E5%AD%97&action=edit&redlink=1" class="new" title="婆羅米數字（页面不存在）">婆羅米數字</a></span><span class="noprint ilh-comment"><span class="ilh-paren">（</span><span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Brahmi_numerals" class="extiw" title="en:Brahmi numerals"><span lang="en" dir="auto">Brahmi numerals</span></a></span><span class="ilh-paren">）</span></span></span></li> <li><span class="ilh-all" data-orig-title="楚瓦什數字" data-lang-code="en" data-lang-name="英语" data-foreign-title="Chuvash numerals"><span class="ilh-page"><a href="/w/index.php?title=%E6%A5%9A%E7%93%A6%E4%BB%80%E6%95%B8%E5%AD%97&action=edit&redlink=1" class="new" title="楚瓦什數字（页面不存在）">楚瓦什數字</a></span><span class="noprint ilh-comment"><span class="ilh-paren">（</span><span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Chuvash_numerals" class="extiw" title="en:Chuvash numerals"><span lang="en" dir="auto">Chuvash numerals</span></a></span><span class="ilh-paren">）</span></span></span></li> <li><a href="/wiki/%E5%8F%A4%E5%9F%83%E5%8F%8A%E6%95%B0%E5%AD%97" title="古埃及数字">古埃及数字</a></li> <li><span class="ilh-all" data-orig-title="伊特拉斯坎數字" data-lang-code="en" data-lang-name="英语" data-foreign-title="Etruscan numerals"><span class="ilh-page"><a href="/w/index.php?title=%E4%BC%8A%E7%89%B9%E6%8B%89%E6%96%AF%E5%9D%8E%E6%95%B8%E5%AD%97&action=edit&redlink=1" class="new" title="伊特拉斯坎數字（页面不存在）">伊特拉斯坎數字</a></span><span class="noprint ilh-comment"><span class="ilh-paren">（</span><span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Etruscan_numerals" class="extiw" title="en:Etruscan numerals"><span lang="en" dir="auto">Etruscan numerals</span></a></span><span class="ilh-paren">）</span></span></span></li> <li><a href="/wiki/%E4%BD%89%E5%8D%A2%E6%96%87" title="佉卢文">佉卢文</a></li> <li><span class="ilh-all" data-orig-title="史前計數" data-lang-code="en" data-lang-name="英语" data-foreign-title="Prehistoric counting"><span class="ilh-page"><a href="/w/index.php?title=%E5%8F%B2%E5%89%8D%E8%A8%88%E6%95%B8&action=edit&redlink=1" class="new" title="史前計數（页面不存在）">史前計數</a></span><span class="noprint ilh-comment"><span class="ilh-paren">（</span><span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Prehistoric_counting" class="extiw" title="en:Prehistoric counting"><span lang="en" dir="auto">Prehistoric counting</span></a></span><span class="ilh-paren">）</span></span></span></li> <li><span class="ilh-all" data-orig-title="原始楔形文字" data-lang-code="en" data-lang-name="英语" data-foreign-title="Proto-cuneiform"><span class="ilh-page"><a href="/w/index.php?title=%E5%8E%9F%E5%A7%8B%E6%A5%94%E5%BD%A2%E6%96%87%E5%AD%97&action=edit&redlink=1" class="new" title="原始楔形文字（页面不存在）">原始楔形文字</a></span><span class="noprint ilh-comment"><span class="ilh-paren">（</span><span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Proto-cuneiform" class="extiw" title="en:Proto-cuneiform"><span lang="en" dir="auto">Proto-cuneiform</span></a></span><span class="ilh-paren">）</span></span></span></li> <li><a href="/wiki/%E7%BD%97%E9%A9%AC%E6%95%B0%E5%AD%97" title="罗马数字">罗马数字</a></li> <li><a href="/wiki/%E8%AE%A1%E6%95%B0%E7%AC%A6%E5%8F%B7" title="计数符号">计数符号</a></li></ul> <hr /> <dl><dt><span class="ilh-all" data-orig-title="字母數制" data-lang-code="en" data-lang-name="英语" data-foreign-title="Alphabetic numeral system"><span class="ilh-page"><a href="/w/index.php?title=%E5%AD%97%E6%AF%8D%E6%95%B8%E5%88%B6&action=edit&redlink=1" class="new" title="字母數制（页面不存在）">字母數制</a></span><span class="noprint ilh-comment"><span class="ilh-paren">（</span><span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Alphabetic_numeral_system" class="extiw" title="en:Alphabetic numeral system"><span lang="en" dir="auto">Alphabetic numeral system</span></a></span><span class="ilh-paren">）</span></span></span></dt></dl> <ul><li><span class="ilh-all" data-orig-title="Abjad numerals" data-lang-code="en" data-lang-name="英语" data-foreign-title="Abjad numerals"><span class="ilh-page"><a href="/w/index.php?title=Abjad_numerals&action=edit&redlink=1" class="new" title="Abjad numerals（页面不存在）">Abjad</a></span><span class="noprint ilh-comment"><span class="ilh-paren">（</span><span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Abjad_numerals" class="extiw" title="en:Abjad numerals"><span lang="en" dir="auto">Abjad numerals</span></a></span><span class="ilh-paren">）</span></span></span></li> <li><a href="/wiki/%E4%BA%9A%E7%BE%8E%E5%B0%BC%E4%BA%9A%E6%95%B0%E5%AD%97" title="亚美尼亚数字">亚美尼亚数字</a></li> <li><span class="ilh-all" data-orig-title="Alphasyllabic numeral system" data-lang-code="en" data-lang-name="英语" data-foreign-title="Alphasyllabic numeral system"><span class="ilh-page"><a href="/w/index.php?title=Alphasyllabic_numeral_system&action=edit&redlink=1" class="new" title="Alphasyllabic numeral system（页面不存在）">Alphasyllabic</a></span><span class="noprint ilh-comment"><span class="ilh-paren">（</span><span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Alphasyllabic_numeral_system" class="extiw" title="en:Alphasyllabic numeral system"><span lang="en" dir="auto">Alphasyllabic numeral system</span></a></span><span class="ilh-paren">）</span></span></span> <ul><li><span class="ilh-all" data-orig-title="Aksharapalli" data-lang-code="en" data-lang-name="英语" data-foreign-title="Aksharapalli"><span class="ilh-page"><a href="/w/index.php?title=Aksharapalli&action=edit&redlink=1" class="new" title="Aksharapalli（页面不存在）">Akṣarapallī</a></span><span class="noprint ilh-comment"><span class="ilh-paren">（</span><span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Aksharapalli" class="extiw" title="en:Aksharapalli"><span lang="en" dir="auto">Aksharapalli</span></a></span><span class="ilh-paren">）</span></span></span></li> <li><span class="ilh-all" data-orig-title="Āryabhaṭa numeration" data-lang-code="en" data-lang-name="英语" data-foreign-title="Āryabhaṭa numeration"><span class="ilh-page"><a href="/w/index.php?title=%C4%80ryabha%E1%B9%ADa_numeration&action=edit&redlink=1" class="new" title="Āryabhaṭa numeration（页面不存在）">Āryabhaṭa</a></span><span class="noprint ilh-comment"><span class="ilh-paren">（</span><span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/%C4%80ryabha%E1%B9%ADa_numeration" class="extiw" title="en:Āryabhaṭa