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cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet edit-page menu__item--page-actions-viewsource"> View source </a> </li> </ul> </nav>  <div id="mw-content-subtitle"></div> </div> <div id="bodyContent" class="content"> <div id="mw-content-text" class="mw-body-content"><script>function mfTempOpenSection(id){var block=document.getElementById("mf-section-"+id);block.className+=" open-block";block.previousSibling.className+=" open-block";}</script><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><section class="mf-section-0" id="mf-section-0"> <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">"Zero" redirects here. For other uses, see <a href="/wiki/0_(disambiguation)" class="mw-disambig" title="0 (disambiguation)">0 (disambiguation)</a> and <a href="/wiki/Zero_(disambiguation)" class="mw-disambig" title="Zero (disambiguation)">Zero (disambiguation)</a>.</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Not to be confused with the letter <a href="/wiki/O" title="O">O</a>.</div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1092331828">@media(min-width:720px){.mw-parser-output .contains-special-characters{width:22em}}</style><div class="side-box metadata side-box-right contains-special-characters noprint selfref"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-text plainlist">This article contains <a href="/wiki/Help:Special_characters" title="Help:Special characters">special characters</a>. Without proper <a href="/wiki/Help:Special_characters" title="Help:Special characters">rendering support</a>, you may see <a href="/wiki/Specials_(Unicode_block)#Replacement_character" title="Specials (Unicode block)">question marks, boxes, or other symbols</a>.</div></div> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable plainlinks selfreference noprint selfref">For <a href="/wiki/Wikipedia:Naming_conventions_(technical_restrictions)" title="Wikipedia:Naming conventions (technical restrictions)">technical reasons</a>, "0#" redirects here. For the concept in set theory, see <a href="/wiki/Zero_sharp" title="Zero sharp">Zero sharp</a>.</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable plainlinks selfreference noprint selfref">For <a href="/wiki/Wikipedia:Naming_conventions_(technical_restrictions)" title="Wikipedia:Naming conventions (technical restrictions)">technical reasons</a>, ":0" redirects here. For the keyboard symbols, see <a href="/wiki/List_of_emoticons" title="List of emoticons">List of emoticons</a>.</div> <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>0 (zero) is a <a href="/wiki/Number" title="Number">number</a> representing an empty <a href="/wiki/Quantity" title="Quantity">quantity</a>. Adding (or subtracting) 0 to any number leaves that number unchanged; in mathematical terminology, 0 is the <a href="/wiki/Additive_identity" title="Additive identity">additive identity</a> of the <a href="/wiki/Integer" title="Integer">integers</a>, <a href="/wiki/Rational_numbers" class="mw-redirect" title="Rational numbers">rational numbers</a>, <a href="/wiki/Real_number" title="Real number">real numbers</a>, and <a href="/wiki/Complex_numbers" class="mw-redirect" title="Complex numbers">complex numbers</a>, as well as other <a href="/wiki/Algebraic_structures" class="mw-redirect" title="Algebraic structures">algebraic structures</a>. Multiplying any number by 0 results in 0, and consequently <a href="/wiki/Division_by_zero" title="Division by zero">division by zero</a> has <a href="/wiki/Undefined_(mathematics)" title="Undefined (mathematics)">no meaning</a> in <a href="/wiki/Arithmetic" title="Arithmetic">arithmetic</a>. <table class="infobox" style="line-height: 1.0em"><tbody><tr><th colspan="2" class="infobox-above" style="font-size: 150%"><table style="width:100%; margin:0"><tbody><tr> <td style="width:15%; text-align:left; white-space: nowrap; font-size:smaller"><a href="/wiki/%E2%88%921_(number)" class="mw-redirect" title="−1 (number)">← −1 </a></td> <td style="width:70%; padding-left:1em; padding-right:1em; text-align: center;">0</td> <td style="width:15%; text-align:right; white-space: nowrap; font-size:smaller"><a href="/wiki/1_(number)" class="mw-redirect" title="1 (number)"> 1 →</a></td> </tr></tbody></table></th></tr><tr><td colspan="2" class="infobox-subheader" style="font-size:100%;"><div style="text-align:center;"> <a href="/wiki/%E2%88%921" title="−1">−1</a> <a class="mw-selflink selflink">0</a> <a href="/wiki/1" title="1">1</a> <a href="/wiki/2" title="2">2</a> <a href="/wiki/3" title="3">3</a> <a href="/wiki/4" title="4">4</a> <a href="/wiki/5" title="5">5</a> <a href="/wiki/6" title="6">6</a> <a href="/wiki/7" title="7">7</a> <a href="/wiki/8" title="8">8</a> <a href="/wiki/9" title="9">9</a> <a href="/wiki/10_(number)" class="mw-redirect" title="10 (number)">→</a></div><div style="text-align:center;"> <style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist li:last-child::after{content:none}.mw-parser-output .hlist dd dd:first-child::before,.mw-parser-output .hlist dd dt:first-child::before,.mw-parser-output .hlist dd li:first-child::before,.mw-parser-output .hlist dt dd:first-child::before,.mw-parser-output .hlist dt dt:first-child::before,.mw-parser-output .hlist dt li:first-child::before,.mw-parser-output .hlist li dd:first-child::before,.mw-parser-output .hlist li dt:first-child::before,.mw-parser-output .hlist li li:first-child::before{content:" (";font-weight:normal}.mw-parser-output .hlist dd dd:last-child::after,.mw-parser-output .hlist dd dt:last-child::after,.mw-parser-output .hlist dd li:last-child::after,.mw-parser-output .hlist dt dd:last-child::after,.mw-parser-output .hlist dt dt:last-child::after,.mw-parser-output .hlist dt li:last-child::after,.mw-parser-output .hlist li dd:last-child::after,.mw-parser-output .hlist li dt:last-child::after,.mw-parser-output .hlist li li:last-child::after{content:")";font-weight:normal}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:" ("counter(listitem)"\a0 "}</style><div class="hlist"><ul><li><a href="/wiki/List_of_numbers" title="List of numbers">List of numbers</a></li><li><a href="/wiki/Integer" title="Integer">Integers</a></li></ul></div></div><div style="text-align:center;"><a href="/wiki/Negative_number" title="Negative number">←</a> <a class="mw-selflink selflink">0</a> <a href="/wiki/10" title="10">10</a> <a href="/wiki/20_(number)" title="20 (number)">20</a> <a href="/wiki/30_(number)" title="30 (number)">30</a> <a href="/wiki/40_(number)" title="40 (number)">40</a> <a href="/wiki/50_(number)" title="50 (number)">50</a> <a href="/wiki/60_(number)" title="60 (number)">60</a> <a href="/wiki/70_(number)" title="70 (number)">70</a> <a href="/wiki/80_(number)" title="80 (number)">80</a> <a href="/wiki/90_(number)" title="90 (number)">90</a> <a href="/wiki/100_(number)" class="mw-redirect" title="100 (number)">→</a></div></td></tr><tr><th scope="row" class="infobox-label" style="font-weight:normal"><a href="/wiki/Cardinal_numeral" title="Cardinal numeral">Cardinal</a></th><td class="infobox-data">0, zero, "oh" (<a href="/wiki/Help:IPA/English" title="Help:IPA/English">/oʊ/</a>), nought, naught, nil</td></tr><tr><th scope="row" class="infobox-label" style="font-weight:normal"><a href="/wiki/Ordinal_numeral" title="Ordinal numeral">Ordinal</a></th><td class="infobox-data">Zeroth, noughth, 0th</td></tr><tr><th scope="row" class="infobox-label" style="font-weight:normal"><a href="/wiki/Latin" title="Latin">Latin</a> <a href="/wiki/Numeral_prefix" title="Numeral prefix">prefix</a></th><td class="infobox-data">nulli-</td></tr><tr><th scope="row" class="infobox-label" style="font-weight:normal"><a href="/wiki/Binary_number" title="Binary number">Binary</a></th><td class="infobox-data">02</td></tr><tr><th scope="row" class="infobox-label" style="font-weight:normal"><a href="/wiki/Ternary_numeral_system" title="Ternary numeral system">Ternary</a></th><td class="infobox-data">03</td></tr><tr><th scope="row" class="infobox-label" style="font-weight:normal"><a href="/wiki/Senary" title="Senary">Senary</a></th><td class="infobox-data">06</td></tr><tr><th scope="row" class="infobox-label" style="font-weight:normal"><a href="/wiki/Octal" title="Octal">Octal</a></th><td class="infobox-data">08</td></tr><tr><th scope="row" class="infobox-label" style="font-weight:normal"><a href="/wiki/Duodecimal" title="Duodecimal">Duodecimal</a></th><td class="infobox-data">012</td></tr><tr><th scope="row" class="infobox-label" style="font-weight:normal"><a href="/wiki/Hexadecimal" title="Hexadecimal">Hexadecimal</a></th><td class="infobox-data">016</td></tr><tr><th scope="row" class="infobox-label" style="font-weight:normal"><a href="/wiki/Eastern_Arabic_numerals" title="Eastern Arabic numerals">Arabic</a>, <a href="/wiki/Central_Kurdish" class="mw-redirect" title="Central Kurdish">Kurdish</a>, <a href="/wiki/Persian_language" title="Persian language">Persian</a>, <a href="/wiki/Sindhi_language" title="Sindhi language">Sindhi</a>, <a href="/wiki/Urdu_numerals" class="mw-redirect" title="Urdu numerals">Urdu</a></th><td class="infobox-data">٠</td></tr><tr><th scope="row" class="infobox-label" style="font-weight:normal"><a href="/wiki/Indian_numerals" class="mw-redirect" title="Indian numerals">Hindu numerals</a></th><td class="infobox-data">०</td></tr><tr><th scope="row" class="infobox-label" style="font-weight:normal"><a href="/wiki/Chinese_numerals" title="Chinese numerals">Chinese</a></th><td class="infobox-data">零, 〇</td></tr><tr><th scope="row" class="infobox-label" style="font-weight:normal"><a href="/wiki/Burmese_numerals" title="Burmese numerals">Burmese</a></th><td class="infobox-data">၀</td></tr><tr><th scope="row" class="infobox-label" style="font-weight:normal"><a href="/wiki/Khmer_numerals" title="Khmer numerals">Khmer</a></th><td class="infobox-data">០</td></tr><tr><th scope="row" class="infobox-label" style="font-weight:normal"><a href="/wiki/Thai_numerals" title="Thai numerals">Thai</a></th><td class="infobox-data">๐</td></tr><tr><th scope="row" class="infobox-label" style="font-weight:normal"><a href="/wiki/Bengali-Assamese_numerals" class="mw-redirect" title="Bengali-Assamese numerals">Assamese, Bengali</a></th><td class="infobox-data">০</td></tr><tr><th scope="row" class="infobox-label" style="font-weight:normal"><a href="/wiki/Maya_numerals" title="Maya numerals">Maya numerals</a></th><td class="infobox-data">𝋠</td></tr><tr><th scope="row" class="infobox-label" style="font-weight:normal"><a href="/wiki/Morse_code" title="Morse code">Morse code</a></th><td class="infobox-data">_ _ _ _ _</td></tr></tbody></table> As a <a href="/wiki/Numerical_digit" title="Numerical digit">numerical digit</a>, 0 plays a crucial role in <a href="/wiki/Decimal" title="Decimal">decimal</a> notation: it indicates that the <a href="/wiki/Power_of_ten" class="mw-redirect" title="Power of ten">power of ten</a> corresponding to the place containing a 0 does not contribute to the total. For example, "205" in decimal means two hundreds, no tens, and five ones. The same principle applies in <a href="/wiki/Place-value_notation" class="mw-redirect" title="Place-value notation">place-value notations</a> that uses a base other than ten, such as <a href="/wiki/Binary_number" title="Binary number">binary</a> and <a href="/wiki/Hexadecimal" title="Hexadecimal">hexadecimal</a>. The modern use of 0 in this manner derives from <a href="/wiki/Indian_mathematics" title="Indian mathematics">Indian mathematics</a> that was transmitted to Europe via <a href="/wiki/Mathematics_in_the_medieval_Islamic_world" title="Mathematics in the medieval Islamic world">medieval Islamic mathematicians</a> and popularized by <a href="/wiki/Fibonacci" title="Fibonacci">Fibonacci</a>. It was independently used by the <a href="/wiki/Maya_civilization" title="Maya civilization">Maya</a>. Common <a href="/wiki/Names_for_the_number_0_in_English" title="Names for the number 0 in English">names for the number 0 in English</a> include zero, nought, naught (<a href="/wiki/Help:IPA/English" title="Help:IPA/English">/nɔːt/</a>), and nil. In contexts where at least one adjacent digit distinguishes it from the <a href="/wiki/O" title="O">letter O</a>, the number is sometimes pronounced as oh or o (<a href="/wiki/Help:IPA/English" title="Help:IPA/English">/oʊ/</a>). Informal or <a href="/wiki/Slang" title="Slang">slang</a> terms for 0 include zilch and zip. Historically, ought, aught (<a href="/wiki/Help:IPA/English" title="Help:IPA/English">/ɔːt/</a>), and cipher have also been used. <div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"><input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none"><div class="toctitle" lang="en" dir="ltr"><h2 id="mw-toc-heading">Contents</h2><label class="toctogglelabel" for="toctogglecheckbox"></label></div> <ul> <li class="toclevel-1 tocsection-1"><a href="#Etymology">1 Etymology</a> <ul> <li class="toclevel-2 tocsection-2"><a href="#Modern_usage">1.1 Modern usage</a></li> </ul> </li> <li class="toclevel-1 tocsection-3"><a href="#History">2 History</a> <ul> <li class="toclevel-2 tocsection-4"><a href="#Ancient_Near_East">2.1 Ancient Near East</a></li> <li class="toclevel-2 tocsection-5"><a href="#Pre-Columbian_Americas">2.2 Pre-Columbian Americas</a></li> <li class="toclevel-2 tocsection-6"><a href="#Classical_antiquity">2.3 Classical antiquity</a></li> <li class="toclevel-2 tocsection-7"><a href="#China">2.4 China</a></li> <li class="toclevel-2 tocsection-8"><a href="#India">2.5 India</a> <ul> <li class="toclevel-3 tocsection-9"><a href="#Epigraphy">2.5.1 Epigraphy</a></li> </ul> </li> <li class="toclevel-2 tocsection-10"><a href="#Middle_Ages">2.6 Middle Ages</a> <ul> <li class="toclevel-3 tocsection-11"><a href="#Transmission_to_Islamic_culture">2.6.1 Transmission to Islamic culture</a></li> <li class="toclevel-3 tocsection-12"><a href="#Transmission_to_Europe">2.6.2 Transmission to Europe</a></li> </ul> </li> </ul> </li> <li class="toclevel-1 tocsection-13"><a href="#Symbols_and_representations">3 Symbols and representations</a></li> <li class="toclevel-1 tocsection-14"><a href="#Mathematics">4 Mathematics</a> <ul> <li class="toclevel-2 tocsection-15"><a href="#As_a_digit">4.1 As a digit</a></li> <li class="toclevel-2 tocsection-16"><a href="#Elementary_algebra">4.2 Elementary algebra</a></li> <li class="toclevel-2 tocsection-17"><a href="#Other_uses_in_mathematics">4.3 Other uses in mathematics</a></li> </ul> </li> <li class="toclevel-1 tocsection-18"><a href="#Physics">5 Physics</a></li> <li class="toclevel-1 tocsection-19"><a href="#Computer_science">6 Computer science</a></li> <li class="toclevel-1 tocsection-20"><a href="#Other_fields">7 Other fields</a> <ul> <li class="toclevel-2 tocsection-21"><a href="#Biology">7.1 Biology</a></li> <li class="toclevel-2 tocsection-22"><a href="#Dating_systems">7.2 Dating systems</a></li> </ul> </li> <li class="toclevel-1 tocsection-23"><a href="#See_also">8 See also</a></li> <li class="toclevel-1 tocsection-24"><a href="#Notes">9 Notes</a></li> <li class="toclevel-1 tocsection-25"><a href="#References">10 References</a></li> <li class="toclevel-1 tocsection-26"><a href="#Bibliography">11 Bibliography</a> <ul> <li class="toclevel-2 tocsection-27"><a href="#Historical_studies">11.1 Historical studies</a></li> </ul> </li> <li class="toclevel-1 tocsection-28"><a href="#External_links">12 External links</a></li> </ul> </div> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(1)"><h2 id="Etymology">Etymology</h2></div><section class="mf-section-1 collapsible-block" id="mf-section-1"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Names_for_the_number_0" title="Names for the number 0">Names for the number 0</a> and <a href="/wiki/Names_for_the_number_0_in_English" title="Names for the number 0 in English">Names for the number 0 in English</a></div> The word zero came into the English language via French zéro from the <a href="/wiki/Italian_language" title="Italian language">Italian</a> zero, a contraction of the Venetian zevero form of Italian zefiro via ṣafira or ṣifr.<a href="#cite_note-1">[1]</a> In pre-Islamic time the word ṣifr (Arabic صفر) had the meaning "empty".<a href="#cite_note-smithsonian-2">[2]</a> Sifr evolved to mean zero when it was used to translate śūnya (<a href="/wiki/Sanskrit_language" class="mw-redirect" title="Sanskrit language">Sanskrit</a>: शून्य) from India.<a href="#cite_note-smithsonian-2">[2]</a> The first known English use of zero was in 1598.<a href="#cite_note-3">[3]</a> The Italian mathematician <a href="/wiki/Fibonacci" title="Fibonacci">Fibonacci</a> (<abbr title="circa">c.</abbr> 1170 – c. 1250), who grew up in North Africa and is credited with introducing the decimal system to Europe, used the term zephyrum. This became zefiro in Italian, and was then contracted to zero in Venetian. The Italian word <a href="https://en.wiktionary.org/wiki/zefiro" class="extiw" title="wikt:zefiro">zefiro</a> was already in existence (meaning "west wind" from Latin and Greek <a href="/wiki/Zephyrus" title="Zephyrus">Zephyrus</a>) and may have influenced the spelling when transcribing Arabic ṣifr.<a href="#cite_note-ifrah-4">[4]</a> <div class="mw-heading mw-heading3"><h3 id="Modern_usage">Modern usage</h3></div> Depending on the context, there may be different words used for the number zero, or the concept of zero. For the simple notion of lacking, the words "<a href="/wiki/Nothing" title="Nothing">nothing</a>" (although this is not accurate) and "none" are often used. The British English words <a href="/wiki/Names_for_the_number_0_in_English#%22Nought%22_and_%22naught%22_versus_%22ought%22_and_%22aught%22" title="Names for the number 0 in English">"nought" or "naught"</a>, and "<a href="https://en.wiktionary.org/wiki/nil" class="extiw" title="wikt:nil">nil</a>" are also synonymous.<a href="#cite_note-5">[5]</a><a href="#cite_note-6">[6]</a> It is often called "oh" in the context of reading out a string of digits, such as <a href="/wiki/Telephone_number" title="Telephone number">telephone numbers</a>, <a href="/wiki/Street_address" class="mw-redirect" title="Street address">street addresses</a>, <a href="/wiki/Credit_card_number" class="mw-redirect" title="Credit card number">credit card numbers</a>, <a href="/wiki/Military_time" class="mw-redirect" title="Military time">military time</a>, or years. For example, the <a href="/wiki/Area_code" class="mw-redirect" title="Area code">area code</a> 201 may be pronounced "two oh one", and the year 1907 is often pronounced "nineteen oh seven". The presence of other digits, indicating that the string contains only numbers, avoids confusion with the letter O. For this reason, systems that include strings with both letters and numbers (such as <a href="/wiki/Canadian_postal_code" class="mw-redirect" title="Canadian postal code">Canadian postal codes</a>) may exclude the use of the letter O.[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] Slang words for zero include "zip", "zilch", "nada", and "scratch".<a href="#cite_note-thesaurus-7">[7]</a> In the context of sports, "nil" is sometimes used, especially in <a href="/wiki/British_English" title="British English">British English</a>. Several sports have specific words for a score of zero, such as "<a href="/wiki/Love_(tennis)" class="mw-redirect" title="Love (tennis)">love</a>" in <a href="/wiki/Tennis" title="Tennis">tennis</a> – from French l'œuf, "the egg" – and "<a href="/wiki/Duck_(cricket)" title="Duck (cricket)">duck</a>" in <a href="/wiki/Cricket_(sport)" class="mw-redirect" title="Cricket (sport)">cricket</a>, a shortening of "duck's egg". "Goose egg" is another general slang term used for zero.<a href="#cite_note-thesaurus-7">[7]</a> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(2)"><h2 id="History">History</h2></div><section class="mf-section-2 collapsible-block" id="mf-section-2"> <div class="mw-heading mw-heading3"><h3 id="Ancient_Near_East">Ancient Near East</h3></div> <table style="float:right; clear:right; text-align:center; border: 1px solid" align="right" cellspacing="0" cellpadding="8"> <tbody><tr> <th>nfr </th> <td>heart with <a href="/wiki/Trachea" title="Trachea">trachea</a> beautiful, pleasant, good </td> <td><table class="mw-hiero-table mw-hiero-outer" dir="ltr"><tbody><tr><td> <table class="mw-hiero-table"><tbody><tr> <td><img class="skin-invert" style="margin: 1px;" src="/w/extensions/wikihiero/img/hiero_F35.png?9f378" height="38" title="F35" alt="F35"></td></tr></tbody></table> </td></tr></tbody></table> </td></tr></tbody></table> Ancient <a href="/wiki/Egyptian_numerals" title="Egyptian numerals">Egyptian numerals</a> were of <a href="/wiki/Decimal" title="Decimal">base 10</a>.<a href="#cite_note-8">[8]</a> They used <a href="/wiki/Hieroglyphs" class="mw-redirect" title="Hieroglyphs">hieroglyphs</a> for the digits and were not <a href="/wiki/Positional_notation" title="Positional notation">positional</a>. In <a href="/wiki/Papyrus_Boulaq_18" title="Papyrus Boulaq 18">one papyrus</a> written around 1770 BC, a scribe recorded daily incomes and expenditures for the <a href="/wiki/Pharaoh" title="Pharaoh">pharaoh</a>'s court, using the <a href="/wiki/Nefer" title="Nefer">nfr</a> hieroglyph to indicate cases where the amount of a foodstuff received was exactly equal to the amount disbursed. Egyptologist <a href="/wiki/Alan_Gardiner" title="Alan Gardiner">Alan Gardiner</a> suggested that the nfr hieroglyph was being used as a symbol for zero. The same symbol was also used to indicate the base level in drawings of tombs and pyramids, and distances were measured relative to the base line as being above or below this line.<a href="#cite_note-9">[9]</a> By the middle of the 2nd millennium BC, <a href="/wiki/Babylonian_mathematics" title="Babylonian mathematics">Babylonian mathematics</a> had a sophisticated <a href="/wiki/Sexagesimal" title="Sexagesimal">base 60</a> positional numeral system. The lack of a positional value (or zero) was indicated by a space between <a href="/wiki/Sexagesimal" title="Sexagesimal">sexagesimal</a> numerals. In a tablet unearthed at <a href="/wiki/Kish_(Sumer)" title="Kish (Sumer)">Kish</a> (dating to as early as 700 BC), the scribe Bêl-bân-aplu used three hooks as a <a href="/wiki/Free_variables_and_bound_variables" title="Free variables and bound variables">placeholder</a> in the same <a href="/wiki/Babylonian_numerals" class="mw-redirect" title="Babylonian numerals">Babylonian system</a>.<a href="#cite_note-FOOTNOTEKaplan2000-10">[10]</a> By 300 BC, a punctuation symbol (two slanted wedges) was repurposed as a placeholder.<a href="#cite_note-11">[11]</a><a href="#cite_note-12">[12]</a> The Babylonian positional numeral system differed from the later <a href="/wiki/Hindu%E2%80%93Arabic_numeral_system" title="Hindu–Arabic numeral system">Hindu–Arabic system</a> in that it did not explicitly specify the magnitude of the leading sexagesimal digit, so that for example the lone digit 1 (<a href="/wiki/File:Babylonian_1.svg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5d/Babylonian_1.svg/20px-Babylonian_1.svg.png" decoding="async" width="20" height="19" class="mw-file-element" data-file-width="144" data-file-height="135"></noscript><span class="lazy-image-placeholder" style="width: 20px;height: 19px;" data-mw-src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5d/Babylonian_1.svg/20px-Babylonian_1.svg.png" data-width="20" data-height="19" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5d/Babylonian_1.svg/30px-Babylonian_1.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5d/Babylonian_1.svg/40px-Babylonian_1.svg.png 2x" data-class="mw-file-element"> </a>) might represent any of 1, 60, 3600 = 602, etc., similar to the significand of a <a href="/wiki/Floating-point_number" class="mw-redirect" title="Floating-point number">floating-point number</a> but without an explicit exponent, and so only distinguished implicitly from context. The zero-like placeholder mark was only ever used in between digits, but never alone or at the end of a number.<a href="#cite_note-FOOTNOTEReimer2014172-13">[13]</a> <div class="mw-heading mw-heading3"><h3 id="Pre-Columbian_Americas">Pre-Columbian Americas</h3></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Cero_maya.svg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7b/Cero_maya.svg/220px-Cero_maya.svg.png" decoding="async" width="220" height="109" class="mw-file-element" data-file-width="303" data-file-height="150"></noscript><span class="lazy-image-placeholder" style="width: 220px;height: 109px;" data-mw-src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7b/Cero_maya.svg/220px-Cero_maya.svg.png" data-width="220" data-height="109" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7b/Cero_maya.svg/330px-Cero_maya.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7b/Cero_maya.svg/440px-Cero_maya.svg.png 2x" data-class="mw-file-element"> </a><figcaption>Maya numeral zero</figcaption></figure> The <a href="/wiki/Mesoamerican_Long_Count_calendar" title="Mesoamerican Long Count calendar">Mesoamerican Long Count calendar</a> developed in south-central Mexico and Central America required the use of zero as a placeholder within its <a href="/wiki/Vigesimal" title="Vigesimal">vigesimal</a> (base-20) positional numeral system. Many different glyphs, including the partial <a href="/wiki/Quatrefoil" title="Quatrefoil">quatrefoil</a> were used as a zero symbol for these Long Count dates, the earliest of which (on Stela 2 at Chiapa de Corzo, <a href="/wiki/Chiapas" title="Chiapas">Chiapas</a>) has a date of 36 BC.<a href="#cite_note-14">[a]</a><a href="#cite_note-15">[14]</a> Since the eight earliest Long Count dates appear outside the Maya homeland,<a href="#cite_note-FOOTNOTEDiehl2004186-16">[15]</a> it is generally believed that the use of zero in the Americas predated the Maya and was possibly the invention of the <a href="/wiki/Olmec" class="mw-redirect" title="Olmec">Olmecs</a>.<a href="#cite_note-17">[16]</a> Many of the earliest Long Count dates were found within the Olmec heartland, although the Olmec civilization ended by the 4th century BC,<a href="#cite_note-18">[17]</a> several centuries before the earliest known Long Count dates.<a href="#cite_note-19">[18]</a> Although zero became an integral part of <a href="/wiki/Maya_numerals" title="Maya numerals">Maya numerals</a>, with a different, empty <a href="/wiki/Tortoise" title="Tortoise">tortoise</a>-like "<a href="/wiki/Plastron" class="mw-redirect" title="Plastron">shell shape</a>" used for many depictions of the "zero" numeral, it is assumed not to have influenced <a href="/wiki/Old_World" title="Old World">Old World</a> numeral systems.[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] <a href="/wiki/Quipu" title="Quipu">Quipu</a>, a knotted cord device, used in the <a href="/wiki/Inca_Empire" title="Inca Empire">Inca Empire</a> and its predecessor societies in the <a href="/wiki/Andes" title="Andes">Andean</a> region to record accounting and other digital data, is encoded in a <a href="/wiki/Decimal" title="Decimal">base ten</a> positional system. Zero is represented by the absence of a knot in the appropriate position.<a href="#cite_note-20">[19]</a> <div class="mw-heading mw-heading3"><h3 id="Classical_antiquity">Classical antiquity</h3></div> The <a href="/wiki/Ancient_Greece" title="Ancient Greece">ancient Greeks</a> had no symbol for zero (μηδέν, pronounced 'midén'), and did not use a digit placeholder for it.<a href="#cite_note-21">[20]</a> According to mathematician <a href="/wiki/Charles_Seife" title="Charles Seife">Charles Seife</a>, the ancient Greeks did begin to adopt the Babylonian placeholder zero for their work in <a href="/wiki/Ancient_Greek_astronomy" title="Ancient Greek astronomy">astronomy</a> after 500 BC, representing it with the lowercase Greek letter ό (όμικρον: <a href="/wiki/Omicron" title="Omicron">omicron</a>). However, after using the Babylonian placeholder zero for astronomical calculations they would typically convert the numbers back into <a href="/wiki/Greek_numerals" title="Greek numerals">Greek numerals</a>. Greeks seemed to have a philosophical opposition to using zero as a number.<a href="#cite_note-Seife2000-22">[21]</a> Other scholars give the Greek partial adoption of the Babylonian zero a later date, with neuroscientist Andreas Nieder giving a date of after 400 BC and mathematician Robert Kaplan dating it after the <a href="/wiki/Wars_of_Alexander_the_Great" title="Wars of Alexander the Great">conquests of Alexander</a>.