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href="https://es.wikipedia.org/wiki/Numeraci%C3%B3n_con_varillas" title="Numeración con varillas – Spanish" lang="es" hreflang="es" data-title="Numeración con varillas" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</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 – French" lang="fr" hreflang="fr" data-title="Baguettes à calculer" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</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 – Korean" lang="ko" hreflang="ko" data-title="산가지" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Tongkat_penghitung" title="Tongkat penghitung – Indonesian" lang="id" hreflang="id" data-title="Tongkat penghitung" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</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 – Hebrew" lang="he" hreflang="he" data-title="מוטות מנייה" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Batang_pembilang" title="Batang pembilang – Malay" lang="ms" hreflang="ms" data-title="Batang pembilang" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</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 – Japanese" lang="ja" hreflang="ja" data-title="算木" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Patyczki_liczbowe" title="Patyczki liczbowe – Polish" lang="pl" hreflang="pl" data-title="Patyczki liczbowe" data-language-autonym="Polski" data-language-local-name="Polish" 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 – Portuguese" lang="pt" hreflang="pt" data-title="Sistema numérico de varas" data-language-autonym="Português" data-language-local-name="Portuguese" 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 – Russian" lang="ru" hreflang="ru" data-title="Счётные палочки" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Puikkolaskenta" title="Puikkolaskenta – Finnish" lang="fi" hreflang="fi" data-title="Puikkolaskenta" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</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 – Ukrainian" lang="uk" hreflang="uk" data-title="Рахункові палички" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</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 – Literary Chinese" lang="lzh" hreflang="lzh" data-title="算籌" data-language-autonym="文言" data-language-local-name="Literary Chinese" class="interlanguage-link-target"><span>文言</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 – Wu" lang="wuu" hreflang="wuu" data-title="算筹" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target"><span>吴语</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 – Cantonese" lang="yue" 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.mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330" /><table class="sidebar sidebar-collapse nomobile nowraplinks hlist"><tbody><tr><td class="sidebar-pretitle">Part of <a href="/wiki/Category:Numeral_systems" title="Category:Numeral systems">a series</a> on</td></tr><tr><th class="sidebar-title-with-pretitle"><a href="/wiki/Numeral_system" title="Numeral system">Numeral systems</a></th></tr><tr><td class="sidebar-content-with-subgroup"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c"><a href="/wiki/Positional_notation" title="Positional notation">Place-value notation</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" style="color: var(--color-base)"><div class="sidebar-list-title-c"><a href="/wiki/Hindu%E2%80%93Arabic_numeral_system" title="Hindu–Arabic numeral system">Hindu–Arabic numerals</a></div></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Arabic_numerals" title="Arabic numerals">Western Arabic</a></li> <li><a href="/wiki/Eastern_Arabic_numerals" title="Eastern Arabic numerals">Eastern Arabic</a></li></ul> <hr /> <ul><li><a href="/wiki/Bengali_numerals" title="Bengali numerals">Bengali</a></li> <li><a href="/wiki/Devanagari_numerals" title="Devanagari numerals">Devanagari</a></li> <li><a href="/wiki/Gujarati_numerals" title="Gujarati numerals">Gujarati</a></li> <li><a href="/wiki/Gurmukhi_numerals" class="mw-redirect" title="Gurmukhi numerals">Gurmukhi</a></li> <li><a href="/wiki/Odia_numerals" title="Odia numerals">Odia</a></li> <li><a href="/wiki/Sinhala_numerals" title="Sinhala numerals">Sinhala</a></li> <li><a href="/wiki/Tamil_numerals" title="Tamil numerals">Tamil</a></li> <li><a href="/wiki/Malayalam_numerals" title="Malayalam numerals">Malayalam</a></li> <li><a href="/wiki/Telugu_script#Numerals" title="Telugu script">Telugu</a></li> <li><a href="/wiki/Kannada_script#Numerals" title="Kannada script">Kannada</a></li> <li><a href="/wiki/Dzongkha_numerals" title="Dzongkha numerals">Dzongkha</a></li></ul> <hr /> <ul><li><a href="/wiki/Tibetan_numerals" title="Tibetan numerals">Tibetan</a></li> <li><a href="/wiki/Balinese_numerals" title="Balinese numerals">Balinese</a></li> <li><a href="/wiki/Burmese_numerals" title="Burmese numerals">Burmese</a></li> <li><a href="/wiki/Javanese_numerals" title="Javanese numerals">Javanese</a></li> <li><a href="/wiki/Khmer_numerals" title="Khmer numerals">Khmer</a></li> <li><a href="/wiki/Lao_script#Numerals" title="Lao script">Lao</a></li> <li><a href="/wiki/Mongolian_numerals" title="Mongolian numerals">Mongolian</a></li> <li><a href="/wiki/Sundanese_numerals" title="Sundanese numerals">Sundanese</a></li> <li><a href="/wiki/Thai_numerals" title="Thai numerals">Thai</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" style="color: var(--color-base)"><div class="sidebar-list-title-c">East Asian systems</div></div><div class="sidebar-list-content mw-collapsible-content"> <dl><dt>Contemporary</dt></dl> <ul><li><a href="/wiki/Chinese_numerals" title="Chinese numerals">Chinese</a> <ul><li><a href="/wiki/Hokkien_numerals" title="Hokkien numerals">Hokkien</a></li> <li><a href="/wiki/Suzhou_numerals" title="Suzhou numerals">Suzhou</a></li></ul></li> <li><a href="/wiki/Japanese_numerals" title="Japanese numerals">Japanese</a></li> <li><a href="/wiki/Korean_numerals" title="Korean numerals">Korean</a></li> <li><a href="/wiki/Vietnamese_numerals" title="Vietnamese numerals">Vietnamese</a></li></ul> <hr /> <dl><dt>Historic</dt></dl> <ul><li><a class="mw-selflink selflink">Counting rods</a></li> <li><a href="/wiki/Tangut_numerals" title="Tangut numerals">Tangut</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" style="color: var(--color-base)"><div class="sidebar-list-title-c">Other systems</div></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/History_of_ancient_numeral_systems" title="History of ancient numeral systems">History</a></li></ul> <hr /> <dl><dt><a href="/wiki/Ancient_history" title="Ancient history">Ancient</a></dt></dl> <ul><li><a href="/wiki/Babylonian_cuneiform_numerals" title="Babylonian cuneiform numerals">Babylonian</a></li></ul> <hr /> <dl><dt><a href="/wiki/Post-classical_history" title="Post-classical history">Post-classical</a></dt></dl> <ul><li><a href="/wiki/Cistercian_numerals" title="Cistercian numerals">Cistercian</a></li> <li><a href="/wiki/Maya_numerals" title="Maya numerals">Mayan</a></li> <li><a href="/wiki/Muisca_numerals" title="Muisca numerals">Muisca</a></li> <li><a href="/wiki/Pentadic_numerals" title="Pentadic numerals">Pentadic</a></li> <li><a href="/wiki/Quipu" title="Quipu">Quipu</a></li> <li><a href="/wiki/Rumi_Numeral_Symbols" title="Rumi Numeral Symbols">Rumi</a></li></ul> <hr /> <dl><dt>Contemporary</dt></dl> <ul><li><a href="/wiki/Cherokee_syllabary#Numerals" title="Cherokee syllabary">Cherokee</a></li> <li><a href="/wiki/Kaktovik_numerals" title="Kaktovik numerals">Kaktovik</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" style="color: var(--color-base)"><div class="sidebar-list-title-c">By <a href="/wiki/Radix" title="Radix">radix/base</a></div></div><div class="sidebar-list-content mw-collapsible-content"> <dl><dt>Common radices/bases</dt></dl> <ul><li><a href="/wiki/Binary_number" title="Binary number">2</a></li> <li><a href="/wiki/Ternary_numeral_system" title="Ternary numeral system">3</a></li> <li><a href="/wiki/Quaternary_numeral_system" title="Quaternary numeral system">4</a></li> <li><a href="/wiki/Quinary" title="Quinary">5</a></li> <li><a href="/wiki/Senary" title="Senary">6</a></li> <li><a href="/wiki/Octal" title="Octal">8</a></li> <li><a href="/wiki/Decimal" title="Decimal">10</a></li> <li><a href="/wiki/Duodecimal" title="Duodecimal">12</a></li> <li><a href="/wiki/Hexadecimal" title="Hexadecimal">16</a></li> <li><a href="/wiki/Vigesimal" title="Vigesimal">20</a></li> <li><a href="/wiki/Sexagesimal" title="Sexagesimal">60</a></li></ul> <hr /> <dl><dt><a href="/wiki/Non-standard_positional_numeral_systems" title="Non-standard positional numeral systems">Non-standard radices/bases</a></dt></dl> <ul><li><a href="/wiki/Bijective_numeration" title="Bijective numeration">Bijective</a><span class="nowrap"> </span>(<a href="/wiki/Unary_numeral_system" title="Unary numeral system">1</a>)</li> <li><a href="/wiki/Signed-digit_representation" title="Signed-digit representation">Signed-digit</a><span class="nowrap"> </span>(<a href="/wiki/Balanced_ternary" title="Balanced ternary">balanced ternary</a>)</li> <li><a href="/wiki/Mixed_radix" title="Mixed radix">Mixed</a><span class="nowrap"> </span>(<a href="/wiki/Factorial_number_system" title="Factorial number system">factorial</a>)</li> <li><a href="/wiki/Negative_base" title="Negative base">Negative</a></li> <li><a href="/wiki/Complex-base_system" title="Complex-base system">Complex</a><span class="nowrap"> </span>(<a href="/wiki/Quater-imaginary_base" title="Quater-imaginary