numeration"><span lang="en" dir="auto">Āryabhaṭa numeration</span></a></span><span class="ilh-paren">）</span></span></span></li> <li><span class="ilh-all" data-orig-title="Katapayadi system" data-lang-code="en" data-lang-name="英语" data-foreign-title="Katapayadi system"><span class="ilh-page"><a href="/w/index.php?title=Katapayadi_system&action=edit&redlink=1" class="new" title="Katapayadi system（页面不存在）">Kaṭapayādi</a></span><span class="noprint ilh-comment"><span class="ilh-paren">（</span><span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Katapayadi_system" class="extiw" title="en:Katapayadi system"><span lang="en" dir="auto">Katapayadi system</span></a></span><span class="ilh-paren">）</span></span></span></li></ul></li> <li><a href="/wiki/%E7%A7%91%E6%99%AE%E7%89%B9%E5%AD%97%E6%AF%8D" title="科普特字母">科普特字母</a></li> <li><a href="/wiki/%E8%A5%BF%E9%87%8C%E7%88%BE%E6%95%B8%E5%AD%97" title="西里爾數字">西里爾數字</a></li> <li><a href="/wiki/%E5%90%89%E8%8C%B2%E5%AD%97%E6%AF%8D" title="吉茲字母">吉茲字母</a></li> <li><span class="ilh-all" data-orig-title="Georgian numerals" data-lang-code="en" data-lang-name="英语" data-foreign-title="Georgian numerals"><span class="ilh-page"><a href="/w/index.php?title=Georgian_numerals&action=edit&redlink=1" class="new" title="Georgian numerals（页面不存在）">Georgian</a></span><span class="noprint ilh-comment"><span class="ilh-paren">（</span><span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Georgian_numerals" class="extiw" title="en:Georgian numerals"><span lang="en" dir="auto">Georgian numerals</span></a></span><span class="ilh-paren">）</span></span></span></li> <li><span class="ilh-all" data-orig-title="Glagolitic_numerals" data-lang-code="en" data-lang-name="英语" data-foreign-title="Glagolitic_numerals"><span class="ilh-page"><a href="/w/index.php?title=Glagolitic_numerals&action=edit&redlink=1" class="new" title="Glagolitic numerals（页面不存在）">Glagolitic</a></span><span class="noprint ilh-comment"><span class="ilh-paren">（</span><span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Glagolitic_numerals" class="extiw" title="en:Glagolitic numerals"><span lang="en" dir="auto">Glagolitic_numerals</span></a></span><span class="ilh-paren">）</span></span></span></li> <li><a href="/wiki/%E5%B8%8C%E8%85%8A%E6%95%B0%E5%AD%97" title="希腊数字">希腊数字</a></li> <li><a href="/wiki/%E5%B8%8C%E4%BC%AF%E4%BE%86%E6%95%B8%E5%AD%97" title="希伯來數字">希伯來數字</a></li></ul></div></div></td> </tr><tr><td class="sidebar-below" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"> <span class="ilh-all" data-orig-title="數制列表" data-lang-code="en" data-lang-name="英语" data-foreign-title="List of numeral systems"><span class="ilh-page"><a href="/w/index.php?title=%E6%95%B8%E5%88%B6%E5%88%97%E8%A1%A8&action=edit&redlink=1" class="new" title="數制列表（页面不存在）">數制列表</a></span><span class="noprint ilh-comment"><span class="ilh-paren">（</span><span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/List_of_numeral_systems" class="extiw" title="en:List of numeral systems"><span lang="en" dir="auto">List of numeral systems</span></a></span><span class="ilh-paren">）</span></span></span></td></tr><tr><td class="sidebar-navbar" style="line-height:1.6"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84265675" /><style data-mw-deduplicate="TemplateStyles:r84244141">.mw-parser-output .navbar{display:inline;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:110%;margin:0 8em}.mw-parser-output .navbar-ct-mini{font-size:110%;margin:0 5em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:%E8%AE%B0%E6%95%B0%E7%B3%BB%E7%BB%9F" title="Template:记数系统"><abbr title="查看该模板">查</abbr></a></li><li class="nv-talk"><a href="/w/index.php?title=Template_talk:%E8%AE%B0%E6%95%B0%E7%B3%BB%E7%BB%9F&action=edit&redlink=1" class="new" title="Template talk:记数系统（页面不存在）"><abbr title="讨论该模板">论</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:%E7%BC%96%E8%BE%91%E9%A1%B5%E9%9D%A2/Template:%E8%AE%B0%E6%95%B0%E7%B3%BB%E7%BB%9F" title="Special:编辑页面/Template:记数系统"><abbr title="编辑该模板">编</abbr></a></li></ul></div></td></tr></tbody></table> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Replica_of_Han_Dynasty_Counting_rods_in_National_Museum_of_Natural_Science_in_Taiwan.JPG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Replica_of_Han_Dynasty_Counting_rods_in_National_Museum_of_Natural_Science_in_Taiwan.JPG/250px-Replica_of_Han_Dynasty_Counting_rods_in_National_Museum_of_Natural_Science_in_Taiwan.JPG" decoding="async" width="200" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Replica_of_Han_Dynasty_Counting_rods_in_National_Museum_of_Natural_Science_in_Taiwan.JPG/330px-Replica_of_Han_Dynasty_Counting_rods_in_National_Museum_of_Natural_Science_in_Taiwan.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Replica_of_Han_Dynasty_Counting_rods_in_National_Museum_of_Natural_Science_in_Taiwan.JPG/500px-Replica_of_Han_Dynasty_Counting_rods_in_National_Museum_of_Natural_Science_in_Taiwan.JPG 2x" data-file-width="4000" data-file-height="3000" /></a><figcaption><a href="/wiki/%E5%9C%8B%E7%AB%8B%E8%87%AA%E7%84%B6%E7%A7%91%E5%AD%B8%E5%8D%9A%E7%89%A9%E9%A4%A8" title="國立自然科學博物館">國立自然科學博物館</a>內藏的漢朝骨製算籌複製品。</figcaption></figure> <p><b>算筹</b>或称<b>筭子</b>、<b>算子</b>，是<a href="/wiki/%E6%BC%A2%E5%AD%97%E6%96%87%E5%8C%96%E5%9C%88" title="漢字文化圈">漢字文化圈</a>古代一种<a href="/wiki/%E5%8D%81%E8%BF%9B%E5%88%B6" title="十进制">十进制</a>计算工具。起源于中國<a href="/wiki/%E5%95%86%E6%9C%9D" title="商朝">商朝</a>的<a href="/wiki/%E5%8D%A0%E5%8D%9C" title="占卜">占卜</a>，占卜用现成的小木棍做计算，就是最早的算筹。<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>古代筹、策、算三字都带竹头，表示用竹制成。策为束字加竹头，表示手握一束竖立的算策，作为占卜之用。筹可能代表周易八卦横向排列时用的阴阳竹，算筹横竖二式，可能来源于此<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> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="算筹的材料和形状"><span id=".E7.AE.97.E7.AD.B9.E7.9A.84.E6.9D.90.E6.96.99.E5.92.8C.E5.BD.A2.E7.8A.B6"></span>算筹的材料和形状</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%AE%97%E7%AD%B9&action=edit&section=1" title="编辑章节：算筹的材料和形状"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/%E5%91%A8%E6%9C%9D" title="周朝">周朝</a>用木枝制成算筹<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>，汉代用竹<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>、骨、象牙<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>、玉石、铁<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>等材料制作，长一般在12厘米左右，直径为2至4毫米。