<a href="#cite_note-Nieder2019-23">[22]</a><a href="#cite_note-FOOTNOTEKaplan200017-24">[23]</a> Greeks seemed unsure about the status of zero as a number. Some of them asked themselves, "How can not being be?", leading to philosophical and, by the <a href="/wiki/Medieval" class="mw-redirect" title="Medieval">medieval</a> period, religious arguments about the nature and existence of zero and the <a href="/wiki/Vacuum" title="Vacuum">vacuum</a>. The <a href="/wiki/Zeno%27s_paradoxes" title="Zeno's paradoxes">paradoxes</a> of <a href="/wiki/Zeno_of_Elea" title="Zeno of Elea">Zeno of Elea</a> depend in large part on the uncertain interpretation of zero.<a href="#cite_note-25">[24]</a> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:P._Lund,_Inv._35a.jpg" class="mw-file-description"><noscript><img alt="Fragment of papyrus with clear Greek script, lower-right corner suggests a tiny zero with a double-headed arrow shape above it" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1b/P._Lund%2C_Inv._35a.jpg/310px-P._Lund%2C_Inv._35a.jpg" decoding="async" width="310" height="169" class="mw-file-element" data-file-width="550" data-file-height="300"></noscript><span class="lazy-image-placeholder" style="width: 310px;height: 169px;" data-mw-src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1b/P._Lund%2C_Inv._35a.jpg/310px-P._Lund%2C_Inv._35a.jpg" data-alt="Fragment of papyrus with clear Greek script, lower-right corner suggests a tiny zero with a double-headed arrow shape above it" data-width="310" data-height="169" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1b/P._Lund%2C_Inv._35a.jpg/465px-P._Lund%2C_Inv._35a.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/1/1b/P._Lund%2C_Inv._35a.jpg 2x" data-class="mw-file-element"> </a><figcaption>Example of the early Greek symbol for zero (lower right corner) from a 2nd-century papyrus</figcaption></figure> By AD 150, <a href="/wiki/Ptolemy" title="Ptolemy">Ptolemy</a>, influenced by <a href="/wiki/Hipparchus" title="Hipparchus">Hipparchus</a> and the <a href="/wiki/Babylonia" title="Babylonia">Babylonians</a>, was using a symbol for zero (—°)<a href="#cite_note-26">[25]</a><a href="#cite_note-Mercier-27">[26]</a> in his work on <a href="/wiki/Mathematical_astronomy" class="mw-redirect" title="Mathematical astronomy">mathematical astronomy</a> called the Syntaxis Mathematica, also known as the <a href="/wiki/Almagest" title="Almagest">Almagest</a>.<a href="#cite_note-Ptolemy-28">[27]</a> This <a href="/wiki/Greek_numerals#Hellenistic_zero" title="Greek numerals">Hellenistic zero</a> was perhaps the earliest documented use of a numeral representing zero in the Old World.<a href="#cite_note-29">[28]</a> Ptolemy used it many times in his Almagest (VI.8) for the magnitude of <a href="/wiki/Solar_eclipse" title="Solar eclipse">solar</a> and <a href="/wiki/Lunar_eclipse" title="Lunar eclipse">lunar eclipses</a>. It represented the value of both <a href="/wiki/Digit_(unit)" title="Digit (unit)">digits</a> and <a href="/wiki/Minute_and_second_of_arc" title="Minute and second of arc">minutes</a> of immersion at first and last contact. Digits varied continuously from 0 to 12 to 0 as the Moon passed over the Sun (a triangular pulse), where twelve digits was the <a href="/wiki/Angular_diameter" title="Angular diameter">angular diameter</a> of the Sun. Minutes of immersion was tabulated from 0′0″ to 31′20″ to 0′0″, where 0′0″ used the symbol as a placeholder in two positions of his <a href="/wiki/Sexagesimal" title="Sexagesimal">sexagesimal</a> positional numeral system,<a href="#cite_note-30">[b]</a> while the combination meant a zero angle. Minutes of immersion was also a continuous function <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>⁠1/12⁠ 31′20″ √d(24−d) (a triangular pulse with <a href="/wiki/Convex_lens" class="mw-redirect" title="Convex lens">convex</a> sides), where d was the digit function and 31′20″ was the sum of the radii of the Sun's and Moon's discs.<a href="#cite_note-Pedersen-31">[29]</a> Ptolemy's symbol was a placeholder as well as a number used by two continuous mathematical functions, one within another, so it meant zero, not none. Over time, Ptolemy's zero tended to increase in size and lose the <a href="/wiki/Overline" title="Overline">overline</a>, sometimes depicted as a large elongated 0-like omicron "Ο" or as omicron with overline "ō" instead of a dot with overline.<a href="#cite_note-32">[30]</a> The earliest use of zero in the calculation of the <a href="/wiki/Computus" class="mw-redirect" title="Computus">Julian Easter</a> occurred before AD 311, at the first entry in a table of <a href="/wiki/Epact" title="Epact">epacts</a> as preserved in an <a href="/wiki/Ethiopia" title="Ethiopia">Ethiopic</a> document for the years 311 to 369, using a <a href="/wiki/Ge%27ez" class="mw-redirect" title="Ge'ez">Ge'ez</a> word for "none" (English translation is "0" elsewhere) alongside Ge'ez numerals (based on Greek numerals), which was translated from an equivalent table published by the <a href="/wiki/Church_of_Alexandria" title="Church of Alexandria">Church of Alexandria</a> in <a href="/wiki/Medieval_Greek" title="Medieval Greek">Medieval Greek</a>.<a href="#cite_note-Neugebauer-33">[31]</a> This use was repeated in 525 in an equivalent table, that was translated via the Latin nulla ("none") by <a href="/wiki/Dionysius_Exiguus" title="Dionysius Exiguus">Dionysius Exiguus</a>, alongside <a href="/wiki/Roman_numerals#Zero" title="Roman numerals">Roman numerals</a>.<a href="#cite_note-Dionysius-34">[32]</a> When division produced zero as a remainder, nihil, meaning "nothing", was used. These medieval zeros were used by all future medieval <a href="/wiki/Computus" class="mw-redirect" title="Computus">calculators of Easter</a>. The initial "N" was used as a zero symbol in a table of Roman numerals by <a href="/wiki/Bede" title="Bede">Bede</a>—or his colleagues—around AD 725.<a href="#cite_note-zero-35">[33]</a> In most <a href="/wiki/History_of_mathematics" title="History of mathematics">cultures</a>, 0 was identified before the idea of negative things (i.e., quantities less than zero) was accepted.[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] <div class="mw-heading mw-heading3"><h3 id="China">China</h3></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Zero_in_Rod_Calculus.png" class="mw-file-description"><noscript><img alt="Five illustrated boxes from left to right contain a T-shape, an empty box, three vertical bars, three lower horizontal bars with an inverted wide T-shape above, and another empty box. Numerals underneath left to right are six, zero, three, nine, and zero" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/17/Zero_in_Rod_Calculus.png/220px-Zero_in_Rod_Calculus.png" decoding="async" width="220" height="80" class="mw-file-element" data-file-width="804" data-file-height="293"></noscript><span class="lazy-image-placeholder" style="width: 220px;height: 80px;" data-mw-src="//upload.wikimedia.org/wikipedia/commons/thumb/1/17/Zero_in_Rod_Calculus.png/220px-Zero_in_Rod_Calculus.png" data-alt="Five illustrated boxes from left to right contain a T-shape, an empty box, three vertical bars, three lower horizontal bars with an inverted wide T-shape above, and another empty box. Numerals underneath left to right are six, zero, three, nine, and zero" data-width="220" data-height="80" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/17/Zero_in_Rod_Calculus.png/330px-Zero_in_Rod_Calculus.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/17/Zero_in_Rod_Calculus.png/440px-Zero_in_Rod_Calculus.png 2x" data-class="mw-file-element"> </a><figcaption>This is a depiction of zero expressed in Chinese <a href="/wiki/Counting_rods" title="Counting rods">counting rods</a>, based on the example provided by A History of Mathematics. An empty space is used to represent zero.<a href="#cite_note-Hodgkin-36">[34]</a></figcaption></figure> The <a href="/wiki/Sunzi_Suanjing" title="Sunzi Suanjing">Sūnzĭ Suànjīng</a>, of unknown date but estimated to be dated from the 1st to 5th centuries AD, describe how the 4th century BC Chinese <a href="/wiki/Counting_rods" title="Counting rods">counting rods</a> system enabled one to perform decimal calculations. As noted in the <a href="/wiki/Xiahou_Yang_Suanjing" title="Xiahou Yang Suanjing">Xiahou Yang Suanjing</a> (425–468 AD), to multiply or divide a number by 10, 100, 1000, or 10000, all one needs to do, with rods on the counting board, is to move them forwards, or back, by 1, 2, 3, or 4 places.<a href="#cite_note-37">[35]</a> The rods gave the decimal representation of a number, with an empty space denoting zero.<a href="#cite_note-Hodgkin-36">[34]</a><a href="#cite_note-38">[36]</a> The counting rod system is a <a href="/wiki/Positional_notation" title="Positional notation">positional notation</a> system.<a href="#cite_note-39">[37]</a><a href="#cite_note-40">[38]</a> Zero was not treated as a number at that time, but as a "vacant position".<a href="#cite_note-Crossley-41">[39]</a> <a href="/wiki/Qin_Jiushao" title="Qin Jiushao">Qín Jiǔsháo</a>'s 1247 <a href="/wiki/Mathematical_Treatise_in_Nine_Sections" title="Mathematical Treatise in Nine Sections">Mathematical Treatise in Nine Sections</a> is the oldest surviving Chinese mathematical text using a round symbol ‘〇’ for zero.<a href="#cite_note-Qin-42">[40]</a> The origin of this symbol is unknown; it may have been produced by modifying a square symbol.<a href="#cite_note-43">[41]</a> Chinese authors had been familiar with the idea of negative numbers by the <a href="/wiki/Han_dynasty" title="Han dynasty">Han dynasty</a> (2nd century AD), as seen in <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>.<a href="#cite_note-struik33-44">[42]</a> <div class="mw-heading mw-heading3"><h3 id="India">India</h3></div> <a href="/wiki/Pingala" title="Pingala">Pingala</a> (<abbr title="circa">c.</abbr> 3rd or 2nd century BC),<a href="#cite_note-plofker-45">[43]</a> a <a href="/wiki/Sanskrit_prosody" title="Sanskrit prosody">Sanskrit prosody</a> scholar,<a href="#cite_note-46">[44]</a> used <a href="/wiki/Binary_numeral_system" class="mw-redirect" title="Binary numeral system">binary sequences</a>, in the form of short and long syllables (the latter equal in length to two short syllables), to identify the possible valid Sanskrit <a href="/wiki/Metre_(poetry)#Disyllables" title="Metre (poetry)">meters</a>, a notation similar to <a href="/wiki/Morse_code" title="Morse code">Morse code</a>.<a href="#cite_note-47">[45]</a> Pingala used the <a href="/wiki/Sanskrit" title="Sanskrit">Sanskrit</a> word <a href="/wiki/%C5%9A%C5%ABnyat%C4%81" title="Śūnyatā">śūnya</a> explicitly to refer to zero.<a href="#cite_note-plofker-45">[43]</a> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Bakhshali_manuscript_zero_detail.jpg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/Bakhshali_manuscript_zero_detail.jpg/220px-Bakhshali_manuscript_zero_detail.jpg" decoding="async" width="220" height="89" class="mw-file-element" data-file-width="498" data-file-height="202"></noscript><span class="lazy-image-placeholder" style="width: 220px;height: 89px;" data-mw-src="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/Bakhshali_manuscript_zero_detail.jpg/220px-Bakhshali_manuscript_zero_detail.jpg" data-width="220" data-height="89" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/Bakhshali_manuscript_zero_detail.jpg/330px-Bakhshali_manuscript_zero_detail.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/83/Bakhshali_manuscript_zero_detail.jpg/440px-Bakhshali_manuscript_zero_detail.jpg 2x" data-class="mw-file-element"> </a><figcaption>Bakhshali manuscript, with the numeral "zero" represented by a black dot; its date is uncertain.<a href="#cite_note-Devlin_2017-48">[46]</a></figcaption></figure> The concept of zero as a written digit in the decimal place value notation was developed in <a href="/wiki/Indian_subcontinent" title="Indian subcontinent">India</a>.<a href="#cite_note-bourbaki46-49">[47]</a> A symbol for zero, a large dot likely to be the precursor of the still-current hollow symbol, is used throughout the <a href="/wiki/Bakhshali_manuscript" title="Bakhshali manuscript">Bakhshali manuscript</a>, a practical manual on arithmetic for merchants.<a href="#cite_note-Weiss-50">[48]</a> In 2017, researchers at the <a href="/wiki/Bodleian_Library" title="Bodleian Library">Bodleian Library</a> reported <a href="/wiki/Radiocarbon_dating" title="Radiocarbon dating">radiocarbon dating</a> results for three samples from the manuscript, indicating that they came from three different centuries: from AD 224–383, AD 680–779, and AD 885–993. It is not known how the <a href="/wiki/Birch" title="Birch">birch</a> bark fragments from different centuries forming the manuscript came to be packaged together. If the writing on the oldest birch bark fragments is as old as those fragments, it represents South Asia's oldest recorded use of a zero symbol. However, it is possible that the writing dates instead to the time period of the youngest fragments, AD 885–993. The latter dating has been argued to be more consistent with the sophisticated use of zero within the document, as portions of it appear to show zero being employed as a number in its own right, rather than only as a positional placeholder.<a href="#cite_note-Devlin_2017-48">[46]</a><a href="#cite_note-51">[49]</a><a href="#cite_note-52">[50]</a> The <a href="/wiki/Lokavibhaga" title="Lokavibhaga">Lokavibhāga</a>, a <a href="/wiki/Jain" class="mw-redirect" title="Jain">Jain</a> text on <a href="/wiki/Cosmology" title="Cosmology">cosmology</a> surviving in a medieval Sanskrit translation of the <a href="/wiki/Prakrit" title="Prakrit">Prakrit</a> original, which is internally dated to AD 458 (<a href="/wiki/Saka_era" class="mw-redirect" title="Saka era">Saka era</a> 380), uses a decimal <a href="/wiki/Positional_notation" title="Positional notation">place-value system</a>, including a zero. In this text, <a href="/wiki/%C5%9A%C5%ABnyat%C4%81" title="Śūnyatā">śūnya</a> ("void, empty") is also used to refer to zero.<a href="#cite_note-FOOTNOTEIfrah2000416-53">[51]</a> The <a href="/wiki/Aryabhatiya" title="Aryabhatiya">Aryabhatiya</a> (<abbr title="circa">c.</abbr> 499), states sthānāt sthānaṁ daśaguṇaṁ syāt "from place to place each is ten times the preceding".<a href="#cite_note-Aryab-54">[52]</a><a href="#cite_note-55">[53]</a><a href="#cite_note-56">[54]</a> Rules governing the use of zero appeared in <a href="/wiki/Brahmagupta" title="Brahmagupta">Brahmagupta</a>'s <a href="/wiki/Br%C4%81hmasphu%E1%B9%ADasiddh%C4%81nta" title="Brāhmasphuṭasiddhānta">Brahmasputha Siddhanta</a> (7th century), which states the sum of zero with itself as zero, and incorrectly describes <a href="/wiki/Division_by_zero" title="Division by zero">division by zero</a> in the following way:<a href="#cite_note-brahmagupta-57">[55]</a><a href="#cite_note-FOOTNOTEKaplan2000%5Bhttpsarchiveorgdetailsnothingthatisnat00kaplpage68_68%E2%80%9375%5D-58">[56]</a> <blockquote>A positive or negative number when divided by zero is a fraction with the zero as denominator. Zero divided by a negative or positive number is either zero or is expressed as a fraction with zero as numerator and the finite quantity as denominator. Zero divided by zero is zero.</blockquote> <div class="mw-heading mw-heading4"><h4 id="Epigraphy">Epigraphy</h4></div> <style data-mw-deduplicate="TemplateStyles:r1273380762/mw-parser-output/.tmulti">.mw-parser-output .tmulti .multiimageinner{display:flex;flex-direction:column}.mw-parser-output .tmulti .trow{display:flex;flex-direction:row;clear:left;flex-wrap:wrap;width:100%;box-sizing:border-box}.mw-parser-output .tmulti .tsingle{margin:1px;float:left}.mw-parser-output .tmulti .theader{clear:both;font-weight:bold;text-align:center;align-self:center;background-color:transparent;width:100%}.mw-parser-output .tmulti .thumbcaption{background-color:transparent}.mw-parser-output .tmulti .text-align-left{text-align:left}.mw-parser-output .tmulti .text-align-right{text-align:right}.mw-parser-output .tmulti .text-align-center{text-align:center}@media all and (max-width:720px){.mw-parser-output .tmulti .thumbinner{width:100%!important;box-sizing:border-box;max-width:none!important;align-items:center}.mw-parser-output .tmulti .trow{justify-content:center}.mw-parser-output .tmulti .tsingle{float:none!important;max-width:100%!important;box-sizing:border-box;text-align:center}.mw-parser-output .tmulti .tsingle .thumbcaption{text-align:left}.mw-parser-output .tmulti .trow>.thumbcaption{text-align:center}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .tmulti .multiimageinner span:not(.skin-invert-image):not(.skin-invert):not(.bg-transparent) img{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .tmulti .multiimageinner span:not(.skin-invert-image):not(.skin-invert):not(.bg-transparent) img{background-color:white}}</style><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:322px;max-width:322px"><div class="trow"><div class="theader">Sambor Inscription</div></div><div class="trow"><div class="tsingle" style="width:147px;max-width:147px"><div class="thumbimage" style="height:108px;overflow:hidden"><a href="/wiki/File:First_zero_1.jpg" class="mw-file-description"><noscript><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/81/First_zero_1.jpg/145px-First_zero_1.jpg" decoding="async" width="145" height="109" class="mw-file-element" data-file-width="4096" data-file-height="3072"></noscript><span class="lazy-image-placeholder" style="width: 145px;height: 109px;" data-mw-src="//upload.wikimedia.org/wikipedia/commons/thumb/8/81/First_zero_1.jpg/145px-First_zero_1.jpg" data-alt="" data-width="145" data-height="109" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/81/First_zero_1.jpg/218px-First_zero_1.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/81/First_zero_1.jpg/290px-First_zero_1.jpg 2x" data-class="mw-file-element"> </a></div></div><div class="tsingle" style="width:171px;max-width:171px"><div class="thumbimage" style="height:108px;overflow:hidden"><a href="/wiki/File:Khmer_Numerals_-_605_from_the_Sambor_inscriptions.jpg" class="mw-file-description"><noscript><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fc/Khmer_Numerals_-_605_from_the_Sambor_inscriptions.jpg/169px-Khmer_Numerals_-_605_from_the_Sambor_inscriptions.jpg" decoding="async" width="169" height="109" class="mw-file-element" data-file-width="244" data-file-height="157"></noscript><span class="lazy-image-placeholder" style="width: 169px;height: 109px;" data-mw-src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fc/Khmer_Numerals_-_605_from_the_Sambor_inscriptions.jpg/169px-Khmer_Numerals_-_605_from_the_Sambor_inscriptions.jpg" data-alt="" data-width="169" data-height="109" data-srcset="//upload.wikimedia.org/wikipedia/commons/f/fc/Khmer_Numerals_-_605_from_the_Sambor_inscriptions.jpg 1.5x" data-class="mw-file-element"> </a></div></div></div><div class="trow"><div class="tsingle" style="width:320px;max-width:320px"><div class="thumbimage" style="height:559px;overflow:hidden"><a href="/wiki/File:03-National_Museum_of_Cambodia-nX-1.jpg" class="mw-file-description"><noscript><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/90/03-National_Museum_of_Cambodia-nX-1.jpg/318px-03-National_Museum_of_Cambodia-nX-1.jpg" decoding="async" width="318" height="559" class="mw-file-element" data-file-width="2220" data-file-height="3903"></noscript><span class="lazy-image-placeholder" style="width: 318px;height: 559px;" data-mw-src="//upload.wikimedia.org/wikipedia/commons/thumb/9/90/03-National_Museum_of_Cambodia-nX-1.jpg/318px-03-National_Museum_of_Cambodia-nX-1.jpg" data-alt="" data-width="318" data-height="559" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/90/03-National_Museum_of_Cambodia-nX-1.jpg/477px-03-National_Museum_of_Cambodia-nX-1.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/90/03-National_Museum_of_Cambodia-nX-1.jpg/636px-03-National_Museum_of_Cambodia-nX-1.jpg 2x" data-class="mw-file-element"> </a></div></div></div><div class="trow" style="display:flex"><div class="thumbcaption">The oldest, firmly dated use of zero as a decimal figure, found on the Sambor Inscription. The number "605" is written in <a href="/wiki/Khmer_numerals" title="Khmer numerals">Khmer numerals</a> (top), referring to the year it was made: <a href="/wiki/Shaka_era" title="Shaka era">605 Saka era</a> (683 CE). The fragment, inscribed in <a href="/wiki/Old_Khmer" title="Old Khmer">Old Khmer</a>, was once part of a temple doorway, and was found in <a href="/wiki/Krati%C3%A9_province" title="Kratié province">Kratié province</a>, <a href="/wiki/Cambodia" title="Cambodia">Cambodia</a>.</div></div></div></div> A black dot is used as a decimal placeholder in the <a href="/wiki/Bakhshali_manuscript" title="Bakhshali manuscript">Bakhshali manuscript</a>, portions of which date from AD 224–993.<a href="#cite_note-Devlin_2017-48">[46]</a> There are numerous copper plate inscriptions, with the same small O in them, some of them possibly dated to the 6th century, but their date or authenticity may be open to doubt.<a href="#cite_note-FOOTNOTEKaplan2000-10">[10]</a> A stone tablet found in the ruins of a temple near Sambor on the <a href="/wiki/Mekong" title="Mekong">Mekong</a>, <a href="/wiki/Krati%C3%A9_Province" class="mw-redirect" title="Kratié Province">Kratié Province</a>, <a href="/wiki/Cambodia" title="Cambodia">Cambodia</a>, includes the inscription of "605" in <a href="/wiki/Khmer_numerals" title="Khmer numerals">Khmer numerals</a> (a set of numeral glyphs for the <a href="/wiki/Hindu%E2%80%93Arabic_numeral_system" title="Hindu–Arabic numeral system">Hindu–Arabic numeral system</a>). The number is the year of the inscription in the <a href="/wiki/Saka_era" class="mw-redirect" title="Saka era">Saka era</a>, corresponding to a date of AD 683.<a href="#cite_note-Cambodia-59">[57]</a> The first known use of special <a href="/wiki/Glyph" title="Glyph">glyphs</a> for the decimal digits that includes the indubitable appearance of a symbol for the digit zero, a small circle, appears on a stone inscription found at the <a href="/wiki/Chaturbhuj_Temple,_Gwalior" title="Chaturbhuj Temple, Gwalior">Chaturbhuj Temple, Gwalior</a>, in India, dated AD 876.<a href="#cite_note-60">[58]</a><a href="#cite_note-FOOTNOTEIfrah2000400-61">[59]</a> <div class="mw-heading mw-heading3"><h3 id="Middle_Ages">Middle Ages</h3></div> <div class="mw-heading mw-heading4"><h4 id="Transmission_to_Islamic_culture">Transmission to Islamic culture</h4></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/History_of_the_Hindu%E2%80%93Arabic_numeral_system" title="History of the Hindu–Arabic numeral system">History of the Hindu–Arabic numeral system</a></div> The <a href="/wiki/Arabic" title="Arabic">Arabic</a>-language inheritance of science was largely <a href="/wiki/Greece" title="Greece">Greek</a>,<a href="#cite_note-62">[60]</a> followed by Hindu influences.<a href="#cite_note-Durant-63">[61]</a> In 773, at <a href="/wiki/Al-Mansur" title="Al-Mansur">Al-Mansur</a>'s behest, translations were made of many ancient treatises including Greek, Roman, Indian, and others. In AD 813, astronomical tables were prepared by a <a href="/wiki/Persian_people" class="mw-redirect" title="Persian people">Persian</a> mathematician, <a href="/wiki/Mu%E1%B8%A5ammad_ibn_M%C5%ABs%C4%81_al-Khw%C4%81rizm%C4%AB" class="mw-redirect" title="Muḥammad ibn Mūsā al-Khwārizmī">Muḥammad ibn Mūsā al-Khwārizmī</a>, using Hindu numerals;<a href="#cite_note-Durant-63">[61]</a> and about 825, he published a book synthesizing Greek and Hindu knowledge and also contained his own contribution to mathematics including an explanation of the use of zero.<a href="#cite_note-64">[62]</a> This book was later translated into <a href="/wiki/Latin" title="Latin">Latin</a> in the 12th century under the title Algoritmi de numero Indorum. This title means "al-Khwarizmi on the Numerals of the Indians". The word "Algoritmi" was the translator's Latinization of Al-Khwarizmi's name, and the word "<a href="/wiki/Algorithm" title="Algorithm">Algorithm</a>" or "<a href="/wiki/Algorism" title="Algorism">Algorism</a>" started to acquire a meaning of any arithmetic based on decimals.<a href="#cite_note-Durant-63">[61]</a> <a href="/wiki/Muhammad_ibn_Ahmad_al-Khwarizmi" title="Muhammad ibn Ahmad al-Khwarizmi">Muhammad ibn Ahmad al-Khwarizmi</a>, in 976, stated that if no number appears in the place of tens in a calculation, a little circle should be used "to keep the rows". This circle was called ṣifr.<a href="#cite_note-65">[63]</a> <div class="mw-heading mw-heading4"><h4 id="Transmission_to_Europe">Transmission to Europe</h4></div> The <a href="/wiki/Hindu%E2%80%93Arabic_numeral_system" title="Hindu–Arabic numeral system">Hindu–Arabic numeral system</a> (base 10) reached Western Europe in the 11th century, via <a href="/wiki/Al-Andalus" title="Al-Andalus">Al-Andalus</a>, through Spanish <a href="/wiki/Muslim" class="mw-redirect" title="Muslim">Muslims</a>, the <a href="/wiki/Moors" title="Moors">Moors</a>, together with knowledge of <a href="/wiki/Classical_astronomy" class="mw-redirect" title="Classical astronomy">classical astronomy</a> and instruments like the <a href="/wiki/Astrolabe" title="Astrolabe">astrolabe</a>. <a href="/wiki/Pope_Sylvester_II" title="Pope Sylvester II">Gerbert of Aurillac</a> is credited with reintroducing the lost teachings into Catholic Europe. For this reason, the numerals came to be known in Europe as "Arabic numerals". The Italian mathematician <a href="/wiki/Fibonacci" title="Fibonacci">Fibonacci</a> or Leonardo of Pisa was instrumental in bringing the system into European mathematics in 1202, stating: <blockquote>After my father's appointment by <a href="/wiki/Republic_of_Pisa" title="Republic of Pisa">his homeland</a> as state official in the customs house of <a href="/wiki/B%C3%A9ja%C3%AFa" title="Béjaïa">Bugia</a> for the Pisan merchants who thronged to it, he took charge; and in view of its future usefulness and convenience, had me in my boyhood come to him and there wanted me to devote myself to and be instructed in the study of calculation for some days. There, following my introduction, as a consequence of marvelous instruction in the art, to the nine digits of the Hindus, the knowledge of the art very much appealed to me before all others, and for it I realized that all its aspects were studied in Egypt, Syria, Greece, Sicily, and Provence, with their varying methods; and at these places thereafter, while on business. I pursued my study in depth and learned the give-and-take of disputation. But all this even, and the <a href="/wiki/Algorism" title="Algorism">algorism</a>, as well as the art of <a href="/wiki/Pythagoras" title="Pythagoras">Pythagoras</a>, I considered as almost a mistake in respect to the method of the <a href="/wiki/Hinduism" title="Hinduism">Hindus</a> [Modus Indorum]. Therefore, embracing more stringently that method of the Hindus, and taking stricter pains in its study, while adding certain things from my own understanding and inserting also certain things from the niceties of <a href="/wiki/Euclid" title="Euclid">Euclid</a>'s geometric art. I have striven to compose this book in its entirety as understandably as I could, dividing it into fifteen chapters. Almost everything which I have introduced I have displayed with exact proof, in order that those further seeking this knowledge, with its pre-eminent method, might be instructed, and further, in order that the <a href="/wiki/Latin_people" class="mw-redirect" title="Latin people">Latin people</a> might not be discovered to be without it, as they have been up to now. If I have perchance omitted anything more or less proper or necessary, I beg indulgence, since there is no one who is blameless and utterly provident in all things. The nine Indian figures are: 9 8 7 6 5 4 3 2 1. With these nine figures, and with the sign 0 ... any number may be written.<a href="#cite_note-66">[64]</a></blockquote> From the 13th century, manuals on calculation (adding, multiplying, extracting roots, etc.) became common in Europe where they were called algorismus after the Persian mathematician <a href="/wiki/Al-Khw%C4%81rizm%C4%AB" class="mw-redirect" title="Al-Khwārizmī">al-Khwārizmī</a>. One popular manual was written by <a href="/wiki/Johannes_de_Sacrobosco" title="Johannes de Sacrobosco">Johannes de Sacrobosco</a> in the early 1200s and was one of the earliest scientific books to be <a href="/wiki/History_of_printing" title="History of printing">printed</a>, in 1488.<a href="#cite_note-Karpinski1911-67">[65]</a><a href="#cite_note-68">[66]</a> The practice of calculating on paper using Hindu–Arabic numerals only gradually displaced calculation by abacus and recording with <a href="/wiki/Roman_numerals" title="Roman numerals">Roman numerals</a>.<a href="#cite_note-FOOTNOTEIfrah2000588%E2%80%93590-69">[67]</a> In the 16th century, Hindu–Arabic numerals became the predominant numerals used in Europe.