base">2<i>i</i></a>)</li> <li><a href="/wiki/Non-integer_base_of_numeration" title="Non-integer base of numeration">Non-integer</a><span class="nowrap"> </span>(<a href="/wiki/Golden_ratio_base" title="Golden ratio base">φ</a>)</li> <li><a href="/wiki/Asymmetric_numeral_systems" title="Asymmetric numeral systems">Asymmetric</a></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" style="color: var(--color-base)"><div class="sidebar-list-title-c"><a href="/wiki/Sign-value_notation" title="Sign-value notation">Sign-value notation</a></div></div><div class="sidebar-list-content mw-collapsible-content"> <dl><dt>Non-alphabetic</dt></dl> <ul><li><a href="/wiki/Aegean_numerals" title="Aegean numerals">Aegean</a></li> <li><a href="/wiki/Attic_numerals" title="Attic numerals">Attic</a></li> <li><a href="/wiki/Aztec_script#Numerals" title="Aztec script">Aztec</a></li> <li><a href="/wiki/Brahmi_numerals" title="Brahmi numerals">Brahmi</a></li> <li><a href="/wiki/Chuvash_numerals" title="Chuvash numerals">Chuvash</a></li> <li><a href="/wiki/Egyptian_numerals" title="Egyptian numerals">Egyptian</a></li> <li><a href="/wiki/Etruscan_numerals" title="Etruscan numerals">Etruscan</a></li> <li><a href="/wiki/Kharosthi_numerals" class="mw-redirect" title="Kharosthi numerals">Kharosthi</a></li> <li><a href="/wiki/Prehistoric_counting" class="mw-redirect" title="Prehistoric counting">Prehistoric counting</a></li> <li><a href="/wiki/Proto-cuneiform" title="Proto-cuneiform">Proto-cuneiform</a></li> <li><a href="/wiki/Roman_numerals" title="Roman numerals">Roman</a></li> <li><a href="/wiki/Tally_marks" title="Tally marks">Tally marks</a></li></ul> <hr /> <dl><dt><a href="/wiki/Alphabetic_numeral_system" title="Alphabetic numeral system">Alphabetic</a></dt></dl> <ul><li><a href="/wiki/Abjad_numerals" title="Abjad numerals">Abjad</a></li> <li><a href="/wiki/Armenian_numerals" title="Armenian numerals">Armenian</a></li> <li><a href="/wiki/Alphasyllabic_numeral_system" title="Alphasyllabic numeral system">Alphasyllabic</a> <ul><li><a href="/wiki/Aksharapalli" title="Aksharapalli">Akṣarapallī</a></li> <li><a href="/wiki/%C4%80ryabha%E1%B9%ADa_numeration" title="Āryabhaṭa numeration">Āryabhaṭa</a></li> <li><a href="/wiki/Katapayadi_system" title="Katapayadi system">Kaṭapayādi</a></li></ul></li> <li><a href="/wiki/Coptic_numerals" class="mw-redirect" title="Coptic numerals">Coptic</a></li> <li><a href="/wiki/Cyrillic_numerals" title="Cyrillic numerals">Cyrillic</a></li> <li><a href="/wiki/Ge%CA%BDez_script#Numerals" title="Geʽez script">Geʽez</a></li> <li><a href="/wiki/Georgian_numerals" title="Georgian numerals">Georgian</a></li> <li><a href="/wiki/Glagolitic_numerals" title="Glagolitic numerals">Glagolitic</a></li> <li><a href="/wiki/Greek_numerals" title="Greek numerals">Greek</a></li> <li><a href="/wiki/Hebrew_numerals" title="Hebrew numerals">Hebrew</a></li></ul></div></div></td> </tr><tr><td class="sidebar-below" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"> <a href="/wiki/List_of_numeral_systems" title="List of numeral systems">List of numeral systems</a></td></tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;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:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}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:Numeral_systems" title="Template:Numeral systems"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Numeral_systems" title="Template talk:Numeral systems"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Numeral_systems" title="Special:EditPage/Template:Numeral systems"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Yanghui_triangle.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ea/Yanghui_triangle.gif/250px-Yanghui_triangle.gif" decoding="async" width="200" height="311" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ea/Yanghui_triangle.gif/330px-Yanghui_triangle.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ea/Yanghui_triangle.gif/500px-Yanghui_triangle.gif 2x" data-file-width="704" data-file-height="1095" /></a><figcaption><a href="/wiki/Yang_Hui" title="Yang Hui">Yang Hui</a> (Pascal's) <a href="/wiki/Pascal%27s_triangle" title="Pascal's triangle">triangle</a>, as depicted by <a href="/wiki/Zhu_Shijie" title="Zhu Shijie">Zhu Shijie</a> in 1303, using rod numerals</figcaption></figure><style data-mw-deduplicate="TemplateStyles:r1257001546">.mw-parser-output .infobox-subbox{padding:0;border:none;margin:-3px;width:auto;min-width:100%;font-size:100%;clear:none;float:none;background-color:transparent}.mw-parser-output .infobox-3cols-child{margin:auto}.mw-parser-output .infobox .navbar{font-size:100%}@media screen{html.skin-theme-clientpref-night .mw-parser-output .infobox-full-data:not(.notheme)>div:not(.notheme)[style]{background:#1f1f23!important;color:#f8f9fa}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .infobox-full-data:not(.notheme) div:not(.notheme){background:#1f1f23!important;color:#f8f9fa}}@media(min-width:640px){body.skin--responsive .mw-parser-output .infobox-table{display:table!important}body.skin--responsive .mw-parser-output .infobox-table>caption{display:table-caption!important}body.skin--responsive .mw-parser-output .infobox-table>tbody{display:table-row-group}body.skin--responsive .mw-parser-output .infobox-table tr{display:table-row!important}body.skin--responsive .mw-parser-output .infobox-table th,body.skin--responsive .mw-parser-output .infobox-table td{padding-left:inherit;padding-right:inherit}}</style><table class="infobox"><tbody><tr><th colspan="2" class="infobox-above" style="color: #202122;background-color:#b0c4de">Counting rods</th></tr><tr style="display:none;"><td colspan="2" class="infobox-full-data"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1257001546" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1257001546" /></td></tr><tr><th colspan="2" class="infobox-header" style="color: #202122;background-color: #b0c4de;">Chinese name</th></tr><tr><th scope="row" class="infobox-label" style="font-weight:normal;"><a href="/wiki/Traditional_Chinese_characters" title="Traditional Chinese characters">Traditional Chinese</a></th><td class="infobox-data"><span title="Chinese-language text"><span lang="zh-Hant" style="font-size: 1rem;">算籌</span></span></td></tr><tr><th scope="row" class="infobox-label" style="font-weight:normal;"><a href="/wiki/Simplified_Chinese_characters" title="Simplified Chinese characters">Simplified Chinese</a></th><td class="infobox-data"><span title="Chinese-language text"><span lang="zh-Hans" style="font-size: 1rem;">算筹</span></span></td></tr><tr><td colspan="2" class="infobox-full-data"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1257001546" /><table class="infobox-subbox mw-collapsible mw-collapsed" style="display:inline-table; text-align: left;"><tbody><tr><th colspan="2" class="infobox-above" style="font-size: 100%; text-align: left;color: #202122; background-color: #f9ffbc;">Transcriptions</th></tr><tr><th colspan="2" class="infobox-header" style="color: #202122;background-color: #dcffc9;"><a href="/wiki/Standard_Chinese" title="Standard Chinese">Standard Mandarin</a></th></tr><tr><th scope="row" class="infobox-label" style="font-weight:normal;"><a href="/wiki/Hanyu_Pinyin" class="mw-redirect" title="Hanyu Pinyin">Hanyu Pinyin</a></th><td class="infobox-data"><span title="Chinese-language romanization"><span style="font-style: normal" lang="zh-Latn">suànchóu</span></span></td></tr><tr><th scope="row" class="infobox-label" style="font-weight:normal;"><a href="/wiki/Help:IPA/Mandarin" title="Help:IPA/Mandarin">IPA</a></th><td class="infobox-data">su̯än⁵¹ ʈ͡ʂʰoʊ̯³⁵</td></tr></tbody></table></td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1257001546" /></tr><tr><th colspan="2" class="infobox-header" style="color: #202122;background-color: #b0c4de;">Alternative Chinese name</th></tr><tr><th scope="row" class="infobox-label" style="font-weight:normal;"><a href="/wiki/Chinese_language" title="Chinese language">Chinese</a></th><td class="infobox-data"><span title="Chinese-language text"><span lang="zh-Hani" style="font-size: 1rem;">算子</span></span></td></tr><tr><td colspan="2" class="infobox-full-data"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1257001546" /><table class="infobox-subbox mw-collapsible mw-collapsed" style="display:inline-table; text-align: left;"><tbody><tr><th colspan="2" class="infobox-above" style="font-size: 100%; text-align: left;color: #202122; background-color: #f9ffbc;">Transcriptions</th></tr><tr><th colspan="2" class="infobox-header" style="color: #202122;background-color: #dcffc9;"><a href="/wiki/Standard_Chinese" title="Standard Chinese">Standard Mandarin</a></th></tr><tr><th scope="row" class="infobox-label" style="font-weight:normal;"><a href="/wiki/Hanyu_Pinyin" class="mw-redirect" title="Hanyu Pinyin">Hanyu Pinyin</a></th><td class="infobox-data"><span title="Chinese-language romanization"><span style="font-style: normal" lang="zh-Latn">suànzǐ</span></span></td></tr><tr><th scope="row" class="infobox-label" style="font-weight:normal;"><a href="/wiki/Help:IPA/Mandarin" title="Help:IPA/Mandarin">IPA</a></th><td class="infobox-data">su̯än⁵¹ t͡sz̩²¹⁴⁻²¹⁽⁴⁾</td></tr></tbody></table></td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1257001546" /></tr><tr><th colspan="2" class="infobox-header" style="color: #202122;background-color: #b0c4de;">Vietnamese name</th></tr><tr><th scope="row" class="infobox-label" style="font-weight:normal;"><a