最初的算筹的截面是圆形的，后来变成三角、四角形。 </p> <ul><li>汉朝：《汉书·律历志》记载<style data-mw-deduplicate="TemplateStyles:r61209892">.mw-parser-output .templatequote{margin-top:0;overflow:hidden}.mw-parser-output .templatequote .templatequotecite{line-height:1em;text-align:left;padding-left:2em;margin-top:0}.mw-parser-output .templatequote .templatequotecite cite{font-size:small}</style><blockquote class="templatequote"><p>其算法用竹。直径一分，长六寸；二百七十一枚而成觚，为一握。</p></blockquote></li> <li>隋朝：《隋书·律历志》记载<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r61209892" /><blockquote class="templatequote"><p>其算法用竹。广二分，长三寸，正策三廉（三角形）积二百一十六枚成六觚；乾之策也。负策四廉（四方形），积一百四十四枚，成方，坤之策也</p></blockquote></li></ul> <p>日本算家所用算筹，少用竹制，多用小木片制作，因此称为“算木”，长者叫策，短者叫筹。<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup>。 </p> <div class="mw-heading mw-heading2"><h2 id="文獻記載"><span id=".E6.96.87.E7.8D.BB.E8.A8.98.E8.BC.89"></span>文獻記載</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%AE%97%E7%AD%B9&action=edit&section=2" title="编辑章节：文獻記載"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/%E8%80%81%E5%AD%90" title="老子">老子</a>：“善數者不用籌策”。</li> <li><a href="/wiki/%E8%AE%B8%E6%85%8E" class="mw-redirect" title="许慎">许慎</a>《<a href="/wiki/%E8%AF%B4%E6%96%87%E8%A7%A3%E5%AD%97" title="说文解字">说文解字</a>》“算，长六寸，计历数者。从竹”。</li> <li><a href="/wiki/%E6%B1%89%E4%B9%A6" title="汉书">汉书</a>律历志：“数者，一、十、百、千、万也，所以算数事物，顺性命之理也。…卡悟空…其算法用竹，径一分，长六寸，二百七十一枚而成六觚，为一握”。</li> <li>魏<a href="/wiki/%E5%8A%89%E5%BE%BD" class="mw-redirect" title="劉徽">劉徽</a>注《<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>》称：“正算赤，负算黑”。</li> <li>汉末三国时期徐岳撰、北周汉中郎甄鸾注《数术记遗》：“今之常算也，以竹为之”。</li> <li>《<a href="/wiki/%E9%9A%8B%E4%B9%A6" title="隋书">隋书</a>·律历志》：“其算用竹，广二分，长三寸。正策三廉，积二百一十六枚成六觚，乾之策也。负策四廉，积一百四十四枚成方，坤之策也”。</li> <li>《<a href="/wiki/%E6%A2%A6%E6%BA%AA%E7%AC%94%E8%B0%88" title="梦溪笔谈">梦溪笔谈</a>》卷八称：“算法用<a href="/wiki/%E8%B5%A4%E5%AD%97" title="赤字">赤筹</a>、黑筹，以别正负之数”，“运筹如飞，人眼不能逐”。</li> <li>《资治通鉴》卷八十二：“每自执牙筹昼夜会计”。</li> <li>北宋陶谷撰《清异录》：“铸铁为算子”。</li></ul> <div class="mw-heading mw-heading2"><h2 id="考古实物"><span id=".E8.80.83.E5.8F.A4.E5.AE.9E.E7.89.A9"></span>考古实物</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%AE%97%E7%AD%B9&action=edit&section=3" title="编辑章节：考古实物"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>秦朝的货币中有<span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_h6.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/c/c1/Counting_rod_h6.png" decoding="async" width="29" height="31" class="mw-file-element" data-file-width="29" data-file-height="31" /></a></span>；王莽货币上有<span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v6.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/1/19/Counting_rod_v6.png" decoding="async" width="29" height="32" class="mw-file-element" data-file-width="29" data-file-height="32" /></a></span>,<span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v7.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/d/d1/Counting_rod_v7.png" decoding="async" width="29" height="32" class="mw-file-element" data-file-width="29" data-file-height="32" /></a></span>,<span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v8.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/6/64/Counting_rod_v8.png" decoding="async" width="29" height="32" class="mw-file-element" data-file-width="29" data-file-height="32" /></a></span>,<span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v9.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/7/7a/Counting_rod_v9.png" decoding="async" width="29" height="32" class="mw-file-element" data-file-width="29" data-file-height="32" /></a></span><sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup>。</li> <li>1954年考古学家在<a href="/wiki/%E6%B9%96%E5%8D%97" class="mw-redirect" title="湖南">湖南</a><a href="/wiki/%E9%95%BF%E6%B2%99" class="mw-redirect" title="长沙">长沙</a>左家公山出土<a href="/wiki/%E6%88%98%E5%9B%BD%E6%97%B6%E4%BB%A3" class="mw-redirect" title="战国时代">战国时代</a>古墓，内藏竹算筹40根，每根长12厘米。</li> <li>1975年在<a href="/wiki/%E6%B9%96%E5%8C%97" class="mw-redirect" title="湖北">湖北</a><a href="/wiki/%E6%B1%9F%E9%99%B5" class="mw-disambig" title="江陵">江陵</a><a href="/wiki/%E5%87%A4%E5%87%B0%E5%B1%B1" class="mw-redirect mw-disambig" title="凤凰山">凤凰山</a>发现竹制算筹。</li> <li>1971年<a href="/wiki/%E8%80%83%E5%8F%A4%E5%AD%A6%E5%AE%B6" class="mw-redirect" title="考古学家">考古学家</a>从<a href="/wiki/%E9%99%95%E8%A5%BF" class="mw-redirect" title="陕西">陕西</a><a href="/wiki/%E5%8D%83%E9%98%B3" class="mw-redirect" title="千阳">千阳</a>县一座出土的<a href="/wiki/%E8%A5%BF%E6%B1%89" title="西汉">西汉</a>古墓中发现一束在一个丝袋内（算筹袋）保存完好的兽骨算筹，长短不一，最长的13.8厘米，最短的12.6厘米，截面呈圆形，直径在2-4毫米间。</li> <li>1980年从<a href="/wiki/%E6%B2%B3%E5%8C%97" class="mw-redirect" title="河北">河北</a><a href="/wiki/%E7%9F%B3%E5%AE%B6%E5%BA%84" class="mw-redirect" title="石家庄">石家庄</a>出土30根<a href="/wiki/%E4%B8%9C%E6%B1%89" title="东汉">东汉</a>骨算筹，长7.8-8.9厘米，截面方形，边长约0.4厘米。此外在<a href="/wiki/%E9%99%95%E8%A5%BF" class="mw-redirect" title="陕西">陕西</a><a href="/wiki/%E6%97%AC%E9%98%B3" class="mw-redirect" title="旬阳">旬阳</a>汉墓中出土象牙筹27根。<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup></li> <li>1983-1984，湖北江陵张家山西汉古墓中出土竹算筹。</li> <li>1973年9月，湖北省江陵县凤凰山出土的十号汉代古木木牍，其中有一片记有“当利二月定筭<span typeof="mw:File"><a href="/wiki/File:Counting_rod_v6.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/19/Counting_rod_v6.png/15px-Counting_rod_v6.png" decoding="async" width="15" height="17" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/19/Counting_rod_v6.png/23px-Counting_rod_v6.png 1.5x, //upload.wikimedia.org/wikipedia/commons/1/19/Counting_rod_v6.png 2x" data-file-width="29" data-file-height="32" /></a></span>”，这是文献中最早出现以筹码代替文字记数例子之一<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup></li></ul> <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=%E7%AE%97%E7%AD%B9&action=edit&section=4" title="编辑章节：計算方法"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r85100532">.mw-parser-output .hatnote{font-size:small}.mw-parser-output div.hatnote{padding-left:2em;margin-bottom:0.8em;margin-top:0.8em}.mw-parser-output .hatnote-notice-img::after{content:"\202f \202f \202f \202f "}.mw-parser-output .hatnote-notice-img-small::after{content:"\202f \202f "}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}body.skin-minerva .mw-parser-output .hatnote-notice-img,body.skin-minerva .mw-parser-output .hatnote-notice-img-small{display:none}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">主条目：<a href="/wiki/%E7%B1%8C%E7%AE%97" class="mw-redirect" title="籌算">籌算</a></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Rod_numeral_positioning.JPG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0c/Rod_numeral_positioning.JPG/300px-Rod_numeral_positioning.JPG" decoding="async" width="300" height="264" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0c/Rod_numeral_positioning.JPG/450px-Rod_numeral_positioning.