<a href="#cite_note-Karpinski1911-67">[65]</a> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(3)"><h2 id="Symbols_and_representations">Symbols and representations</h2></div><section class="mf-section-3 collapsible-block" id="mf-section-3"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Symbols_for_zero" title="Symbols for zero">Symbols for zero</a></div> <figure class="mw-halign-left" typeof="mw:File"><a href="/wiki/File:Text_figures_036.svg" class="mw-file-description"><noscript><img alt="horizontal guidelines with a zero touching top and bottom, a three dipping below, and a six cresting above the guidelines, from left to right" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c4/Text_figures_036.svg/71px-Text_figures_036.svg.png" decoding="async" width="71" height="34" class="mw-file-element" data-file-width="85" data-file-height="41"></noscript><span class="lazy-image-placeholder" style="width: 71px;height: 34px;" data-mw-src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c4/Text_figures_036.svg/71px-Text_figures_036.svg.png" data-alt="horizontal guidelines with a zero touching top and bottom, a three dipping below, and a six cresting above the guidelines, from left to right" data-width="71" data-height="34" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c4/Text_figures_036.svg/107px-Text_figures_036.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c4/Text_figures_036.svg/142px-Text_figures_036.svg.png 2x" data-class="mw-file-element"> </a><figcaption></figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Oslo_airport_train_station,_Platform_0.jpg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ad/Oslo_airport_train_station%2C_Platform_0.jpg/220px-Oslo_airport_train_station%2C_Platform_0.jpg" decoding="async" width="220" height="165" class="mw-file-element" data-file-width="3648" data-file-height="2736"></noscript><span class="lazy-image-placeholder" style="width: 220px;height: 165px;" data-mw-src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ad/Oslo_airport_train_station%2C_Platform_0.jpg/220px-Oslo_airport_train_station%2C_Platform_0.jpg" data-width="220" data-height="165" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ad/Oslo_airport_train_station%2C_Platform_0.jpg/330px-Oslo_airport_train_station%2C_Platform_0.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ad/Oslo_airport_train_station%2C_Platform_0.jpg/440px-Oslo_airport_train_station%2C_Platform_0.jpg 2x" data-class="mw-file-element"> </a><figcaption>Oslo airport train station, Platform 0</figcaption></figure> Today, the numerical digit 0 is usually written as a circle or ellipse. Traditionally, many print <a href="/wiki/Typeface" title="Typeface">typefaces</a> made the capital letter <a href="/wiki/O" title="O">O</a> more rounded than the narrower, elliptical digit 0.<a href="#cite_note-bemer-70">[68]</a> <a href="/wiki/Typewriter" title="Typewriter">Typewriters</a> originally made no distinction in shape between O and 0; some models did not even have a separate key for the digit 0. The distinction came into prominence on modern character <a href="/wiki/Visual_display_unit" class="mw-redirect" title="Visual display unit">displays</a>.<a href="#cite_note-bemer-70">[68]</a> A <a href="/wiki/Slashed_zero" title="Slashed zero">slashed zero</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\!\!\!{/}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\!\!\!{/}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4059f8b4ddbe8f89a13eb4ccc1e87cf3e776d8db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.164ex; height:2.843ex;" alt="{\displaystyle 0\!\!\!{/}}"></noscript><span class="lazy-image-placeholder" style="width: 1.164ex;height: 2.843ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4059f8b4ddbe8f89a13eb4ccc1e87cf3e776d8db" data-alt="{\displaystyle 0\!\!\!{/}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> ) is often used to distinguish the number from the letter (mostly in computing, navigation and in the military, for example). The digit 0 with a dot in the center seems to have originated as an option on <a href="/wiki/IBM_3270" title="IBM 3270">IBM 3270</a> displays and has continued with some modern computer typefaces such as <a href="/wiki/Andal%C3%A9_Mono" title="Andalé Mono">Andalé Mono</a>, and in some airline reservation systems. One variation uses a short vertical bar instead of the dot. Some fonts designed for use with computers made the "0" character more squared at the edges, like a rectangle, and the "O" character more rounded. A further distinction is made in <a href="/wiki/FE-Schrift" title="FE-Schrift">falsification-hindering typeface</a> as used on <a href="/wiki/Vehicle_registration_plates_of_Germany" title="Vehicle registration plates of Germany">German car number plates</a> by slitting open the digit 0 on the upper right side. In some systems either the letter O or the numeral 0, or both, are excluded from use, to avoid confusion. </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(4)"><h2 id="Mathematics">Mathematics</h2></div><section class="mf-section-4 collapsible-block" id="mf-section-4"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Null_(mathematics)" title="Null (mathematics)">Null (mathematics)</a></div> The concept of zero plays multiple roles in mathematics: as a digit, it is an important part of positional notation for representing numbers, while it also plays an important role as a number in its own right in many algebraic settings. <div class="mw-heading mw-heading3"><h3 id="As_a_digit">As a digit</h3></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/Positional_notation" title="Positional notation">Positional notation</a></div> In positional number systems (such as the usual <a href="/wiki/Decimal_notation" class="mw-redirect" title="Decimal notation">decimal notation</a> for representing numbers), the digit 0 plays the role of a placeholder, indicating that certain powers of the base do not contribute. For example, the decimal number 205 is the sum of two hundreds and five ones, with the 0 digit indicating that no tens are added. The digit plays the same role in <a href="/wiki/Decimal_fractions" class="mw-redirect" title="Decimal fractions">decimal fractions</a> and in the <a href="/wiki/Decimal_representation" title="Decimal representation">decimal representation</a> of other real numbers (indicating whether any tenths, hundredths, thousandths, etc., are present) and in bases other than 10 (for example, in binary, where it indicates which powers of 2 are omitted).<a href="#cite_note-FOOTNOTEReimer2014156,_199%E2%80%93204-71">[69]</a> <div class="mw-heading mw-heading3"><h3 id="Elementary_algebra">Elementary algebra</h3></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Number_line_with_numbers_-3_to_3.svg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/19/Number_line_with_numbers_-3_to_3.svg/310px-Number_line_with_numbers_-3_to_3.svg.png" decoding="async" width="310" height="37" class="mw-file-element" data-file-width="457" data-file-height="54"></noscript><span class="lazy-image-placeholder" style="width: 310px;height: 37px;" data-mw-src="//upload.wikimedia.org/wikipedia/commons/thumb/1/19/Number_line_with_numbers_-3_to_3.svg/310px-Number_line_with_numbers_-3_to_3.svg.png" data-width="310" data-height="37" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/19/Number_line_with_numbers_-3_to_3.svg/465px-Number_line_with_numbers_-3_to_3.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/19/Number_line_with_numbers_-3_to_3.svg/620px-Number_line_with_numbers_-3_to_3.svg.png 2x" data-class="mw-file-element"> </a><figcaption>A <a href="/wiki/Number_line" title="Number line">number line</a> from −3 to 3, with 0 in the middle</figcaption></figure> The number 0 is the smallest <a href="/wiki/Nonnegative_integer" class="mw-redirect" title="Nonnegative integer">nonnegative integer</a>, and the largest nonpositive integer. The <a href="/wiki/Natural_number" title="Natural number">natural number</a> following 0 is 1 and no natural number precedes 0. The number 0 <a href="/wiki/Natural_number" title="Natural number">may or may not be considered a natural number</a>,<a href="#cite_note-72">[70]</a><a href="#cite_note-FOOTNOTECheng201732-73">[71]</a> but it is an <a href="/wiki/Integer" title="Integer">integer</a>, and hence a <a href="/wiki/Rational_number" title="Rational number">rational number</a> and a <a href="/wiki/Real_number" title="Real number">real number</a>.<a href="#cite_note-FOOTNOTECheng201741,_48%E2%80%9353-74">[72]</a> All rational numbers are <a href="/wiki/Algebraic_number" title="Algebraic number">algebraic numbers</a>, including 0. When the real numbers are extended to form the <a href="/wiki/Complex_number" title="Complex number">complex numbers</a>, 0 becomes the <a href="/wiki/Origin_(mathematics)" title="Origin (mathematics)">origin</a> of the complex plane. The number 0 can be regarded as neither positive nor negative<a href="#cite_note-75">[73]</a> or, alternatively, both positive and negative<a href="#cite_note-76">[74]</a> and is usually displayed as the central number in a <a href="/wiki/Number_line" title="Number line">number line</a>. Zero is <a href="/wiki/Parity_(mathematics)" title="Parity (mathematics)">even</a><a href="#cite_note-77">[75]</a> (that is, a multiple of 2), and is also an <a href="/wiki/Integer_multiple" class="mw-redirect" title="Integer multiple">integer multiple</a> of any other integer, rational, or real number. It is neither a <a href="/wiki/Prime_number" title="Prime number">prime number</a> nor a <a href="/wiki/Composite_number" title="Composite number">composite number</a>: it is not prime because prime numbers are greater than 1 by definition, and it is not composite because it cannot be expressed as the product of two smaller natural numbers.<a href="#cite_note-78">[76]</a> (However, the <a href="/wiki/Singleton_set" class="mw-redirect" title="Singleton set">singleton set</a> {0} is a <a href="/wiki/Prime_ideal" title="Prime ideal">prime ideal</a> in the <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">ring</a> of the integers.) The following are some basic rules for dealing with the number 0. These rules apply for any real or complex number x, unless otherwise stated. <ul><li><a href="/wiki/Addition" title="Addition">Addition</a>: x + 0 = 0 + x = x. That is, 0 is an <a href="/wiki/Identity_element" title="Identity element">identity element</a> (or neutral element) with respect to addition.</li> <li><a href="/wiki/Subtraction" title="Subtraction">Subtraction</a>: x − 0 = x and 0 − x = −x.</li> <li><a href="/wiki/Multiplication" title="Multiplication">Multiplication</a>: x · 0 = 0 · x = 0.</li> <li><a href="/wiki/Division_(mathematics)" title="Division (mathematics)">Division</a>: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠0/x⁠ = 0, for nonzero x. But <a href="/wiki/Division_by_zero" title="Division by zero"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠x/0⁠</a> is <a href="/wiki/Defined_and_undefined" class="mw-redirect" title="Defined and undefined">undefined</a>, because 0 has no <a href="/wiki/Multiplicative_inverse" title="Multiplicative inverse">multiplicative inverse</a> (no real number multiplied by 0 produces 1), a consequence of the previous rule.<a href="#cite_note-FOOTNOTECheng201747-79">[77]</a></li> <li><a href="/wiki/Exponentiation" title="Exponentiation">Exponentiation</a>: x0 = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠x/x⁠ = 1, except that <a href="/wiki/Zero_to_the_power_of_zero" title="Zero to the power of zero">the case x = 0</a> is considered undefined in some contexts. For all positive real x, 0x = 0.</li></ul> The expression <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠0/0⁠, which may be obtained in an attempt to determine the limit of an expression of the form <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠f(x)/g(x)⁠ as a result of applying the <a href="/wiki/Limit_of_a_function" title="Limit of a function">lim</a> operator independently to both operands of the fraction, is a so-called "<a href="/wiki/Indeterminate_form" title="Indeterminate form">indeterminate form</a>". That does not mean that the limit sought is necessarily undefined; rather, it means that the limit of <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠f(x)/g(x)⁠, if it exists, must be found by another method, such as <a href="/wiki/L%27H%C3%B4pital%27s_rule" title="L'Hôpital's rule">l'Hôpital's rule</a>.<a href="#cite_note-80">[78]</a> The sum of 0 numbers (the <a href="/wiki/Empty_sum" title="Empty sum">empty sum</a>) is 0, and the product of 0 numbers (the <a href="/wiki/Empty_product" title="Empty product">empty product</a>) is 1. The <a href="/wiki/Factorial" title="Factorial">factorial</a> 0! evaluates to 1, as a special case of the empty product.<a href="#cite_note-gkp-81">[79]</a> <div class="mw-heading mw-heading3"><h3 id="Other_uses_in_mathematics">Other uses in mathematics</h3></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Nullset.svg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6a/Nullset.svg/90px-Nullset.svg.png" decoding="async" width="90" height="116" class="mw-file-element" data-file-width="126" data-file-height="163"></noscript><span class="lazy-image-placeholder" style="width: 90px;height: 116px;" data-mw-src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6a/Nullset.svg/90px-Nullset.svg.png" data-width="90" data-height="116" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6a/Nullset.svg/135px-Nullset.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6a/Nullset.svg/180px-Nullset.svg.png 2x" data-class="mw-file-element"> </a><figcaption>The empty set has zero elements</figcaption></figure>The role of 0 as the smallest counting number can be generalized or extended in various ways. In <a href="/wiki/Set_theory" title="Set theory">set theory</a>, 0 is the <a href="/wiki/Cardinality" title="Cardinality">cardinality</a> of the <a href="/wiki/Empty_set" title="Empty set">empty set</a> (notated as "{ }", "<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \emptyset }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∅</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \emptyset }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6af50205f42bb2ec3c666b7b847d2c7f96e464c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.162ex; height:2.509ex;" alt="{\displaystyle \emptyset }"></noscript><span class="lazy-image-placeholder" style="width: 1.162ex;height: 2.509ex;vertical-align: -0.505ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6af50205f42bb2ec3c666b7b847d2c7f96e464c7" data-alt="{\displaystyle \emptyset }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> ", or "∅"): if one does not have any apples, then one has 0 apples. In fact, in certain axiomatic developments of mathematics from set theory, 0 is <a href="/wiki/Definition" title="Definition">defined</a> to be the empty set.<a href="#cite_note-FOOTNOTECheng201760-82">[80]</a> When this is done, the empty set is the <a href="/wiki/Von_Neumann_cardinal_assignment" title="Von Neumann cardinal assignment">von Neumann cardinal assignment</a> for a set with no elements, which is the empty set. The cardinality function, applied to the empty set, returns the empty set as a value, thereby assigning it 0 elements. Also in set theory, 0 is the lowest <a href="/wiki/Ordinal_number" title="Ordinal number">ordinal number</a>, corresponding to the empty set viewed as a <a href="/wiki/Well-order" title="Well-order">well-ordered set</a>. In <a href="/wiki/Order_theory" title="Order theory">order theory</a> (and especially its subfield <a href="/wiki/Lattice_theory" class="mw-redirect" title="Lattice theory">lattice theory</a>), 0 may denote the <a href="/wiki/Least_element" class="mw-redirect" title="Least element">least element</a> of a <a href="/wiki/Lattice_(order)" title="Lattice (order)">lattice</a> or other <a href="/wiki/Partially_ordered_set" title="Partially ordered set">partially ordered set</a>. The role of 0 as additive identity generalizes beyond elementary algebra. In <a href="/wiki/Abstract_algebra" title="Abstract algebra">abstract algebra</a>, 0 is commonly used to denote a <a href="/wiki/Zero_element" title="Zero element">zero element</a>, which is the <a href="/wiki/Identity_element" title="Identity element">identity element</a> for addition (if defined on the structure under consideration) and an <a href="/wiki/Absorbing_element" title="Absorbing element">absorbing element</a> for multiplication (if defined). (Such elements may also be called <a href="/wiki/Zero_element" title="Zero element">zero elements</a>.) Examples include identity elements of <a href="/wiki/Additive_group" title="Additive group">additive groups</a> and <a href="/wiki/Vector_space" title="Vector space">vector spaces</a>. Another example is the zero function (or zero map) on a domain D. This is the <a href="/wiki/Constant_function" title="Constant function">constant function</a> with 0 as its only possible output value, that is, it is the function f defined by f(x) = 0 for all x in D. As a function from the real numbers to the real numbers, the zero function is the only function that is both <a href="/wiki/Even_function" class="mw-redirect" title="Even function">even</a> and <a href="/wiki/Odd_function" class="mw-redirect" title="Odd function">odd</a>. The number 0 is also used in several other ways within various branches of mathematics: <ul><li>A <a href="/wiki/Zero_of_a_function" title="Zero of a function">zero of a function</a> f is a point x in the domain of the function such that f(x) = 0.</li> <li>In <a href="/wiki/Propositional_logic" class="mw-redirect" title="Propositional logic">propositional logic</a>, 0 may be used to denote the <a href="/wiki/Truth_value" title="Truth value">truth value</a> false.</li> <li>In <a href="/wiki/Probability_theory" title="Probability theory">probability theory</a>, 0 is the smallest allowed value for the probability of any event.<a href="#cite_note-FOOTNOTEKardar200735-83">[81]</a></li> <li><a href="/wiki/Category_theory" title="Category theory">Category theory</a> introduces the idea of a <a href="/wiki/Zero_object" class="mw-redirect" title="Zero object">zero object</a>, often denoted 0, and the related concept of <a href="/wiki/Zero_morphism" title="Zero morphism">zero morphisms</a>, which generalize the zero function.<a href="#cite_note-84">[82]</a></li></ul> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(5)"><h2 id="Physics">Physics</h2></div><section class="mf-section-5 collapsible-block" id="mf-section-5"> The value zero plays a special role for many physical quantities. For some quantities, the zero level is naturally distinguished from all other levels, whereas for others it is more or less arbitrarily chosen. For example, for an <a href="/wiki/Thermodynamic_temperature" title="Thermodynamic temperature">absolute temperature</a> (typically measured in <a href="/wiki/Kelvin" title="Kelvin">kelvins</a>), <a href="/wiki/Absolute_zero" title="Absolute zero">zero</a> is the lowest possible value. (<a href="/wiki/Negative_temperature" title="Negative temperature">Negative temperatures</a> can be defined for some physical systems, but negative-temperature systems are not actually colder.) This is in contrast to temperatures on the Celsius scale, for example, where zero is arbitrarily defined to be at the <a href="/wiki/Melting_point" title="Melting point">freezing point</a> of water.<a href="#cite_note-85">[83]</a><a href="#cite_note-FOOTNOTEKardar20074%E2%80%935,_103%E2%80%93104-86">[84]</a> Measuring sound intensity in <a href="/wiki/Decibel" title="Decibel">decibels</a> or <a href="/wiki/Phon" title="Phon">phons</a>, the zero level is arbitrarily set at a reference value—for example, at a value for the threshold of hearing. In <a href="/wiki/Physics" title="Physics">physics</a>, the <a href="/wiki/Zero-point_energy" title="Zero-point energy">zero-point energy</a> is the lowest possible energy that a <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanical</a> <a href="/wiki/Physical_system" title="Physical system">physical system</a> may possess and is the energy of the <a href="/wiki/Stationary_state" title="Stationary state">ground state</a> of the system. </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(6)"><h2 id="Computer_science">Computer science</h2></div><section class="mf-section-6 collapsible-block" id="mf-section-6"> Modern computers store information in <a href="/wiki/Binary_code" title="Binary code">binary</a>, that is, using an "alphabet" that contains only two symbols, usually chosen to be "0" and "1". Binary coding is convenient for <a href="/wiki/Digital_electronics" title="Digital electronics">digital electronics</a>, where "0" and "1" can stand for the absence or presence of electrical current in a wire.<a href="#cite_note-FOOTNOTEWoodford20069-87">[85]</a> <a href="/wiki/Computer_programming" title="Computer programming">Computer programmers</a> typically use <a href="/wiki/High-level_programming_language" title="High-level programming language">high-level programming languages</a> that are more easily intelligible to humans than the <a href="/wiki/Machine_code" title="Machine code">binary instructions</a> that are directly executed by the <a href="/wiki/Central_processing_unit" title="Central processing unit">central processing unit</a>. 0 plays various important roles in high-level languages. For example, a <a href="/wiki/Boolean_data_type" title="Boolean data type">Boolean variable</a> stores a value that is either true or false, and 0 is often the numerical representation of false.<a href="#cite_note-FOOTNOTEHill202020-88">[86]</a> 0 also plays a role in <a href="/wiki/Array_(data_type)" title="Array (data type)">array</a> indexing. The most common practice throughout human history has been to start counting at one, and this is the practice in early classic programming languages such as <a href="/wiki/Fortran" title="Fortran">Fortran</a> and <a href="/wiki/COBOL" title="COBOL">COBOL</a>.<a href="#cite_note-89">[87]</a> However, in the late 1950s <a href="/wiki/LISP" class="mw-redirect" title="LISP">LISP</a> introduced <a href="/wiki/Zero-based_numbering" title="Zero-based numbering">zero-based numbering</a> for arrays while <a href="/wiki/Algol_58" class="mw-redirect" title="Algol 58">Algol 58</a> introduced completely flexible basing for array subscripts (allowing any positive, negative, or zero integer as base for array subscripts), and most subsequent programming languages adopted one or other of these positions.[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] For example, the elements of an array are numbered starting from 0 in <a href="/wiki/C_(computer_language)" class="mw-redirect" title="C (computer language)">C</a>, so that for an array of n items the sequence of array indices runs from 0 to n−1.<a href="#cite_note-90">[88]</a> There can be confusion between 0- and 1-based indexing; for example, Java's <a href="/wiki/JDBC" class="mw-redirect" title="JDBC">JDBC</a> indexes parameters from 1 although <a href="/wiki/Java_(programming_language)" title="Java (programming language)">Java</a> itself uses 0-based indexing.<a href="#cite_note-91">[89]</a> In C, a <a href="/wiki/Byte" title="Byte">byte</a> containing the value 0 serves to indicate where a <a href="/wiki/String_(computer_science)" title="String (computer science)">string</a> of characters ends. Also, 0 is a standard way to refer to a <a href="/wiki/Null_pointer" title="Null pointer">null pointer</a> in code.<a href="#cite_note-92">[90]</a> In databases, it is possible for a field not to have a value. It is then said to have a <a href="/wiki/Null_(SQL)" title="Null (SQL)">null value</a>.<a href="#cite_note-93">[91]</a> For numeric fields it is not the value zero. For text fields this is not blank nor the empty string. The presence of null values leads to <a href="/wiki/Ternary_logic" class="mw-redirect" title="Ternary logic">three-valued logic</a>. No longer is a condition either true or false, but it can be undetermined. Any computation including a null value delivers a null result.<a href="#cite_note-94">[92]</a> In mathematics, there is no "positive zero" or "negative zero" distinct from zero; both −0 and +0 represent exactly the same number. However, in some computer hardware <a href="/wiki/Signed_number_representations" title="Signed number representations">signed number representations</a>, zero has two distinct representations, a positive one grouped with the positive numbers and a negative one grouped with the negatives. This kind of dual representation is known as <a href="/wiki/Signed_zero" title="Signed zero">signed zero</a>, with the latter form sometimes called negative zero. These representations include the <a href="/wiki/Signed_magnitude" class="mw-redirect" title="Signed magnitude">signed magnitude</a> and <a href="/wiki/Ones%27_complement" title="Ones' complement">ones' complement</a> binary integer representations (but not the <a href="/wiki/Two%27s_complement" title="Two's complement">two's complement</a> binary form used in most modern computers), and most <a href="/wiki/Floating-point_arithmetic" title="Floating-point arithmetic">floating-point</a> number representations (such as <a href="/wiki/IEEE_floating_point" class="mw-redirect" title="IEEE floating point">IEEE 754</a> and <a href="/wiki/IBM_hexadecimal_floating-point" title="IBM hexadecimal floating-point">IBM S/390</a> floating-point formats).[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] An <a href="/wiki/Epoch_(computing)" title="Epoch (computing)">epoch</a>, in computing terminology, is the date and time associated with a zero timestamp. The <a href="/wiki/Unix_epoch" class="mw-redirect" title="Unix epoch">Unix epoch</a> begins the midnight before the first of January 1970.<a href="#cite_note-95">[93]</a><a href="#cite_note-96">[94]</a><a href="#cite_note-97">[95]</a> The <a href="/wiki/Classic_Mac_OS" title="Classic Mac OS">Classic Mac OS</a> epoch and <a href="/wiki/Palm_OS" title="Palm OS">Palm OS</a> epoch begin the midnight before the first of January 1904.<a href="#cite_note-98">[96]</a> Many <a href="/wiki/Application_programming_interface" class="mw-redirect" title="Application programming interface">APIs</a> and <a href="/wiki/Operating_system" title="Operating system">operating systems</a> that require applications to return an integer value as an <a href="/wiki/Exit_status" title="Exit status">exit status</a> typically use zero to indicate success and non-zero values to indicate specific <a href="/wiki/Error_code" title="Error code">error</a> or warning conditions.<a href="#cite_note-99">[97]</a>[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] Programmers often use a <a href="/wiki/Slashed_zero#Usage" title="Slashed zero">slashed zero</a> to avoid confusion with the letter "<a href="/wiki/O" title="O">O</a>".<a href="#cite_note-100">[98]</a> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(7)"><h2 id="Other_fields">Other fields</h2></div><section class="mf-section-7 collapsible-block" id="mf-section-7"> <div class="mw-heading mw-heading3"><h3 id="Biology">Biology</h3></div> In <a href="/wiki/Zoology" title="Zoology">comparative zoology</a> and <a href="/wiki/Cognitive_science" title="Cognitive science">cognitive science</a>, recognition that some animals display awareness of the concept of zero leads to the conclusion that the capability for numerical abstraction arose early in the <a href="/wiki/Evolution" title="Evolution">evolution</a> of species.<a href="#cite_note-101">[99]</a> <div class="mw-heading mw-heading3"><h3 id="Dating_systems">Dating systems</h3></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/Year_zero" title="Year zero">Year zero</a></div> In the <a href="/wiki/Before_Christ" class="mw-redirect" title="Before Christ">BC</a> <a href="/wiki/Calendar_era" title="Calendar era">calendar era</a>, the year 1 BC is the first year before AD 1; there is not a <a href="/wiki/Year_zero" title="Year zero">year zero</a>. By contrast, in <a href="/wiki/Astronomical_year_numbering" title="Astronomical year numbering">astronomical year numbering</a>, the year 1 BC is numbered 0, the year 2 BC is numbered −1, and so forth.