SherbyNet - /page/wikibook/entry.mu

<!DOCTYPE html>

<title>Positional notation</title>

</head>

Positional notation

</h1>

</header>

/* start https://en.wikipedia.org/ */

.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}}

/* end https://en.wikipedia.org/ */

</style><div role="note" class="hatnote navigation-not-searchable">"Positional system" redirects here. For the voting rule, see <a href="Positional_voting" title="Positional voting">positional voting</a>.</div>

/* start https://en.wikipedia.org/ */

.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 "}

/* end https://en.wikipedia.org/ */

</style><style data-mw-deduplicate="TemplateStyles:r1246091330">

/* start https://en.wikipedia.org/ */

.mw-parser-output .sidebar{width:22em;float:right;clear:right;margin:0.5em 0 1em 1em;background:var(--background-color-neutral-subtle,#f8f9fa);border:1px solid var(--border-color-base,#a2a9b1);padding:0.2em;text-align:center;line-height:1.4em;font-size:88%;border-collapse:collapse;display:table}body.skin-minerva .mw-parser-output .sidebar{display:table!important;float:right!important;margin:0.5em 0 1em 1em!important}.mw-parser-output .sidebar-subgroup{width:100%;margin:0;border-spacing:0}.mw-parser-output .sidebar-left{float:left;clear:left;margin:0.5em 1em 1em 0}.mw-parser-output .sidebar-none{float:none;clear:both;margin:0.5em 1em 1em 0}.mw-parser-output .sidebar-outer-title{padding:0 0.4em 0.2em;font-size:125%;line-height:1.2em;font-weight:bold}.mw-parser-output .sidebar-top-image{padding:0.4em}.mw-parser-output .sidebar-top-caption,.mw-parser-output .sidebar-pretitle-with-top-image,.mw-parser-output .sidebar-caption{padding:0.2em 0.4em 0;line-height:1.2em}.mw-parser-output .sidebar-pretitle{padding:0.4em 0.4em 0;line-height:1.2em}.mw-parser-output .sidebar-title,.mw-parser-output .sidebar-title-with-pretitle{padding:0.2em 0.8em;font-size:145%;line-height:1.2em}.mw-parser-output .sidebar-title-with-pretitle{padding:0.1em 0.4em}.mw-parser-output .sidebar-image{padding:0.2em 0.4em 0.4em}.mw-parser-output .sidebar-heading{padding:0.1em 0.4em}.mw-parser-output .sidebar-content{padding:0 0.5em 0.4em}.mw-parser-output .sidebar-content-with-subgroup{padding:0.1em 0.4em 0.2em}.mw-parser-output .sidebar-above,.mw-parser-output .sidebar-below{padding:0.3em 0.8em;font-weight:bold}.mw-parser-output .sidebar-collapse .sidebar-above,.mw-parser-output .sidebar-collapse .sidebar-below{border-top:1px solid #aaa;border-bottom:1px solid #aaa}.mw-parser-output .sidebar-navbar{text-align:right;font-size:115%;padding:0 0.4em 0.4em}.mw-parser-output .sidebar-list-title{padding:0 0.4em;text-align:left;font-weight:bold;line-height:1.6em;font-size:105%}.mw-parser-output .sidebar-list-title-c{padding:0 0.4em;text-align:center;margin:0 3.3em}@media(max-width:640px){body.mediawiki .mw-parser-output .sidebar{width:100%!important;clear:both;float:none!important;margin-left:0!important;margin-right:0!important}}body.skin--responsive .mw-parser-output .sidebar a>img{max-width:none!important}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}

/* end https://en.wikipedia.org/ */

</style><table class="sidebar sidebar-collapse nomobile nowraplinks hlist"><tbody><tr><td class="sidebar-pretitle">Part of a series on</td></tr><tr><th class="sidebar-title-with-pretitle"><a href="Numeral_system" title="Numeral system">Numeral systems</a></th></tr><tr><td class="sidebar-content-with-subgroup">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c"><a href="Hindu%E2%80%93Arabic_numeral_system" title="Hindu–Arabic numeral system">Hindu–Arabic numerals</a></div></div><div class="sidebar-list-content mw-collapsible-content">

<ul><li><a href="Arabic_numerals" title="Arabic numerals">Western Arabic</a></li>

<li><a href="Eastern_Arabic_numerals" title="Eastern Arabic numerals">Eastern Arabic</a></li></ul>

<hr>

<ul><li><a href="Bengali_numerals" title="Bengali numerals">Bengali</a></li>

<li><a href="Devanagari_numerals" title="Devanagari numerals">Devanagari</a></li>

<li><a href="Gujarati_numerals" title="Gujarati numerals">Gujarati</a></li>

<li><a href="Gurmukhi_numerals" class="mw-redirect" title="Gurmukhi numerals">Gurmukhi</a></li>