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/0/0c/Rod_numeral_positioning.JPG 2x" data-file-width="556" data-file-height="490" /></a><figcaption>永乐大典算筹布位图</figcaption></figure> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Counting_board.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/47/Counting_board.jpg/300px-Counting_board.jpg" decoding="async" width="300" height="134" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/4/47/Counting_board.jpg 1.5x" data-file-width="450" data-file-height="201" /></a><figcaption><a href="/wiki/%E6%97%A5%E6%9C%AC" title="日本">日本</a>带格算筹板</figcaption></figure> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Checker_counting_board.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Checker_counting_board.jpg/330px-Checker_counting_board.jpg" decoding="async" width="300" height="304" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Checker_counting_board.jpg/500px-Checker_counting_board.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/63/Checker_counting_board.jpg/960px-Checker_counting_board.jpg 2x" data-file-width="1559" data-file-height="1580" /></a><figcaption>十八世纪日本格子算筹版</figcaption></figure> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Seki_Kowa_Katsuyo_Sampo_Bernoulli_numbers.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Seki_Kowa_Katsuyo_Sampo_Bernoulli_numbers.png/300px-Seki_Kowa_Katsuyo_Sampo_Bernoulli_numbers.png" decoding="async" width="300" height="395" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Seki_Kowa_Katsuyo_Sampo_Bernoulli_numbers.png/450px-Seki_Kowa_Katsuyo_Sampo_Bernoulli_numbers.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Seki_Kowa_Katsuyo_Sampo_Bernoulli_numbers.png/600px-Seki_Kowa_Katsuyo_Sampo_Bernoulli_numbers.png 2x" data-file-width="1614" data-file-height="2124" /></a><figcaption>日本古算术中带格子的算筹码</figcaption></figure> <p>使用算筹进行计算的方法，则称为<a href="/wiki/%E7%AD%B9%E7%AE%97" title="筹算">筹算</a>。算筹一般布置在地面、桌上等平面运算；南宋黄伯思著《燕几图》中列举布算桌，长7尺，宽5尺许，小布算长宽为5尺余。清代数学家劳乃宣说：“盖古者席地而坐，布算于地，后世施于几案”<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup>。日本古算书中有带方格子的算筹板图。此外北宋数学家<a href="/wiki/%E8%B4%BE%E5%AE%AA" title="贾宪">贾宪</a>将万、千、百、十、步作为算筹位值标记：他将七万一千八百二十四写成 </p> <dl><dd>七一八二四</dd> <dd>万千百十步</dd></dl> <p>然后将算筹按位值布位。 </p> <table border="0" width="500" align="centre" style="border: 5px solid #999; background-color:#FFFFFF;"> <tbody><tr align="center" bgcolor="#EFEFEF"> <th>七 </th> <th>一 </th> <th>八 </th> <th>二 </th> <th>四 </th></tr> <tr align="center" bgcolor="#EFEFEF"> <td>万 </td> <td>千 </td> <td>百 </td> <td>十 </td> <td>步 </td></tr> <tr align="center" bgcolor="#EFEFEF"> <td><span typeof="mw:File"><a href="/wiki/File:Counting_rod_v7.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d1/Counting_rod_v7.png/20px-Counting_rod_v7.png" decoding="async" width="20" height="22" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/d/d1/Counting_rod_v7.png 1.5x" data-file-width="29" data-file-height="32" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Counting_rod_h1.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/17/Counting_rod_h1.png/20px-Counting_rod_h1.png" decoding="async" width="20" height="5" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/1/17/Counting_rod_h1.png 1.5x" data-file-width="29" data-file-height="7" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Counting_rod_v8.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/64/Counting_rod_v8.png/20px-Counting_rod_v8.png" decoding="async" width="20" height="22" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/6/64/Counting_rod_v8.png 1.5x" data-file-width="29" data-file-height="32" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Counting_rod_h2.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/51/Counting_rod_h2.png/20px-Counting_rod_h2.png" decoding="async" width="20" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/5/51/Counting_rod_h2.png 1.5x" data-file-width="29" data-file-height="13" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Counting_rod_v4.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/67/Counting_rod_v4.png/20px-Counting_rod_v4.png" decoding="async" width="20" height="23" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/6/67/Counting_rod_v4.png 1.5x" data-file-width="25" data-file-height="29" /></a></span> </td></tr></tbody></table> <p>一本十八世纪日本数学书籍中有一幅格子筹算版，其上方标记千百十一分厘毛等数量级<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup>。 </p><p>算筹出现的具体时间无从可考，但是早在中国<a href="/wiki/%E6%98%A5%E7%A7%8B%E6%88%98%E5%9B%BD" class="mw-redirect" title="春秋战国">春秋战国</a>的时期就已普遍使用了。算筹采用的是<a href="/wiki/%E5%8D%81%E8%BF%9B%E4%BD%8D%E5%88%B6" class="mw-redirect" title="十进位制">十进位制</a>的记数方法，同一个数字在不同的数位上，数值也就相应不同，每进一位数值乘10，并且在算筹盘上，以空位表示0。为了不使<a href="/wiki/%E6%95%B0%E5%AD%97" class="mw-redirect" title="数字">数字</a>和<a href="/wiki/%E6%95%B8%E4%BD%8D" class="mw-redirect" title="數位">数位</a>混淆，算筹采用<b>纵式</b>和<b>横式</b>两种方法记数。中国古代算筹记数，采用<a href="/wiki/%E5%8D%81%E8%BF%9B%E4%BD%8D%E5%88%B6" class="mw-redirect" title="十进位制">十进位制</a>，个位用纵式，十位用横式，百位再用纵式……这样纵横交替摆放，就可以摆出任意大的数字来了。<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>记载： </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r61209892" /><blockquote class="templatequote"><p>凡算之法，先识其位，一纵十横，百立千僵，千十相望，百万相当</p></blockquote><p>，此外又说明用空位表示零。<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup>。在进行乘法时，“言十即过，不满自如”，即“逢十进一”。 </p><p>《<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>》习惯用“置”表示放置算筹：“置一十九道”，“置周三百六十”，而不用“写下”，说明算筹是运算器具，不是书面计算。 </p><p>成书于前202年至前186年之间<a href="/wiki/%E7%AE%97%E6%95%B0%E4%B9%A6" title="算数书">算数书</a>一样用“置”。 </p><p>南宋数学家开始用书写算草代替算筹算板，为了减少书写的笔划，将4、5、9的筹码简化；为了算草需要，引入〇符号。 </p> <table class="wikitable" border="1" style="text-align:center"> <caption>算筹正数 </caption> <tbody><tr> <th>  </th> <th style="width:50px">0 </th> <th style="width:50px">1 </th> <th style="width:50px">2 </th> <th style="width:50px">3 </th> <th style="width:50px">4 </th> <th style="width:50px">5 </th> <th style="width:50px">6 </th> <th style="width:50px">7 </th> <th style="width:50px">8 </th> <th style="width:50px">9 </th></tr> <tr> <th style="height:50px">直式 </th> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_0.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/5/50/Counting_rod_0.png" decoding="async" width="23" height="23" class="mw-file-element" data-file-width="23" data-file-height="23" /></a></span> </td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v1.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/1/19/Counting_rod_v1.png" decoding="async" width="7" height="29" class="mw-file-element" data-file-width="7" data-file-height="29" /></a></span> </td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v2.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/f/ff/Counting_rod_v2.png" decoding="async" width="13" height="29" class="mw-file-element" data-file-width="13" data-file-height="29" /></a></span> </td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v3.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/b/b5/Counting_rod_v3.png" decoding="async" width="19" height="29" class="mw-file-element" data-file-width="19" data-file-height="29" /></a></span> </td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v4.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/6/67/Counting_rod_v4.png" decoding="async" width="25" height="29" class="mw-file-element" data-file-width="25" data-file-height="29" /></a></span> </td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v5.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/e/e4/Counting_rod_v5.png" decoding="async" width="31" height="29" class="mw-file-element" data-file-width="31" data-file-height="29" /></a></span> </td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v6.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/1/19/Counting_rod_v6.png" decoding="async" width="29" height="32" class="mw-file-element" data-file-width="29" data-file-height="32" /></a></span> </td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v7.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/d/d1/Counting_rod_v7.png" decoding="async" width="29" height="32" class="mw-file-element" data-file-width="29" data-file-height="32" /></a></span> </td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v8.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/6/64/Counting_rod_v8.png" decoding="async" width="29" height="32" class="mw-file-element" data-file-width="29" data-file-height="32" /></a></span> </td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v9.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/7/7a/Counting_rod_v9.png" decoding="async" width="29" height="32" class="mw-file-element" data-file-width="29" data-file-height="32" /></a></span> </td></tr> <tr> <th style="height:50px">横式 </th> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_0.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/5/50/Counting_rod_0.png" decoding="async" width="23" height="23" class="mw-file-element" data-file-width="23" data-file-height="23" /></a></span> </td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_h1.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/1/17/Counting_rod_h1.png" decoding="async" width="29" height="7" class="mw-file-element" data-file-width="29" data-file-height="7" /></a></span> </td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_h2.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/5/51/Counting_rod_h2.png" decoding="async" width="29" height="13" class="mw-file-element" data-file-width="29" data-file-height="13" /></a></span> </td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_h3.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/6/6b/Counting_rod_h3.png" decoding="async" width="29" height="19" class="mw-file-element" data-file-width="29" data-file-height="19" /></a></span> </td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_h4.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/c/ce/Counting_rod_h4.png" decoding="async" width="29" height="25" class="mw-file-element" data-file-width="29" data-file-height="25" /></a></span> </td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_h5.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/7/74/Counting_rod_h5.png" decoding="async" width="29" height="31" class="mw-file-element" data-file-width="29" data-file-height="31" /></a></span> </td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_h6.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/c/c1/Counting_rod_h6.png" decoding="async" width="29" height="31" class="mw-file-element" data-file-width="29" data-file-height="31" /></a></span> </td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_h7_num.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/3/3c/Counting_rod_h7_num.png" decoding="async" width="29" height="31" class="mw-file-element" data-file-width="29" data-file-height="31" /></a></span> </td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_h8_num.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/7/7e/Counting_rod_h8_num.png" decoding="async" width="29" height="31" class="mw-file-element" data-file-width="29" data-file-height="31" /></a></span> </td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_h9_num.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/e/e4/Counting_rod_h9_num.png" decoding="async" width="29" height="31" class="mw-file-element" data-file-width="29" data-file-height="31" /></a></span> </td></tr></tbody></table> <table class="wikitable" border="1" style="text-align:center"> <caption>负数 </caption> <tbody><tr> <th>  </th> <th style="width:50px">-0 </th> <th style="width:50px">-1 </th> <th style="width:50px">-2 </th> <th style="width:50px">-3 </th> <th style="width:50px">-4 </th> <th style="width:50px">-5 </th> <th style="width:50px">-6 </th> <th style="width:50px">-7 </th> <th style="width:50px">-8 </th> <th style="width:50px">-9 </th></tr> <tr> <th style="height:50px">直式 </th> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_-0.