href="/wiki/Vietnamese_alphabet" title="Vietnamese alphabet">Vietnamese alphabet</a></th><td class="infobox-data"><span title="Vietnamese-language text"><i lang="vi">que tính / toán trù</i></span></td></tr><tr><th scope="row" class="infobox-label" style="font-weight:normal;"><a href="/wiki/H%C3%A1n-N%C3%B4m" class="mw-redirect" title="Hán-Nôm">Hán-Nôm</a></th><td class="infobox-data"><span title="Vietnamese-language text"><span lang="vi-Hani">𣠗併 / 算籌</span></span></td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1257001546" /></tr><tr><th colspan="2" class="infobox-header" style="color: #202122;background-color: #b0c4de;">Korean name</th></tr><tr><th scope="row" class="infobox-label" style="font-weight:normal;"><a href="/wiki/Hangul" title="Hangul">Hangul</a></th><td class="infobox-data"><span title="Korean-language text"><span lang="ko" style="font-size: 1rem;">산가지 / 산목</span></span></td></tr><tr><th scope="row" class="infobox-label" style="font-weight:normal;"><a href="/wiki/Hanja" title="Hanja">Hanja</a></th><td class="infobox-data"><span title="Korean-language text"><span lang="ko" style="font-size: 1rem;">算가지 / 算木</span></span></td></tr><tr><td colspan="2" class="infobox-full-data"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1257001546" /><table class="infobox-subbox mw-collapsible mw-collapsed" style="display:inline-table; text-align: left;"><tbody><tr><th colspan="2" class="infobox-above" style="font-size: 100%; text-align: left;color: #202122; background-color: #f9ffbc;">Transcriptions</th></tr><tr><th scope="row" class="infobox-label" style="font-weight:normal;"><a href="/wiki/Revised_Romanization_of_Korean" title="Revised Romanization of Korean">Revised Romanization</a></th><td class="infobox-data"><span title="Korean-language romanization"><span style="font-style: normal" lang="ko-Latn">sangaji / sanmok</span></span></td></tr></tbody></table></td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1257001546" /></tr><tr><th colspan="2" class="infobox-header" style="color: #202122;background-color: #b0c4de;">Japanese name</th></tr><tr><th scope="row" class="infobox-label" style="font-weight:normal;"><a href="/wiki/Kanji" title="Kanji">Kanji</a></th><td class="infobox-data"><span title="Japanese-language text"><span lang="ja">算木 / 算籌</span></span></td></tr><tr><th scope="row" class="infobox-label" style="font-weight:normal;"><a href="/wiki/Hiragana" title="Hiragana">Hiragana</a></th><td class="infobox-data"><span title="Japanese-language text"><span lang="ja">さんぎ / さんちゅう</span></span></td></tr><tr><td colspan="2" class="infobox-full-data"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1257001546" /><table class="infobox-subbox mw-collapsible mw-collapsed" style="display:inline-table; text-align: left;"><tbody><tr><th colspan="2" class="infobox-above" style="font-size: 100%; text-align: left;color: #202122; background-color: #f9ffbc;">Transcriptions</th></tr><tr><th scope="row" class="infobox-label" style="font-weight:normal;"><a href="/wiki/Romanization_of_Japanese" title="Romanization of Japanese">Romanization</a></th><td class="infobox-data"><span title="Japanese-language romanization"><span style="font-style: normal" lang="ja-Latn">sangi / sanchū</span></span></td></tr></tbody></table></td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1257001546" /></tr><tr style="display:none"><td colspan="2"> </td></tr></tbody></table> <p><b>Counting rods</b> (筭) are small bars, typically 3–14 cm (1" to 6") long, that were used by mathematicians for calculation in ancient <a href="/wiki/East_Asia" title="East Asia">East Asia</a>. They are placed either horizontally or vertically to represent any <a href="/wiki/Integer" title="Integer">integer</a> or <a href="/wiki/Rational_number" title="Rational number">rational number</a>. </p><p>The written forms based on them are called <b>rod numerals</b>. They are a true <a href="/wiki/Positional_notation" title="Positional notation">positional numeral system</a> with <a href="/wiki/Numerical_digit" title="Numerical digit">digits</a> for 1–9 and a blank for 0, from the <a href="/wiki/Warring_States_period" title="Warring States period">Warring states</a> period (circa 475 BCE)<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> to the 16th century. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Counting_rods&action=edit&section=1" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Chinese arithmeticians used counting rods well over two thousand years ago. </p><p>In 1954, forty-odd counting rods of the <a href="/wiki/Warring_States_period" title="Warring States period">Warring States period</a> (5th century BCE to 221 BCE) were found in <a href="/w/index.php?title=Zu%C7%92ji%C4%81g%C5%8Dngsh%C4%81n&action=edit&redlink=1" class="new" title="Zuǒjiāgōngshān (page does not exist)">Zuǒjiāgōngshān</a> (左家公山) <a href="/wiki/Chu_(state)" title="Chu (state)">Chu</a> Grave No.15 in <a href="/wiki/Changsha" title="Changsha">Changsha</a>, <a href="/wiki/Hunan" title="Hunan">Hunan</a>.<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><span class="failed-verification-content" style="padding-left:0.1em; padding-right:0.1em; color:var(--color-subtle, #54595d); border:1px solid var(--border-color-subtle, #c8ccd1);"><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></span><sup class="noprint Inline-Template Template-Fact" style="margin-left:0.1em; white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability"><span title="The material highlighted is not verifiable using the citation(s) presented. (March 2018)">failed verification</span></a></i>]</sup> </p><p>In 1973, archeologists unearthed a number of wood scripts from a tomb in Hubei dating from the period of the <a href="/wiki/Han_dynasty" title="Han dynasty">Han dynasty</a> (206 BCE to 220 CE). On one of the wooden scripts was written: "当利二月定算𝍥".<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (March 2018)">citation needed</span></a></i>]</sup> This is one of the earliest examples of using counting-rod numerals in writing. </p><p>A square lacquer box, dating from c. 168 BCE, containing a square chess board with the TLV patterns, chessmen, counting rods, and other items, was excavated in 1972, from <a href="/wiki/Mawangdui" title="Mawangdui">Mawangdui</a> M3, Changsha, Hunan Province.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>In 1976, a bundle of <a href="/wiki/Western_Han" class="mw-redirect" title="Western Han">Western Han</a>-era (202 BCE to 9 CE) counting rods made of bones was unearthed from <a href="/wiki/Qianyang_County" title="Qianyang County">Qianyang County</a> in <a href="/wiki/Shaanxi" title="Shaanxi">Shaanxi</a>.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><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> The use of counting rods must predate it; <a href="/wiki/Sun_Tzu" title="Sun Tzu">Sunzi</a> (<abbr title="circa">c.</abbr> 544 to <abbr title="circa">c.</abbr> 496 BCE), a military strategist at the end of <a href="/wiki/Spring_and_Autumn_period" title="Spring and Autumn period">Spring and Autumn period</a> of 771 BCE to 5th century BCE, mentions their use to make calculations to win wars before going into the battle;<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> <a href="/wiki/Laozi" title="Laozi">Laozi</a> (died 531 BCE), writing in the Warring States period, said "a good calculator doesn't use counting rods".<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> The <i><a href="/wiki/Book_of_Han" title="Book of Han">Book of Han</a></i> (finished 111 CE) recorded: "they calculate with bamboo, diameter one fen, length six cun, arranged into a hexagonal bundle of two hundred seventy one pieces".<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p><p>At first, calculating rods were round in cross-section, but by the time of the <a href="/wiki/Sui_dynasty" title="Sui dynasty">Sui dynasty</a> (581 to 618 CE) mathematicians used triangular rods to represent positive numbers and rectangular rods for <a href="/wiki/Negative_number" title="Negative number">negative numbers</a>.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (March 2018)">citation needed</span></a></i>]</sup> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:To%C3%A1n_tr%C3%B9_%E7%AE%97%E7%B1%8C.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f4/To%C3%A1n_tr%C3%B9_%E7%AE%97%E7%B1%8C.jpg/220px-To%C3%A1n_tr%C3%B9_%E7%AE%97%E7%B1%8C.jpg" decoding="async" width="220" height="153" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f4/To%C3%A1n_tr%C3%B9_%E7%AE%97%E7%B1%8C.jpg/330px-To%C3%A1n_tr%C3%B9_%E7%AE%97%E7%B1%8C.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f4/To%C3%A1n_tr%C3%B9_%E7%AE%97%E7%B1%8C.jpg/440px-To%C3%A1n_tr%C3%B9_%E7%AE%97%E7%B1%8C.jpg 2x" data-file-width="2000" data-file-height="1389" /></a><figcaption>Toán trù 算籌 (counting rods) in a Vietnamese mathematics textbook, Cửu chương lập thành toán pháp 九章立成算法 is shown at the bottom of the page.</figcaption></figure> <p>After the <a href="/wiki/Abacus" title="Abacus">abacus</a> flourished<sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Manual_of_Style/Dates_and_numbers#Chronological_items" title="Wikipedia:Manual of Style/Dates and numbers"><span title="The time period mentioned near this tag is ambiguous. (March 2018)">when?