<a href="#cite_note-102">[100]</a> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(8)"><h2 id="See_also">See also</h2></div><section class="mf-section-8 collapsible-block" id="mf-section-8"> <ul><li><a href="/wiki/Grammatical_number" title="Grammatical number">Grammatical number</a></li> <li><a href="/wiki/Mathematical_constant" title="Mathematical constant">Mathematical constant</a></li> <li><a href="/wiki/Number_theory" title="Number theory">Number theory</a></li> <li><a href="/wiki/Peano_axioms" title="Peano axioms">Peano axioms</a></li></ul> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(9)"><h2 id="Notes">Notes</h2></div><section class="mf-section-9 collapsible-block" id="mf-section-9"> <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 reflist-lower-alpha"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-14"><a href="#cite_ref-14">^</a> No long count date actually using the number 0 has been found before the 3rd century AD, but since the long count system would make no sense without some placeholder, and since Mesoamerican glyphs do not typically leave empty spaces, these earlier dates are taken as indirect evidence that the concept of 0 already existed at the time. </li> <li id="cite_note-30"><a href="#cite_ref-30">^</a> Each place in Ptolemy's sexagesimal system was written in <a href="/wiki/Greek_numerals" title="Greek numerals">Greek numerals</a> from 0 to 59, where 31 was written λα meaning 30+1, and 20 was written κ meaning 20. </li> </ol></div></div> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(10)"><h2 id="References">References</h2></div><section class="mf-section-10 collapsible-block" id="mf-section-10"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><div class="plainlist" style="display:inline;"><ul style="display:inline;"><li style="margin-bottom:.5em; display:block;;display:inline; margin:0;"><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="CITEREFHarper2011" class="citation encyclopaedia cs1">Harper, Douglas (2011). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170703014638/http://www.etymonline.com/index.php?allowed_in_frame=0&search=zero&searchmode=none">"Zero"</a>. Etymonline. Archived from <a rel="nofollow" class="external text" href="https://www.etymonline.com/index.php?allowed_in_frame=0&search=zero&searchmode=none">the original</a> on 3 July 2017. <q>"figure which stands for naught in the Arabic notation," also "the absence of all quantity considered as quantity," c. 1600, from French zéro or directly from Italian zero, from Medieval Latin zephirum, from Arabic sifr "cipher," translation of Sanskrit sunya-m "empty place, desert, naught</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Zero&rft.btitle=Etymonline&rft.date=2011&rft.aulast=Harper&rft.aufirst=Douglas&rft_id=https%3A%2F%2Fwww.etymonline.com%2Findex.php%3Fallowed_in_frame%3D0%26search%3Dzero%26searchmode%3Dnone&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988">.</li><li style="margin-bottom:.5em; display:block;;margin-top:.5em;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMenninger1992" class="citation book cs1">Menninger, Karl (1992). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=BFJHzSIj2u0C">Number Words and Number Symbols: A cultural history of numbers</a>. Courier Dover Publications. pp. 399–404. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-27096-8" title="Special:BookSources/978-0-486-27096-8"><bdi>978-0-486-27096-8</bdi></a>. Retrieved 5 January 2016.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Number+Words+and+Number+Symbols%3A+A+cultural+history+of+numbers&rft.pages=%3Cspan+class%3D%22nowrap%22%3E399-%3C%2Fspan%3E404&rft.pub=Courier+Dover+Publications&rft.date=1992&rft.isbn=978-0-486-27096-8&rft.aulast=Menninger&rft.aufirst=Karl&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DBFJHzSIj2u0C&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988"></li><li style="margin-bottom:.5em; display:block;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.oed.com/view/Entry/232803?rskey=zGcSoq&result=1&isAdvanced=false">"zero, n."</a> <a href="/wiki/Oxford_English_Dictionary" title="Oxford English Dictionary">OED</a> Online. <a href="/wiki/Oxford_University_Press" title="Oxford University Press">Oxford University Press</a>. December 2011. <a rel="nofollow" class="external text" href="https://www.webcitation.org/65yd7ur9u?url=http://www.oed.com/view/Entry/232803?rskey=zGcSoq&result=1&isAdvanced=false">Archived</a> from the original on 7 March 2012. Retrieved 4 March 2012. <q>French zéro (1515 in Hatzfeld & Darmesteter) or its source Italian zero, for *zefiro, < Arabic çifr</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=OED+Online&rft.atitle=zero%2C+n.&rft.date=2011-12&rft_id=http%3A%2F%2Fwww.oed.com%2Fview%2FEntry%2F232803%3Frskey%3DzGcSoq%26result%3D1%26isAdvanced%3Dfalse&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988">.</li></ul></div> </li> <li id="cite_note-smithsonian-2">^ <a href="#cite_ref-smithsonian_2-0">a</a> <a href="#cite_ref-smithsonian_2-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><div class="plainlist" style="display:inline;"><ul style="display:inline;"><li style="margin-bottom:.5em; display:block;;display:inline; margin:0;">Smithsonian Institution. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=0_UyAQAAMAAJ&pg=PA518">Oriental Elements of Culture in the Occident</a>, p. 518, at <a href="/wiki/Google_Books" title="Google Books">Google Books</a>. Annual Report of the Board of Regents of the Smithsonian Institution; Harvard University Archives. "Sifr occurs in the meaning of "empty" even in the pre-Islamic time. ... Arabic sifr in the meaning of zero is a translation of the corresponding India sunya."</li><li style="margin-bottom:.5em; display:block;;margin-top:.5em;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGullberg1997" class="citation book cs1">Gullberg, Jan (1997). Mathematics: From the Birth of Numbers. <a href="/wiki/W.W._Norton_%26_Co." class="mw-redirect" title="W.W. Norton & Co.">W.W. Norton & Co.</a> <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-393-04002-9" title="Special:BookSources/978-0-393-04002-9"><bdi>978-0-393-04002-9</bdi></a>. p. 26: <q>Zero derives from Hindu sunya – meaning void, emptiness – via Arabic sifr, Latin cephirum, Italian zevero.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematics%3A+From+the+Birth+of+Numbers&rft.pages=26&rft.pub=W.W.+Norton+%26+Co.&rft.date=1997&rft.isbn=978-0-393-04002-9&rft.aulast=Gullberg&rft.aufirst=Jan&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988"></li><li style="margin-bottom:.5em; display:block;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLogan2010" class="citation book cs1">Logan, Robert (2010). The Poetry of Physics and the Physics of Poetry. World Scientific. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-4295-92-5" title="Special:BookSources/978-981-4295-92-5"><bdi>978-981-4295-92-5</bdi></a>. p. 38: <q>The idea of sunya and place numbers was transmitted to the Arabs who translated sunya or "leave a space" into their language as sifr.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Poetry+of+Physics+and+the+Physics+of+Poetry&rft.pages=38&rft.pub=World+Scientific&rft.date=2010&rft.isbn=978-981-4295-92-5&rft.aulast=Logan&rft.aufirst=Robert&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988"></li></ul></div> </li> <li id="cite_note-3"><a href="#cite_ref-3">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation encyclopaedia cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20171206230446/https://www.merriam-webster.com/dictionary/zero">"Zero"</a>. Merriam Webster online Dictionary. Archived from <a rel="nofollow" class="external text" href="http://www.merriam-webster.com/dictionary/zero">the original</a> on 6 December 2017.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Zero&rft.btitle=Merriam+Webster+online+Dictionary&rft_id=http%3A%2F%2Fwww.merriam-webster.com%2Fdictionary%2Fzero&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988"> </li> <li id="cite_note-ifrah-4"><a href="#cite_ref-ifrah_4-0">^</a> <a href="#CITEREFIfrah2000">Ifrah 2000</a>, p. 589. </li> <li id="cite_note-5"><a href="#cite_ref-5">^</a> <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.collinsdictionary.com/dictionary/english/nought">"Collins – Free online dictionary"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Collins+%E2%80%93+Free+online+dictionary&rft_id=https%3A%2F%2Fwww.collinsdictionary.com%2Fdictionary%2Fenglish%2Fnought&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988"> </li> <li id="cite_note-6"><a href="#cite_ref-6">^</a> <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.collinsdictionary.com/dictionary/english/nil">"Collins – Free online dictionary, thesaurus and reference materials – nill"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Collins+%E2%80%93+Free+online+dictionary%2C+thesaurus+and+reference+materials+%E2%80%93+nill&rft_id=https%3A%2F%2Fwww.collinsdictionary.com%2Fdictionary%2Fenglish%2Fnil&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988"> </li> <li id="cite_note-thesaurus-7">^ <a href="#cite_ref-thesaurus_7-0">a</a> <a href="#cite_ref-thesaurus_7-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20140823071642/http://thesaurus.com/browse/aught#visualthesaurus">"'Aught' synonyms"</a>. Thesaurus.com. Archived from <a rel="nofollow" class="external text" href="http://thesaurus.com/browse/aught#visualthesaurus">the original</a> on 23 August 2014. Retrieved 23 April 2013.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Thesaurus.com&rft.atitle=%27Aught%27+synonyms&rft_id=http%3A%2F%2Fthesaurus.com%2Fbrowse%2Faught%23visualthesaurus&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988"> </li> <li id="cite_note-8"><a href="#cite_ref-8">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFO'ConnorRobertson2000" class="citation web cs1">O'Connor, J. J.; Robertson, E. F. (2000). <a rel="nofollow" class="external text" href="http://mathshistory.st-andrews.ac.uk/HistTopics/Egyptian_numerals.html">"Egyptian numerals"</a>. <a href="/wiki/Mathshistory.st-andrews.ac.uk" class="mw-redirect" title="Mathshistory.st-andrews.ac.uk">mathshistory.st-andrews.ac.uk</a>. University of St Andrews. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191115221313/http://mathshistory.st-andrews.ac.uk/HistTopics/Egyptian_numerals.html">Archived</a> from the original on 15 November 2019. Retrieved 21 December 2019.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=mathshistory.st-andrews.ac.uk&rft.atitle=Egyptian+numerals&rft.date=2000&rft.aulast=O%27Connor&rft.aufirst=J.+J.&rft.au=Robertson%2C+E.+F.&rft_id=http%3A%2F%2Fmathshistory.st-andrews.ac.uk%2FHistTopics%2FEgyptian_numerals.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988"> </li> <li id="cite_note-9"><a href="#cite_ref-9">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLumpkin2002" class="citation journal cs1">Lumpkin, Beatrice (2002). "Mathematics Used in Egyptian Construction and Bookkeeping". The Mathematical Intelligencer. 24 (2): 20–25. <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%2FBF03024613">10.1007/BF03024613</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:120648746">120648746</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Mathematical+Intelligencer&rft.atitle=Mathematics+Used+in+Egyptian+Construction+and+Bookkeeping&rft.volume=24&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E20-%3C%2Fspan%3E25&rft.date=2002&rft_id=info%3Adoi%2F10.1007%2FBF03024613&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A120648746%23id-name%3DS2CID&rft.aulast=Lumpkin&rft.aufirst=Beatrice&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988"> </li> <li id="cite_note-FOOTNOTEKaplan2000-10">^ <a href="#cite_ref-FOOTNOTEKaplan2000_10-0">a</a> <a href="#cite_ref-FOOTNOTEKaplan2000_10-1">b</a> <a href="#CITEREFKaplan2000">Kaplan 2000</a>. </li> <li id="cite_note-11"><a href="#cite_ref-11">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFO'ConnorRobertson2000" class="citation web cs1">O'Connor, J. J.; Robertson, E. F. (2000). <a rel="nofollow" class="external text" href="https://mathshistory.st-andrews.ac.uk/HistTopics/Zero/">"Zero"</a>. Maths History. University of St Andrews. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20210921191118/https://mathshistory.st-andrews.ac.uk/HistTopics/Zero/">Archived</a> from the original on 21 September 2021. Retrieved 7 September 2021.</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=Zero&rft.date=2000&rft.aulast=O%27Connor&rft.aufirst=J.+J.&rft.au=Robertson%2C+E.+F.&rft_id=https%3A%2F%2Fmathshistory.st-andrews.ac.uk%2FHistTopics%2FZero%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988"> </li> <li id="cite_note-12"><a href="#cite_ref-12">^</a> <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.open.edu/openlearn/ocw/mod/oucontent/view.php?printable=1&id=1976">"Babylonian mathematics"</a>. The Open University. 2016. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20210907135356/https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?printable=1&id=1976">Archived</a> from the original on 7 September 2021. Retrieved 7 September 2021.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=The+Open+University&rft.atitle=Babylonian+mathematics&rft.date=2016&rft_id=https%3A%2F%2Fwww.open.edu%2Fopenlearn%2Focw%2Fmod%2Foucontent%2Fview.php%3Fprintable%3D1%26id%3D1976&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988"> </li> <li id="cite_note-FOOTNOTEReimer2014172-13"><a href="#cite_ref-FOOTNOTEReimer2014172_13-0">^</a> <a href="#CITEREFReimer2014">Reimer 2014</a>, p. 172. </li> <li id="cite_note-15"><a href="#cite_ref-15">^</a> <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.mexicolore.co.uk/aztecs/calendar/cyclical-views-of-time">"Cyclical views of time"</a>. www.mexicolore.co.uk. Retrieved 20 January 2024.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.mexicolore.co.uk&rft.atitle=Cyclical+views+of+time&rft_id=https%3A%2F%2Fwww.mexicolore.co.uk%2Faztecs%2Fcalendar%2Fcyclical-views-of-time&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988"> </li> <li id="cite_note-FOOTNOTEDiehl2004186-16"><a href="#cite_ref-FOOTNOTEDiehl2004186_16-0">^</a> <a href="#CITEREFDiehl2004">Diehl (2004)</a>, p. 186. </li> <li id="cite_note-17"><a href="#cite_ref-17">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMortaigne2014" class="citation news cs1">Mortaigne, Véronique (28 November 2014). <a rel="nofollow" class="external text" href="https://www.theguardian.com/culture/2014/nov/28/mayan-civilisation-paris-exhibition">"The golden age of Mayan civilisation – exhibition review"</a>. <a href="/wiki/The_Guardian" title="The Guardian">The Guardian</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20141128222215/http://www.theguardian.com/culture/2014/nov/28/mayan-civilisation-paris-exhibition">Archived</a> from the original on 28 November 2014. Retrieved 10 October 2015.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Guardian&rft.atitle=The+golden+age+of+Mayan+civilisation+%E2%80%93+exhibition+review&rft.date=2014-11-28&rft.aulast=Mortaigne&rft.aufirst=V%C3%A9ronique&rft_id=https%3A%2F%2Fwww.theguardian.com%2Fculture%2F2014%2Fnov%2F28%2Fmayan-civilisation-paris-exhibition&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988"> </li> <li id="cite_note-18"><a href="#cite_ref-18">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCyphers2014" class="citation cs2">Cyphers, Ann (2014), Renfrew, Colin; Bahn, Paul (eds.), <a rel="nofollow" class="external text" href="https://www.cambridge.org/core/books/cambridge-world-prehistory/olmec-1800400-bce/2C66AF7B3D041260EE2BFC94DF085029">"The Olmec, 1800–400 BCE"</a>, The Cambridge World Prehistory, Cambridge: Cambridge University Press, pp. 1005–1025, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-11993-1" title="Special:BookSources/978-0-521-11993-1"><bdi>978-0-521-11993-1</bdi></a>, retrieved 13 August 2024</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Cambridge+World+Prehistory&rft.atitle=The+Olmec%2C+1800%E2%80%93400+BCE&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1005-%3C%2Fspan%3E1025&rft.date=2014&rft.isbn=978-0-521-11993-1&rft.aulast=Cyphers&rft.aufirst=Ann&rft_id=https%3A%2F%2Fwww.cambridge.org%2Fcore%2Fbooks%2Fcambridge-world-prehistory%2Folmec-1800400-bce%2F2C66AF7B3D041260EE2BFC94DF085029&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988">. </li> <li id="cite_note-19"><a href="#cite_ref-19">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation magazine cs1"><a rel="nofollow" class="external text" href="https://www.penn.museum/sites/expedition/time-kingship-and-the-maya-universe-maya-calendars/">"Expedition Magazine | Time, Kingship, and the Maya Universe Maya Calendars"</a>. Expedition Magazine. Retrieved 13 August 2024.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Expedition+Magazine&rft.atitle=Expedition+Magazine+%7C+Time%2C+Kingship%2C+and+the+Maya+Universe+Maya+Calendars&rft_id=https%3A%2F%2Fwww.penn.museum%2Fsites%2Fexpedition%2Ftime-kingship-and-the-maya-universe-maya-calendars%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988"> </li> <li id="cite_note-20"><a href="#cite_ref-20">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLeon2022" class="citation web cs1">Leon, Manuel de (20 December 2022). <a rel="nofollow" class="external text" href="https://english.elpais.com/science-tech/2022-12-20/knots-representing-numbers-the-mathematics-of-the-incas.html">"Knots representing numbers: The mathematics of the Incas"</a>. EL PAÍS English. Retrieved 5 June 2024.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=EL+PA%C3%8DS+English&rft.atitle=Knots+representing+numbers%3A+The+mathematics+of+the+Incas&rft.date=2022-12-20&rft.aulast=Leon&rft.aufirst=Manuel+de&rft_id=https%3A%2F%2Fenglish.elpais.com%2Fscience-tech%2F2022-12-20%2Fknots-representing-numbers-the-mathematics-of-the-incas.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988"> </li> <li id="cite_note-21"><a href="#cite_ref-21">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWallin2002" class="citation web cs1">Wallin, Nils-Bertil (19 November 2002). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160825124525/http://yaleglobal.yale.edu/about/zero.jsp">"The History of Zero"</a>. YaleGlobal online. The Whitney and Betty Macmillan Center for International and Area Studies at Yale. Archived from <a rel="nofollow" class="external text" href="http://yaleglobal.yale.edu/about/zero.jsp">the original</a> on 25 August 2016. Retrieved 1 September 2016.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=YaleGlobal+online&rft.atitle=The+History+of+Zero&rft.date=2002-11-19&rft.aulast=Wallin&rft.aufirst=Nils-Bertil&rft_id=http%3A%2F%2Fyaleglobal.yale.edu%2Fabout%2Fzero.jsp&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988"> </li> <li id="cite_note-Seife2000-22"><a href="#cite_ref-Seife2000_22-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSeife2000" class="citation book cs1"><a href="/wiki/Charles_Seife" title="Charles Seife">Seife, Charles</a> (1 September 2000). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=obJ70nxVYFUC">Zero: The Biography of a Dangerous Idea</a>. Penguin. p. 39. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-14-029647-1" title="Special:BookSources/978-0-14-029647-1"><bdi>978-0-14-029647-1</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/1005913932">1005913932</a>. Retrieved 30 April 2022.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Zero%3A+The+Biography+of+a+Dangerous+Idea&rft.pages=39&rft.pub=Penguin&rft.date=2000-09-01&rft_id=info%3Aoclcnum%2F1005913932&rft.isbn=978-0-14-029647-1&rft.aulast=Seife&rft.aufirst=Charles&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DobJ70nxVYFUC&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988"> </li> <li id="cite_note-Nieder2019-23"><a href="#cite_ref-Nieder2019_23-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNieder2019" class="citation book cs1">Nieder, Andreas (19 November 2019). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=x4y5DwAAQBAJ&pg=PA286">A Brain for Numbers: The Biology of the Number Instinct</a>. MIT Press. p. 286. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-262-35432-5" title="Special:BookSources/978-0-262-35432-5"><bdi>978-0-262-35432-5</bdi></a>. Retrieved 30 April 2022.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Brain+for+Numbers%3A+The+Biology+of+the+Number+Instinct&rft.pages=286&rft.pub=MIT+Press&rft.date=2019-11-19&rft.isbn=978-0-262-35432-5&rft.aulast=Nieder&rft.aufirst=Andreas&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dx4y5DwAAQBAJ%26pg%3DPA286&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988"> </li> <li id="cite_note-FOOTNOTEKaplan200017-24"><a href="#cite_ref-FOOTNOTEKaplan200017_24-0">^</a> <a href="#CITEREFKaplan2000">Kaplan 2000</a>, p. 17. </li> <li id="cite_note-25"><a href="#cite_ref-25">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHuggett2019" class="citation encyclopaedia cs1">Huggett, Nick (2019). <a rel="nofollow" class="external text" href="https://plato.stanford.edu/archives/win2019/entries/paradox-zeno/">"Zeno's Paradoxes"</a>. In Zalta, Edward N. (ed.). The Stanford Encyclopedia of Philosophy (Winter 2019 ed.). Metaphysics Research Lab, Stanford University. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20210110135804/https://plato.stanford.edu/archives/win2019/entries/paradox-zeno/">Archived</a> from the original on 10 January 2021. Retrieved 9 August 2020.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Zeno%27s+Paradoxes&rft.btitle=The+Stanford+Encyclopedia+of+Philosophy&rft.edition=Winter+2019&rft.pub=Metaphysics+Research+Lab%2C+Stanford+University&rft.date=2019&rft.aulast=Huggett&rft.aufirst=Nick&rft_id=https%3A%2F%2Fplato.stanford.edu%2Farchives%2Fwin2019%2Fentries%2Fparadox-zeno%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988"> </li> <li id="cite_note-26"><a href="#cite_ref-26">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNeugebauer1969" class="citation book cs1"><a href="/wiki/Otto_E._Neugebauer" title="Otto E. Neugebauer">Neugebauer, Otto</a> (1969) [1957]. <a rel="nofollow" class="external text" href="https://archive.org/details/exactsciencesant00neug">The Exact Sciences in Antiquity</a> (2 ed.). <a href="/wiki/Dover_Publications" title="Dover Publications">Dover Publications</a>. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/exactsciencesant00neug/page/n30">13</a>–14, plate 2. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-22332-2" title="Special:BookSources/978-0-486-22332-2"><bdi>978-0-486-22332-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Exact+Sciences+in+Antiquity&rft.pages=13-14%2C+plate+2&rft.edition=2&rft.pub=Dover+Publications&rft.date=1969&rft.isbn=978-0-486-22332-2&rft.aulast=Neugebauer&rft.aufirst=Otto&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fexactsciencesant00neug&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988"> </li> <li id="cite_note-Mercier-27"><a href="#cite_ref-Mercier_27-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMercier" class="citation web cs1">Mercier, Raymond. <a rel="nofollow" class="external text" href="http://www.raymondm.co.uk/prog/GreekZeroSign.pdf">"Consideration of the Greek symbol 'zero'"</a> (PDF). Home of Kairos. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20201105113109/http://www.raymondm.co.uk/prog/GreekZeroSign.pdf">Archived</a> (PDF) from the original on 5 November 2020. Retrieved 28 March 2020.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Home+of+Kairos&rft.atitle=Consideration+of+the+Greek+symbol+%27zero%27&rft.aulast=Mercier&rft.aufirst=Raymond&rft_id=http%3A%2F%2Fwww.raymondm.co.uk%2Fprog%2FGreekZeroSign.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988">[<a href="/wiki/Wikipedia:Verifiability#Self-published_sources" title="Wikipedia:Verifiability">self-published source?</a>] </li> <li id="cite_note-Ptolemy-28"><a href="#cite_ref-Ptolemy_28-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPtolemy1998" class="citation book cs1"><a href="/wiki/Ptolemy" title="Ptolemy">Ptolemy</a> (1998) [1984, <abbr title="circa">c.</abbr>150]. <a href="/wiki/Almagest" title="Almagest">Ptolemy's Almagest</a>. Translated by <a href="/wiki/Gerald_J._Toomer" title="Gerald J. Toomer">Toomer, G. J.</a> <a href="/wiki/Princeton_University_Press" title="Princeton University Press">Princeton University Press</a>. pp. 306–307. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-691-00260-6" title="Special:BookSources/0-691-00260-6"><bdi>0-691-00260-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Ptolemy%27s+Almagest&rft.pages=%3Cspan+class%3D%22nowrap%22%3E306-%3C%2Fspan%3E307&rft.pub=Princeton+University+Press&rft.date=1998&rft.isbn=0-691-00260-6&rft.au=Ptolemy&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988"> </li> <li id="cite_note-29"><a href="#cite_ref-29">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFO'ConnorRobertson" class="citation web cs1">O'Connor, J. J.; Robertson, E. F. <a rel="nofollow" class="external text" href="http://mathshistory.st-andrews.ac.uk/HistTopics/Zero.html">"A history of Zero"</a>. MacTutor History of Mathematics. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20200407074239/http://mathshistory.st-andrews.ac.uk/HistTopics/Zero.html">Archived</a> from the original on 7 April 2020. Retrieved 28 March 2020.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=A+history+of+Zero&rft.pub=MacTutor+History+of+Mathematics&rft.aulast=O%27Connor&rft.aufirst=J.+J.&rft.au=Robertson%2C+E.+F.&rft_id=http%3A%2F%2Fmathshistory.st-andrews.ac.uk%2FHistTopics%2FZero.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988"> </li> <li id="cite_note-Pedersen-31"><a href="#cite_ref-Pedersen_31-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPedersen2010" class="citation book cs1"><a href="/wiki/Olaf_Pedersen" title="Olaf Pedersen">Pedersen, Olaf</a> (2010) [1974]. Alexander Jones (ed.). A Survey of the Almagest. Sources and Studies in the History of Mathematics and Physical Sciences. Springer. pp. 232–235. <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%2F978-0-387-84826-6_7">10.1007/978-0-387-84826-6_7</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-84825-9" title="Special:BookSources/978-0-387-84825-9"><bdi>978-0-387-84825-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Survey+of+the+Almagest&rft.series=Sources+and+Studies+in+the+History+of+Mathematics+and+Physical+Sciences&rft.pages=%3Cspan+class%3D%22nowrap%22%3E232-%3C%2Fspan%3E235&rft.pub=Springer&rft.date=2010&rft_id=info%3Adoi%2F10.1007%2F978-0-387-84826-6_7&rft.isbn=978-0-387-84825-9&rft.aulast=Pedersen&rft.aufirst=Olaf&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988"> </li> <li id="cite_note-32"><a href="#cite_ref-32">^</a> <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.unicode.org/L2/L2004/04054-greek-zero.pdf">"Proposal to encode the Greek Zero in the UCS"</a> (PDF). 31 July 2024. <a rel="nofollow" class="external text" href="http://web.archive.org/web/20221007235444/https://unicode.org/L2/L2004/04054-greek-zero.pdf">Archived</a> (PDF) from the original on 7 October 2022.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Proposal+to+encode+the+Greek+Zero+in+the+UCS&rft.date=2024-07-31&rft_id=https%3A%2F%2Fwww.unicode.org%2FL2%2FL2004%2F04054-greek-zero.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988"> </li> <li id="cite_note-Neugebauer-33"><a href="#cite_ref-Neugebauer_33-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNeugebauer2016" class="citation book cs1"><a href="/wiki/Otto_Neugebauer" class="mw-redirect" title="Otto Neugebauer">Neugebauer, Otto</a> (2016) [1979]. Ethiopic Astronomy and Computus (Red Sea Press ed.). Red Sea Press. pp. 25, 53, 93, 183, Plate I. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-56902-440-9" title="Special:BookSources/978-1-56902-440-9"><bdi>978-1-56902-440-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Ethiopic+Astronomy+and+Computus&rft.pages=25%2C+53%2C+93%2C+183%2C+Plate+I&rft.edition=Red+Sea+Press&rft.pub=Red+Sea+Press&rft.date=2016&rft.isbn=978-1-56902-440-9&rft.aulast=Neugebauer&rft.aufirst=Otto&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988">. The pages in this edition have numbers six less than the same pages in the original edition. </li> <li id="cite_note-Dionysius-34"><a href="#cite_ref-Dionysius_34-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDeckers2003" class="citation web cs1">Deckers, Michael (2003) [525]. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20190115083618/http://hbar.phys.msu.ru/gorm/chrono/paschata.htm">"Cyclus Decemnovennalis Dionysii"</a> [Nineteen Year Cycle of Dionysius]. Archived from <a rel="nofollow" class="external text" href="http://hbar.phys.msu.ru/gorm/chrono/paschata.htm">the original</a> on 15 January 2019.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Cyclus+Decemnovennalis+Dionysii&rft.date=2003&rft.aulast=Deckers&rft.aufirst=Michael&rft_id=http%3A%2F%2Fhbar.phys.msu.ru%2Fgorm%2Fchrono%2Fpaschata.