<li><a href="Sinhala_numerals" title="Sinhala numerals">Sinhala</a></li>

<li><a href="Tamil_numerals" title="Tamil numerals">Tamil</a></li>

<li><a href="Malayalam_numerals" title="Malayalam numerals">Malayalam</a></li>

<li><a href="Telugu_script#Numerals" title="Telugu script">Telugu</a></li>

<li><a href="Kannada_script#Numerals" title="Kannada script">Kannada</a></li>

<li><a href="Dzongkha_numerals" title="Dzongkha numerals">Dzongkha</a></li></ul>

<hr>

<ul><li><a href="Tibetan_numerals" title="Tibetan numerals">Tibetan</a></li>

<li><a href="Balinese_numerals" title="Balinese numerals">Balinese</a></li>

<li><a href="Burmese_numerals" title="Burmese numerals">Burmese</a></li>

<li><a href="Javanese_numerals" title="Javanese numerals">Javanese</a></li>

<li><a href="Khmer_numerals" title="Khmer numerals">Khmer</a></li>

<li><a href="Mongolian_numerals" title="Mongolian numerals">Mongolian</a></li>

<li><a href="Sundanese_numerals" title="Sundanese numerals">Sundanese</a></li>

</tr><tr><td class="sidebar-content hlist">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c">East Asian systems</div></div><div class="sidebar-list-content mw-collapsible-content">

<dl><dt>Contemporary</dt></dl>

<ul><li><a href="Chinese_numerals" title="Chinese numerals">Chinese</a>

<ul><li><a href="Hokkien_numerals" title="Hokkien numerals">Hokkien</a></li>

<li><a href="Suzhou_numerals" title="Suzhou numerals">Suzhou</a></li></ul></li>

<li><a href="Japanese_numerals" title="Japanese numerals">Japanese</a></li>

<li><a href="Korean_numerals" title="Korean numerals">Korean</a></li>

<li><a href="Vietnamese_numerals" title="Vietnamese numerals">Vietnamese</a></li></ul>

<hr>

<dl><dt>Historic</dt></dl>

<ul><li><a href="Counting_rods" title="Counting rods">Counting rods</a></li>

<li><a href="Tangut_numerals" title="Tangut numerals">Tangut</a></li></ul></div></div></td>

</tr><tr><td class="sidebar-content hlist">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c">Other systems</div></div><div class="sidebar-list-content mw-collapsible-content">

<ul><li><a href="History_of_ancient_numeral_systems" title="History of ancient numeral systems">History</a></li></ul>

<hr>

<dl><dt><a href="Ancient_history" title="Ancient history">Ancient</a></dt></dl>

<ul><li><a href="Babylonian_cuneiform_numerals" title="Babylonian cuneiform numerals">Babylonian</a></li></ul>

<hr>

<dl><dt><a href="Post-classical_history" title="Post-classical history">Post-classical</a></dt></dl>

<ul><li><a href="Cistercian_numerals" title="Cistercian numerals">Cistercian</a></li>

<li><a href="Maya_numerals" title="Maya numerals">Mayan</a></li>

<li><a href="Muisca_numerals" title="Muisca numerals">Muisca</a></li>

<li><a href="Pentadic_numerals" title="Pentadic numerals">Pentadic</a></li>

<li><a href="Quipu" title="Quipu">Quipu</a></li>

<hr>

<dl><dt>Contemporary</dt></dl>

<ul><li><a href="Cherokee_syllabary#Numerals" title="Cherokee syllabary">Cherokee</a></li>

<li><a href="Kaktovik_numerals" title="Kaktovik numerals">Kaktovik</a> (Iñupiaq)</li></ul></div></div></td>

</tr><tr><td class="sidebar-content hlist">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c">By <a href="Radix" title="Radix">radix/base</a></div></div><div class="sidebar-list-content mw-collapsible-content">

<dl><dt>Common radices/bases</dt></dl>

<hr>

<dl><dt><a href="Non-standard_positional_numeral_systems" title="Non-standard positional numeral systems">Non-standard radices/bases</a></dt></dl>

<ul><li><a href="Bijective_numeration" title="Bijective numeration">Bijective</a> (<a href="Unary_numeral_system" title="Unary numeral system">1</a>)</li>

<li><a href="Signed-digit_representation" title="Signed-digit representation">Signed-digit</a> (<a href="Balanced_ternary" title="Balanced ternary">balanced ternary</a>)</li>

<li><a href="Mixed_radix" title="Mixed radix">Mixed</a> (<a href="Factorial_number_system" title="Factorial number system">factorial</a>)</li>

<li><a href="Negative_base" title="Negative base">Negative</a></li>

<li><a href="Complex-base_system" title="Complex-base system">Complex</a> (<a href="Quater-imaginary_base" title="Quater-imaginary base">2i</a>)</li>