</span></a></i>]</sup>, counting rods were abandoned except in Japan, where rod numerals developed into a symbolic notation for <a href="/wiki/Algebra" title="Algebra">algebra</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Using_counting_rods">Using counting rods</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Counting_rods&action=edit&section=2" title="Edit section: Using counting rods"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" 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/220px-Rod_numeral_positioning.JPG" decoding="async" width="220" height="194" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0c/Rod_numeral_positioning.JPG/330px-Rod_numeral_positioning.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0c/Rod_numeral_positioning.JPG/440px-Rod_numeral_positioning.JPG 2x" data-file-width="556" data-file-height="490" /></a><figcaption>Rod numeral place value from <a href="/wiki/Yongle_Encyclopedia" title="Yongle Encyclopedia">Yongle Encyclopedia</a>: 71,824</figcaption></figure> <figure class="mw-default-size" 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/220px-Counting_board.jpg" decoding="async" width="220" height="98" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/47/Counting_board.jpg/330px-Counting_board.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/47/Counting_board.jpg/440px-Counting_board.jpg 2x" data-file-width="450" data-file-height="201" /></a><figcaption>Japanese counting board with grids</figcaption></figure> <figure class="mw-default-size" 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="260" height="264" 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>A checker counting board diagram in an 18th-century Japanese mathematics textbook</figcaption></figure> <figure class="mw-default-size" 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/260px-Seki_Kowa_Katsuyo_Sampo_Bernoulli_numbers.png" decoding="async" width="260" height="342" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Seki_Kowa_Katsuyo_Sampo_Bernoulli_numbers.png/390px-Seki_Kowa_Katsuyo_Sampo_Bernoulli_numbers.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Seki_Kowa_Katsuyo_Sampo_Bernoulli_numbers.png/520px-Seki_Kowa_Katsuyo_Sampo_Bernoulli_numbers.png 2x" data-file-width="1614" data-file-height="2124" /></a><figcaption>Counting rod numerals in grids in a Japanese mathematic book</figcaption></figure> <p>Counting rods represent digits by the number of rods, and the <a href="/wiki/Perpendicular" title="Perpendicular">perpendicular</a> rod represents five. To avoid confusion, vertical and horizontal forms are alternately used. Generally, vertical rod numbers are used for the position for the units, hundreds, ten thousands, etc., while horizontal rod numbers are used for the tens, thousands, hundred thousands etc. It is written in <i><a href="/wiki/Sunzi_Suanjing" title="Sunzi Suanjing">Sunzi Suanjing</a></i> that "one is vertical, ten is horizontal".<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> </p><p>Red rods represent <a href="/wiki/Positive_number" class="mw-redirect" title="Positive number">positive numbers</a> and black rods represent <a href="/wiki/Negative_number" title="Negative number">negative numbers</a>.<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> Ancient Chinese clearly understood negative numbers and zero (leaving a blank space for it), though they had no symbol for the latter. <a href="/wiki/The_Nine_Chapters_on_the_Mathematical_Art" title="The Nine Chapters on the Mathematical Art">The Nine Chapters on the Mathematical Art</a>, which was mainly composed in the first century CE, stated "(when using subtraction) subtract same signed numbers, add different signed numbers, subtract a positive number from zero to make a negative number, and subtract a negative number from zero to make a positive number".<sup id="cite_ref-wang_13-0" class="reference"><a href="#cite_note-wang-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> Later, a <a href="/wiki/Go_(game)" title="Go (game)">go</a> stone was sometimes used to represent zero. </p><p>This alternation of vertical and horizontal rod numeral form is very important to understanding written transcription of rod numerals on manuscripts correctly. For instance, in Licheng suanjin, 81 was transcribed as <span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_h8.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/e/e9/Counting_rod_h8.png" decoding="async" width="29" height="43" class="mw-file-element" data-file-width="29" data-file-height="43" /></a></span><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>, and 108 was transcribed as <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><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>; it is clear that the latter clearly had a blank zero on the "counting board" (i.e., floor or mat), even though on the written transcription, there was no blank. In the same manuscript, 405 was transcribed as <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><span class="nowrap"> </span><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>, with a blank space in between for obvious reasons, and could in no way be interpreted as "45"<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_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>. In other words, transcribed rod numerals may not be positional, but on the counting board, they are positional. <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><span class="nowrap"> </span><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> is an exact image of the counting rod number 405 on a table top or floor. </p> <div class="mw-heading mw-heading3"><h3 id="Place_value">Place value</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Counting_rods&action=edit&section=3" title="Edit section: Place value"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The value of a number depends on its physical position on the counting board. A 9 at the rightmost position on the board stands for 9. Moving the batch of rods representing 9 to the left one position (i.e., to the tens place) gives 9[] or 90. Shifting left again to the third position (to the hundreds place) gives 9[][] or 900. Each time one shifts a number one position to the left, it is multiplied by 10. Each time one shifts a number one position to the right, it is divided by 10. This applies to single-digit numbers or multiple-digit numbers. </p><p>Song dynasty mathematician <a href="/wiki/Jia_Xian" title="Jia Xian">Jia Xian</a> used hand-written Chinese decimal orders 步十百千萬 as rod numeral place value, as evident from a facsimile from a page of <a href="/wiki/Yongle_Encyclopedia" title="Yongle Encyclopedia">Yongle Encyclopedia</a>. He arranged 七萬一千八百二十四 as </p> <dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd><dl><dd>七一八二四</dd> <dd>萬千百十步</dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl> <p>He treated the Chinese order numbers as place value markers, and 七一八二四 became place value decimal number. He then wrote the rod numerals according to their place value: </p> <table border="0" width="500" align="center" 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>萬 </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>In Japan, mathematicians put counting rods on a counting board, a sheet of cloth with grids, and used only vertical forms relying on the grids. An 18th-century Japanese mathematics book has a checker counting board diagram, with the order of magnitude symbols "千百十一分厘毛" (thousand, hundred, ten, unit, tenth, hundredth, thousandth).<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p> <table class="wikitable" border="1" style="text-align:center"> <caption>Zero and positive integers </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">Vertical </th> <td>  </td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v1r.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/Counting_rod_v1r.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_v2r.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/1/16/Counting_rod_v2r.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_v3r.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/2/2d/Counting_rod_v3r.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_v4r.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/3/3a/Counting_rod_v4r.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_v5r.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/8/88/Counting_rod_v5r.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_v6r.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/2/20/Counting_rod_v6r.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_v7r.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/4/49/Counting_rod_v7r.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_v8r.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/d/df/Counting_rod_v8r.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_v9r.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/2/2a/Counting_rod_v9r.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">Horizontal </th> <td>  </td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_h1r.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/e/e8/Counting_rod_h1r.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_h2r.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/9/9a/Counting_rod_h2r.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_h3r.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/7/7f/Counting_rod_h3r.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_h4r.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/2/2d/Counting_rod_h4r.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_h5r.