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988"> </li> <li id="cite_note-zero-35"><a href="#cite_ref-zero_35-0">^</a> C. W. Jones, ed., Opera Didascalica, vol. 123C in Corpus Christianorum, Series Latina. </li> <li id="cite_note-Hodgkin-36">^ <a href="#cite_ref-Hodgkin_36-0">a</a> <a href="#cite_ref-Hodgkin_36-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHodgkin2005" class="citation book cs1">Hodgkin, Luke (2005). <a rel="nofollow" class="external text" href="https://archive.org/details/historyofmathema0000hodg">A History of Mathematics: From Mesopotamia to Modernity</a>. Oxford University Press. p. <a rel="nofollow" class="external text" href="https://archive.org/details/historyofmathema0000hodg/page/85">85</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-152383-0" title="Special:BookSources/978-0-19-152383-0"><bdi>978-0-19-152383-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+History+of+Mathematics%3A+From+Mesopotamia+to+Modernity&rft.pages=85&rft.pub=Oxford+University+Press&rft.date=2005&rft.isbn=978-0-19-152383-0&rft.aulast=Hodgkin&rft.aufirst=Luke&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fhistoryofmathema0000hodg&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988"> </li> <li id="cite_note-37"><a href="#cite_ref-37">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFO'ConnorRobertson2004" class="citation cs2">O'Connor, John J.; <a href="/wiki/Edmund_F._Robertson" class="mw-redirect" title="Edmund F. Robertson">Robertson, Edmund F.</a> (January 2004), <a rel="nofollow" class="external text" href="https://mathshistory.st-andrews.ac.uk/HistTopics/Chinese_numerals.html">"Chinese numerals"</a>, <a href="/wiki/MacTutor_History_of_Mathematics_Archive" title="MacTutor History of Mathematics Archive">MacTutor History of Mathematics Archive</a>, <a href="/wiki/University_of_St_Andrews" title="University of St Andrews">University of St Andrews</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Chinese+numerals&rft.btitle=MacTutor+History+of+Mathematics+Archive&rft.pub=University+of+St+Andrews&rft.date=2004-01&rft.aulast=O%27Connor&rft.aufirst=John+J.&rft.au=Robertson%2C+Edmund+F.&rft_id=https%3A%2F%2Fmathshistory.st-andrews.ac.uk%2FHistTopics%2FChinese_numerals.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988"> </li> <li id="cite_note-38"><a href="#cite_ref-38">^</a> <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>. Maths History. Retrieved 28 April 2024.</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%3A0" class="Z3988"> </li> <li id="cite_note-39"><a href="#cite_ref-39">^</a> <a href="#CITEREFShenCrossleyLun1999">Shen, Crossley & Lun 1999</a>, p. 12: "the ancient Chinese system is a place notation system" </li> <li id="cite_note-40"><a href="#cite_ref-40">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEberhard-Bréard2008" class="citation cs2">Eberhard-Bréard, Andrea (2008), "Mathematics in China", in Selin, Helaine (ed.), Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures, Dordrecht: Springer Netherlands, pp. 1371–1378, <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%2F978-1-4020-4425-0_9453">10.1007/978-1-4020-4425-0_9453</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4020-4425-0" title="Special:BookSources/978-1-4020-4425-0"><bdi>978-1-4020-4425-0</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Mathematics+in+China&rft.btitle=Encyclopaedia+of+the+History+of+Science%2C+Technology%2C+and+Medicine+in+Non-Western+Cultures&rft.place=Dordrecht&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1371-%3C%2Fspan%3E1378&rft.pub=Springer+Netherlands&rft.date=2008&rft_id=info%3Adoi%2F10.1007%2F978-1-4020-4425-0_9453&rft.isbn=978-1-4020-4425-0&rft.aulast=Eberhard-Br%C3%A9ard&rft.aufirst=Andrea&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988">. </li> <li id="cite_note-Crossley-41"><a href="#cite_ref-Crossley_41-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFShenCrossleyLun1999" class="citation book cs1">Shen Kanshen Crossley, John N.; Lun, Anthony W.-C. (1999). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=eiTJHRGTG6YC">The Nine Chapters on the Mathematical Art: Companion and Commentary</a>. Oxford University Press. p. 35. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-853936-0" title="Special:BookSources/978-0-19-853936-0"><bdi>978-0-19-853936-0</bdi></a>. <q>zero was regarded as a number in India ... whereas the Chinese employed a vacant position</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Nine+Chapters+on+the+Mathematical+Art%3A+Companion+and+Commentary&rft.pages=35&rft.pub=Oxford+University+Press&rft.date=1999&rft.isbn=978-0-19-853936-0&rft.aulast=Shen&rft.aufirst=Kangshen&rft.au=Crossley%2C+John+N.&rft.au=Lun%2C+Anthony+W.-C.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DeiTJHRGTG6YC&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988"> </li> <li id="cite_note-Qin-42"><a href="#cite_ref-Qin_42-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://grmath4.phpnet.us/istoria/the_history_of%20math_greece/the_history_of%20math_greece_3-5.pdf">"Mathematics in the Near and Far East"</a> (PDF). grmath4.phpnet.us. p. 262. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20131104120005/http://grmath4.phpnet.us/istoria/the_history_of%20math_greece/the_history_of%20math_greece_3-5.pdf">Archived</a> (PDF) from the original on 4 November 2013. Retrieved 7 June 2012.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=grmath4.phpnet.us&rft.atitle=Mathematics+in+the+Near+and+Far+East&rft.pages=262&rft_id=http%3A%2F%2Fgrmath4.phpnet.us%2Fistoria%2Fthe_history_of%2520math_greece%2Fthe_history_of%2520math_greece_3-5.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988"> </li> <li id="cite_note-43"><a href="#cite_ref-43">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMartzloff2007" class="citation book cs1">Martzloff, Jean-Claude (2007). A History of Chinese Mathematics. Translated by Wilson, Stephen S. Springer. p. 208. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-33783-6" title="Special:BookSources/978-3-540-33783-6"><bdi>978-3-540-33783-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+History+of+Chinese+Mathematics&rft.pages=208&rft.pub=Springer&rft.date=2007&rft.isbn=978-3-540-33783-6&rft.aulast=Martzloff&rft.aufirst=Jean-Claude&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988"> </li> <li id="cite_note-struik33-44"><a href="#cite_ref-struik33_44-0">^</a> Struik, Dirk J. (1987). A Concise History of Mathematics. New York: Dover Publications. pp. 32–33. "In these matrices we find negative numbers, which appear here for the first time in history." </li> <li id="cite_note-plofker-45">^ <a href="#cite_ref-plofker_45-0">a</a> <a href="#cite_ref-plofker_45-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPlofker2009" class="citation book cs1"><a href="/wiki/Kim_Plofker" title="Kim Plofker">Plofker, Kim</a> (2009). <a href="/wiki/Mathematics_in_India_(book)" title="Mathematics in India (book)">Mathematics in India</a>. Princeton University Press. pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=DHvThPNp9yMC&pg=PA54">54–56</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-12067-6" title="Special:BookSources/978-0-691-12067-6"><bdi>978-0-691-12067-6</bdi></a>. <q>In the Chandah-sutra of Pingala, dating perhaps the third or second century BC, [ ...] Pingala's use of a zero symbol [śūnya] as a marker seems to be the first known explicit reference to zero. ... In the Chandah-sutra of Pingala, dating perhaps the third or second century BC, there are five questions concerning the possible meters for any value "n". [ ...] The answer is (2)7 = 128, as expected, but instead of seven doublings, the process (explained by the sutra) required only three doublings and two squarings – a handy time saver where "n" is large. Pingala's use of a zero symbol as a marker seems to be the first known explicit reference to zero.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematics+in+India&rft.pages=54-56&rft.pub=Princeton+University+Press&rft.date=2009&rft.isbn=978-0-691-12067-6&rft.aulast=Plofker&rft.aufirst=Kim&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988"> </li> <li id="cite_note-46"><a href="#cite_ref-46">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVaman_Shivaram_Apte1970" class="citation book cs1">Vaman Shivaram Apte (1970). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=4ArxvCxV1l4C&pg=PA648">"Sanskrit Prosody and Important Literary and Geographical Names in the Ancient History of India"</a>. The Student's Sanskrit-English Dictionary. Motilal Banarsidass. pp. 648–649. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-81-208-0045-8" title="Special:BookSources/978-81-208-0045-8"><bdi>978-81-208-0045-8</bdi></a>. Retrieved 21 April 2017.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Sanskrit+Prosody+and+Important+Literary+and+Geographical+Names+in+the+Ancient+History+of+India&rft.btitle=The+Student%27s+Sanskrit-English+Dictionary&rft.pages=%3Cspan+class%3D%22nowrap%22%3E648-%3C%2Fspan%3E649&rft.pub=Motilal+Banarsidass&rft.date=1970&rft.isbn=978-81-208-0045-8&rft.au=Vaman+Shivaram+Apte&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D4ArxvCxV1l4C%26pg%3DPA648&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988"> </li> <li id="cite_note-47"><a href="#cite_ref-47">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHall2005" class="citation web cs1">Hall, Rachel (15 February 2005). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20190122014628/http://people.sju.edu/~rhall/Rhythms/Poets/arcadia.pdf">"Math for Poets and Drummers: The Mathematics of Rhythm"</a> (PDF) (slideshow). Saint Joseph's University. Archived from <a rel="nofollow" class="external text" href="http://people.sju.edu/~rhall/Rhythms/Poets/arcadia.pdf">the original</a> (PDF) on 22 January 2019. Retrieved 20 December 2015.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Math+for+Poets+and+Drummers%3A+The+Mathematics+of+Rhythm&rft.pub=Saint+Joseph%27s+University&rft.date=2005-02-15&rft.aulast=Hall&rft.aufirst=Rachel&rft_id=http%3A%2F%2Fpeople.sju.edu%2F~rhall%2FRhythms%2FPoets%2Farcadia.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988"> </li> <li id="cite_note-Devlin_2017-48">^ <a href="#cite_ref-Devlin_2017_48-0">a</a> <a href="#cite_ref-Devlin_2017_48-1">b</a> <a href="#cite_ref-Devlin_2017_48-2">c</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDevlin2017" class="citation news cs1"><a href="/wiki/Hannah_Devlin" title="Hannah Devlin">Devlin, Hannah</a> (13 September 2017). <a rel="nofollow" class="external text" href="https://www.theguardian.com/science/2017/sep/14/much-ado-about-nothing-ancient-indian-text-contains-earliest-zero-symbol">"Much ado about nothing: ancient Indian text contains earliest zero symbol"</a>. The Guardian. <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/0261-3077">0261-3077</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20171120225416/https://www.theguardian.com/science/2017/sep/14/much-ado-about-nothing-ancient-indian-text-contains-earliest-zero-symbol">Archived</a> from the original on 20 November 2017. Retrieved 14 September 2017.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Guardian&rft.atitle=Much+ado+about+nothing%3A+ancient+Indian+text+contains+earliest+zero+symbol&rft.date=2017-09-13&rft.issn=0261-3077&rft.aulast=Devlin&rft.aufirst=Hannah&rft_id=https%3A%2F%2Fwww.theguardian.com%2Fscience%2F2017%2Fsep%2F14%2Fmuch-ado-about-nothing-ancient-indian-text-contains-earliest-zero-symbol&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988"> </li> <li id="cite_note-bourbaki46-49"><a href="#cite_ref-bourbaki46_49-0">^</a> <a href="#CITEREFBourbaki1998">Bourbaki 1998</a>, p. 46. </li> <li id="cite_note-Weiss-50"><a href="#cite_ref-Weiss_50-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeiss2017" class="citation news cs1">Weiss, Ittay (20 September 2017). <a rel="nofollow" class="external text" href="https://theconversation.com/nothing-matters-how-the-invention-of-zero-helped-create-modern-mathematics-84232">"Nothing matters: How India's invention of zero helped create modern mathematics"</a>. The Conversation. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20180712124031/https://theconversation.com/nothing-matters-how-the-invention-of-zero-helped-create-modern-mathematics-84232">Archived</a> from the original on 12 July 2018. Retrieved 12 July 2018.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Conversation&rft.atitle=Nothing+matters%3A+How+India%27s+invention+of+zero+helped+create+modern+mathematics&rft.date=2017-09-20&rft.aulast=Weiss&rft.aufirst=Ittay&rft_id=https%3A%2F%2Ftheconversation.com%2Fnothing-matters-how-the-invention-of-zero-helped-create-modern-mathematics-84232&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988"> </li> <li id="cite_note-51"><a href="#cite_ref-51">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation news cs1"><a rel="nofollow" class="external text" href="http://www.bodleian.ox.ac.uk/bodley/news/2017/sep-14">"Carbon dating finds Bakhshali manuscript contains oldest recorded origins of the symbol 'zero'"</a>. Bodleian Library. 14 September 2017. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170914215604/http://www.bodleian.ox.ac.uk/bodley/news/2017/sep-14">Archived</a> from the original on 14 September 2017. Retrieved 25 October 2017.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Bodleian+Library&rft.atitle=Carbon+dating+finds+Bakhshali+manuscript+contains+oldest+recorded+origins+of+the+symbol+%27zero%27&rft.date=2017-09-14&rft_id=http%3A%2F%2Fwww.bodleian.ox.ac.uk%2Fbodley%2Fnews%2F2017%2Fsep-14&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988"> </li> <li id="cite_note-52"><a href="#cite_ref-52">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPlofkerKellerHayashiMontelle2017" class="citation journal cs1"><a href="/wiki/Kim_Plofker" title="Kim Plofker">Plofker, Kim</a>; Keller, Agathe; <a href="/wiki/Takao_Hayashi" title="Takao Hayashi">Hayashi, Takao</a>; <a href="/wiki/Clemency_Montelle" title="Clemency Montelle">Montelle, Clemency</a>; Wujastyk, Dominik (6 October 2017). <a rel="nofollow" class="external text" href="https://journals.library.ualberta.ca/hssa/index.php/hssa/article/view/22">"The Bakhshālī Manuscript: A Response to the Bodleian Library's Radiocarbon Dating"</a>. History of Science in South Asia. 5 (1): 134–150. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.18732%2FH2XT07">10.18732/H2XT07</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=History+of+Science+in+South+Asia&rft.atitle=The+Bakhsh%C4%81l%C4%AB+Manuscript%3A+A+Response+to+the+Bodleian+Library%27s+Radiocarbon+Dating&rft.volume=5&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E134-%3C%2Fspan%3E150&rft.date=2017-10-06&rft_id=info%3Adoi%2F10.18732%2FH2XT07&rft.aulast=Plofker&rft.aufirst=Kim&rft.au=Keller%2C+Agathe&rft.au=Hayashi%2C+Takao&rft.au=Montelle%2C+Clemency&rft.au=Wujastyk%2C+Dominik&rft_id=https%3A%2F%2Fjournals.library.ualberta.ca%2Fhssa%2Findex.php%2Fhssa%2Farticle%2Fview%2F22&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988"> </li> <li id="cite_note-FOOTNOTEIfrah2000416-53"><a href="#cite_ref-FOOTNOTEIfrah2000416_53-0">^</a> <a href="#CITEREFIfrah2000">Ifrah (2000)</a>, p. 416. </li> <li id="cite_note-Aryab-54"><a href="#cite_ref-Aryab_54-0">^</a> Aryabhatiya of Aryabhata, translated by <a href="/wiki/Walter_Eugene_Clark" title="Walter Eugene Clark">Walter Eugene Clark</a>. </li> <li id="cite_note-55"><a href="#cite_ref-55">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFO'ConnorRobertson2000" class="citation web cs1">O'Connor, J. J.; Robertson, E. F. (2000). <a rel="nofollow" class="external text" href="http://www-history.mcs.st-andrews.ac.uk/Biographies/Aryabhata_I.html">"Aryabhata the Elder"</a>. School of Mathematics and Statistics, University of St. Andrews. Scotland. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20150711055702/http://www-history.mcs.st-andrews.ac.uk/Biographies/Aryabhata_I.html">Archived</a> from the original on 11 July 2015. Retrieved 26 May 2013.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=School+of+Mathematics+and+Statistics%2C+University+of+St.+Andrews&rft.atitle=Aryabhata+the+Elder&rft.date=2000&rft.aulast=O%27Connor&rft.aufirst=J.+J.&rft.au=Robertson%2C+E.+F.&rft_id=http%3A%2F%2Fwww-history.mcs.st-andrews.ac.uk%2FBiographies%2FAryabhata_I.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988"> </li> <li id="cite_note-56"><a href="#cite_ref-56">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHosch2010" class="citation book cs1">Hosch, William L., ed. (15 August 2010). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=cuN7rH6RzikC&pg=PA97">The Britannica Guide to Numbers and Measurement (Math Explained)</a>. The Rosen Publishing Group. pp. 97–98. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-61530-108-9" title="Special:BookSources/978-1-61530-108-9"><bdi>978-1-61530-108-9</bdi></a>. Retrieved 26 September 2016.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Britannica+Guide+to+Numbers+and+Measurement+%28Math+Explained%29&rft.pages=%3Cspan+class%3D%22nowrap%22%3E97-%3C%2Fspan%3E98&rft.pub=The+Rosen+Publishing+Group&rft.date=2010-08-15&rft.isbn=978-1-61530-108-9&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DcuN7rH6RzikC%26pg%3DPA97&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988"> </li> <li id="cite_note-brahmagupta-57"><a href="#cite_ref-brahmagupta_57-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1"><a rel="nofollow" class="external text" href="https://archive.org/details/algebrawitharith00brahuoft">Algebra, with Arithmetic and Mensuration from the Sanscrit of Brahmegupta and Bháscara</a>. Translated by Henry Thomas Colebrooke. London, England: John Murray. 1817. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/1039515732">1039515732</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebra%2C+with+Arithmetic+and+Mensuration+from+the+Sanscrit+of+Brahmegupta+and+Bh%C3%A1scara&rft.place=London%2C+England&rft.pub=John+Murray&rft.date=1817&rft_id=info%3Aoclcnum%2F1039515732&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Falgebrawitharith00brahuoft&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988"> </li> <li id="cite_note-FOOTNOTEKaplan2000[httpsarchiveorgdetailsnothingthatisnat00kaplpage68_68–75]-58"><a href="#cite_ref-FOOTNOTEKaplan2000%5Bhttpsarchiveorgdetailsnothingthatisnat00kaplpage68_68%E2%80%9375%5D_58-0">^</a> <a href="#CITEREFKaplan2000">Kaplan 2000</a>, p. <a rel="nofollow" class="external text" href="https://archive.org/details/nothingthatisnat00kapl/page/68">68–75</a>. </li> <li id="cite_note-Cambodia-59"><a href="#cite_ref-Cambodia_59-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><div class="plainlist" style="display:inline;"><ul style="display:inline;"><li style="margin-bottom:.5em; display:block;;display:inline; margin:0;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCœdès1931" class="citation journal cs1 cs1-prop-foreign-lang-source"><a href="/wiki/George_C%C5%93d%C3%A8s" title="George Cœdès">Cœdès, George</a> (1931). "A propos de l'origine des chiffres arabes". Bulletin of the School of Oriental Studies, University of London (in French). 6 (2). Cambridge University Press: 323–328. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FS0041977X00092806">10.1017/S0041977X00092806</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/607661">607661</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:130482979">130482979</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Bulletin+of+the+School+of+Oriental+Studies%2C+University+of+London&rft.atitle=A+propos+de+l%27origine+des+chiffres+arabes&rft.volume=6&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E323-%3C%2Fspan%3E328&rft.date=1931&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A130482979%23id-name%3DS2CID&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F607661%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.1017%2FS0041977X00092806&rft.aulast=C%C5%93d%C3%A8s&rft.aufirst=George&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988"></li><li style="margin-bottom:.5em; display:block;;margin-top:.5em;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDiller1996" class="citation journal cs1">Diller, Anthony (1996). <a rel="nofollow" class="external text" href="http://sealang.net/sala/archives/pdf8/diller1996new.pdf">"New Zeros and Old Khmer"</a> (PDF). <a href="/wiki/Mon-Khmer_Studies" title="Mon-Khmer Studies">Mon-Khmer Studies</a>. 25: 125–132.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mon-Khmer+Studies&rft.atitle=New+Zeros+and+Old+Khmer&rft.volume=25&rft.pages=%3Cspan+class%3D%22nowrap%22%3E125-%3C%2Fspan%3E132&rft.date=1996&rft.aulast=Diller&rft.aufirst=Anthony&rft_id=http%3A%2F%2Fsealang.net%2Fsala%2Farchives%2Fpdf8%2Fdiller1996new.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988"></li></ul></div> </li> <li id="cite_note-60"><a href="#cite_ref-60">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCasselman" class="citation web cs1"><a href="/wiki/Bill_Casselman_(mathematician)" class="mw-redirect" title="Bill Casselman (mathematician)">Casselman, Bill</a>. <a rel="nofollow" class="external text" href="http://www.ams.org/samplings/feature-column/fcarc-india-zero">"All for Nought"</a>. ams.org. University of British Columbia), American Mathematical Society. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20151206184352/http://www.ams.org/samplings/feature-column/fcarc-india-zero">Archived</a> from the original on 6 December 2015. Retrieved 20 December 2015.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=ams.org&rft.atitle=All+for+Nought&rft.aulast=Casselman&rft.aufirst=Bill&rft_id=http%3A%2F%2Fwww.ams.org%2Fsamplings%2Ffeature-column%2Ffcarc-india-zero&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988"> </li> <li id="cite_note-FOOTNOTEIfrah2000400-61"><a href="#cite_ref-FOOTNOTEIfrah2000400_61-0">^</a> <a href="#CITEREFIfrah2000">Ifrah (2000)</a>, p. 400. </li> <li id="cite_note-62"><a href="#cite_ref-62">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPannekoek1961" class="citation book cs1"><a href="/wiki/Anton_Pannekoek" title="Anton Pannekoek">Pannekoek, Anton</a> (1961). A History of Astronomy. George Allen & Unwin. p. 165. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780045200023" title="Special:BookSources/9780045200023"><bdi>9780045200023</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/840043">840043</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+History+of+Astronomy&rft.pages=165&rft.pub=George+Allen+%26+Unwin&rft.date=1961&rft_id=info%3Aoclcnum%2F840043&rft.isbn=9780045200023&rft.aulast=Pannekoek&rft.aufirst=Anton&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988"> </li> <li id="cite_note-Durant-63">^ <a href="#cite_ref-Durant_63-0">a</a> <a href="#cite_ref-Durant_63-1">b</a> <a href="#cite_ref-Durant_63-2">c</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDurant1950" class="citation book cs1"><a href="/wiki/Will_Durant" title="Will Durant">Durant, Will</a> (1950). The Story of Civilization, Volume 4, The Age of Faith: Constantine to Dante – A.D. 325–1300. Simon & Schuster. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-9650007-5-8" title="Special:BookSources/978-0-9650007-5-8"><bdi>978-0-9650007-5-8</bdi></a>. p. 241: <q>The Arabic inheritance of science was overwhelmingly Greek, but Hindu influences ranked next. In 773, at Mansur's behest, translations were made of the Siddhantas – Indian astronomical treatises dating as far back as 425 BC; these versions may have the vehicle through which the "Arabic" numerals and the zero were brought from India into Islam. In 813, al-Khwarizmi used the Hindu numerals in his astronomical tables.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Story+of+Civilization%2C+Volume+4%2C+The+Age+of+Faith%3A+Constantine+to+Dante+%E2%80%93+A.D.+325%E2%80%931300&rft.pages=241&rft.pub=Simon+%26+Schuster&rft.date=1950&rft.isbn=978-0-9650007-5-8&rft.aulast=Durant&rft.aufirst=Will&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988"> </li> <li id="cite_note-64"><a href="#cite_ref-64">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBrezina2006" class="citation book cs1">Brezina, Corona (2006). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=955jPgAACAAJ">Al-Khwarizmi: The Inventor of Algebra</a>. The Rosen Publishing Group. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4042-0513-0" title="Special:BookSources/978-1-4042-0513-0"><bdi>978-1-4042-0513-0</bdi></a>. Retrieved 26 September 2016.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Al-Khwarizmi%3A+The+Inventor+of+Algebra&rft.pub=The+Rosen+Publishing+Group&rft.date=2006&rft.isbn=978-1-4042-0513-0&rft.aulast=Brezina&rft.aufirst=Corona&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D955jPgAACAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988"> </li> <li id="cite_note-65"><a href="#cite_ref-65">^</a> <a href="#CITEREFDurant1950">Durant 1950</a>, p. 241: "In 976, Muhammad ibn Ahmad, in his Keys of the Sciences, remarked that if, in a calculation, no number appears in the place of tens, a little circle should be used "to keep the rows". This circle the Mosloems called ṣifr, "empty" whence our cipher." </li> <li id="cite_note-66"><a href="#cite_ref-66">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><div class="plainlist" style="display:inline;"><ul style="display:inline;"><li style="margin-bottom:.5em; display:block;;display:inline; margin:0;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSigler2003" class="citation book cs1">Sigler, Laurence (2003). Fibonacci's Liber Abaci: A Translation into Modern English of Leonardo Pisano's Book of Calculation. Sources and Studies in the History of Mathematics and Physical Sciences. Translated by Sigler, Laurence E. Springer. <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%2F978-1-4613-0079-3">10.1007/978-1-4613-0079-3</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4613-0079-3" title="Special:BookSources/978-1-4613-0079-3"><bdi>978-1-4613-0079-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fibonacci%27s+Liber+Abaci%3A+A+Translation+into+Modern+English+of+Leonardo+Pisano%27s+Book+of+Calculation&rft.series=Sources+and+Studies+in+the+History+of+Mathematics+and+Physical+Sciences&rft.pub=Springer&rft.date=2003&rft_id=info%3Adoi%2F10.1007%2F978-1-4613-0079-3&rft.isbn=978-1-4613-0079-3&rft.aulast=Sigler&rft.aufirst=Laurence&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988"></li><li style="margin-bottom:.5em; display:block;;margin-top:.5em;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGrimm1973" class="citation magazine cs1">Grimm, Richard E. (February 1973). <a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=318a17253f745e2af400eb2ebb4dc4e762560a5b">"The Autobiography of Leonardo Pisano"</a>. <a href="/wiki/Fibonacci_Quarterly" title="Fibonacci Quarterly">Fibonacci Quarterly</a>. Vol. 11, no. 1. pp. 99–104. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20231126180044/https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=318a17253f745e2af400eb2ebb4dc4e762560a5b">Archived</a> from the original on 26 November 2023.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Fibonacci+Quarterly&rft.atitle=The+Autobiography+of+Leonardo+Pisano&rft.volume=11&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E99-%3C%2Fspan%3E104&rft.date=1973-02&rft.aulast=Grimm&rft.aufirst=Richard+E.&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fdocument%3Frepid%3Drep1%26type%3Dpdf%26doi%3D318a17253f745e2af400eb2ebb4dc4e762560a5b&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988"></li><li style="margin-bottom:.5em; display:block;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHansen2008" class="citation book cs1">Hansen, Alice (2008). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=COJsbuUI1h8C&pg=PT31">Primary Mathematics: Extending Knowledge in Practice</a>. SAGE. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.4135%2F9781446276532">10.4135/9781446276532</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-85725-233-3" title="Special:BookSources/978-0-85725-233-3"><bdi>978-0-85725-233-3</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20210307234959/https://books.google.com/books?id=COJsbuUI1h8C&q=%22Therefore%2C+embracing+more+stringently+that+method+of+the+Hindus%2C+and+taking+stricter+pains+in+its+study%2C+while+adding+certain+things+from+my+own+understanding+and+inserting+also+certain+things+from+the+niceties+of+Euclid%27s+geometric+art.%22&pg=PT31">Archived</a> from the original on 7 March 2021. Retrieved 7 November 2020.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Primary+Mathematics%3A+Extending+Knowledge+in+Practice&rft.pub=SAGE&rft.date=2008&rft_id=info%3Adoi%2F10.4135%2F9781446276532&rft.isbn=978-0-85725-233-3&rft.aulast=Hansen&rft.aufirst=Alice&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DCOJsbuUI1h8C%26pg%3DPT31&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988"></li></ul></div> </li> <li id="cite_note-Karpinski1911-67">^ <a href="#cite_ref-Karpinski1911_67-0">a</a> <a href="#cite_ref-Karpinski1911_67-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSmithKarpinski1911" class="citation book cs1">Smith, D. E.; Karpinski, L. C. (1911). <a rel="nofollow" class="external text" href="https://archive.org/stream/hinduarabicnume02karpgoog#page/n145/mode/1up">"The spread of the [Hindu–Arabic] numerals in Europe"</a>. The Hindu–Arabic Numerals. Ginn and Company. pp. 134–136 – via Internet Archive.