<li><a href="Non-integer_base_of_numeration" title="Non-integer base of numeration">Non-integer</a> (<a href="Golden_ratio_base" title="Golden ratio base">φ</a>)</li>

<li><a href="Asymmetric_numeral_systems" title="Asymmetric numeral systems">Asymmetric</a></li></ul></div></div></td>

</tr></tbody></table></div></div></td>

</tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c"><a href="Sign-value_notation" title="Sign-value notation">Sign-value notation</a></div></div><div class="sidebar-list-content mw-collapsible-content">

<dl><dt>Non-alphabetic</dt></dl>

<ul><li><a href="Aegean_numerals" title="Aegean numerals">Aegean</a></li>

<li><a href="Attic_numerals" title="Attic numerals">Attic</a></li>

<li><a href="Aztec_script#Numerals" title="Aztec script">Aztec</a></li>

<li><a href="Brahmi_numerals" title="Brahmi numerals">Brahmi</a></li>

<li><a href="Chuvash_numerals" title="Chuvash numerals">Chuvash</a></li>

<li><a href="Egyptian_numerals" title="Egyptian numerals">Egyptian</a></li>

<li><a href="Etruscan_numerals" title="Etruscan numerals">Etruscan</a></li>

<li><a href="Kharosthi_numerals" class="mw-redirect" title="Kharosthi numerals">Kharosthi</a></li>

<li><a href="Prehistoric_counting" class="mw-redirect" title="Prehistoric counting">Prehistoric counting</a></li>

<li><a href="Proto-cuneiform" title="Proto-cuneiform">Proto-cuneiform</a></li>

<li><a href="Roman_numerals" title="Roman numerals">Roman</a></li>

<li><a href="Tally_marks" title="Tally marks">Tally marks</a></li></ul>

<hr>

<dl><dt><a href="Alphabetic_numeral_system" title="Alphabetic numeral system">Alphabetic</a></dt></dl>

<ul><li><a href="Abjad_numerals" title="Abjad numerals">Abjad</a></li>

<li><a href="Armenian_numerals" title="Armenian numerals">Armenian</a></li>

<li><a href="Alphasyllabic_numeral_system" title="Alphasyllabic numeral system">Alphasyllabic</a>

<ul><li><a href="Aksharapalli" title="Aksharapalli">Akṣarapallī</a></li>

<li><a href="%C4%80ryabha%E1%B9%ADa_numeration" title="Āryabhaṭa numeration">Āryabhaṭa</a></li>

<li><a href="Katapayadi_system" title="Katapayadi system">Kaṭapayādi</a></li></ul></li>

<li><a href="Coptic_numerals" class="mw-redirect" title="Coptic numerals">Coptic</a></li>

<li><a href="Cyrillic_numerals" title="Cyrillic numerals">Cyrillic</a></li>

<li><a href="Ge%CA%BDez_script#Numerals" title="Geʽez script">Geʽez</a></li>

<li><a href="Georgian_numerals" title="Georgian numerals">Georgian</a></li>

<li><a href="Glagolitic_numerals" title="Glagolitic numerals">Glagolitic</a></li>

<li><a href="Greek_numerals" title="Greek numerals">Greek</a></li>

<li><a href="Hebrew_numerals" title="Hebrew numerals">Hebrew</a></li></ul></div></div></td>

</tr><tr><td class="sidebar-below" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;">

<a href="List_of_numeral_systems" title="List of numeral systems">List of numeral systems</a></td></tr><tr><td class="sidebar-navbar"><style data-mw-deduplicate="TemplateStyles:r1239400231">

/* start https://en.wikipedia.org/ */

.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}

/* end https://en.wikipedia.org/ */

</style></td></tr></tbody></table>

Positional notation, also known as place-value notation, positional numeral system, or simply place value, usually denotes the extension to any <a href="Radix" title="Radix">base</a> of the <a href="Hindu%E2%80%93Arabic_numeral_system" title="Hindu–Arabic numeral system">Hindu–Arabic numeral system</a> (or <a href="Decimal" title="Decimal">decimal system</a>). More generally, a positional system is a <a href="Numeral_system" title="Numeral system">numeral system</a> in which the contribution of a digit to the value of a number is the value of the digit multiplied by a factor determined by the position of the digit. In early <a href="Numeral_system" title="Numeral system">numeral systems</a>, such as <a href="Roman_numerals" title="Roman numerals">Roman numerals</a>, a digit has only one value: I means one, X means ten and C a hundred (however, the values may be modified when combined). In modern positional systems, such as the <a href="Decimal" title="Decimal">decimal system</a>, the position of the digit means that its value must be multiplied by some value: in 555, the three identical symbols represent five hundreds, five tens, and five units, respectively, due to their different positions in the digit string.