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/3/37/Counting_rod_-0.png" decoding="async" width="23" height="23" class="mw-file-element" data-file-width="23" data-file-height="23" /></a></span> </td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v-1.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/6/6f/Counting_rod_v-1.png" decoding="async" width="23" height="29" class="mw-file-element" data-file-width="23" data-file-height="29" /></a></span> </td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v-2.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/6/64/Counting_rod_v-2.png" decoding="async" width="23" height="29" class="mw-file-element" data-file-width="23" data-file-height="29" /></a></span> </td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v-3.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/3/3f/Counting_rod_v-3.png" decoding="async" width="25" height="29" class="mw-file-element" data-file-width="25" data-file-height="29" /></a></span> </td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v-4.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/b/ba/Counting_rod_v-4.png" decoding="async" width="31" height="29" class="mw-file-element" data-file-width="31" data-file-height="29" /></a></span> </td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v-5.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/6/66/Counting_rod_v-5.png" decoding="async" width="35" height="29" class="mw-file-element" data-file-width="35" data-file-height="29" /></a></span> </td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v-6.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/1/11/Counting_rod_v-6.png" decoding="async" width="29" height="32" class="mw-file-element" data-file-width="29" data-file-height="32" /></a></span> </td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v-7.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/0/04/Counting_rod_v-7.png" decoding="async" width="29" height="32" class="mw-file-element" data-file-width="29" data-file-height="32" /></a></span> </td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v-8.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/7/72/Counting_rod_v-8.png" decoding="async" width="29" height="32" class="mw-file-element" data-file-width="29" data-file-height="32" /></a></span> </td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v-9.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/0/0d/Counting_rod_v-9.png" decoding="async" width="31" height="32" class="mw-file-element" data-file-width="31" data-file-height="32" /></a></span> </td></tr></tbody></table> <table class="wikitable" border="1" style="text-align:center"> <caption>南宋正筹码 </caption> <tbody><tr> <th>  </th> <th style="width:50px">0 </th> <th style="width:50px">1 </th> <th style="width:50px">2 </th> <th style="width:50px">3 </th> <th style="width:50px">4 </th> <th style="width:50px">5 </th> <th style="width:50px">6 </th> <th style="width:50px">7 </th> <th style="width:50px">8 </th> <th style="width:50px">9 </th></tr> <tr> <th style="height:50px">直式 </th> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_0.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/5/50/Counting_rod_0.png" decoding="async" width="23" height="23" class="mw-file-element" data-file-width="23" data-file-height="23" /></a></span> </td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v1.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/1/19/Counting_rod_v1.png" decoding="async" width="7" height="29" class="mw-file-element" data-file-width="7" data-file-height="29" /></a></span> </td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v2.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/f/ff/Counting_rod_v2.png" decoding="async" width="13" height="29" class="mw-file-element" data-file-width="13" data-file-height="29" /></a></span> </td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v3.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/b/b5/Counting_rod_v3.png" decoding="async" width="19" height="29" class="mw-file-element" data-file-width="19" data-file-height="29" /></a></span> </td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_4_song.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/1/10/Counting_rod_4_song.png" decoding="async" width="23" height="24" class="mw-file-element" data-file-width="23" data-file-height="24" /></a></span> </td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v5_song.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/5/57/Counting_rod_v5_song.png" decoding="async" width="29" height="29" class="mw-file-element" data-file-width="29" data-file-height="29" /></a></span> </td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v6.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/1/19/Counting_rod_v6.png" decoding="async" width="29" height="32" class="mw-file-element" data-file-width="29" data-file-height="32" /></a></span> </td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v7.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/d/d1/Counting_rod_v7.png" decoding="async" width="29" height="32" class="mw-file-element" data-file-width="29" data-file-height="32" /></a></span> </td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v8.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/6/64/Counting_rod_v8.png" decoding="async" width="29" height="32" class="mw-file-element" data-file-width="29" data-file-height="32" /></a></span> </td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v9_song.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/d/d1/Counting_rod_v9_song.png" decoding="async" width="29" height="29" class="mw-file-element" data-file-width="29" data-file-height="29" /></a></span> </td></tr> <tr> <th style="height:50px">横式 </th> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_0.