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/7/73/Counting_rod_h5r.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_h6r.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/2/2c/Counting_rod_h6r.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_h7r.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/f/fe/Counting_rod_h7r.png" decoding="async" width="29" height="37" class="mw-file-element" data-file-width="29" data-file-height="37" /></a></span> </td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_h8r.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/e/e4/Counting_rod_h8r.png" decoding="async" width="29" height="43" class="mw-file-element" data-file-width="29" data-file-height="43" /></a></span> </td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_h9r.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/a/ac/Counting_rod_h9r.png" decoding="async" width="29" height="47" class="mw-file-element" data-file-width="29" data-file-height="47" /></a></span> </td></tr></tbody></table> <table class="wikitable" border="1" style="text-align:center"> <caption>Zero and negative numbers </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">Vertical </th> <td>  </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">Horizontal </th> <td>  </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.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/5/5d/Counting_rod_h7.png" decoding="async" width="29" height="37" class="mw-file-element" data-file-width="29" data-file-height="37" /></a></span> </td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_h8.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/e/e9/Counting_rod_h8.png" decoding="async" width="29" height="43" class="mw-file-element" data-file-width="29" data-file-height="43" /></a></span> </td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_h9.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/f/ff/Counting_rod_h9.png" decoding="async" width="29" height="47" class="mw-file-element" data-file-width="29" data-file-height="47" /></a></span> </td></tr></tbody></table> <p>Examples: </p> <table class="wikitable" border="1" style="text-align:center"> <tbody><tr> <th style="text-align:right;height:50px">231 </th> <td style="width:50px">  </td> <td style="width:50px"><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v2r.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/1/16/Counting_rod_v2r.png" decoding="async" width="13" height="29" class="mw-file-element" data-file-width="13" data-file-height="29" /></a></span> </td> <td style="width:50px"><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_h3r.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/7/7f/Counting_rod_h3r.png" decoding="async" width="29" height="19" class="mw-file-element" data-file-width="29" data-file-height="19" /></a></span> </td> <td style="width:50px"><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v1r.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/Counting_rod_v1r.png" decoding="async" width="7" height="29" class="mw-file-element" data-file-width="7" data-file-height="29" /></a></span> </td></tr> <tr> <th style="text-align:right;height:50px">5089 </th> <td style="width:50px"><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_h5r.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/7/73/Counting_rod_h5r.png" decoding="async" width="29" height="31" class="mw-file-element" data-file-width="29" data-file-height="31" /></a></span> </td> <td style="width:50px">  </td> <td style="width:50px"><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_h8r.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/e/e4/Counting_rod_h8r.png" decoding="async" width="29" height="43" class="mw-file-element" data-file-width="29" data-file-height="43" /></a></span> </td> <td style="width:50px"><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_v9r.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/2/2a/Counting_rod_v9r.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="text-align:right;height:50px">−407 </th> <td style="width:50px">  </td> <td style="width:50px"><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 style="width:50px">  </td> <td style="width:50px"><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></tr> <tr> <th style="text-align:right;height:50px">−6720 </th> <td style="width:50px"><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 style="width:50px"><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 style="width:50px"><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 style="width:50px">  </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Rod_numerals">Rod numerals</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Counting_rods&action=edit&section=4" title="Edit section: Rod numerals"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Rod numerals are a positional numeral system made from shapes of counting rods. Positive numbers are written as they are and the negative numbers are written with a slant bar at the last digit. The vertical bar in the horizontal forms 6–9 are drawn shorter to have the same character height. </p><p>A circle (〇) is used for 0. Many historians think it was imported from <a href="/wiki/Indian_numerals" class="mw-redirect" title="Indian numerals">Indian numerals</a> by <a href="/wiki/Gautama_Siddha" title="Gautama Siddha">Gautama Siddha</a> in 718,<sup id="cite_ref-wang_13-1" class="reference"><a href="#cite_note-wang-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> but some think it was created from the Chinese text space filler "□", and others think that the Indians acquired it from China, because it resembles a Confucian philosophical symbol for "nothing".<sup id="cite_ref-qian_16-0" class="reference"><a href="#cite_note-qian-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p><p>In the 13th century, <a href="/wiki/Song_dynasty" title="Song dynasty">Southern Song</a> mathematicians changed digits for 4, 5, and 9 to reduce strokes.<sup id="cite_ref-qian_16-1" class="reference"><a href="#cite_note-qian-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> The new horizontal forms eventually transformed into <a href="/wiki/Suzhou_numerals" title="Suzhou numerals">Suzhou numerals</a>. Japanese continued to use the traditional forms. </p> <table class="wikitable" border="1" style="text-align:center"> <caption>Positive numbers (traditional) </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">Vertical </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">Horizontal </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>Negative numbers (traditional) </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">Vertical </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> <tr> <th style="height:50px">Horizontal </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_h-1.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/b/bf/Counting_rod_h-1.png" decoding="async" width="29" height="23" class="mw-file-element" data-file-width="29" data-file-height="23" /></a></span> </td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_h-2.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/8/86/Counting_rod_h-2.png" decoding="async" width="29" height="23" class="mw-file-element" data-file-width="29" data-file-height="23" /></a></span> </td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_h-3.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/0/06/Counting_rod_h-3.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_h-4.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/9/98/Counting_rod_h-4.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_h-5.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/4/41/Counting_rod_h-5.png" decoding="async" width="29" height="35" class="mw-file-element" data-file-width="29" data-file-height="35" /></a></span> </td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_h-6.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/a/a2/Counting_rod_h-6.png" decoding="async" width="40" height="40" class="mw-file-element" data-file-width="40" data-file-height="40" /></a></span> </td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_h-7.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/0/01/Counting_rod_h-7.png" decoding="async" width="29" height="38" class="mw-file-element" data-file-width="29" data-file-height="38" /></a></span> </td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_h-8.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/b/b5/Counting_rod_h-8.png" decoding="async" width="29" height="44" class="mw-file-element" data-file-width="29" data-file-height="44" /></a></span> </td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Counting_rod_h-9.