The <a href="Babylonian_Numerals" class="mw-redirect" title="Babylonian Numerals">Babylonian numeral system</a>, base 60, was the first positional system to be developed, and its influence is present today in the way time and angles are counted in tallies related to 60, such as 60 minutes in an hour and 360 degrees in a circle. Today, the Hindu–Arabic numeral system (<a href="Base_ten" class="mw-redirect" title="Base ten">base ten</a>) is the most commonly used system globally. However, the <a href="Binary_numeral_system" class="mw-redirect" title="Binary numeral system">binary numeral system</a> (base two) is used in almost all <a href="Computer" title="Computer">computers</a> and <a href="Electronic_device" class="mw-redirect" title="Electronic device">electronic devices</a> because it is easier to implement efficiently in <a href="Electronic_circuit" title="Electronic circuit">electronic circuits</a>.

Systems with negative base, <a href="Complex_number" title="Complex number">complex</a> base or negative digits have been described. Most of them do not require a minus sign for designating negative numbers.

The use of a <a href="Radix_point" class="mw-redirect" title="Radix point">radix point</a> (decimal point in base ten), extends to include <a href="Fraction_(mathematics)" class="mw-redirect" title="Fraction (mathematics)">fractions</a> and allows the representation of any <a href="Real_number" title="Real number">real number</a> with arbitrary accuracy. With positional notation, <a href="Arithmetic" title="Arithmetic">arithmetical computations</a> are much simpler than with any older numeral system; this led to the rapid spread of the notation when it was introduced in western Europe.

<div class="mw-heading mw-heading2"><h2 id="History">History</h2></div>

Today, the base-10 (<a href="Decimal" title="Decimal">decimal</a>) system, which is presumably motivated by counting with the ten <a href="Finger" title="Finger">fingers</a>, is ubiquitous. Other bases have been used in the past, and some continue to be used today. For example, the <a href="Babylonian_numerals" class="mw-redirect" title="Babylonian numerals">Babylonian numeral system</a>, credited as the first positional numeral system, was <a href="Base-60" class="mw-redirect" title="Base-60">base-60</a>. However, it lacked a real <a href="Zero" class="mw-redirect" title="Zero">zero</a>. Initially inferred only from context, later, by about 700 BC, zero came to be indicated by a "space" or a "punctuation symbol" (such as two slanted wedges) between numerals.<a href="#cite_note-multiref1-1">[1]</a> It was a <a href="Variable_(mathematics)" title="Variable (mathematics)">placeholder</a> rather than a true zero because it was not used alone or at the end of a number. Numbers like 2 and 120 (2×60) looked the same because the larger number lacked a final placeholder. Only context could differentiate them.

The polymath <a href="Archimedes" title="Archimedes">Archimedes</a> (ca. 287–212 BC) invented a decimal positional system based on 108 in his <a href="The_Sand_Reckoner" title="The Sand Reckoner">Sand Reckoner</a>;<a href="#cite_note-Greek_numerals-2">[2]</a> 19th century German mathematician <a href="Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Carl Gauss</a> lamented how science might have progressed had Archimedes only made the leap to something akin to the modern decimal system.<a href="#cite_note-3">[3]</a> <a href="Hellenistic_period" title="Hellenistic period">Hellenistic</a> and <a href="Roman_Empire" title="Roman Empire">Roman</a> astronomers used a base-60 system based on the Babylonian model (see <a href="Greek_numerals#Zero" title="Greek numerals">Greek numerals § Zero</a>).

Before positional notation became standard, simple additive systems (<a href="Sign-value_notation" title="Sign-value notation">sign-value notation</a>) such as <a href="Roman_numerals" title="Roman numerals">Roman numerals</a> or <a href="Chinese_numerals" title="Chinese numerals">Chinese numerals</a> were used, and accountants in the past used the <a href="Abacus" title="Abacus">abacus</a> or stone counters to do arithmetic until the introduction of positional notation.<a href="#cite_note-4">[4]</a>

<a href="Counting_rods" title="Counting rods">Counting rods</a> and most abacuses have been used to represent numbers in a positional numeral system. With counting rods or <a href="Abacus" title="Abacus">abacus</a> to perform arithmetic operations, the writing of the starting, intermediate and final values of a calculation could easily be done with a simple additive system in each position or column. This approach required no memorization of tables (as does positional notation) and could produce practical results quickly.

The oldest extant positional notation system is either that of Chinese <a href="Rod_numerals" class="mw-redirect" title="Rod numerals">rod numerals</a>, used from at least the early 8th century, or perhaps <a href="Khmer_numerals" title="Khmer numerals">Khmer numerals</a>, showing possible usages of positional-numbers in the 7th century. Khmer numerals and other <a href="Indian_numerals" class="mw-redirect" title="Indian numerals">Indian numerals</a> originate with the <a href="Brahmi_numerals" title="Brahmi numerals">Brahmi numerals</a> of about the 3rd century BC, which symbols were, at the time, not used positionally. Medieval Indian numerals are positional, as are the derived <a href="Arabic_numerals" title="Arabic numerals">Arabic numerals</a>, recorded from the 10th century.