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/5/50/Counting_rod_0.png" decoding="async" width="23" height="23" class="mw-file-element" data-file-width="23" data-file-height="23" /></a></span> </td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_h1.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/1/17/Counting_rod_h1.png" decoding="async" width="29" height="7" class="mw-file-element" data-file-width="29" data-file-height="7" /></a></span> </td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_h2.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/5/51/Counting_rod_h2.png" decoding="async" width="29" height="13" class="mw-file-element" data-file-width="29" data-file-height="13" /></a></span> </td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_h3.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/6/6b/Counting_rod_h3.png" decoding="async" width="29" height="19" class="mw-file-element" data-file-width="29" data-file-height="19" /></a></span> </td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_4_song.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/1/10/Counting_rod_4_song.png" decoding="async" width="23" height="24" class="mw-file-element" data-file-width="23" data-file-height="24" /></a></span> </td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_h5_song.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/c/c4/Counting_rod_h5_song.png" decoding="async" width="23" height="29" class="mw-file-element" data-file-width="23" data-file-height="29" /></a></span> </td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_h6.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/c/c1/Counting_rod_h6.png" decoding="async" width="29" height="31" class="mw-file-element" data-file-width="29" data-file-height="31" /></a></span> </td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_h7_num.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/3/3c/Counting_rod_h7_num.png" decoding="async" width="29" height="31" class="mw-file-element" data-file-width="29" data-file-height="31" /></a></span> </td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_h8_num.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/7/7e/Counting_rod_h8_num.png" decoding="async" width="29" height="31" class="mw-file-element" data-file-width="29" data-file-height="31" /></a></span> </td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_h9_song.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/f/f2/Counting_rod_h9_song.png" decoding="async" width="23" height="29" class="mw-file-element" data-file-width="23" data-file-height="29" /></a></span> </td></tr></tbody></table> <p>日本数学家<a href="/wiki/%E5%85%B3%E5%AD%9D%E5%92%8C" title="关孝和">关孝和</a>将算筹用于代数。后来日本数学家发明用汉字代替算筹。 </p> <table class="wikitable" border="1" style="text-align:center"> <tbody><tr> <th>现代</th> <th>关孝和</th> <th>后关孝和 </th></tr> <tr> <td><i>x</i> + <i>y</i> + 246 </td> <td><big><b><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v1.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/1/19/Counting_rod_v1.png" decoding="async" width="7" height="29" class="mw-file-element" data-file-width="7" data-file-height="29" /></a></span>甲<span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v1.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/1/19/Counting_rod_v1.png" decoding="async" width="7" height="29" class="mw-file-element" data-file-width="7" data-file-height="29" /></a></span>乙<span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v2.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/f/ff/Counting_rod_v2.png" decoding="async" width="13" height="29" class="mw-file-element" data-file-width="13" data-file-height="29" /></a></span><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_h4.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/c/ce/Counting_rod_h4.png" decoding="async" width="29" height="25" class="mw-file-element" data-file-width="29" data-file-height="25" /></a></span><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v6.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/1/19/Counting_rod_v6.png" decoding="async" width="29" height="32" class="mw-file-element" data-file-width="29" data-file-height="32" /></a></span></b> </big></td> <td><big><b><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v1.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/1/19/Counting_rod_v1.png" decoding="async" width="7" height="29" class="mw-file-element" data-file-width="7" data-file-height="29" /></a></span>甲<span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v1.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/1/19/Counting_rod_v1.png" decoding="async" width="7" height="29" class="mw-file-element" data-file-width="7" data-file-height="29" /></a></span>乙<span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v1.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/1/19/Counting_rod_v1.png" decoding="async" width="7" height="29" class="mw-file-element" data-file-width="7" data-file-height="29" /></a></span>二四六</b> </big></td></tr> <tr> <td>5<i>x</i> - 6<i>y</i> </td> <td><big><b><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v5.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/e/e4/Counting_rod_v5.png" decoding="async" width="31" height="29" class="mw-file-element" data-file-width="31" data-file-height="29" /></a></span>甲<span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v-6.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/1/11/Counting_rod_v-6.png" decoding="async" width="29" height="32" class="mw-file-element" data-file-width="29" data-file-height="32" /></a></span>乙</b> </big></td> <td><big><b><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v1.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/1/19/Counting_rod_v1.png" decoding="async" width="7" height="29" class="mw-file-element" data-file-width="7" data-file-height="29" /></a></span>五甲<span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v-1.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/6/6f/Counting_rod_v-1.png" decoding="async" width="23" height="29" class="mw-file-element" data-file-width="23" data-file-height="29" /></a></span>六乙</b> </big></td></tr> <tr> <td>7<i>xy</i> </td> <td><big><b><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v7.