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=The+spread+of+the+%5BHindu%E2%80%93Arabic%5D+numerals+in+Europe&rft.btitle=The+Hindu%E2%80%93Arabic+Numerals&rft.pages=%3Cspan+class%3D%22nowrap%22%3E134-%3C%2Fspan%3E136&rft.pub=Ginn+and+Company&rft.date=1911&rft.aulast=Smith&rft.aufirst=D.+E.&rft.au=Karpinski%2C+L.+C.&rft_id=https%3A%2F%2Farchive.org%2Fstream%2Fhinduarabicnume02karpgoog%23page%2Fn145%2Fmode%2F1up&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988"> </li> <li id="cite_note-68"><a href="#cite_ref-68">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPedersen1985" class="citation journal cs1">Pedersen, Olaf (1985). "In Quest of Sacrobosco". Journal for the History of Astronomy. 16 (3): 175–221. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1985JHA....16..175P">1985JHA....16..175P</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1177%2F002182868501600302">10.1177/002182868501600302</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:118227787">118227787</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+for+the+History+of+Astronomy&rft.atitle=In+Quest+of+Sacrobosco&rft.volume=16&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E175-%3C%2Fspan%3E221&rft.date=1985&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A118227787%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1177%2F002182868501600302&rft_id=info%3Abibcode%2F1985JHA....16..175P&rft.aulast=Pedersen&rft.aufirst=Olaf&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988"> </li> <li id="cite_note-FOOTNOTEIfrah2000588–590-69"><a href="#cite_ref-FOOTNOTEIfrah2000588%E2%80%93590_69-0">^</a> <a href="#CITEREFIfrah2000">Ifrah 2000</a>, pp. 588–590. </li> <li id="cite_note-bemer-70">^ <a href="#cite_ref-bemer_70-0">a</a> <a href="#cite_ref-bemer_70-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBemer1967" class="citation journal cs1">Bemer, R. W. (1967). "Towards standards for handwritten zero and oh: much ado about nothing (and a letter), or a partial dossier on distinguishing between handwritten zero and oh". Communications of the ACM. 10 (8): 513–518. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F363534.363563">10.1145/363534.363563</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:294510">294510</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Communications+of+the+ACM&rft.atitle=Towards+standards+for+handwritten+zero+and+oh%3A+much+ado+about+nothing+%28and+a+letter%29%2C+or+a+partial+dossier+on+distinguishing+between+handwritten+zero+and+oh&rft.volume=10&rft.issue=8&rft.pages=%3Cspan+class%3D%22nowrap%22%3E513-%3C%2Fspan%3E518&rft.date=1967&rft_id=info%3Adoi%2F10.1145%2F363534.363563&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A294510%23id-name%3DS2CID&rft.aulast=Bemer&rft.aufirst=R.+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988"> </li> <li id="cite_note-FOOTNOTEReimer2014156,_199–204-71"><a href="#cite_ref-FOOTNOTEReimer2014156,_199%E2%80%93204_71-0">^</a> <a href="#CITEREFReimer2014">Reimer 2014</a>, pp. 156, 199–204. </li> <li id="cite_note-72"><a href="#cite_ref-72">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBuntJonesBedient1976" class="citation book cs1">Bunt, Lucas Nicolaas Hendrik; Jones, Phillip S.; Bedient, Jack D. (1976). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=7xArILpcndYC">The historical roots of elementary mathematics</a>. Courier Dover Publications. pp. 254–255. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-13968-5" title="Special:BookSources/978-0-486-13968-5"><bdi>978-0-486-13968-5</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160623174716/https://books.google.com/books?id=7xArILpcndYC">Archived</a> from the original on 23 June 2016. Retrieved 5 January 2016.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+historical+roots+of+elementary+mathematics&rft.pages=%3Cspan+class%3D%22nowrap%22%3E254-%3C%2Fspan%3E255&rft.pub=Courier+Dover+Publications&rft.date=1976&rft.isbn=978-0-486-13968-5&rft.aulast=Bunt&rft.aufirst=Lucas+Nicolaas+Hendrik&rft.au=Jones%2C+Phillip+S.&rft.au=Bedient%2C+Jack+D.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D7xArILpcndYC&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988">, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=7xArILpcndYC&pg=PA255">Extract of pp. 254–255</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160510195505/https://books.google.com/books?id=7xArILpcndYC&pg=PA255">Archived</a> 10 May 2016 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> </li> <li id="cite_note-FOOTNOTECheng201732-73"><a href="#cite_ref-FOOTNOTECheng201732_73-0">^</a> <a href="#CITEREFCheng2017">Cheng 2017</a>, p. 32. </li> <li id="cite_note-FOOTNOTECheng201741,_48–53-74"><a href="#cite_ref-FOOTNOTECheng201741,_48%E2%80%9353_74-0">^</a> <a href="#CITEREFCheng2017">Cheng 2017</a>, pp. 41, 48–53. </li> <li id="cite_note-75"><a href="#cite_ref-75">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein,_Eric_W." class="citation web cs1">Weisstein, Eric W. <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/Zero.html">"Zero"</a>. Wolfram. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20130601190920/http://mathworld.wolfram.com/Zero.html">Archived</a> from the original on 1 June 2013. Retrieved 4 April 2018.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Wolfram&rft.atitle=Zero&rft.au=Weisstein%2C+Eric+W.&rft_id=http%3A%2F%2Fmathworld.wolfram.com%2FZero.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988"> </li> <li id="cite_note-76"><a href="#cite_ref-76">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeil2012" class="citation book cs1"><a href="/wiki/Andr%C3%A9_Weil" title="André Weil">Weil, André</a> (6 December 2012). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=NEHaBwAAQBAJ&pg=PA3">Number Theory for Beginners</a>. Springer Science & Business Media. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4612-9957-8" title="Special:BookSources/978-1-4612-9957-8"><bdi>978-1-4612-9957-8</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20210614182810/https://books.google.com/books?id=NEHaBwAAQBAJ&pg=PA3">Archived</a> from the original on 14 June 2021. Retrieved 6 April 2021.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Number+Theory+for+Beginners&rft.pub=Springer+Science+%26+Business+Media&rft.date=2012-12-06&rft.isbn=978-1-4612-9957-8&rft.aulast=Weil&rft.aufirst=Andr%C3%A9&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DNEHaBwAAQBAJ%26pg%3DPA3&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988"> </li> <li id="cite_note-77"><a href="#cite_ref-77">^</a> <a href="/wiki/Lemma_(mathematics)" title="Lemma (mathematics)">Lemma</a> B.2.2, The integer 0 is even and is not odd, in <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPenner1999" class="citation book cs1">Penner, Robert C. (1999). <a rel="nofollow" class="external text" href="https://archive.org/details/discretemathemat0000penn">Discrete Mathematics: Proof Techniques and Mathematical Structures</a>. World Scientific. p. <a rel="nofollow" class="external text" href="https://archive.org/details/discretemathemat0000penn/page/34">34</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-02-4088-2" title="Special:BookSources/978-981-02-4088-2"><bdi>978-981-02-4088-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Discrete+Mathematics%3A+Proof+Techniques+and+Mathematical+Structures&rft.pages=34&rft.pub=World+Scientific&rft.date=1999&rft.isbn=978-981-02-4088-2&rft.aulast=Penner&rft.aufirst=Robert+C.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fdiscretemathemat0000penn&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988"> </li> <li id="cite_note-78"><a href="#cite_ref-78">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFReid1992" class="citation book cs1">Reid, Constance (1992). <a href="/wiki/From_Zero_to_Infinity" title="From Zero to Infinity">From zero to infinity: what makes numbers interesting</a> (4th ed.). <a href="/wiki/Mathematical_Association_of_America" title="Mathematical Association of America">Mathematical Association of America</a>. p. 23. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-88385-505-8" title="Special:BookSources/978-0-88385-505-8"><bdi>978-0-88385-505-8</bdi></a>. <q>zero neither prime nor composite</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=From+zero+to+infinity%3A+what+makes+numbers+interesting&rft.pages=23&rft.edition=4th&rft.pub=Mathematical+Association+of+America&rft.date=1992&rft.isbn=978-0-88385-505-8&rft.aulast=Reid&rft.aufirst=Constance&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988"> </li> <li id="cite_note-FOOTNOTECheng201747-79"><a href="#cite_ref-FOOTNOTECheng201747_79-0">^</a> <a href="#CITEREFCheng2017">Cheng 2017</a>, p. 47. </li> <li id="cite_note-80"><a href="#cite_ref-80">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHermanStrang2017" class="citation book cs1">Herman, Edwin; <a href="/wiki/Gilbert_Strang" title="Gilbert Strang">Strang, Gilbert</a>; et al. (2017). <a rel="nofollow" class="external text" href="https://openstax.org/details/books/calculus-volume-1">Calculus</a>. Vol. 1. Houston, Texas: OpenStax. pp. 454–459. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-938168-02-4" title="Special:BookSources/978-1-938168-02-4"><bdi>978-1-938168-02-4</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/1022848630">1022848630</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20220923230919/https://openstax.org/details/books/calculus-volume-1">Archived</a> from the original on 23 September 2022. Retrieved 26 July 2022.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Calculus&rft.place=Houston%2C+Texas&rft.pages=%3Cspan+class%3D%22nowrap%22%3E454-%3C%2Fspan%3E459&rft.pub=OpenStax&rft.date=2017&rft_id=info%3Aoclcnum%2F1022848630&rft.isbn=978-1-938168-02-4&rft.aulast=Herman&rft.aufirst=Edwin&rft.au=Strang%2C+Gilbert&rft_id=https%3A%2F%2Fopenstax.org%2Fdetails%2Fbooks%2Fcalculus-volume-1&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988"> </li> <li id="cite_note-gkp-81"><a href="#cite_ref-gkp_81-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGrahamKnuthPatashnik1988" class="citation book cs1"><a href="/wiki/Ronald_Graham" title="Ronald Graham">Graham, Ronald L.</a>; <a href="/wiki/Donald_Knuth" title="Donald Knuth">Knuth, Donald E.</a>; <a href="/wiki/Oren_Patashnik" title="Oren Patashnik">Patashnik, Oren</a> (1988). <a href="/wiki/Concrete_Mathematics" title="Concrete Mathematics">Concrete Mathematics</a>. Reading, MA: Addison-Wesley. p. 111. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-201-14236-8" title="Special:BookSources/0-201-14236-8"><bdi>0-201-14236-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Concrete+Mathematics&rft.place=Reading%2C+MA&rft.pages=111&rft.pub=Addison-Wesley&rft.date=1988&rft.isbn=0-201-14236-8&rft.aulast=Graham&rft.aufirst=Ronald+L.&rft.au=Knuth%2C+Donald+E.&rft.au=Patashnik%2C+Oren&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988"> </li> <li id="cite_note-FOOTNOTECheng201760-82"><a href="#cite_ref-FOOTNOTECheng201760_82-0">^</a> <a href="#CITEREFCheng2017">Cheng 2017</a>, p. 60. </li> <li id="cite_note-FOOTNOTEKardar200735-83"><a href="#cite_ref-FOOTNOTEKardar200735_83-0">^</a> <a href="#CITEREFKardar2007">Kardar 2007</a>, p. 35. </li> <li id="cite_note-84"><a href="#cite_ref-84">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRiehl2016" class="citation book cs1"><a href="/wiki/Emily_Riehl" title="Emily Riehl">Riehl, Emily</a> (2016). <a rel="nofollow" class="external text" href="https://math.jhu.edu/~eriehl/context/">Category Theory in Context</a>. Dover. p. 103. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-80903-8" title="Special:BookSources/978-0-486-80903-8"><bdi>978-0-486-80903-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Category+Theory+in+Context&rft.pages=103&rft.pub=Dover&rft.date=2016&rft.isbn=978-0-486-80903-8&rft.aulast=Riehl&rft.aufirst=Emily&rft_id=https%3A%2F%2Fmath.jhu.edu%2F~eriehl%2Fcontext%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988"> </li> <li id="cite_note-85"><a href="#cite_ref-85">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRexFinn2017" class="citation book cs1">Rex, Andrew; Finn, C. B. P. (2017). Finn's Thermal Physics (3rd ed.). CRC Press. pp. 8–16. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4987-1887-5" title="Special:BookSources/978-1-4987-1887-5"><bdi>978-1-4987-1887-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Finn%27s+Thermal+Physics&rft.pages=%3Cspan+class%3D%22nowrap%22%3E8-%3C%2Fspan%3E16&rft.edition=3rd&rft.pub=CRC+Press&rft.date=2017&rft.isbn=978-1-4987-1887-5&rft.aulast=Rex&rft.aufirst=Andrew&rft.au=Finn%2C+C.+B.+P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988"> </li> <li id="cite_note-FOOTNOTEKardar20074–5,_103–104-86"><a href="#cite_ref-FOOTNOTEKardar20074%E2%80%935,_103%E2%80%93104_86-0">^</a> <a href="#CITEREFKardar2007">Kardar 2007</a>, pp. 4–5, 103–104. </li> <li id="cite_note-FOOTNOTEWoodford20069-87"><a href="#cite_ref-FOOTNOTEWoodford20069_87-0">^</a> <a href="#CITEREFWoodford2006">Woodford 2006</a>, p. 9. </li> <li id="cite_note-FOOTNOTEHill202020-88"><a href="#cite_ref-FOOTNOTEHill202020_88-0">^</a> <a href="#CITEREFHill2020">Hill 2020</a>, p. 20. </li> <li id="cite_note-89"><a href="#cite_ref-89">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFOverland2004" class="citation book cs1">Overland, Brian (14 September 2004). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=bW6MiHxPULUC&dq=cobol+array+index&pg=PT132">C++ Without Fear: A Beginner's Guide That Makes You Feel Smart</a>. Pearson Education. p. 132. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7686-8488-9" title="Special:BookSources/978-0-7686-8488-9"><bdi>978-0-7686-8488-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=C%2B%2B+Without+Fear%3A+A+Beginner%27s+Guide+That+Makes+You+Feel+Smart&rft.pages=132&rft.pub=Pearson+Education&rft.date=2004-09-14&rft.isbn=978-0-7686-8488-9&rft.aulast=Overland&rft.aufirst=Brian&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DbW6MiHxPULUC%26dq%3Dcobol%2Barray%2Bindex%26pg%3DPT132&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988"> </li> <li id="cite_note-90"><a href="#cite_ref-90">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFOliveiraStewart2006" class="citation book cs1">Oliveira, Suely; Stewart, David E. (7 September 2006). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=E6a8oZOS8noC&dq=C+array+index+zero&pg=PA64">Writing Scientific Software: A Guide to Good Style</a>. Cambridge University Press. p. 64. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-139-45862-7" title="Special:BookSources/978-1-139-45862-7"><bdi>978-1-139-45862-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Writing+Scientific+Software%3A+A+Guide+to+Good+Style&rft.pages=64&rft.pub=Cambridge+University+Press&rft.date=2006-09-07&rft.isbn=978-1-139-45862-7&rft.aulast=Oliveira&rft.aufirst=Suely&rft.au=Stewart%2C+David+E.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DE6a8oZOS8noC%26dq%3DC%2Barray%2Bindex%2Bzero%26pg%3DPA64&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988"> </li> <li id="cite_note-91"><a href="#cite_ref-91">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://docs.oracle.com/javase/8/docs/api/java/sql/ResultSet.html">"ResultSet (Java Platform SE 8 )"</a>. docs.oracle.com. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20220509185749/https://docs.oracle.com/javase/8/docs/api/java/sql/ResultSet.html">Archived</a> from the original on 9 May 2022. Retrieved 9 May 2022.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=docs.oracle.com&rft.atitle=ResultSet+%28Java+Platform+SE+8+%29&rft_id=https%3A%2F%2Fdocs.oracle.com%2Fjavase%2F8%2Fdocs%2Fapi%2Fjava%2Fsql%2FResultSet.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988"> </li> <li id="cite_note-92"><a href="#cite_ref-92">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFReese2013" class="citation book cs1">Reese, Richard M. (2013). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=-U155tRMLJgC&dq=C%20%22null%20pointer%22%200&pg=PT26">Understanding and Using C Pointers: Core Techniques for Memory Management</a>. O'Reilly Media. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-449-34455-9" title="Special:BookSources/978-1-449-34455-9"><bdi>978-1-449-34455-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Understanding+and+Using+C+Pointers%3A+Core+Techniques+for+Memory+Management&rft.pub=O%27Reilly+Media&rft.date=2013&rft.isbn=978-1-449-34455-9&rft.aulast=Reese&rft.aufirst=Richard+M.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D-U155tRMLJgC%26dq%3DC%2520%2522null%2520pointer%2522%25200%26pg%3DPT26&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988"> </li> <li id="cite_note-93"><a href="#cite_ref-93">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWuIchikawaCercone1996" class="citation book cs1">Wu, X.; Ichikawa, T.; Cercone, N. (25 October 1996). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=SdLsCgAAQBAJ&q=%C2%A0In+databases%2C+it+is+possible+for+a+field+not+to+have+a+value+%28null%29&pg=PT197">Knowledge-Base Assisted Database Retrieval Systems</a>. World Scientific. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-4501-75-0" title="Special:BookSources/978-981-4501-75-0"><bdi>978-981-4501-75-0</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20220331032618/https://books.google.com/books?id=SdLsCgAAQBAJ&q=%C2%A0In+databases%2C+it+is+possible+for+a+field+not+to+have+a+value+%28null%29&pg=PT197">Archived</a> from the original on 31 March 2022. Retrieved 7 November 2020.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Knowledge-Base+Assisted+Database+Retrieval+Systems&rft.pub=World+Scientific&rft.date=1996-10-25&rft.isbn=978-981-4501-75-0&rft.aulast=Wu&rft.aufirst=X.&rft.au=Ichikawa%2C+T.&rft.au=Cercone%2C+N.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DSdLsCgAAQBAJ%26q%3D%25C2%25A0In%2Bdatabases%252C%2Bit%2Bis%2Bpossible%2Bfor%2Ba%2Bfield%2Bnot%2Bto%2Bhave%2Ba%2Bvalue%2B%2528null%2529%26pg%3DPT197&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988"> </li> <li id="cite_note-94"><a href="#cite_ref-94">^</a> <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.ibm.com/docs/en/rbd/9.5.1?topic=parts-null-values-nullable-type">"Null values and the nullable type"</a>. IBM. 12 December 2018. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20211123185142/https://www.ibm.com/docs/en/rbd/9.5.1?topic=parts-null-values-nullable-type">Archived</a> from the original on 23 November 2021. Retrieved 23 November 2021. <q>In regard to services, sending a null value as an argument in a remote service call means that no data is sent. Because the receiving parameter is nullable, the receiving function creates a new, uninitialized value for the missing data then passes it to the requested service function.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=IBM&rft.atitle=Null+values+and+the+nullable+type&rft.date=2018-12-12&rft_id=https%3A%2F%2Fwww.ibm.com%2Fdocs%2Fen%2Frbd%2F9.5.1%3Ftopic%3Dparts-null-values-nullable-type&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988"> </li> <li id="cite_note-95"><a href="#cite_ref-95">^</a> Paul DuBois. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=lFsaBAAAQBAJ">"MySQL Cookbook: Solutions for Database Developers and Administrators"</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170224134429/https://books.google.com/books?id=lFsaBAAAQBAJ">Archived</a> 24 February 2017 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>, 2014. p. 204. </li> <li id="cite_note-96"><a href="#cite_ref-96">^</a> Arnold Robbins; Nelson Beebe. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=J9WbAgAAQBAJ">"Classic Shell Scripting"</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170224134147/https://books.google.com/books?id=J9WbAgAAQBAJ">Archived</a> 24 February 2017 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>. 2005. p. 274. </li> <li id="cite_note-97"><a href="#cite_ref-97">^</a> Iztok Fajfar. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=eHq9CgAAQBAJ">"Start Programming Using HTML, CSS, and JavaScript"</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170224134155/https://books.google.com/books?id=eHq9CgAAQBAJ">Archived</a> 24 February 2017 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>. 2015. p. 160. </li> <li id="cite_note-98"><a href="#cite_ref-98">^</a> Darren R. Hayes. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=0qPfBQAAQBAJ">"A Practical Guide to Computer Forensics Investigations"</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170224134341/https://books.google.com/books?id=0qPfBQAAQBAJ">Archived</a> 24 February 2017 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>. 2014. p. 399. </li> <li id="cite_note-99"><a href="#cite_ref-99">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRochkind1985" class="citation book cs1">Rochkind, Marc J. (1985). Advanced UNIX Programming. Prentice-Hall Software Series. Englewood Cliffs, New Jersey: Prentice Hall. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-13-011818-4" title="Special:BookSources/0-13-011818-4"><bdi>0-13-011818-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Advanced+UNIX+Programming&rft.place=Englewood+Cliffs%2C+New+Jersey&rft.series=Prentice-Hall+Software+Series&rft.pub=Prentice+Hall&rft.date=1985&rft.isbn=0-13-011818-4&rft.aulast=Rochkind&rft.aufirst=Marc+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988"> Section 5.5, "Exit system call", p.114. </li> <li id="cite_note-100"><a href="#cite_ref-100">^</a> <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.codeproject.com/Articles/30040/Font-Survey-42-of-the-Best-Monospaced-Programming">"Font Survey: 42 of the Best Monospaced Programming Fonts"</a>. codeproject.com. 18 August 2010. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20120124051919/http://www.codeproject.com:80/Articles/30040/Font-Survey-42-of-the-Best-Monospaced-Programming">Archived</a> from the original on 24 January 2012. Retrieved 22 July 2021.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=codeproject.com&rft.atitle=Font+Survey%3A+42+of+the+Best+Monospaced+Programming+Fonts&rft.date=2010-08-18&rft_id=https%3A%2F%2Fwww.codeproject.com%2FArticles%2F30040%2FFont-Survey-42-of-the-Best-Monospaced-Programming&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988"> </li> <li id="cite_note-101"><a href="#cite_ref-101">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCepelewicz2021" class="citation magazine cs1">Cepelewicz, Jordana (9 August 2021). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20210818042926/https://www.quantamagazine.org/animals-can-count-and-use-zero-how-far-does-their-number-sense-go-20210809/">"Animals Count and Use Zero. How Far Does Their Number Sense Go?"</a>. <a href="/wiki/Quanta_Magazine" title="Quanta Magazine">Quanta Magazine</a>. Archived from <a rel="nofollow" class="external text" href="https://www.quantamagazine.org/animals-can-count-and-use-zero-how-far-does-their-number-sense-go-20210809/">the original</a> on 18 August 2021.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Quanta+Magazine&rft.atitle=Animals+Count+and+Use+Zero.+How+Far+Does+Their+Number+Sense+Go%3F&rft.date=2021-08-09&rft.aulast=Cepelewicz&rft.aufirst=Jordana&rft_id=https%3A%2F%2Fwww.quantamagazine.org%2Fanimals-can-count-and-use-zero-how-far-does-their-number-sense-go-20210809%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988"> </li> <li id="cite_note-102"><a href="#cite_ref-102">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSteel2000" class="citation book cs1">Steel, Duncan (2000). <a rel="nofollow" class="external text" href="https://archive.org/details/markingtimeepicq00stee_0/page/113">Marking Time: The epic quest to invent the perfect calendar</a>. John Wiley & Sons. p. <a rel="nofollow" class="external text" href="https://archive.org/details/markingtimeepicq00stee_0/page/113">113</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-29827-4" title="Special:BookSources/978-0-471-29827-4"><bdi>978-0-471-29827-4</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/1135427740">1135427740</a>. <q>In the B.C./A.D. scheme there is no year zero. After 31 December 1 BC came 1 January AD 1. ... If you object to that no-year-zero scheme, then don't use it: use the astronomer's counting scheme, with negative year numbers.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Marking+Time%3A+The+epic+quest+to+invent+the+perfect+calendar&rft.pages=113&rft.pub=John+Wiley+%26+Sons&rft.date=2000&rft_id=info%3Aoclcnum%2F1135427740&rft.isbn=978-0-471-29827-4&rft.aulast=Steel&rft.aufirst=Duncan&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fmarkingtimeepicq00stee_0%2Fpage%2F113&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988"> </li> </ol></div></div> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(11)"><h2 id="Bibliography">Bibliography</h2></div><section class="mf-section-11 collapsible-block" id="mf-section-11"> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin" style=""> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAczel2015" class="citation book cs1"><a href="/wiki/Amir_D._Aczel" class="mw-redirect" title="Amir D. Aczel">Aczel, Amir D.</a> (2015). <a rel="nofollow" class="external text" href="https://archive.org/details/findingzeromathe0000acze">Finding Zero</a>. New York: Palgrave Macmillan. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-137-27984-2" title="Special:BookSources/978-1-137-27984-2"><bdi>978-1-137-27984-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Finding+Zero&rft.place=New+York&rft.pub=Palgrave+Macmillan&rft.date=2015&rft.isbn=978-1-137-27984-2&rft.aulast=Aczel&rft.aufirst=Amir+D.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Ffindingzeromathe0000acze&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAsimov1978" class="citation book cs1"><a href="/wiki/Isaac_Asimov" title="Isaac Asimov">Asimov, Isaac</a> (1978). "Nothing Counts". <a rel="nofollow" class="external text" href="https://archive.org/details/asimovonnumbers00isaa">Asimov on Numbers</a>. New York: Pocket Books. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-671-82134-0" title="Special:BookSources/978-0-671-82134-0"><bdi>978-0-671-82134-0</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/1105483009">1105483009</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Nothing+Counts&rft.btitle=Asimov+on+Numbers&rft.place=New+York&rft.pub=Pocket+Books&rft.date=1978&rft_id=info%3Aoclcnum%2F1105483009&rft.isbn=978-0-671-82134-0&rft.aulast=Asimov&rft.aufirst=Isaac&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fasimovonnumbers00isaa&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBarrow2001" class="citation book cs1"><a href="/wiki/John_D._Barrow" title="John D. Barrow">Barrow, John D.</a> (2001). <a rel="nofollow" class="external text" href="https://archive.org/details/bookofnothing0000barr">The Book of Nothing</a>. Vintage. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-09-928845-1" title="Special:BookSources/0-09-928845-1"><bdi>0-09-928845-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Book+of+Nothing&rft.pub=Vintage&rft.date=2001&rft.isbn=0-09-928845-1&rft.aulast=Barrow&rft.aufirst=John+D.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fbookofnothing0000barr&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCheng2017" class="citation book cs1"><a href="/wiki/Eugenia_Cheng" title="Eugenia Cheng">Cheng, Eugenia</a> (2017). Beyond Infinity: An Expedition to the Outer Limits of Mathematics. Basic Books. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-5416-4413-7" title="Special:BookSources/978-1-5416-4413-7"><bdi>978-1-5416-4413-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Beyond+Infinity%3A+An+Expedition+to+the+Outer+Limits+of+Mathematics&rft.pub=Basic+Books&rft.date=2017&rft.isbn=978-1-5416-4413-7&rft.aulast=Cheng&rft.aufirst=Eugenia&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKardar2007" class="citation book cs1"><a href="/wiki/Mehran_Kardar" title="Mehran Kardar">Kardar, Mehran</a> (2007). <a href="/wiki/Statistical_Physics_of_Particles" title="Statistical Physics of Particles">Statistical Physics of Particles</a>. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-87342-0" title="Special:BookSources/978-0-521-87342-0"><bdi>978-0-521-87342-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Statistical+Physics+of+Particles&rft.pub=Cambridge+University+Press&rft.date=2007&rft.isbn=978-0-521-87342-0&rft.aulast=Kardar&rft.aufirst=Mehran&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFReimer2014" class="citation book cs1">Reimer, David (2014). Count Like an Egyptian. Princeton University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-16012-2" title="Special:BookSources/978-0-691-16012-2"><bdi>978-0-691-16012-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Count+Like+an+Egyptian&rft.pub=Princeton+University+Press&rft.date=2014&rft.isbn=978-0-691-16012-2&rft.aulast=Reimer&rft.aufirst=David&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWoodford2006" class="citation book cs1"><a href="/wiki/Chris_Woodford_(author)" class="mw-redirect" title="Chris Woodford (author)">Woodford, Chris</a> (2006). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=My7Zr0aP2L8C&pg=PA9">Digital Technology</a>. Evans Brothers. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-237-52725-9" title="Special:BookSources/978-0-237-52725-9"><bdi>978-0-237-52725-9</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20190817150242/https://books.google.com/books?id=My7Zr0aP2L8C&pg=PA9">Archived</a> from the original on 17 August 2019. Retrieved 24 March 2016.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Digital+Technology&rft.pub=Evans+Brothers&rft.date=2006&rft.isbn=978-0-237-52725-9&rft.aulast=Woodford&rft.aufirst=Chris&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DMy7Zr0aP2L8C%26pg%3DPA9&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHill2020" class="citation book cs1">Hill, Christian (2020). Learning Scientific Programming with Python (2nd ed.). Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-10707541-2" title="Special:BookSources/978-1-10707541-2"><bdi>978-1-10707541-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Learning+Scientific+Programming+with+Python&rft.edition=2nd&rft.pub=Cambridge+University+Press&rft.date=2020&rft.isbn=978-1-10707541-2&rft.aulast=Hill&rft.aufirst=Christian&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988"></li></ul> </div> <div class="mw-heading mw-heading3"><h3 id="Historical_studies">Historical studies</h3></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239549316"><div class="refbegin" style=""> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBourbaki1998" class="citation book cs1"><a href="/wiki/Nicolas_Bourbaki" title="Nicolas Bourbaki">Bourbaki, Nicolas</a> (1998). Elements of the History of Mathematics. Berlin, Heidelberg, and New York: Springer-Verlag. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3-540-64767-8" title="Special:BookSources/3-540-64767-8"><bdi>3-540-64767-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elements+of+the+History+of+Mathematics&rft.place=Berlin%2C+Heidelberg%2C+and+New+York&rft.pub=Springer-Verlag&rft.date=1998&rft.isbn=3-540-64767-8&rft.aulast=Bourbaki&rft.aufirst=Nicolas&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDiehl2004" class="citation book cs1">Diehl, Richard A. (2004). The Olmecs: America's First Civilization. London, England: Thames & Hudson. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-500-28503-9" title="Special:BookSources/978-0-500-28503-9"><bdi>978-0-500-28503-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Olmecs%3A+America%27s+First+Civilization&rft.place=London%2C+England&rft.pub=Thames+%26+Hudson&rft.date=2004&rft.isbn=978-0-500-28503-9&rft.aulast=Diehl&rft.aufirst=Richard+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFIfrah2000" class="citation book cs1">Ifrah, Georges (2000). The Universal History of Numbers: From Prehistory to the Invention of the Computer. Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-39340-1" title="Special:BookSources/0-471-39340-1"><bdi>0-471-39340-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Universal+History+of+Numbers%3A+From+Prehistory+to+the+Invention+of+the+Computer&rft.pub=Wiley&rft.date=2000&rft.isbn=0-471-39340-1&rft.aulast=Ifrah&rft.aufirst=Georges&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKaplan2000" class="citation book cs1">Kaplan, Robert (2000). The Nothing That Is: A Natural History of Zero. Oxford University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-198-02945-8" title="Special:BookSources/978-0-198-02945-8"><bdi>978-0-198-02945-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Nothing+That+Is%3A+A+Natural+History+of+Zero&rft.pub=Oxford+University+Press&rft.date=2000&rft.isbn=978-0-198-02945-8&rft.aulast=Kaplan&rft.aufirst=Robert&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSeife2000" class="citation book cs1"><a href="/wiki/Charles_Seife" title="Charles Seife">Seife, Charles</a> (2000). Zero: The Biography of a Dangerous Idea. Penguin USA. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-14-029647-6" title="Special:BookSources/0-14-029647-6"><bdi>0-14-029647-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Zero%3A+The+Biography+of+a+Dangerous+Idea&rft.pub=Penguin+USA&rft.date=2000&rft.isbn=0-14-029647-6&rft.aulast=Seife&rft.aufirst=Charles&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988"></li></ul> </div> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(12)"><h2 id="External_links">External links</h2></div><section class="mf-section-12 collapsible-block" id="mf-section-12"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1235681985"><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media 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data-mw-src="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/40px-Wiktionary-logo-en-v2.svg.png" data-alt="" data-width="40" data-height="40" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/60px-Wiktionary-logo-en-v2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/80px-Wiktionary-logo-en-v2.svg.png 2x" data-class="mw-file-element"> </a></div> <div class="side-box-text plainlist">Look up <a href="https://en.wiktionary.org/wiki/zero" class="extiw" title="wiktionary:zero">zero</a> in Wiktionary, the free dictionary.</div></div> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1235681985"><style data-mw-deduplicate="TemplateStyles:r1250146164">.mw-parser-output .sister-box .side-box-abovebelow{padding:0.75em 0;text-align:center}.mw-parser-output .sister-box .side-box-abovebelow>b{display:block}.mw-parser-output .sister-box .side-box-text>ul{border-top:1px solid #aaa;padding:0.75em 0;width:217px;margin:0 auto}.mw-parser-output .sister-box .side-box-text>ul>li{min-height:31px}.mw-parser-output .sister-logo{display:inline-block;width:31px;line-height:31px;vertical-align:middle;text-align:center}.mw-parser-output .sister-link{display:inline-block;margin-left:4px;width:182px;vertical-align:middle}@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-v2.svg"]{background-color:white}}</style><div role="navigation" aria-labelledby="sister-projects" class="side-box metadata side-box-right sister-box sistersitebox plainlinks"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"> <div class="side-box-abovebelow"> 0 at Wikipedia's <a href="/wiki/Wikipedia:Wikimedia_sister_projects" title="Wikipedia:Wikimedia sister projects">sister projects</a></div> <div class="side-box-flex"> <div class="side-box-text plainlist"><ul><li><noscript><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/0/06/Wiktionary-logo-v2.svg/27px-Wiktionary-logo-v2.svg.png" decoding="async" width="27" height="27" class="mw-file-element" data-file-width="391" data-file-height="391"></noscript><span class="lazy-image-placeholder" style="width: 27px;height: 27px;" data-mw-src="//upload.wikimedia.org/wikipedia/en/thumb/0/06/Wiktionary-logo-v2.svg/27px-Wiktionary-logo-v2.svg.png" data-alt="" data-width="27" data-height="27" data-srcset="//upload.wikimedia.org/wikipedia/en/thumb/0/06/Wiktionary-logo-v2.svg/41px-Wiktionary-logo-v2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/0/06/Wiktionary-logo-v2.svg/54px-Wiktionary-logo-v2.svg.png 2x" data-class="mw-file-element"> <a href="https://en.wiktionary.org/wiki/zero" class="extiw" title="wikt:zero">Definitions</a> from Wiktionary</li><li><noscript><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/20px-Commons-logo.svg.png" decoding="async" width="20" height="27" class="mw-file-element" data-file-width="1024" data-file-height="1376"></noscript><span class="lazy-image-placeholder" style="width: 20px;height: 27px;" data-mw-src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/20px-Commons-logo.svg.png" data-alt="" data-width="20" data-height="27" data-srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/40px-Commons-logo.svg.png 2x" data-class="mw-file-element"> <a href="https://commons.wikimedia.org/wiki/Category:0_(number)" class="extiw" title="c:Category:0 (number)">Media</a> from Commons</li><li><noscript><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/23px-Wikiquote-logo.svg.png" decoding="async" width="23" height="27" class="mw-file-element" data-file-width="300" data-file-height="355"></noscript><span class="lazy-image-placeholder" style="width: 23px;height: 27px;" data-mw-src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/23px-Wikiquote-logo.svg.png" data-alt="" data-width="23" data-height="27" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/35px-Wikiquote-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/46px-Wikiquote-logo.svg.png 2x" data-class="mw-file-element"> <a href="https://en.wikiquote.org/wiki/Zero" class="extiw" title="q:Zero">Quotations</a> from Wikiquote</li></ul></div></div> </div> <ul><li><a rel="nofollow" class="external text" href="http://www.huffingtonpost.com/amir-aczel/worlds-first-zero_b_3276709.html">Searching for the World's First Zero</a></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20081204042054/http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Zero.html">A History of Zero</a></li> <li><a rel="nofollow" class="external text" href="http://home.ubalt.edu/ntsbarsh/zero/ZERO.HTM">Zero Saga</a></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20141009100628/http://www.ucs.louisiana.edu/~sxw8045/history.htm">The History of Algebra</a></li> <li><a href="/wiki/Edsger_W._Dijkstra" title="Edsger W. Dijkstra">Edsger W. Dijkstra</a>: <a rel="nofollow" class="external text" href="http://www.cs.utexas.edu/users/EWD/ewd08xx/EWD831.PDF">Why numbering should start at zero</a>, EWD831 (<a href="/wiki/Portable_Document_Format" class="mw-redirect" title="Portable Document Format">PDF</a> of a handwritten manuscript)</li> <li><a rel="nofollow" class="external text" href="https://www.bbc.co.uk/programmes/p004y254">Zero</a> on <a href="/wiki/In_Our_Time_(radio_series)" title="In Our Time (radio series)">In Our Time</a> at the <a href="/wiki/BBC" title="BBC">BBC</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/0.html">"0"</a>. <a href="/wiki/MathWorld" title="MathWorld">MathWorld</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=0&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2F0.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988"></li> <li><a href="/wiki/File:Wikisource-logo.svg" class="mw-file-description"><noscript><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/15px-Wikisource-logo.svg.png" decoding="async" width="15" height="16" class="mw-file-element" data-file-width="410" data-file-height="430"></noscript><span class="lazy-image-placeholder" style="width: 15px;height: 16px;" data-mw-src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/15px-Wikisource-logo.svg.png" data-alt="" data-width="15" data-height="16" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/23px-Wikisource-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/30px-Wikisource-logo.svg.png 2x" data-class="mw-file-element"> </a> Texts on Wikisource: <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation encyclopaedia cs1">"<a href="https://en.wikisource.org/wiki/1911_Encyclop%C3%A6dia_Britannica/Zero" class="extiw" title="s:1911 Encyclopædia Britannica/Zero">Zero</a>". <a href="/wiki/Encyclop%C3%A6dia_Britannica_Eleventh_Edition" title="Encyclopædia Britannica Eleventh Edition">Encyclopædia Britannica</a> (11th ed.). 1911.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Zero&rft.btitle=Encyclop%C3%A6dia+Britannica&rft.edition=11th&rft.date=1911&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation encyclopaedia cs1">"<a href="https://en.wikisource.org/wiki/The_Encyclopedia_Americana_(1920)/Zero" class="extiw" title="s:The Encyclopedia Americana (1920)/Zero">Zero</a>". <a href="/wiki/Encyclopedia_Americana" title="Encyclopedia Americana">Encyclopedia Americana</a>. 1920.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Zero&rft.btitle=Encyclopedia+Americana&rft.date=1920&rfr_id=info%3Asid%2Fen.wikipedia.org%3A0" class="Z3988"></li></ul></li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output .navbox-list{line-height:1.5em;border-color:#fdfdfd}.mw-parser-output .navbox-list-with-group{text-align:left;border-left-width:2px;border-left-style:solid}.mw-parser-output tr+tr>.navbox-abovebelow,.mw-parser-output tr+tr>.navbox-group,.mw-parser-output tr+tr>.navbox-image,.mw-parser-output tr+tr>.navbox-list{border-top:2px solid #fdfdfd}.mw-parser-output .navbox-title{background-color:#ccf}.mw-parser-output .navbox-abovebelow,.mw-parser-output .navbox-group,.mw-parser-output .navbox-subgroup .navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"><style data-mw-deduplicate="TemplateStyles:r1038841319">.mw-parser-output .tooltip-dotted{border-bottom:1px dotted;cursor:help}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1038841319"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1038841319"></div>    </section></div> <noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=mobile&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" href="https://en.wikipedia.org/w/index.php?title=0&oldid=1269408534">https://en.wikipedia.org/w/index.php?title=0&oldid=1269408534</a>"</div></div> </div> <div class="post-content" id="page-secondary-actions"> </div> </main> <footer class="mw-footer minerva-footer" role="contentinfo"> <a class="last-modified-bar" href="/w/index.php?title=0&action=history"> <div class="post-content last-modified-bar__content"> Last edited on 14 January 2025, at 14:43 </div> </a> <div class="post-content footer-content"> <div id='mw-data-after-content'> <div class="read-more-container"></div> </div> <div id="p-lang"> <h4>Languages</h4> <section> <ul id="p-variants" class="minerva-languages"></ul> <ul class="minerva-languages"><li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/0_(getal)" title="0 (getal) – Afrikaans" lang="af" hreflang="af" data-title="0 (getal)" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target">Afrikaans</a></li><li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Null" title="Null – Alemannic" lang="gsw" hreflang="gsw" data-title="Null" data-language-autonym="Alemannisch" data-language-local-name="Alemannic" class="interlanguage-link-target">Alemannisch</a></li><li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/0" title="0 – Amharic" lang="am" hreflang="am" data-title="0" data-language-autonym="አማርኛ" data-language-local-name="Amharic" class="interlanguage-link-target">አማርኛ</a></li><li class="interlanguage-link interwiki-ab mw-list-item"><a href="https://ab.wikipedia.org/wiki/%D0%90%D0%BD%D0%BE%D0%BB%D1%8C" title="Аноль – Abkhazian" lang="ab" hreflang="ab" data-title="Аноль" data-language-autonym="Аԥсшәа" data-language-local-name="Abkhazian" class="interlanguage-link-target">Аԥсшәа</a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/0_(%D8%B9%D8%AF%D8%AF)" title="0 (عدد) – Arabic" lang="ar" hreflang="ar" data-title="0 (عدد)" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-arc mw-list-item"><a href="https://arc.wikipedia.org/wiki/0_(%DC%A1%DC%A2%DC%9D%DC%A2%DC%90)" title="0 (ܡܢܝܢܐ) – Aramaic" lang="arc" hreflang="arc" data-title="0 (ܡܢܝܢܐ)" data-language-autonym="ܐܪܡܝܐ" data-language-local-name="Aramaic" class="interlanguage-link-target">ܐܪܡܝܐ</a></li><li class="interlanguage-link interwiki-as mw-list-item"><a href="https://as.wikipedia.org/wiki/%E0%A6%B6%E0%A7%82%E0%A6%A8%E0%A7%8D%E0%A6%AF" title="শূন্য – Assamese" lang="as" hreflang="as" data-title="শূন্য" data-language-autonym="অসমীয়া" data-language-local-name="Assamese" class="interlanguage-link-target">অসমীয়া</a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Cero" title="Cero – Asturian" lang="ast" hreflang="ast" data-title="Cero" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target">Asturianu</a></li><li class="interlanguage-link interwiki-gn mw-list-item"><a href="https://gn.wikipedia.org/wiki/Mba%27eve_(papaha)" title="Mba'eve (papaha) – Guarani" lang="gn" hreflang="gn" data-title="Mba'eve (papaha)" data-language-autonym="Avañe'ẽ" data-language-local-name="Guarani" class="interlanguage-link-target">Avañe'ẽ</a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/0" title="0 – Azerbaijani" lang="az" hreflang="az" data-title="0" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target">Azərbaycanca</a></li><li class="interlanguage-link interwiki-azb mw-list-item"><a href="https://azb.wikipedia.org/wiki/%DB%B0_(%D8%B3%D8%A7%DB%8C%DB%8C)" title="۰ (سایی) – South Azerbaijani" lang="azb" hreflang="azb" data-title="۰ (سایی)" data-language-autonym="تۆرکجه" data-language-local-name="South Azerbaijani" class="interlanguage-link-target">تۆرکجه</a></li><li class="interlanguage-link interwiki-ban mw-list-item"><a href="https://ban.wikipedia.org/wiki/0_(angka)" title="0 (angka) – Balinese" lang="ban" hreflang="ban" data-title="0 (angka)" data-language-autonym="Basa Bali" data-language-local-name="Balinese" class="interlanguage-link-target">Basa Bali</a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A7%A6_(%E0%A6%B8%E0%A6%82%E0%A6%96%E0%A7%8D%E0%A6%AF%E0%A6%BE)" title="০ (সংখ্যা) – Bangla" lang="bn" hreflang="bn" data-title="০ (সংখ্যা)" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target">বাংলা</a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/0" title="0 – Minnan" lang="nan" hreflang="nan" data-title="0" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target">閩南語 / Bân-lâm-gú</a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/0_(%D2%BB%D0%B0%D0%BD)" title="0 (һан) – Bashkir" lang="ba" hreflang="ba" data-title="0 (һан)" data-language-autonym="Башҡортса" data-language-local-name="Bashkir" class="interlanguage-link-target">Башҡортса</a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/0_(%D0%BB%D1%96%D0%BA)" title="0 (лік) – Belarusian" lang="be" hreflang="be" data-title="0 (лік)" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target">Беларуская</a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%9D%D1%83%D0%BB%D1%8C" title="Нуль – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Нуль" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target">Беларуская (тарашкевіца)</a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9D%D1%83%D0%BB%D0%B0" title="Нула – Bulgarian" lang="bg" hreflang="bg" data-title="Нула" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target">Български</a></li><li class="interlanguage-link interwiki-bo mw-list-item"><a href="https://bo.wikipedia.org/wiki/%E0%BC%A0_(%E0%BD%82%E0%BE%B2%E0%BD%84%E0%BD%A6%E0%BC%8B%E0%BD%80%E0%BC%8D)" title="༠ (གྲངས་ཀ།) – Tibetan" lang="bo" hreflang="bo" data-title="༠ (གྲངས་ཀ།)" data-language-autonym="བོད་ཡིག" data-language-local-name="Tibetan" class="interlanguage-link-target">བོད་ཡིག</a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/0_(broj)" title="0 (broj) – Bosnian" lang="bs" hreflang="bs" data-title="0 (broj)" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target">Bosanski</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Zero" title="Zero – Catalan" lang="ca" hreflang="ca" data-title="Zero" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Nula" title="Nula – Czech" lang="cs" hreflang="cs" data-title="Nula" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-sn mw-list-item"><a href="https://sn.wikipedia.org/wiki/Chipasina" title="Chipasina – Shona" lang="sn" hreflang="sn" data-title="Chipasina" data-language-autonym="ChiShona" data-language-local-name="Shona" class="interlanguage-link-target">ChiShona</a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/0" title="0 – Welsh" lang="cy" hreflang="cy" data-title="0" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target">Cymraeg</a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/0_(tal)" title="0 (tal) – Danish" lang="da" hreflang="da" data-title="0 (tal)" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target">Dansk</a></li><li class="interlanguage-link interwiki-ary mw-list-item"><a href="https://ary.wikipedia.org/wiki/0" title="0 – Moroccan Arabic" lang="ary" hreflang="ary" data-title="0" data-language-autonym="الدارجة" data-language-local-name="Moroccan Arabic" class="interlanguage-link-target">الدارجة</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Null" title="Null – German" lang="de" hreflang="de" data-title="Null" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Null" title="Null – Estonian" lang="et" hreflang="et" data-title="Null" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target">Eesti</a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/0_(%CE%B1%CF%81%CE%B9%CE%B8%CE%BC%CF%8C%CF%82)" title="0 (αριθμός) – Greek" lang="el" hreflang="el" data-title="0 (αριθμός)" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target">Ελληνικά</a></li><li class="interlanguage-link interwiki-myv mw-list-item"><a href="https://myv.wikipedia.org/wiki/%D0%A7%D0%B0%D0%B2%D0%BE" title="Чаво – Erzya" lang="myv" hreflang="myv" data-title="Чаво" data-language-autonym="Эрзянь" data-language-local-name="Erzya" class="interlanguage-link-target">Эрзянь</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Cero" title="Cero – Spanish" lang="es" hreflang="es" data-title="Cero" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Nulo" title="Nulo – Esperanto" lang="eo" hreflang="eo" data-title="Nulo" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target">Esperanto</a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Zero" title="Zero – Basque" lang="eu" hreflang="eu" data-title="Zero" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target">Euskara</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%B5%D9%81%D8%B1" title="صفر – Persian" lang="fa" hreflang="fa" data-title="صفر" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fo mw-list-item"><a href="https://fo.wikipedia.org/wiki/0_(tal)" title="0 (tal) – Faroese" lang="fo" hreflang="fo" data-title="0 (tal)" data-language-autonym="Føroyskt" data-language-local-name="Faroese" class="interlanguage-link-target">Føroyskt</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Z%C3%A9ro" title="Zéro – French" lang="fr" hreflang="fr" data-title="Zéro" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-fur mw-list-item"><a href="https://fur.wikipedia.org/wiki/0_(numar)" title="0 (numar) – Friulian" lang="fur" hreflang="fur" data-title="0 (numar)" data-language-autonym="Furlan" data-language-local-name="Friulian" class="interlanguage-link-target">Furlan</a></li><li class="interlanguage-link interwiki-gv mw-list-item"><a href="https://gv.wikipedia.org/wiki/0_(earroo)" title="0 (earroo) – Manx" lang="gv" hreflang="gv" data-title="0 (earroo)" data-language-autonym="Gaelg" data-language-local-name="Manx" class="interlanguage-link-target">Gaelg</a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Cero" title="Cero – Galician" lang="gl" hreflang="gl" data-title="Cero" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target">Galego</a></li><li class="interlanguage-link interwiki-inh mw-list-item"><a href="https://inh.wikipedia.org/wiki/%D0%9A%D3%80%D0%BE%D0%BB%D1%85" title="КӀолх – Ingush" lang="inh" hreflang="inh" data-title="КӀолх" data-language-autonym="ГӀалгӀай" data-language-local-name="Ingush" class="interlanguage-link-target">ГӀалгӀай</a></li><li class="interlanguage-link interwiki-ki mw-list-item"><a href="https://ki.wikipedia.org/wiki/K%C4%A9b%C5%A9g%C5%A9_(mathabu)" title="Kĩbũgũ (mathabu) – Kikuyu" lang="ki" hreflang="ki" data-title="Kĩbũgũ (mathabu)" data-language-autonym="Gĩkũyũ" data-language-local-name="Kikuyu" class="interlanguage-link-target">Gĩkũyũ</a></li><li class="interlanguage-link interwiki-gu mw-list-item"><a href="https://gu.wikipedia.org/wiki/%E0%AB%A6_(%E0%AA%B6%E0%AB%82%E0%AA%A8%E0%AB%8D%E0%AA%AF)" title="૦ (શૂન્ય) – Gujarati" lang="gu" hreflang="gu" data-title="૦ (શૂન્ય)" data-language-autonym="ગુજરાતી" data-language-local-name="Gujarati" class="interlanguage-link-target">ગુજરાતી</a></li><li class="interlanguage-link interwiki-hak mw-list-item"><a href="https://hak.wikipedia.org/wiki/0" title="0 – Hakka Chinese" lang="hak" hreflang="hak" data-title="0" data-language-autonym="客家語 / Hak-kâ-ngî" data-language-local-name="Hakka Chinese" class="interlanguage-link-target">客家語 / Hak-kâ-ngî</a></li><li class="interlanguage-link interwiki-xal mw-list-item"><a href="https://xal.wikipedia.org/wiki/0" title="0 – Kalmyk" lang="xal" hreflang="xal" data-title="0" data-language-autonym="Хальмг" data-language-local-name="Kalmyk" class="interlanguage-link-target">Хальмг</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/0" title="0 – Korean" lang="ko" hreflang="ko" data-title="0" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-ha mw-list-item"><a href="https://ha.wikipedia.org/wiki/0_(al%C6%99alami)" title="0 (alƙalami) – Hausa" lang="ha" hreflang="ha" data-title="0 (alƙalami)" data-language-autonym="Hausa" data-language-local-name="Hausa" class="interlanguage-link-target">Hausa</a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/0_(%D5%A9%D5%AB%D5%BE)" title="0 (թիվ) – Armenian" lang="hy" hreflang="hy" data-title="0 (թիվ)" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target">Հայերեն</a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B6%E0%A5%82%E0%A4%A8%E0%A5%8D%E0%A4%AF" title="शून्य – Hindi" lang="hi" hreflang="hi" data-title="शून्य" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target">हिन्दी</a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Nula" title="Nula – Croatian" lang="hr" hreflang="hr" data-title="Nula" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target">Hrvatski</a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Zero" title="Zero – Ido" lang="io" hreflang="io" data-title="Zero" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target">Ido</a></li><li class="interlanguage-link interwiki-ig mw-list-item"><a href="https://ig.wikipedia.org/wiki/0_(%C3%93n%C3%BA%E1%BB%8Dg%E1%BB%A5g%E1%BB%A5)" title="0 (Ónúọgụgụ) – Igbo" lang="ig" hreflang="ig" data-title="0 (Ónúọgụgụ)" data-language-autonym="Igbo" data-language-local-name="Igbo" class="interlanguage-link-target">Igbo</a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/0_(angka)" title="0 (angka) – Indonesian" lang="id" hreflang="id" data-title="0 (angka)" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target">Bahasa Indonesia</a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/0_(numero)" title="0 (numero) – Interlingua" lang="ia" hreflang="ia" data-title="0 (numero)" data-language-autonym="Interlingua" data-language-local-name="Interlingua" class="interlanguage-link-target">Interlingua</a></li><li class="interlanguage-link interwiki-ik mw-list-item"><a href="https://ik.wikipedia.org/wiki/Kisitchisa%C4%A1vik" title="Kisitchisaġvik – Inupiaq" lang="ik" hreflang="ik" data-title="Kisitchisaġvik" data-language-autonym="Iñupiatun" data-language-local-name="Inupiaq" class="interlanguage-link-target">Iñupiatun</a></li><li class="interlanguage-link interwiki-os mw-list-item"><a href="https://os.wikipedia.org/wiki/%D0%9D%D1%83%D0%BB%D1%8C" title="Нуль – Ossetic" lang="os" hreflang="os" data-title="Нуль" data-language-autonym="Ирон" data-language-local-name="Ossetic" class="interlanguage-link-target">Ирон</a></li><li class="interlanguage-link interwiki-xh mw-list-item"><a href="https://xh.wikipedia.org/wiki/Iqanda" title="Iqanda – Xhosa" lang="xh" hreflang="xh" data-title="Iqanda" data-language-autonym="IsiXhosa" data-language-local-name="Xhosa" class="interlanguage-link-target">IsiXhosa</a></li><li class="interlanguage-link interwiki-zu mw-list-item"><a href="https://zu.wikipedia.org/wiki/Iqanda(inombolo)" title="Iqanda(inombolo) – Zulu" lang="zu" hreflang="zu" data-title="Iqanda(inombolo)" data-language-autonym="IsiZulu" data-language-local-name="Zulu" class="interlanguage-link-target">IsiZulu</a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/N%C3%BAll" title="Núll – Icelandic" lang="is" hreflang="is" data-title="Núll" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target">Íslenska</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/0_(numero)" title="0 (numero) – Italian" lang="it" hreflang="it" data-title="0 (numero)" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/0_(%D7%9E%D7%A1%D7%A4%D7%A8)" title="0 (מספר) – Hebrew" lang="he" hreflang="he" data-title="0 (מספר)" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%B8%E0%B3%8A%E0%B2%A8%E0%B3%8D%E0%B2%A8%E0%B3%86" title="ಸೊನ್ನೆ – Kannada" lang="kn" hreflang="kn" data-title="ಸೊನ್ನೆ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="Kannada" class="interlanguage-link-target">ಕನ್ನಡ</a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/0_(%E1%83%A0%E1%83%98%E1%83%AA%E1%83%AE%E1%83%95%E1%83%98)" title="0 (რიცხვი) – Georgian" lang="ka" hreflang="ka" data-title="0 (რიცხვი)" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target">ქართული</a></li><li class="interlanguage-link interwiki-ks mw-list-item"><a href="https://ks.wikipedia.org/wiki/0_(%DA%AF%D8%B1%D9%8E%D9%9B%D9%86%D9%9B%D8%AF)" title="0 (گرَٛنٛد) – Kashmiri" lang="ks" hreflang="ks" data-title="0 (گرَٛنٛد)" data-language-autonym="कॉशुर / کٲشُر" data-language-local-name="Kashmiri" class="interlanguage-link-target">कॉशुर / کٲشُر</a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/0_(%D1%81%D0%B0%D0%BD)" title="0 (сан) – Kazakh" lang="kk" hreflang="kk" data-title="0 (сан)" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target">Қазақша</a></li><li class="interlanguage-link interwiki-rw mw-list-item"><a href="https://rw.wikipedia.org/wiki/Ubusa" title="Ubusa – Kinyarwanda" lang="rw" hreflang="rw" data-title="Ubusa" data-language-autonym="Ikinyarwanda" data-language-local-name="Kinyarwanda" class="interlanguage-link-target">Ikinyarwanda</a></li><li class="interlanguage-link interwiki-rn mw-list-item"><a href="https://rn.wikipedia.org/wiki/Ubusa" title="Ubusa – Rundi" lang="rn" hreflang="rn" data-title="Ubusa" data-language-autonym="Ikirundi" data-language-local-name="Rundi" class="interlanguage-link-target">Ikirundi</a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Sifuri" title="Sifuri – Swahili" lang="sw" hreflang="sw" data-title="Sifuri" data-language-autonym="Kiswahili" data-language-local-name="Swahili" class="interlanguage-link-target">Kiswahili</a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/0_(nonm)" title="0 (nonm) – Haitian Creole" lang="ht" hreflang="ht" data-title="0 (nonm)" data-language-autonym="Kreyòl ayisyen" data-language-local-name="Haitian Creole" class="interlanguage-link-target">Kreyòl ayisyen</a></li><li class="interlanguage-link interwiki-ku mw-list-item"><a href="https://ku.wikipedia.org/wiki/Sifir_(hejmar)" title="Sifir (hejmar) – Kurdish" lang="ku" hreflang="ku" data-title="Sifir (hejmar)" data-language-autonym="Kurdî" data-language-local-name="Kurdish" class="interlanguage-link-target">Kurdî</a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%9D%D3%A9%D0%BB" title="Нөл – Kyrgyz" lang="ky" hreflang="ky" data-title="Нөл" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" class="interlanguage-link-target">Кыргызча</a></li><li class="interlanguage-link interwiki-lbe mw-list-item"><a href="https://lbe.wikipedia.org/wiki/%D0%9D%D0%BE%D0%BB%D1%8C" title="Ноль – Lak" lang="lbe" hreflang="lbe" data-title="Ноль" data-language-autonym="Лакку" data-language-local-name="Lak" class="interlanguage-link-target">Лакку</a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/0" title="0 – Latin" lang="la" hreflang="la" data-title="0" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target">Latina</a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Nulle" title="Nulle – Latvian" lang="lv" hreflang="lv" data-title="Nulle" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target">Latviešu</a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/0_(skai%C4%8Dius)" title="0 (skaičius) – Lithuanian" lang="lt" hreflang="lt" data-title="0 (skaičius)" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target">Lietuvių</a></li><li class="interlanguage-link interwiki-li mw-list-item"><a href="https://li.wikipedia.org/wiki/N%C3%B3l" title="Nól – Limburgish" lang="li" hreflang="li" data-title="Nól" data-language-autonym="Limburgs" data-language-local-name="Limburgish" class="interlanguage-link-target">Limburgs</a></li><li class="interlanguage-link interwiki-ln mw-list-item"><a href="https://ln.wikipedia.org/wiki/Lib%C3%BAng%C3%BAtul%C3%BA" title="Libúngútulú – Lingala" lang="ln" hreflang="ln" data-title="Libúngútulú" data-language-autonym="Lingála" data-language-local-name="Lingala" class="interlanguage-link-target">Lingála</a></li><li class="interlanguage-link interwiki-lg mw-list-item"><a href="https://lg.wikipedia.org/wiki/Zeero" title="Zeero – Ganda" lang="lg" hreflang="lg" data-title="Zeero" data-language-autonym="Luganda" data-language-local-name="Ganda" class="interlanguage-link-target">Luganda</a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/N%C3%BCmar_0" title="Nümar 0 – Lombard" lang="lmo" hreflang="lmo" data-title="Nümar 0" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target">Lombard</a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/0_(sz%C3%A1m)" title="0 (szám) – Hungarian" lang="hu" hreflang="hu" data-title="0 (szám)" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target">Magyar</a></li><li class="interlanguage-link interwiki-mai mw-list-item"><a href="https://mai.wikipedia.org/wiki/%E0%A4%B6%E0%A5%82%E0%A4%A8%E0%A5%8D%E0%A4%AF" title="शून्य – Maithili" lang="mai" hreflang="mai" data-title="शून्य" data-language-autonym="मैथिली" data-language-local-name="Maithili" class="interlanguage-link-target">मैथिली</a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/0_(%D0%B1%D1%80%D0%BE%D1%98)" title="0 (број) – Macedonian" lang="mk" hreflang="mk" data-title="0 (број)" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target">Македонски</a></li><li class="interlanguage-link interwiki-mg mw-list-item"><a href="https://mg.wikipedia.org/wiki/Aotra" title="Aotra – Malagasy" lang="mg" hreflang="mg" data-title="Aotra" data-language-autonym="Malagasy" data-language-local-name="Malagasy" class="interlanguage-link-target">Malagasy</a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%AA%E0%B5%82%E0%B4%9C%E0%B5%8D%E0%B4%AF%E0%B4%82" title="പൂജ്യം – Malayalam" lang="ml" hreflang="ml" data-title="പൂജ്യം" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target">മലയാളം</a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A5%A6_(%E0%A4%B8%E0%A4%82%E0%A4%96%E0%A5%8D%E0%A4%AF%E0%A4%BE)" title="० (संख्या) – Marathi" lang="mr" hreflang="mr" data-title="० (संख्या)" data-language-autonym="मराठी" data-language-local-name="Marathi" class="interlanguage-link-target">मराठी</a></li><li class="interlanguage-link interwiki-xmf mw-list-item"><a href="https://xmf.wikipedia.org/wiki/%E1%83%9C%E1%83%A3%E1%83%9A%E1%83%98" title="ნული – Mingrelian" lang="xmf" hreflang="xmf" data-title="ნული" data-language-autonym="მარგალური" data-language-local-name="Mingrelian" class="interlanguage-link-target">მარგალური</a></li><li class="interlanguage-link interwiki-mzn mw-list-item"><a href="https://mzn.wikipedia.org/wiki/%DB%B0" title="۰ – Mazanderani" lang="mzn" hreflang="mzn" data-title="۰" data-language-autonym="مازِرونی" data-language-local-name="Mazanderani" class="interlanguage-link-target">مازِرونی</a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/0_(nombor)" title="0 (nombor) – Malay" lang="ms" hreflang="ms" data-title="0 (nombor)" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target">Bahasa Melayu</a></li><li class="interlanguage-link interwiki-mni mw-list-item"><a href="https://mni.wikipedia.org/wiki/%EA%AF%B0" title="꯰ – Manipuri" lang="mni" hreflang="mni" data-title="꯰" data-language-autonym="ꯃꯤꯇꯩ ꯂꯣꯟ" data-language-local-name="Manipuri" class="interlanguage-link-target">ꯃꯤꯇꯩ ꯂꯣꯟ</a></li><li class="interlanguage-link interwiki-cdo mw-list-item"><a href="https://cdo.wikipedia.org/wiki/0" title="0 – Mindong" lang="cdo" hreflang="cdo" data-title="0" data-language-autonym="閩東語 / Mìng-dĕ̤ng-ngṳ̄" data-language-local-name="Mindong" class="interlanguage-link-target">閩東語 / Mìng-dĕ̤ng-ngṳ̄</a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%81%80" title="၀ – Burmese" lang="my" hreflang="my" data-title="၀" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="Burmese" class="interlanguage-link-target">မြန်မာဘာသာ</a></li><li class="interlanguage-link interwiki-fj mw-list-item"><a href="https://fj.wikipedia.org/wiki/Saiva" title="Saiva – Fijian" lang="fj" hreflang="fj" data-title="Saiva" data-language-autonym="Na Vosa Vakaviti" data-language-local-name="Fijian" class="interlanguage-link-target">Na Vosa Vakaviti</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/0_(getal)" title="0 (getal) – Dutch" lang="nl" hreflang="nl" data-title="0 (getal)" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-nds-nl mw-list-item"><a href="https://nds-nl.wikipedia.org/wiki/0_(getal)" title="0 (getal) – Low Saxon" lang="nds-NL" hreflang="nds-NL" data-title="0 (getal)" data-language-autonym="Nedersaksies" data-language-local-name="Low Saxon" class="interlanguage-link-target">Nedersaksies</a></li><li class="interlanguage-link interwiki-new mw-list-item"><a href="https://new.wikipedia.org/wiki/%E0%A4%B6%E0%A5%82%E0%A4%A8%E0%A5%8D%E0%A4%AF" title="शून्य – Newari" lang="new" hreflang="new" data-title="शून्य" data-language-autonym="नेपाल भाषा" data-language-local-name="Newari" class="interlanguage-link-target">नेपाल भाषा</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/0" title="0 – Japanese" lang="ja" hreflang="ja" data-title="0" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-nap mw-list-item"><a href="https://nap.wikipedia.org/wiki/Zero" title="Zero – Neapolitan" lang="nap" hreflang="nap" data-title="Zero" data-language-autonym="Napulitano" data-language-local-name="Neapolitan" class="interlanguage-link-target">Napulitano</a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Nol" title="Nol – Northern Frisian" lang="frr" hreflang="frr" data-title="Nol" data-language-autonym="Nordfriisk" data-language-local-name="Northern Frisian" class="interlanguage-link-target">Nordfriisk</a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/0_(tall)" title="0 (tall) – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="0 (tall)" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target">Norsk bokmål</a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/0" title="0 – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="0" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target">Norsk nynorsk</a></li><li class="interlanguage-link interwiki-nov mw-list-item"><a href="https://nov.wikipedia.org/wiki/Sero" title="Sero – Novial" lang="nov" hreflang="nov" data-title="Sero" data-language-autonym="Novial" data-language-local-name="Novial" class="interlanguage-link-target">Novial</a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/0_(nombre)" title="0 (nombre) – Occitan" lang="oc" hreflang="oc" data-title="0 (nombre)" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target">Occitan</a></li><li class="interlanguage-link interwiki-or mw-list-item"><a href="https://or.wikipedia.org/wiki/%E0%AD%A6_(%E0%AC%B8%E0%AC%82%E0%AC%96%E0%AD%8D%E0%AD%9F%E0%AC%BE)" title="୦ (ସଂଖ୍ୟା) – Odia" lang="or" hreflang="or" data-title="୦ (ସଂଖ୍ୟା)" data-language-autonym="ଓଡ଼ିଆ" data-language-local-name="Odia" class="interlanguage-link-target">ଓଡ଼ିଆ</a></li><li class="interlanguage-link interwiki-om mw-list-item"><a href="https://om.wikipedia.org/wiki/0" title="0 – Oromo" lang="om" hreflang="om" data-title="0" data-language-autonym="Oromoo" data-language-local-name="Oromo" class="interlanguage-link-target">Oromoo</a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/0_(son)" title="0 (son) – Uzbek" lang="uz" hreflang="uz" data-title="0 (son)" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target">Oʻzbekcha / ўзбекча</a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%B8%E0%A8%BF%E0%A8%AB%E0%A8%BC%E0%A8%B0" title="ਸਿਫ਼ਰ – Punjabi" lang="pa" hreflang="pa" data-title="ਸਿਫ਼ਰ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target">ਪੰਜਾਬੀ</a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D8%B5%D9%81%D8%B1" title="صفر – Western Punjabi" lang="pnb" hreflang="pnb" data-title="صفر" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target">پنجابی</a></li><li class="interlanguage-link interwiki-ps mw-list-item"><a href="https://ps.wikipedia.org/wiki/%DB%B0" title="۰ – Pashto" lang="ps" hreflang="ps" data-title="۰" data-language-autonym="پښتو" data-language-local-name="Pashto" class="interlanguage-link-target">پښتو</a></li><li class="interlanguage-link interwiki-km mw-list-item"><a href="https://km.wikipedia.org/wiki/%E1%9F%A0" title="០ – Khmer" lang="km" hreflang="km" data-title="០" data-language-autonym="ភាសាខ្មែរ" data-language-local-name="Khmer" class="interlanguage-link-target">ភាសាខ្មែរ</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/0" title="0 – Polish" lang="pl" hreflang="pl" data-title="0" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/0_(n%C3%BAmero)" title="0 (número) – Portuguese" lang="pt" hreflang="pt" data-title="0 (número)" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/0_(cifr%C4%83)" title="0 (cifră) – Romanian" lang="ro" hreflang="ro" data-title="0 (cifră)" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target">Română</a></li><li class="interlanguage-link interwiki-qu mw-list-item"><a href="https://qu.wikipedia.org/wiki/Ch%27usaq_yupay" title="Ch'usaq yupay – Quechua" lang="qu" hreflang="qu" data-title="Ch'usaq yupay" data-language-autonym="Runa Simi" data-language-local-name="Quechua" class="interlanguage-link-target">Runa Simi</a></li><li class="interlanguage-link interwiki-rue mw-list-item"><a href="https://rue.wikipedia.org/wiki/0_(%D1%87%D0%B8%D1%81%D0%BB%D0%BE)" title="0 (число) – Rusyn" lang="rue" hreflang="rue" data-title="0 (число)" data-language-autonym="Русиньскый" data-language-local-name="Rusyn" class="interlanguage-link-target">Русиньскый</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9D%D0%BE%D0%BB%D1%8C" title="Ноль – Russian" lang="ru" hreflang="ru" data-title="Ноль" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-sah mw-list-item"><a href="https://sah.wikipedia.org/wiki/%D0%9D%D1%83%D1%83%D0%BB" title="Нуул – Yakut" lang="sah" hreflang="sah" data-title="Нуул" data-language-autonym="Саха тыла" data-language-local-name="Yakut" class="interlanguage-link-target">Саха тыла</a></li><li class="interlanguage-link interwiki-sa mw-list-item"><a href="https://sa.wikipedia.org/wiki/%E0%A4%B6%E0%A5%82%E0%A4%A8%E0%A5%8D%E0%A4%AF%E0%A4%AE%E0%A5%8D_(%E0%A4%B8%E0%A4%99%E0%A5%8D%E0%A4%96%E0%A5%8D%E0%A4%AF%E0%A4%BE)" title="शून्यम् (सङ्ख्या) – Sanskrit" lang="sa" hreflang="sa" data-title="शून्यम् (सङ्ख्या)" data-language-autonym="संस्कृतम्" data-language-local-name="Sanskrit" class="interlanguage-link-target">संस्कृतम्</a></li><li class="interlanguage-link interwiki-sat mw-list-item"><a href="https://sat.wikipedia.org/wiki/%E1%B1%90" title="᱐ – Santali" lang="sat" hreflang="sat" data-title="᱐" data-language-autonym="ᱥᱟᱱᱛᱟᱲᱤ" data-language-local-name="Santali" class="interlanguage-link-target">ᱥᱟᱱᱛᱟᱲᱤ</a></li><li class="interlanguage-link interwiki-sco mw-list-item"><a href="https://sco.wikipedia.org/wiki/0" title="0 – Scots" lang="sco" hreflang="sco" data-title="0" data-language-autonym="Scots" data-language-local-name="Scots" class="interlanguage-link-target">Scots</a></li><li class="interlanguage-link interwiki-nso mw-list-item"><a href="https://nso.wikipedia.org/wiki/0_(nomoro)" title="0 (nomoro) – Northern Sotho" lang="nso" hreflang="nso" data-title="0 (nomoro)" data-language-autonym="Sesotho sa Leboa" data-language-local-name="Northern Sotho" class="interlanguage-link-target">Sesotho sa Leboa</a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/0_(num%C3%ABr)" title="0 (numër) – Albanian" lang="sq" hreflang="sq" data-title="0 (numër)" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target">Shqip</a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Zeru" title="Zeru – Sicilian" lang="scn" hreflang="scn" data-title="Zeru" data-language-autonym="Sicilianu" data-language-local-name="Sicilian" class="interlanguage-link-target">Sicilianu</a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/0" title="0 – Simple English" lang="en-simple" hreflang="en-simple" data-title="0" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target">Simple English</a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/0_(%C4%8D%C3%ADslo)" title="0 (číslo) – Slovak" lang="sk" hreflang="sk" data-title="0 (číslo)" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target">Slovenčina</a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/0" title="0 – Slovenian" lang="sl" hreflang="sl" data-title="0" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target">Slovenščina</a></li><li class="interlanguage-link interwiki-szl mw-list-item"><a href="https://szl.wikipedia.org/wiki/0" title="0 – Silesian" lang="szl" hreflang="szl" data-title="0" data-language-autonym="Ślůnski" data-language-local-name="Silesian" class="interlanguage-link-target">Ślůnski</a></li><li class="interlanguage-link interwiki-so mw-list-item"><a href="https://so.wikipedia.org/wiki/0_(tiro)" title="0 (tiro) – Somali" lang="so" hreflang="so" data-title="0 (tiro)" data-language-autonym="Soomaaliga" data-language-local-name="Somali" class="interlanguage-link-target">Soomaaliga</a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D9%A0_(%DA%98%D9%85%D8%A7%D8%B1%DB%95)" title="٠ (ژمارە) – Central Kurdish" lang="ckb" hreflang="ckb" data-title="٠ (ژمارە)" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target">کوردی</a></li><li class="interlanguage-link interwiki-srn mw-list-item"><a href="https://srn.wikipedia.org/wiki/Numro_0" title="Numro 0 – Sranan Tongo" lang="srn" hreflang="srn" data-title="Numro 0" data-language-autonym="Sranantongo" data-language-local-name="Sranan Tongo" class="interlanguage-link-target">Sranantongo</a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/0_(%D0%B1%D1%80%D0%BE%D1%98)" title="0 (број) – Serbian" lang="sr" hreflang="sr" data-title="0 (број)" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target">Српски / srpski</a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/0_(broj)" title="0 (broj) – Serbo-Croatian" lang="sh" hreflang="sh" data-title="0 (broj)" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target">Srpskohrvatski / српскохрватски</a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/0_(luku)" title="0 (luku) – Finnish" lang="fi" hreflang="fi" data-title="0 (luku)" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target">Suomi</a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/0_(tal)" title="0 (tal) – Swedish" lang="sv" hreflang="sv" data-title="0 (tal)" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target">Svenska</a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/0_(bilang)" title="0 (bilang) – Tagalog" lang="tl" hreflang="tl" data-title="0 (bilang)" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target">Tagalog</a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/0_(%E0%AE%8E%E0%AE%A3%E0%AF%8D)" title="0 (எண்) – Tamil" lang="ta" hreflang="ta" data-title="0 (எண்)" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target">தமிழ்</a></li><li class="interlanguage-link interwiki-shi mw-list-item"><a href="https://shi.wikipedia.org/wiki/0_(am%E1%B8%8Dan)" title="0 (amḍan) – Tachelhit" lang="shi" hreflang="shi" data-title="0 (amḍan)" data-language-autonym="Taclḥit" data-language-local-name="Tachelhit" class="interlanguage-link-target">Taclḥit</a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/0_(%D1%81%D0%B0%D0%BD)" title="0 (сан) – Tatar" lang="tt" hreflang="tt" data-title="0 (сан)" data-language-autonym="Татарча / tatarça" data-language-local-name="Tatar" class="interlanguage-link-target">Татарча / tatarça</a></li><li class="interlanguage-link interwiki-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%B8%E0%B1%81%E0%B0%A8%E0%B1%8D%E0%B0%A8" title="సున్న – Telugu" lang="te" hreflang="te" data-title="సున్న" data-language-autonym="తెలుగు" data-language-local-name="Telugu" class="interlanguage-link-target">తెలుగు</a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/0" title="0 – Thai" lang="th" hreflang="th" data-title="0" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target">ไทย</a></li><li class="interlanguage-link interwiki-tg mw-list-item"><a href="https://tg.wikipedia.org/wiki/%D0%A1%D0%B8%D1%84%D1%80" title="Сифр – Tajik" lang="tg" hreflang="tg" data-title="Сифр" data-language-autonym="Тоҷикӣ" data-language-local-name="Tajik" class="interlanguage-link-target">Тоҷикӣ</a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/0" title="0 – Turkish" lang="tr" hreflang="tr" data-title="0" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target">Türkçe</a></li><li class="interlanguage-link interwiki-kcg mw-list-item"><a href="https://kcg.wikipedia.org/wiki/0" title="0 – Tyap" lang="kcg" hreflang="kcg" data-title="0" data-language-autonym="Tyap" data-language-local-name="Tyap" class="interlanguage-link-target">Tyap</a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/0_(%D1%87%D0%B8%D1%81%D0%BB%D0%BE)" title="0 (число) – Ukrainian" lang="uk" hreflang="uk" data-title="0 (число)" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target">Українська</a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D8%B5%D9%81%D8%B1_(%D8%B9%D8%AF%D8%AF)" title="صفر (عدد) – Urdu" lang="ur" hreflang="ur" data-title="صفر (عدد)" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target">اردو</a></li><li class="interlanguage-link interwiki-vep mw-list-item"><a href="https://vep.wikipedia.org/wiki/0_(lugu)" title="0 (lugu) – Veps" lang="vep" hreflang="vep" data-title="0 (lugu)" data-language-autonym="Vepsän kel’" data-language-local-name="Veps" class="interlanguage-link-target">Vepsän kel’</a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/0_(s%E1%BB%91)" title="0 (số) – Vietnamese" lang="vi" hreflang="vi" data-title="0 (số)" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target">Tiếng Việt</a></li><li class="interlanguage-link interwiki-wa mw-list-item"><a href="https://wa.wikipedia.org/wiki/0" title="0 – Walloon" lang="wa" hreflang="wa" data-title="0" data-language-autonym="Walon" data-language-local-name="Walloon" class="interlanguage-link-target">Walon</a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E3%80%87" title="〇 – Literary Chinese" lang="lzh" hreflang="lzh" data-title="〇" data-language-autonym="文言" data-language-local-name="Literary Chinese" class="interlanguage-link-target">文言</a></li><li class="interlanguage-link interwiki-vls mw-list-item"><a href="https://vls.wikipedia.org/wiki/0_(getal)" title="0 (getal) – West Flemish" lang="vls" hreflang="vls" data-title="0 (getal)" data-language-autonym="West-Vlams" data-language-local-name="West Flemish" class="interlanguage-link-target">West-Vlams</a></li><li class="interlanguage-link interwiki-war mw-list-item"><a href="https://war.wikipedia.org/wiki/0_(ihap)" title="0 (ihap) – Waray" lang="war" hreflang="war" data-title="0 (ihap)" data-language-autonym="Winaray" data-language-local-name="Waray" class="interlanguage-link-target">Winaray</a></li><li class="interlanguage-link interwiki-wo mw-list-item"><a href="https://wo.wikipedia.org/wiki/Tus" title="Tus – Wolof" lang="wo" hreflang="wo" data-title="Tus" data-language-autonym="Wolof" data-language-local-name="Wolof" class="interlanguage-link-target">Wolof</a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/0" title="0 – Wu" lang="wuu" hreflang="wuu" data-title="0" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target">吴语</a></li><li class="interlanguage-link interwiki-ts mw-list-item"><a href="https://ts.wikipedia.org/wiki/Tandza" title="Tandza – Tsonga" lang="ts" hreflang="ts" data-title="Tandza" data-language-autonym="Xitsonga" data-language-local-name="Tsonga" class="interlanguage-link-target">Xitsonga</a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/0_(%D7%A0%D7%95%D7%9E%D7%A2%D7%A8)" title="0 (נומער) – Yiddish" lang="yi" hreflang="yi" data-title="0 (נומער)" data-language-autonym="ייִדיש" data-language-local-name="Yiddish" class="interlanguage-link-target">ייִדיש</a></li><li class="interlanguage-link interwiki-yo mw-list-item"><a href="https://yo.wikipedia.org/wiki/0_(n%E1%BB%8D%CC%81mb%C3%A0)" title="0 (nọ́mbà) – Yoruba" lang="yo" hreflang="yo" data-title="0 (nọ́mbà)" data-language-autonym="Yorùbá" data-language-local-name="Yoruba" class="interlanguage-link-target">Yorùbá</a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/0" title="0 – Cantonese" lang="yue" hreflang="yue" data-title="0" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target">粵語</a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/0" title="0 – Chinese" lang="zh" hreflang="zh" data-title="0" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target">中文</a></li><li class="interlanguage-link interwiki-kge mw-list-item"><a href="https://kge.wikipedia.org/wiki/0" title="0 – Komering" lang="kge" hreflang="kge" data-title="0" data-language-autonym="Kumoring" data-language-local-name="Komering" class="interlanguage-link-target">Kumoring</a></li><li class="interlanguage-link interwiki-zgh mw-list-item"><a href="https://zgh.wikipedia.org/wiki/0_(%E2%B4%B0%E2%B5%8E%E2%B4%B9%E2%B4%B0%E2%B5%8F)" title="0 (ⴰⵎⴹⴰⵏ) – Standard Moroccan Tamazight" lang="zgh" hreflang="zgh" data-title="0 (ⴰⵎⴹⴰⵏ)" data-language-autonym="ⵜⴰⵎⴰⵣⵉⵖⵜ ⵜⴰⵏⴰⵡⴰⵢⵜ" data-language-local-name="Standard Moroccan Tamazight" class="interlanguage-link-target">ⵜⴰⵎⴰⵣⵉⵖⵜ ⵜⴰⵏⴰⵡⴰⵢⵜ</a></li></ul> </section> </div> <div class="minerva-footer-logo"><img src="/static/images/mobile/copyright/wikipedia-wordmark-en.svg" alt="Wikipedia" width="120" height="18" style="width: 7.5em; height: 1.125em;"/> </div> <ul id="footer-info" class="footer-info hlist hlist-separated"> <li id="footer-info-lastmod"> This page was last edited on 14 January 2025, at 14:43 (UTC).</li> <li id="footer-info-copyright">Content is available under <a class="external" rel="nofollow" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">CC BY-SA 4.0</a> unless otherwise noted.</li> </ul> <ul id="footer-places" class="footer-places hlist hlist-separated"> <li id="footer-places-privacy"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy">Privacy policy</a></li> <li id="footer-places-about"><a href="/wiki/Wikipedia:About">About Wikipedia</a></li> <li id="footer-places-disclaimers"><a href="/wiki/Wikipedia:General_disclaimer">Disclaimers</a></li> 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</footer> </div> </div> <div class="mw-notification-area" data-mw="interface"></div>  <script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.main-b766959bd-xjd9v","wgBackendResponseTime":324,"wgPageParseReport":{"limitreport":{"cputime":"1.966","walltime":"2.427","ppvisitednodes":{"value":12811,"limit":1000000},"postexpandincludesize":{"value":505322,"limit":2097152},"templateargumentsize":{"value":15047,"limit":2097152},"expansiondepth":{"value":17,"limit":100},"expensivefunctioncount":{"value":24,"limit":500},"unstrip-depth":{"value":1,"limit":20},"unstrip-size":{"value":379506,"limit":5000000},"entityaccesscount":{"value":1,"limit":400},"timingprofile":["100.00% 1973.472 1 -total"," 32.77% 646.653 2 Template:Reflist"," 13.73% 270.950 49 Template:Cite_book"," 8.88% 175.275 13 Template:Navbox"," 8.15% 160.780 1 Template:Infobox_number"," 7.72% 152.371 4 Template:Multiref2"," 7.62% 150.298 13 Template:Lang"," 7.11% 140.308 1 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1,\n [\"CITEREFGullberg1997\"] = 1,\n [\"CITEREFHall2005\"] = 1,\n [\"CITEREFHansen2008\"] = 1,\n [\"CITEREFHarper2011\"] = 1,\n [\"CITEREFHermanStrang2017\"] = 1,\n [\"CITEREFHill2020\"] = 1,\n [\"CITEREFHodgkin2005\"] = 1,\n [\"CITEREFHosch2010\"] = 1,\n [\"CITEREFHuggett2019\"] = 1,\n [\"CITEREFIfrah2000\"] = 1,\n [\"CITEREFKaplan2000\"] = 1,\n [\"CITEREFKardar2007\"] = 1,\n [\"CITEREFLeon2022\"] = 1,\n [\"CITEREFLogan2010\"] = 1,\n [\"CITEREFLumpkin2002\"] = 1,\n [\"CITEREFMartzloff2007\"] = 1,\n [\"CITEREFMenninger1992\"] = 1,\n [\"CITEREFMercier\"] = 1,\n [\"CITEREFMortaigne2014\"] = 1,\n [\"CITEREFNeugebauer1969\"] = 1,\n [\"CITEREFNeugebauer2016\"] = 1,\n [\"CITEREFNieder2019\"] = 1,\n [\"CITEREFO\u0026#039;ConnorRobertson\"] = 1,\n [\"CITEREFO\u0026#039;ConnorRobertson2000\"] = 3,\n [\"CITEREFOliveiraStewart2006\"] = 1,\n [\"CITEREFOverland2004\"] = 1,\n [\"CITEREFPannekoek1961\"] = 1,\n [\"CITEREFPedersen1985\"] = 1,\n [\"CITEREFPedersen2010\"] = 1,\n [\"CITEREFPenner1999\"] = 1,\n [\"CITEREFPlofker2009\"] = 1,\n [\"CITEREFPlofkerKellerHayashiMontelle2017\"] = 1,\n [\"CITEREFPtolemy1998\"] = 1,\n [\"CITEREFReese2013\"] = 1,\n [\"CITEREFReid1992\"] = 1,\n [\"CITEREFReimer2014\"] = 1,\n [\"CITEREFRexFinn2017\"] = 1,\n [\"CITEREFRiehl2016\"] = 1,\n [\"CITEREFRines1920\"] = 1,\n [\"CITEREFRochkind1985\"] = 1,\n [\"CITEREFSeife2000\"] = 2,\n [\"CITEREFShenCrossleyLun1999\"] = 1,\n [\"CITEREFSigler2003\"] = 1,\n [\"CITEREFSmithKarpinski1911\"] = 1,\n [\"CITEREFSteel2000\"] = 1,\n [\"CITEREFVaman_Shivaram_Apte1970\"] = 1,\n [\"CITEREFWallin2002\"] = 1,\n [\"CITEREFWeil2012\"] = 1,\n [\"CITEREFWeiss2017\"] = 1,\n [\"CITEREFWeisstein,_Eric_W.\"] = 1,\n [\"CITEREFWoodford2006\"] = 1,\n [\"CITEREFWuIchikawaCercone1996\"] = 1,\n}\ntemplate_list = table#1 {\n [\"!\"] = 1,\n [\"=\"] = 3,\n [\"Authority control\"] = 1,\n [\"Circa\"] = 4,\n [\"Citation\"] = 2,\n [\"Citation needed\"] = 6,\n [\"Cite Americana\"] = 1,\n [\"Cite EB1911\"] = 1,\n [\"Cite book\"] = 49,\n 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