After the <a href="French_Revolution" title="French Revolution">French Revolution</a> (1789–1799), the new French government promoted the extension of the decimal system.<a href="#cite_note-5">[5]</a> Some of those pro-decimal efforts—such as <a href="Decimal_time" title="Decimal time">decimal time</a> and the <a href="Decimal_calendar" title="Decimal calendar">decimal calendar</a>—were unsuccessful. Other French pro-decimal efforts—currency <a href="Decimalisation" title="Decimalisation">decimalisation</a> and the <a href="Metrication" title="Metrication">metrication</a> of weights and measures—spread widely out of France to almost the whole world.

<div class="mw-heading mw-heading3"><h3 id="History_of_positional_fractions">History of positional fractions</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Decimal" title="Decimal">Decimal</a></div>

Decimal fractions were first developed and used by the Chinese in the form of <a href="Rod_calculus#Fractions" title="Rod calculus">rod calculus</a> in the 1st century BC, and then spread to the rest of the world. <a href="#cite_note-6">[6]</a><a href="#cite_note-jnfractn1-7">[7]</a> J. Lennart Berggren notes that positional decimal fractions were used being, in <a href="Damascus" title="Damascus">Damascus</a>, by mathematician <a href="Abu'l-Hasan_al-Uqlidisi" title="Abu'l-Hasan al-Uqlidisi">Abu'l-Hasan al-Uqlidisi</a>, in the mid 10th century.<a href="#cite_note-Berggren-8">[8]</a> The Jewish mathematician <a href="Immanuel_Bonfils" title="Immanuel Bonfils">Immanuel Bonfils</a> used decimal fractions around 1350, but did not develop any notation to represent them.<a href="#cite_note-9">[9]</a> The Persian mathematician <a href="Jamsh%C4%ABd_al-K%C4%81sh%C4%AB" class="mw-redirect" title="Jamshīd al-Kāshī">Jamshīd al-Kāshī</a> similarly adopted their use in the 15th century.<a href="#cite_note-Berggren-8">[8]</a> <a href="Al_Khwarizmi" class="mw-redirect" title="Al Khwarizmi">Al Khwarizmi</a> introduced fractions to Islamic countries in the early 9th century; his fraction presentation was similar to the traditional Chinese mathematical fractions from <a href="Sunzi_Suanjing" title="Sunzi Suanjing">Sunzi Suanjing</a>.<a href="#cite_note-Lam-10">[10]</a> This form of fraction with numerator on top and denominator at bottom without a horizontal bar was also used by 10th century <a href="Abu'l-Hasan_al-Uqlidisi" title="Abu'l-Hasan al-Uqlidisi">Abu'l-Hasan al-Uqlidisi</a> and 15th century <a href="Jamsh%C4%ABd_al-K%C4%81sh%C4%AB" class="mw-redirect" title="Jamshīd al-Kāshī">Jamshīd al-Kāshī</a>'s work "Arithmetic Key".<a href="#cite_note-Lam-10">[10]</a><a href="#cite_note-11">[11]</a>

<th style="background: #ececec; color: black; font-weight: bold; vertical-align: middle; text-align: right;" class="table-rh">Number

</th>

<td>184.54290

</td></tr>

<tr>

<th style="background: #ececec; color: black; font-weight: bold; vertical-align: middle; text-align: right;" class="table-rh">Simon Stevin's notation

</th>

<td>184⓪5①4②2③9④0

</td></tr></tbody></table>

The adoption of the <a href="Decimal_representation" title="Decimal representation">decimal representation</a> of numbers less than one, a <a href="Fraction_(mathematics)" class="mw-redirect" title="Fraction (mathematics)">fraction</a>, is often credited to <a href="Simon_Stevin" title="Simon Stevin">Simon Stevin</a> through his textbook <a href="De_Thiende" title="De Thiende">De Thiende</a>;<a href="#cite_note-van-12">[12]</a> but both Stevin and <a href="E._J._Dijksterhuis" class="mw-redirect" title="E. J. Dijksterhuis">E. J. Dijksterhuis</a> indicate that <a href="Regiomontanus" title="Regiomontanus">Regiomontanus</a> contributed to the European adoption of general <a href="Decimal" title="Decimal">decimals</a>:<a href="#cite_note-EJD-13">[13]</a>: 17, 18

/* start https://en.wikipedia.org/ */

.mw-parser-output .templatequote{overflow:hidden;margin:1em 0;padding:0 32px}.mw-parser-output .templatequotecite{line-height:1.5em;text-align:left;margin-top:0}@media(min-width:500px){.mw-parser-output .templatequotecite{padding-left:1.6em}}

/* end https://en.wikipedia.org/ */

</style><blockquote class="templatequote">European mathematicians, when taking over from the Hindus, via the Arabs, the idea of positional value for integers, neglected to extend this idea to fractions. For some centuries they confined themselves to using common and <a href="Sexagesimal" title="Sexagesimal">sexagesimal</a> fractions ... This half-heartedness has never been completely overcome, and sexagesimal fractions still form the basis of our trigonometry, astronomy and measurement of time.