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/d/d1/Counting_rod_v7.png" decoding="async" width="29" height="32" class="mw-file-element" data-file-width="29" data-file-height="32" /></a></span>甲乙</b> </big></td> <td><big><b><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v1.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/1/19/Counting_rod_v1.png" decoding="async" width="7" height="29" class="mw-file-element" data-file-width="7" data-file-height="29" /></a></span>七甲乙</b> </big></td></tr> <tr> <td>8<i>x</i> / <i>y</i> </td> <td>N/A </td> <td><big><b>乙<span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v1.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/1/19/Counting_rod_v1.png" decoding="async" width="7" height="29" class="mw-file-element" data-file-width="7" data-file-height="29" /></a></span>八甲</b></big> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="参考文献"><span id=".E5.8F.82.E8.80.83.E6.96.87.E7.8C.AE"></span>参考文献</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%AE%97%E7%AD%B9&action=edit&section=5" title="编辑章节：参考文献"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></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"><a href="/wiki/%E5%90%B4%E6%96%87%E4%BF%8A" title="吴文俊">吴文俊</a>院士主编《<a href="/wiki/%E4%B8%AD%E5%9B%BD%E6%95%B0%E5%AD%A6%E5%8F%B2%E5%A4%A7%E7%B3%BB" class="mw-disambig" title="中国数学史大系">中国数学史大系</a>》第一卷上古到西汉 164页 <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/7303045554" class="internal mw-magiclink-isbn">ISBN 7-303-04555-4</a>/O</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"><a href="/wiki/%E4%B8%89%E4%B8%8A%E4%B9%89%E5%A4%AB" title="三上义夫">三上义夫</a>《中国算学之特色》第十章《算器算筹》第44-45页商务印书馆《万有文库》#0400 1933</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"><a href="/wiki/%E5%90%B4%E6%96%87%E4%BF%8A" title="吴文俊">吴文俊</a>主编《中国数学史大系》卷一《上古到西汉》第四章《算筹与筹算法》</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">《<a href="/wiki/%E6%B1%89%E4%B9%A6" title="汉书">汉书</a>·律历志》：“其算法用竹”</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">《<a href="/wiki/%E8%B5%84%E6%B2%BB%E9%80%9A%E9%89%B4" class="mw-redirect" title="资治通鉴">资治通鉴</a>》卷八十二“每自执牙筹昼夜会计”</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"><a href="/wiki/%E5%8C%97%E5%AE%8B" title="北宋">北宋</a>陶谷撰《清异录》：“铸铁为算子”。</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><a href="/wiki/%E4%B8%89%E4%B8%8A%E4%B9%89%E5%A4%AB" title="三上义夫">三上义夫</a>《中国算学之特色》第十章算筹四十四页商务印书馆《万有文库》#0400 1933</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><a href="/wiki/%E6%9D%8E%E4%BF%A8_(%E7%8E%B0%E4%BB%A3%E5%AD%A6%E8%80%85)" class="mw-redirect" title="李俨 (现代学者)">李俨</a>《中国算学小史》《李俨.钱宝琮科学史全集》卷1 85页辽宁教育出版社</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text">吴文俊主编中国数学史大系第一卷上古至西汉第一页彩色图板</span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text">吴文俊主编中国数学史大系第一卷325页</span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text">清劳乃宣古筹算考释卷一1886年</span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text">Karl Menninger, Number Words and Number Symbols, p369 MIT Press 1970</span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><a href="/wiki/%E5%90%B4%E6%96%87%E4%BF%8A" title="吴文俊">吴文俊</a>主编《<a href="/wiki/%E4%B8%AD%E5%9B%BD%E6%95%B0%E5%AD%A6%E5%8F%B2%E5%A4%A7%E7%B3%BB" class="mw-disambig" title="中国数学史大系">中国数学史大系</a>》第四卷 43页</span> </li> </ol> </div> <div class="mw-heading mw-heading2"><h2 id="参看"><span id=".E5.8F.82.E7.9C.8B"></span>参看</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%AE%97%E7%AD%B9&action=edit&section=6" title="编辑章节：参看"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/%E7%AD%B9%E7%AE%97" title="筹算">筹算</a></li> <li><a href="/wiki/%E7%AE%97%E7%9B%98" title="算盘">算盘</a></li> <li><a href="/wiki/%E8%8B%8F%E5%B7%9E%E7%A0%81%E5%AD%90" title="苏州码子">苏州码子</a></li> <li><a href="/wiki/%E4%B8%AD%E6%96%87%E6%95%B8%E5%AD%97" class="mw-redirect" title="中文數字">中文數字</a></li> <li><a href="/wiki/%E7%9E%BF%E6%9B%87%E6%82%89%E9%81%94" title="瞿曇悉達">瞿曇悉達</a></li> <li><a href="/wiki/%E5%8D%B0%E5%BA%A6-%E9%98%BF%E6%8B%89%E4%BC%AF%E6%95%B0%E5%AD%97%E7%B3%BB%E7%BB%9F" title="印度-阿拉伯数字系统">印度-阿拉伯數字系統</a></li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84265675" /><style data-mw-deduplicate="TemplateStyles:r84261037">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output 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title="秦九韶算法">秦九韶算法</a></li> <li>《<a href="/wiki/%E6%95%B0%E4%B9%A6%E4%B9%9D%E7%AB%A0" title="数书九章">数书九章</a>》</li> <li><a href="/wiki/%E5%A4%A7%E8%A1%8D%E6%B1%82%E4%B8%80%E6%9C%AF" title="大衍求一术">大衍求一术</a></li></ul></li> <li><a href="/wiki/%E6%9D%A8%E8%BE%89" title="杨辉">杨辉</a> <ul><li><a href="/wiki/%E5%B9%BB%E6%96%B9#杨辉纵横图" title="幻方">杨辉纵横图</a></li> <li><a href="/wiki/%E5%B9%BB%E5%9C%86" title="幻圆">幻圆</a></li> <li>《<a href="/wiki/%E6%9D%A8%E8%BE%89%E7%AE%97%E6%B3%95" title="杨辉算法">杨辉算法</a>》 <ul><li>《<a href="/wiki/%E6%9D%A8%E8%BE%89%E7%AE%97%E6%B3%95#乘除通变算宝" title="杨辉算法">乘除通變算寶</a>》</li> <li>《<a href="/wiki/%E6%9D%A8%E8%BE%89%E7%AE%97%E6%B3%95#田亩比类乘除捷法" title="杨辉算法">田畝比類乘除捷法</a>》</li> <li>《<a href="/wiki/%E6%9D%A8%E8%BE%89%E7%AE%97%E6%B3%95#续古摘奇算法" title="杨辉算法">續古摘奇算法</a>》</li></ul></li> <li>《<a href="/wiki/%E8%A9%B3%E8%A7%A3%E4%B9%9D%E7%AB%A0%E7%AE%97%E6%B3%95" title="詳解九章算法">詳解九章算法</a>》</li> <li>《<a href="/wiki/%E6%97%A5%E7%94%A8%E7%AE%97%E6%B3%95" title="日用算法">日用算法</a>》</li></ul></li> <li>《<a href="/w/index.php?title=%E4%B9%9D%E7%AB%A0%E7%AE%97%E6%B3%95%E7%BA%82%E9%A1%9E&action=edit&redlink=1" class="new" title="九章算法纂類（页面不存在）">九章算法纂類</a>》</li> <li><a href="/w/index.php?title=%E9%80%92%E5%A2%9E%E4%B8%89%E4%B9%98%E5%BC%80%E5%9B%9B%E6%AC%A1%E6%96%B9%E6%9C%AF&action=edit&redlink=1" class="new" title="递增三乘开四次方术（页面不存在）">递增三乘开四次方术</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%E8%BE%BD%E6%9C%9D" title="辽朝">辽</a><a href="/wiki/%E9%87%91%E6%9C%9D" title="金朝">金</a><a href="/wiki/%E5%85%83%E6%9C%9D" title="元朝">元</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E5%BC%A0%E8%A1%8C%E7%AE%80" title="张行简">张行简</a></li> <li><a href="/wiki/%E5%A4%A9%E5%85%83%E6%9C%AF" title="天元术">天元术</a></li> <li><a href="/wiki/%E6%9D%8E%E5%86%B6" title="李冶">李冶</a> <ul><li>《<a 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