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/a/a8/Counting_rod_h-9.png" decoding="async" width="29" height="47" class="mw-file-element" data-file-width="29" data-file-height="47" /></a></span> </td></tr></tbody></table> <table class="wikitable" border="1" style="text-align:center"> <caption>Positive numbers (Southern Song) </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">Vertical </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">Horizontal </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>Examples: </p> <table class="wikitable" border="1" style="text-align:center"> <tbody><tr> <th></th> <th>Traditional</th> <th>Southern Song </th></tr> <tr> <th style="text-align:right;height:50px">231 </th> <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><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><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><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><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></tr> <tr> <th style="text-align:right;height:50px">5089 </th> <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><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><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><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> <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><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><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><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> <tr> <th style="text-align:right;height:50px">−407 </th> <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><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><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_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><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><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></tr> <tr> <th style="text-align:right;height:50px">−6720 </th> <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><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_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><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_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><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_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><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></tr></tbody></table> <p>In Japan, <a href="/wiki/Seki_Takakazu" title="Seki Takakazu">Seki Takakazu</a> developed the rod numerals into symbolic notation for algebra and drastically improved <a href="/wiki/Japanese_mathematics" title="Japanese mathematics">Japanese mathematics</a>.<sup id="cite_ref-wang_13-2" class="reference"><a href="#cite_note-wang-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> After his period, the <a href="/wiki/Positional_notation" title="Positional notation">positional numeral system</a> using <a href="/wiki/Chinese_numerals#Characters_used_to_represent_numbers" title="Chinese numerals">Chinese numeral characters</a> was developed, and the rod numerals were used only for the <a href="/wiki/Plus_and_minus_signs" title="Plus and minus signs">plus and minus signs</a>. </p> <table class="wikitable" border="1" style="text-align:center"> <tbody><tr> <th>Western</th> <th>Seki</th> <th>After Seki </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><sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Accuracy_dispute#Disputed_statement" title="Wikipedia:Accuracy dispute"><span title="seems to read y + 8x (May 2019)">dubious</span></a> – <a href="/wiki/Talk:Counting_rods#Dubious" title="Talk:Counting rods">discuss</a></i>]</sup> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Fractions">Fractions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Counting_rods&action=edit&section=5" title="Edit section: Fractions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Rod_fraction.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2e/Rod_fraction.jpg/200px-Rod_fraction.jpg" decoding="async" width="200" height="116" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2e/Rod_fraction.jpg/300px-Rod_fraction.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/2/2e/Rod_fraction.jpg 2x" data-file-width="381" data-file-height="221" /></a><figcaption> Fraction <style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">7</span></span>⁠</span></figcaption></figure> <p>A fraction was expressed with rod numerals as two rod numerals one on top of another (without any other symbol, like the modern horizontal bar).<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (August 2024)">citation needed</span></a></i>]</sup> </p> <div class="mw-heading mw-heading2"><h2 id="Rod_calculus">Rod calculus</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Counting_rods&action=edit&section=6" title="Edit section: Rod calculus"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Rod_calculus" title="Rod calculus">Rod calculus</a></div> <p>The method for using counting rods for mathematical calculation was called <i>rod calculation</i> or <a href="/wiki/Rod_calculus" title="Rod calculus">rod calculus</a> (筹算). Rod calculus can be used for a wide range of calculations, including finding the value of <a href="/wiki/Pi" title="Pi"><span class="texhtml mvar" style="font-style:italic;">π</span></a>, finding <a href="/wiki/Square_root" title="Square root">square roots</a>, <a href="/wiki/Cube_root" title="Cube root">cube roots</a>, or <a href="/wiki/Nth_root" title="Nth root">higher order roots</a>, and solving a <a href="/wiki/System_of_linear_equations" title="System of linear equations">system of linear equations</a>. </p><p>Before the introduction of a written zero, a space was used to indicate no units, and the rotation of the character in the subsequent unit column, by 90°, adopted, to help reduce the ambiguity in record values calculated on the rods.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> For example 107 (𝍠 𝍧) and 17 (𝍩𝍧) would be distinguished by rotation, though multiple zero units could lead to ambiguity, eg. 1007 (𝍩 𝍧) , and 10007 (𝍠 𝍧). Once written zero came into play, the rod numerals had become independent, and their use indeed outlives the counting rods, after its replacement by <a href="/wiki/Abacus" title="Abacus">abacus</a>. One variation of horizontal rod numerals, the <a href="/wiki/Suzhou_numerals" title="Suzhou numerals">Suzhou numerals</a> is still in use for book-keeping and in herbal medicine prescription in <a href="/wiki/Chinatown" title="Chinatown">Chinatowns</a> in some parts of the world. </p> <div class="mw-heading mw-heading2"><h2 id="Unicode">Unicode</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Counting_rods&action=edit&section=7" title="Edit section: Unicode"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Counting_Rod_Numerals" title="Counting Rod Numerals">Counting Rod Numerals</a></div> <p><a href="/wiki/Unicode" title="Unicode">Unicode</a> 5.0 includes counting rod numerals in their own block in the <a href="/wiki/Plane_(Unicode)#Supplementary_Multilingual_Plane" title="Plane (Unicode)">Supplementary Multilingual Plane</a> (SMP) from U+1D360 to U+1D37F. The <a href="/wiki/Code_point" title="Code point">code points</a> for the horizontal digits 1–9 are U+1D360 to U+1D368 and those for the vertical digits 1–9 are U+1D369 to U+1D371. The former are called <i>unit digits</i> and the latter are called <i>tens digits</i>,<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> which is opposite of the convention described above. The Unicode Standard states that the orientation of the Unicode characters follows Song dynasty convention, which differs from Han dynasty practice which represented digits as vertical lines, and tens as horizontal lines.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> Zero should be represented by U+3007 (〇, ideographic number zero) and the negative sign should be represented by U+20E5 (combining reverse solidus overlay).<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> As these were recently added to the character set and since they are included in the SMP, font support may still be limited. </p> <table class="wikitable nounderlines" style="border-collapse:collapse;background:var(--background-color-base,#fff);color:inherit;font-size:large;text-align:center"> <tbody><tr> <td colspan="17" style="background:var(--background-color-neutral-subtle,#f8f9fa);color:inherit;font-size:small"><b><a href="/wiki/Counting_Rod_Numerals" title="Counting Rod Numerals">Counting Rod Numerals</a></b><style data-mw-deduplicate="TemplateStyles:r1041539562">.mw-parser-output .citation{word-wrap:break-word}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}</style><sup class="citation nobold" id="ref_U1D360_as_of_Unicode_version"><a href="#endnote_U1D360_as_of_Unicode_version">[1]</a></sup><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1041539562" /><sup class="citation nobold" id="ref_U1D360_grey"><a href="#endnote_U1D360_grey">[2]</a></sup><br /><a rel="nofollow" class="external text" href="https://www.unicode.org/charts/PDF/U1D360.pdf">Official Unicode Consortium code chart</a> (PDF) </td></tr> <tr style="background:var(--background-color-neutral-subtle,#f8f9fa);color:inherit;font-size:small"> <td style="width:45pt"> </td> <td style="width:20pt">0</td> <td style="width:20pt">1</td> <td style="width:20pt">2</td> <td style="width:20pt">3</td> <td style="width:20pt">4</td> <td style="width:20pt">5</td> <td style="width:20pt">6</td> <td style="width:20pt">7</td> <td style="width:20pt">8</td> <td style="width:20pt">9</td> <td style="width:20pt">A</td> <td style="width:20pt">B</td> <td style="width:20pt">C</td> <td style="width:20pt">D</td> <td style="width:20pt">E</td> <td style="width:20pt">F </td></tr> <tr> <td style="background:var(--background-color-neutral-subtle,#f8f9fa);color:inherit;font-size:small">U+1D36x </td> <td title="U+1D360: COUNTING ROD UNIT DIGIT ONE"><span style="font-family:Noto Sans Symbols2;">𝍠</span> </td> <td title="U+1D361: COUNTING ROD UNIT DIGIT TWO"><span style="font-family:Noto Sans Symbols2;">𝍡</span> </td> <td title="U+1D362: COUNTING ROD UNIT DIGIT THREE"><span style="font-family:Noto Sans Symbols2;">𝍢</span> </td> <td title="U+1D363: COUNTING ROD UNIT DIGIT FOUR"><span style="font-family:Noto Sans Symbols2;">𝍣</span> </td> <td