... Mathematicians sought to avoid fractions by taking the radius R equal to a number of units of length of the form 10n and then assuming for n so great an integral value that all occurring quantities could be expressed with sufficient accuracy by integers.

The first to apply this method was the German astronomer Regiomontanus. To the extent that he expressed goniometrical line-segments in a unit R/10n, Regiomontanus may be called an anticipator of the doctrine of decimal positional fractions.</blockquote>

In the estimation of Dijksterhuis, "after the publication of <a href="De_Thiende" title="De Thiende">De Thiende</a> only a small advance was required to establish the complete system of decimal positional fractions, and this step was taken promptly by a number of writers ... next to Stevin the most important figure in this development was Regiomontanus." Dijksterhuis noted that [Stevin] "gives full credit to Regiomontanus for his prior contribution, saying that the trigonometric tables of the German astronomer actually contain the whole theory of 'numbers of the tenth progress'."<a href="#cite_note-EJD-13">[13]</a>: 19

<div class="mw-heading mw-heading2"><h2 id="Mathematics">Mathematics</h2></div>

/* start https://en.wikipedia.org/ */

.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}

/* end https://en.wikipedia.org/ */

</style>

<div class="mw-heading mw-heading3"><h3 id="Base_of_the_numeral_system">Base of the numeral system</h3></div>

In <a href="Numeral_system" title="Numeral system">mathematical numeral systems</a> the <a href="Radix" title="Radix">radix</a> r is usually the number of unique <a href="Numerical_digit" title="Numerical digit">digits</a>, including zero, that a positional numeral system uses to represent numbers. In some cases, such as with a <a href="Negative_base" title="Negative base">negative base</a>, the radix is the <a href="Absolute_value" title="Absolute value">absolute value</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=|b|}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r=|b|}</annotation>

</semantics>

</math><img src="./73c965cf2ba34ac24477d0b4a868125d76e36549.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.438ex; height:2.843ex;" alt="{\displaystyle r=|b|}" loading="lazy"> of the base b. For example, for the decimal system the radix (and base) is ten, because it uses the ten digits from 0 through 9. When a number "hits" 9, the next number will not be another different symbol, but a "1" followed by a "0". In binary, the radix is two, since after it hits "1", instead of "2" or another written symbol, it jumps straight to "10", followed by "11" and "100".

The highest symbol of a positional numeral system usually has the value one less than the value of the radix of that numeral system. The standard positional numeral systems differ from one another only in the base they use.

The radix is an integer that is greater than 1, since a radix of zero would not have any digits, and a radix of 1 would only have the zero digit. Negative bases are rarely used. In a system with more than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |b|}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |b|}</annotation>

</semantics>

</math><img src="./881f49e94388a46a05d329251551ce20baf4f05d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.291ex; height:2.843ex;" alt="{\displaystyle |b|}" loading="lazy"> unique digits, numbers may have many different possible representations.

It is important that the radix is finite, from which follows that the number of digits is quite low. Otherwise, the length of a numeral would not necessarily be <a href="Logarithm" title="Logarithm">logarithmic</a> in its size.

(In certain <a href="Non-standard_positional_numeral_systems" title="Non-standard positional numeral systems">non-standard positional numeral systems</a>, including <a href="Bijective_numeration" title="Bijective numeration">bijective numeration</a>, the definition of the base or the allowed digits deviates from the above.)

In standard base-ten (<a href="Decimal" title="Decimal">decimal</a>) positional notation, there are ten <a href="Decimal_digit" class="mw-redirect" title="Decimal digit">decimal digits</a> and the number

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 5305_{\mathrm {dec} }=(5\times 10^{3})+(3\times 10^{2})+(0\times 10^{1})+(5\times 10^{0})}</annotation>

</semantics>

</math><img src="./4c2412cfdd5672ba2cff154d02a151b0900e8b61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:55.641ex; height:3.176ex;" alt="{\displaystyle 5305_{\mathrm {dec} }=(5\times 10^{3})+(3\times 10^{2})+(0\times 10^{1})+(5\times 10^{0})}" loading="lazy">.</dd></dl>

In standard base-sixteen (<a href="Hexadecimal" title="Hexadecimal">hexadecimal</a>), there are the sixteen hexadecimal digits (0–9 and A–F) and the number

</mrow>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 14\mathrm {B} 9_{\mathrm {hex} }=(1\times 16^{3})+(4\times 16^{2})+(\mathrm {B} \times 16^{1})+(9\times 16^{0})\qquad (=5305_{\mathrm {dec} }),}</annotation>

</semantics>

</math><img src="./64644b34e53c20584e9e6ea9e4246fd6dc9a54fb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:73.556ex; height:3.176ex;" alt="{\displaystyle 14\mathrm {B} 9_{\mathrm {hex} }=(1\times 16^{3})+(4\times 16^{2})+(\mathrm {B} \times 16^{1})+(9\times 16^{0})\qquad (=5305_{\mathrm {dec} }),}" loading="lazy"></dd></dl>

where B represents the number eleven as a single symbol.

In general, in base-b, there are b digits <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{d_{1},d_{2},\dotsb ,d_{b}\}=:D}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{d_{1},d_{2},\dotsb ,d_{b}\}=:D}</annotation>

</semantics>

</math><img src="./686483265001276483fefbe7beb42c1ffe3b30c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.879ex; height:2.843ex;" alt="{\displaystyle \{d_{1},d_{2},\dotsb ,d_{b}\}=:D}" loading="lazy"> and the number

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (a_{3}a_{2}a_{1}a_{0})_{b}=(a_{3}\times b^{3})+(a_{2}\times b^{2})+(a_{1}\times b^{1})+(a_{0}\times b^{0})}</annotation>

</semantics>

</math><img src="./36c1fcc47702f566ae82a37047114be13864f583.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:59.445ex; height:3.176ex;" alt="{\displaystyle (a_{3}a_{2}a_{1}a_{0})_{b}=(a_{3}\times b^{3})+(a_{2}\times b^{2})+(a_{1}\times b^{1})+(a_{0}\times b^{0})}" loading="lazy"></dd></dl>

has <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall k\colon a_{k}\in D.}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \forall k\colon a_{k}\in D.}</annotation>

</semantics>

</math><img src="./d317a38701aed6182eab5240f637bdabf5bc1f72.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.268ex; height:2.509ex;" alt="{\displaystyle \forall k\colon a_{k}\in D.}" loading="lazy">

Note that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{3}a_{2}a_{1}a_{0}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a_{3}a_{2}a_{1}a_{0}}</annotation>

</semantics>

</math><img src="./894bf4e81c14ae3394f7b5b054bb8d3352f1daa7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.136ex; height:2.009ex;" alt="{\displaystyle a_{3}a_{2}a_{1}a_{0}}" loading="lazy"> represents a sequence of digits, not <a href="Multiplication" title="Multiplication">multiplication</a>.

<div class="mw-heading mw-heading3"><h3 id="Notation">Notation</h3></div>

When describing base in <a href="Mathematical_notation" title="Mathematical notation">mathematical notation</a>, the letter b is generally used as a <a href="Symbol" title="Symbol">symbol</a> for this concept, so, for a <a href="Binary_numeral_system" class="mw-redirect" title="Binary numeral system">binary</a> system, b <a href="Equality_(mathematics)" title="Equality (mathematics)">equals</a> 2. Another common way of expressing the base is writing it as a decimal subscript after the number that is being represented (this notation is used in this article). 11110112 implies that the number 1111011 is a base-2 number, equal to 12310 (a <a href="Decimal_notation" class="mw-redirect" title="Decimal notation">decimal notation</a> representation), 1738 (<a href="Octal" title="Octal">octal</a>) and 7B16 (<a href="Hexadecimal" title="Hexadecimal">hexadecimal</a>). In books and articles, when using initially the written abbreviations of number bases, the base is not subsequently printed: it is assumed that binary 1111011 is the same as 11110112.

The base b may also be indicated by the phrase "base-b". So binary numbers are "base-2"; octal numbers are "base-8"; decimal numbers are "base-10"; and so on.

To a given radix b the set of digits {0, 1, ..., b−2, b−1} is called the standard set of digits. Thus, binary numbers have digits {0, 1}; decimal numbers have digits {0, 1, 2, ..., 8, 9}; and so on. Therefore, the following are notational errors: 522, 22, 1A9. (In all cases, one or more digits is not in the set of allowed digits for the given base.)

<div class="mw-heading mw-heading3"><h3 id="Exponentiation">Exponentiation</h3></div>

Positional numeral systems work using <a href="Exponentiation" title="Exponentiation">exponentiation</a> of the base. A digit's value is the digit multiplied by the value of its place. Place values are the number of the base raised to the nth power, where n is the number of other digits between a given digit and the <a href="Radix_point" class="mw-redirect" title="Radix point">radix point</a>. If a given digit is on the left hand side of the radix point (i.e. its value is an <a href="Integer" title="Integer">integer</a>) then n is positive or zero; if the digit is on the right hand side of the radix point (i.e., its value is fractional) then n is negative.