title="U+1D364: COUNTING ROD UNIT DIGIT FIVE"><span style="font-family:Noto Sans Symbols2;">𝍤</span> </td> <td title="U+1D365: COUNTING ROD UNIT DIGIT SIX"><span style="font-family:Noto Sans Symbols2;">𝍥</span> </td> <td title="U+1D366: COUNTING ROD UNIT DIGIT SEVEN"><span style="font-family:Noto Sans Symbols2;">𝍦</span> </td> <td title="U+1D367: COUNTING ROD UNIT DIGIT EIGHT"><span style="font-family:Noto Sans Symbols2;">𝍧</span> </td> <td title="U+1D368: COUNTING ROD UNIT DIGIT NINE"><span style="font-family:Noto Sans Symbols2;">𝍨</span> </td> <td title="U+1D369: COUNTING ROD TENS DIGIT ONE"><span style="font-family:Noto Sans Symbols2;">𝍩</span> </td> <td title="U+1D36A: COUNTING ROD TENS DIGIT TWO"><span style="font-family:Noto Sans Symbols2;">𝍪</span> </td> <td title="U+1D36B: COUNTING ROD TENS DIGIT THREE"><span style="font-family:Noto Sans Symbols2;">𝍫</span> </td> <td title="U+1D36C: COUNTING ROD TENS DIGIT FOUR"><span style="font-family:Noto Sans Symbols2;">𝍬</span> </td> <td title="U+1D36D: COUNTING ROD TENS DIGIT FIVE"><span style="font-family:Noto Sans Symbols2;">𝍭</span> </td> <td title="U+1D36E: COUNTING ROD TENS DIGIT SIX"><span style="font-family:Noto Sans Symbols2;">𝍮</span> </td> <td title="U+1D36F: COUNTING ROD TENS DIGIT SEVEN"><span style="font-family:Noto Sans Symbols2;">𝍯</span> </td></tr> <tr> <td style="background:var(--background-color-neutral-subtle,#f8f9fa);color:inherit;font-size:small">U+1D37x </td> <td title="U+1D370: COUNTING ROD TENS DIGIT EIGHT"><span style="font-family:Noto Sans Symbols2;">𝍰</span> </td> <td title="U+1D371: COUNTING ROD TENS DIGIT NINE"><span style="font-family:Noto Sans Symbols2;">𝍱</span> </td> <td title="U+1D372: IDEOGRAPHIC TALLY MARK ONE"><span style="font-family:Noto Sans Symbols2;">𝍲</span> </td> <td title="U+1D373: IDEOGRAPHIC TALLY MARK TWO"><span style="font-family:Noto Sans Symbols2;">𝍳</span> </td> <td title="U+1D374: IDEOGRAPHIC TALLY MARK THREE"><span style="font-family:Noto Sans Symbols2;">𝍴</span> </td> <td title="U+1D375: IDEOGRAPHIC TALLY MARK FOUR"><span style="font-family:Noto Sans Symbols2;">𝍵</span> </td> <td title="U+1D376: IDEOGRAPHIC TALLY MARK FIVE"><span style="font-family:Noto Sans Symbols2;">𝍶</span> </td> <td title="U+1D377: TALLY MARK ONE"><span style="font-family:Noto Sans Symbols2;">𝍷</span> </td> <td title="U+1D378: TALLY MARK FIVE"><span style="font-family:Noto Sans Symbols2;">𝍸</span> </td> <td title="Reserved" style="background-color:var(--background-color-disabled,#c8ccd1);color:inherit;"> </td> <td title="Reserved" style="background-color:var(--background-color-disabled,#c8ccd1);color:inherit;"> </td> <td title="Reserved" style="background-color:var(--background-color-disabled,#c8ccd1);color:inherit;"> </td> <td title="Reserved" style="background-color:var(--background-color-disabled,#c8ccd1);color:inherit;"> </td> <td title="Reserved" style="background-color:var(--background-color-disabled,#c8ccd1);color:inherit;"> </td> <td title="Reserved" style="background-color:var(--background-color-disabled,#c8ccd1);color:inherit;"> </td> <td title="Reserved" style="background-color:var(--background-color-disabled,#c8ccd1);color:inherit;"> </td></tr> <tr> <td colspan="17" style="background:var(--background-color-neutral-subtle,#f8f9fa);color:inherit;font-size:small;text-align:left"><b>Notes</b> <dl><dd>1.<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1041539562" /><span class="citation wikicite" id="endnote_U1D360_as_of_Unicode_version"><b><a href="#ref_U1D360_as_of_Unicode_version">^</a></b></span> As of Unicode version 16.0</dd> <dd>2.<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1041539562" /><span class="citation wikicite" id="endnote_U1D360_grey"><b><a href="#ref_U1D360_grey">^</a></b></span> Grey areas indicate non-assigned code points</dd></dl> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Counting_rods&action=edit&section=8" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col"> <ul><li><a href="/wiki/Abacus" title="Abacus">Abacus</a></li> <li><a href="/wiki/Chinese_mathematics" title="Chinese mathematics">Chinese mathematics</a></li> <li><a href="/wiki/Rod_calculus" title="Rod calculus">Rod calculus</a></li> <li><a href="/wiki/Tally_marks" title="Tally marks">Tally marks</a></li> <li><a href="/wiki/Tian_yuan_shu" title="Tian yuan shu">Tian yuan shu</a></li> <li><a href="/wiki/Unicode_numerals" class="mw-redirect" title="Unicode numerals">Unicode numerals</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Counting_rods&action=edit&section=9" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><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"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFLay-Yong1986" class="citation journal cs1">Lay-Yong, Lam (1986). <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/41133800">"The Conceptual Origins of our Numeral System and the Symbolic Form of Algebra"</a>. <i>Archive for History of Exact Sciences</i>. <b>36</b> (3): <span class="nowrap">183–</span>195. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF00412260">10.1007/BF00412260</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0003-9519">0003-9519</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/41133800">41133800</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Archive+for+History+of+Exact+Sciences&rft.atitle=The+Conceptual+Origins+of+our+Numeral+System+and+the+Symbolic+Form+of+Algebra&rft.volume=36&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E183-%3C%2Fspan%3E195&rft.date=1986&rft.issn=0003-9519&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F41133800%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.1007%2FBF00412260&rft.aulast=Lay-Yong&rft.aufirst=Lam&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F41133800&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACounting+rods" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20071121050945/http://mkd.lyge.cn/zhanzheng/a04/x3/040.htm"><i>中国独特的计算工具</i></a>, archived from <a rel="nofollow" class="external text" href="http://mkd.lyge.cn/zhanzheng/a04/x3/040.htm">the original</a> on 2007-11-21<span class="reference-accessdate">, retrieved <span class="nowrap">2007-12-16</span></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=%E4%B8%AD%E5%9B%BD%E7%8B%AC%E7%89%B9%E7%9A%84%E8%AE%A1%E7%AE%97%E5%B7%A5%E5%85%B7&rft_id=http%3A%2F%2Fmkd.lyge.cn%2Fzhanzheng%2Fa04%2Fx3%2F040.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACounting+rods" class="Z3988"></span>1954年，考古学家在位于长江中游地区的湖南省长沙左家公山发掘了一座距今两千一百多年的战国时期楚墓。墓中出土了一个竹筒，里面盛放着戥秤、砝码、毛笔以及四十余根长度约为十二厘米的竹片。这是最早发现也是最古的算筹实物。In English-In 1954, archaeologists discovered a tomb of the Chu Dynasty from the Warring States Period, which is more than 2,100 years old, at Zuojiagong Mountain in Changsha, Hunan Province, in the middle reaches of the Yangtze River. A bamboo tube was unearthed in the tomb, which contained a balance scale, weights, a brush, and more than 40 bamboo pieces about 12 centimeters long. This is the earliest and oldest abacus discovered.</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 rel="nofollow" class="external text" href="http://www.tutorvista.com/math/ancient-china-math">Ancient China Math</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110126232121/http://www.tutorvista.com/math/ancient-china-math">Archived</a> 2011-01-26 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> – Copyright © 2010 – TutorVista.com, All rights reserved.</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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFWONG2006" class="citation journal cs1">WONG, Pui Yin Marianne (Mar 2006). <a rel="nofollow" class="external text" href="https://eprints.soas.ac.uk/28946/1/10673190.pdf">"The Siling (four cardinal animals) in Han Pictorial Art - vol 1"</a> <span class="cs1-format">(PDF)</span>. <i>SOAS</i>: 370, 416.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=SOAS&rft.atitle=The+Siling+%28four+cardinal+animals%29+in+Han+Pictorial+Art+-+vol+1&rft.pages=370%2C+416&rft.date=2006-03&rft.aulast=WONG&rft.aufirst=Pui+Yin+Marianne&rft_id=https%3A%2F%2Feprints.soas.ac.uk%2F28946%2F1%2F10673190.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACounting+rods" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.babelstone.co.uk/Blog/2011/01/lost-game-of-liubo-part-4-game-boards.html">"BabelStone Blog : The Lost Game of Liubo Part 4 : Game Boards and Equipment"</a>. <i>www.babelstone.co.uk</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2020-08-05</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.babelstone.co.uk&rft.atitle=BabelStone+Blog+%3A+The+Lost+Game+of+Liubo+Part+4+%3A+Game+Boards+and+Equipment&rft_id=https%3A%2F%2Fwww.babelstone.co.uk%2FBlog%2F2011%2F01%2Flost-game-of-liubo-part-4-game-boards.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACounting+rods" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text">Wu Wenjun ed, Grand Series of History of Chinese Mathematics, vol 1, p371</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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFLamAng2004" class="citation book cs1">Lam, Lay Yong; Ang, Tian Se (2004). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=fGYmpWE5UZgC&q=tomb+in+Hubei+rod+numerals&pg=PA45"><i>Fleeting Footsteps: Tracing the Conception of Arithmetic and Algebra in Ancient China</i></a>. World Scientific. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-256-725-3" title="Special:BookSources/978-981-256-725-3"><bdi>978-981-256-725-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fleeting+Footsteps%3A+Tracing+the+Conception+of+Arithmetic+and+Algebra+in+Ancient+China&rft.pub=World+Scientific&rft.date=2004&rft.isbn=978-981-256-725-3&rft.aulast=Lam&rft.aufirst=Lay+Yong&rft.au=Ang%2C+Tian+Se&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DfGYmpWE5UZgC%26q%3Dtomb%2Bin%2BHubei%2Brod%2Bnumerals%26pg%3DPA45&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACounting+rods" class="Z3988"></span></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">孫子: 夫未戰而廟算勝者，得算多也</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">老子: 善數者不用籌策。</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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFZhu2018" class="citation web cs1">Zhu, Yiwen (2018). <a rel="nofollow" class="external text" href="https://www.researchgate.net/publication/326525684">"How were Western written calculations introduced into China? — An analysis of the Tongwen suanzhi (Arithmetic Guidance in the Common Language, 1613)"</a>. <i>ResearchGate</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2020-08-05</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=ResearchGate&rft.atitle=How+were+Western+written+calculations+introduced+into+China%3F+%E2%80%94+An+analysis+of+the+Tongwen+suanzhi+%28Arithmetic+Guidance+in+the+Common+Language%2C+1613%29&rft.date=2018&rft.aulast=Zhu&rft.aufirst=Yiwen&rft_id=https%3A%2F%2Fwww.researchgate.net%2Fpublication%2F326525684&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACounting+rods" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><a class="external text" href="https://zh.wikisource.org/wiki/%E5%AD%AB%E5%AD%90%E7%AE%97%E7%B6%93">Chinese Wikisource</a> 孫子算經: 先識其位，一從十橫，百立千僵，千十相望，萬百相當。</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"><a href="https://en.wikisource.org/wiki/zh:%E5%A4%A2%E6%BA%AA%E7%AD%86%E8%AB%87/%E5%8D%B708" class="extiw" title="s:zh:夢溪筆談/卷08">Chinese Wikisource</a>, 夢溪筆談: 如算法用赤籌、黑籌，以別正負之數。</span> </li> <li id="cite_note-wang-13"><span class="mw-cite-backlink">^ <a href="#cite_ref-wang_13-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-wang_13-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-wang_13-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFWáng1999" class="citation cs2">Wáng, Qīngxiáng (1999), <i>Sangi o koeta otoko (The man who exceeded counting rods)</i>, Tokyo: Tōyō Shoten, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/4-88595-226-3" title="Special:BookSources/4-88595-226-3"><bdi>4-88595-226-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Sangi+o+koeta+otoko+%28The+man+who+exceeded+counting+rods%29&rft.place=Tokyo&rft.pub=T%C5%8Dy%C5%8D+Shoten&rft.date=1999&rft.isbn=4-88595-226-3&rft.aulast=W%C3%A1ng&rft.aufirst=Q%C4%ABngxi%C3%A1ng&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACounting+rods" class="Z3988"></span></span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><a class="external text" href="https://zh.wikisource.org/wiki/%E4%B9%9D%E7%AB%A0%E7%AE%97%E8%A1%93">Chinese Wikisource</a> 正負術曰: 同名相除，異名相益，正無入負之，負無入正之。其異名相除，同名相益，正無入正之，負無入負之。</span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><a href="/wiki/Karl_Menninger_(mathematics)" title="Karl Menninger (mathematics)">Karl Menninger</a>, Number Words and Number Symbols, p 369, MIT Press, 1970</span> </li> <li id="cite_note-qian-16"><span class="mw-cite-backlink">^ <a href="#cite_ref-qian_16-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-qian_16-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFQian1964" class="citation cs2">Qian, Baocong (1964), <i>Zhongguo Shuxue Shi (The history of Chinese mathematics)</i>, Beijing: Kexue Chubanshe</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Zhongguo+Shuxue+Shi+%28The+history+of+Chinese+mathematics%29&rft.place=Beijing&rft.pub=Kexue+Chubanshe&rft.date=1964&rft.aulast=Qian&rft.aufirst=Baocong&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACounting+rods" class="Z3988"></span></span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://mathshistory.st-andrews.ac.uk/HistTopics/Chinese_numerals/">"Chinese numerals"</a>. <i>Maths History</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2024-04-28</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Maths+History&rft.atitle=Chinese+numerals&rft_id=https%3A%2F%2Fmathshistory.st-andrews.ac.uk%2FHistTopics%2FChinese_numerals%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACounting+rods" class="Z3988"></span></span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text">Christopher Cullen et John H. Jenkins, <a rel="nofollow" class="external text" href="https://www.unicode.org/L2/L2004/04227-count-rod.pdf"><i>Proposal to add Chinese counting rod numerals to Unicode and ISO/IEC 10646</i></a>, 2004</span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.unicode.org/charts/PDF/U1D360.pdf"><i>The Unicode Standard, Version 5.0 – Electronic edition</i></a> <span class="cs1-format">(PDF)</span>, Unicode, Inc., 2006, p. 558</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Unicode+Standard%2C+Version+5.0+%E2%80%93+Electronic+edition&rft.pages=558&rft.pub=Unicode%2C+Inc.&rft.date=2006&rft_id=https%3A%2F%2Fwww.unicode.org%2Fcharts%2FPDF%2FU1D360.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACounting+rods" class="Z3988"></span></span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.unicode.org/versions/Unicode15.0.0/ch22.pdf#G12367"><i>The Unicode Standard, Version 15.0 – Electronic edition</i></a> <span class="cs1-format">(PDF)</span>, Unicode, Inc., 2022, pp. <span class="nowrap">869–</span>870</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Unicode+Standard%2C+Version+15.0+%E2%80%93+Electronic+edition&rft.pages=%3Cspan+class%3D%22nowrap%22%3E869-%3C%2Fspan%3E870&rft.pub=Unicode%2C+Inc.&rft.date=2022&rft_id=https%3A%2F%2Fwww.unicode.org%2Fversions%2FUnicode15.0.0%2Fch22.pdf%23G12367&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACounting+rods" class="Z3988"></span></span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.unicode.org/versions/Unicode5.0.0/ch15.pdf#G28213"><i>The Unicode Standard, Version 5.0 – Electronic edition</i></a> <span class="cs1-format">(PDF)</span>, Unicode, Inc., 2006, pp. <span class="nowrap">499–</span>500</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Unicode+Standard%2C+Version+5.0+%E2%80%93+Electronic+edition&rft.pages=%3Cspan+class%3D%22nowrap%22%3E499-%3C%2Fspan%3E500&rft.pub=Unicode%2C+Inc.&rft.date=2006&rft_id=https%3A%2F%2Fwww.unicode.org%2Fversions%2FUnicode5.0.0%2Fch15.pdf%23G28213&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACounting+rods" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Counting_rods&action=edit&section=10" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For a look of the ancient counting rods, and further explanation, you can visit the sites </p> <ul><li><a rel="nofollow" class="external free" href="https://web.archive.org/web/20010217175749/http://www.math.sfu.ca/histmath/China/Beginning/Rod.html">https://web.archive.org/web/20010217175749/http://www.math.sfu.ca/histmath/China/Beginning/Rod.html</a></li> <li><a rel="nofollow" class="external free" href="http://mathforum.org/library/drmath/view/52557.html">http://mathforum.org/library/drmath/view/52557.html</a></li> <li><span class="machinetrans machinetrans_to_zh machinetrans_from_en"><a rel="nofollow" class="external text" href="http://big5.ce.cn/gate/big5/cathay.ce.cn/history/200612/05/t20061205_9676970.shtml">Counting rods in China</a> <span class="languageicon">(in Chinese)</span> (<span title="Caution: Machine translations may be inaccurate." style="border-bottom: 1px dashed #86a1ff;">Translate</span> to English: <a rel="nofollow" class="external text" href="https://translate.google.com/translate?u=http%3A%2F%2Fbig5.ce.cn%2Fgate%2Fbig5%2Fcathay.ce.cn%2Fhistory%2F200612%2F05%2Ft20061205_9676970.shtml&hl=en&ie=UTF-8&sl=zh&tl=en&edit-text=">Google</a>, <a rel="nofollow" class="external text" href="https://www.microsofttranslator.com/bv.aspx?from=zh&to=en&a=http%3A%2F%2Fbig5.ce.cn%2Fgate%2Fbig5%2Fcathay.ce.cn%2Fhistory%2F200612%2F05%2Ft20061205_9676970.shtml">Bing</a>, <a rel="nofollow" class="external text" href="https://translate.yandex.com/translate?url=http%3A%2F%2Fbig5.ce.cn%2Fgate%2Fbig5%2Fcathay.ce.cn%2Fhistory%2F200612%2F05%2Ft20061205_9676970.shtml&lang=zh-en">Yandex</a>)</span></li> <li><span class="machinetrans machinetrans_to_ja machinetrans_from_en"><a rel="nofollow" class="external text" href="http://www.city.saitama.jp/www/contents/1044324363225/index.html">Counting rods and go stones of a Japanese mathematician around 1872</a> <span class="languageicon">(in Japanese)</span> (<span title="Caution: Machine translations may be inaccurate." style="border-bottom: 1px dashed #86a1ff;">Translate</span> to English: <a rel="nofollow" class="external text" href="https://translate.google.com/translate?u=http%3A%2F%2Fwww.city.saitama.jp%2Fwww%2Fcontents%2F1044324363225%2Findex.html&hl=en&ie=UTF-8&sl=ja&tl=en&edit-text=">Google</a>, <a rel="nofollow" class="external text" href="https://www.microsofttranslator.com/bv.aspx?from=ja&to=en&a=http%3A%2F%2Fwww.city.saitama.jp%2Fwww%2Fcontents%2F1044324363225%2Findex.html">Bing</a>, <a rel="nofollow" class="external text" href="https://translate.yandex.com/translate?url=http%3A%2F%2Fwww.city.saitama.jp%2Fwww%2Fcontents%2F1044324363225%2Findex.html&lang=ja-en">Yandex</a>)</span></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" 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