As an example of usage, the number 465 in its respective base b (which must be at least base 7 because the highest digit in it is 6) is equal to:

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 4\times b^{2}+6\times b^{1}+5\times b^{0}}</annotation>

</semantics>

</math><img src="./36471538764f4e53747ed807bc11736cf6b77c64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:23.845ex; height:2.843ex;" alt="{\displaystyle 4\times b^{2}+6\times b^{1}+5\times b^{0}}" loading="lazy"></dd></dl>

If the number 465 was in base-10, then it would equal:

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 465_{10}=4\times 10^{2}+6\times 10^{1}+5\times 10^{0}=4\times 100+6\times 10+5\times 1=465_{10}}</annotation>

</semantics>

</math><img src="./1e1fe0717b276ac5eb654407a1b54135e2e90bdb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:72.513ex; height:3.009ex;" alt="{\displaystyle 465_{10}=4\times 10^{2}+6\times 10^{1}+5\times 10^{0}=4\times 100+6\times 10+5\times 1=465_{10}}" loading="lazy"></dd></dl>

If however, the number were in base 7, then it would equal:

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 465_{7}=4\times 7^{2}+6\times 7^{1}+5\times 7^{0}=4\times 49+6\times 7+5\times 1=243_{10}}</annotation>

</semantics>

</math><img src="./bffae5bb3a58df62aece32422cab8a45113c7af2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:65.879ex; height:3.009ex;" alt="{\displaystyle 465_{7}=4\times 7^{2}+6\times 7^{1}+5\times 7^{0}=4\times 49+6\times 7+5\times 1=243_{10}}" loading="lazy"></dd></dl>

10b = b for any base b, since 10b = 1×b1 + 0×b0. For example, 102 = 2; 103 = 3; 1016 = 1610. Note that the last "16" is indicated to be in base 10. The base makes no difference for one-digit numerals.

This concept can be demonstrated using a diagram. One object represents one unit. When the number of objects is equal to or greater than the base b, then a group of objects is created with b objects. When the number of these groups exceeds b, then a group of these groups of objects is created with b groups of b objects; and so on. Thus the same number in different bases will have different values:

<pre>241 in base 5:

2 groups of 52 (25) 4 groups of 5 1 group of 1

ooooo ooooo

ooooo ooooo ooooo ooooo

ooooo ooooo + + o

ooooo ooooo ooooo ooooo

ooooo ooooo

</pre>

<pre>241 in base 8:

2 groups of 82 (64) 4 groups of 8 1 group of 1

oooooooo oooooooo

oooooooo oooooooo oooooooo oooooooo

oooooooo oooooooo + + o

oooooooo oooooooo

oooooooo oooooooo oooooooo oooooooo

oooooooo oooooooo

</pre>

The notation can be further augmented by allowing a leading minus sign. This allows the representation of negative numbers. For a given base, every representation corresponds to exactly one <a href="Real_number" title="Real number">real number</a> and every real number has at least one representation. The representations of rational numbers are those representations that are finite, use the bar notation, or end with an infinitely repeating cycle of digits.

<div class="mw-heading mw-heading3"><h3 id="Digits_and_numerals">Digits and numerals</h3></div>

A digit is a symbol that is used for positional notation, and a numeral consists of one or more digits used for representing a <a href="Number" title="Number">number</a> with positional notation. Today's most common digits are the <a href="Arabic_numerals" title="Arabic numerals">decimal digits</a> "0", "1", "2", "3", "4", "5", "6", "7", "8", and "9". The distinction between a digit and a numeral is most pronounced in the context of a number base.

A non-zero numeral with more than one digit position will mean a different number in a different number base, but in general, the digits will mean the same.<a href="#cite_note-14">[14]</a> For example, the base-8 numeral 238 contains two digits, "2" and "3", and with a base number (subscripted) "8". When converted to base-10, the 238 is equivalent to 1910, i.e. 238 = 1910. In our notation here, the subscript "8" of the numeral 238 is part of the numeral, but this may not always be the case.

Imagine the numeral "23" as having <a href="#Non-standard_positional_numeral_systems">an ambiguous base</a> number. Then "23" could likely be any base, from base-4 up. In base-4, the "23" means 1110, i.e. 234 = 1110. In base-60, the "23" means the number 12310, i.e. 2360 = 12310. The numeral "23" then, in this case, corresponds to the set of base-10 numbers {11, 13, 15, 17, 19, 21, 23, ..., 121, 123} while its digits "2" and "3" always retain their original meaning: the "2" means "two of", and the "3" means "three of".

In certain applications when a numeral with a fixed number of positions needs to represent a greater number, a higher number-base with more digits per position can be used. A three-digit, decimal numeral can represent only up to 999. But if the number-base is increased to 11, say, by adding the digit "A", then the same three positions, maximized to "AAA", can represent a number as great as 1330. We could increase the number base again and assign "B" to 11, and so on (but there is also a possible encryption between number and digit in the number-digit-numeral hierarchy). A three-digit numeral "ZZZ" in base-60 could mean 215999. If we use the entire collection of our <a href="Alphanumerics" class="mw-redirect" title="Alphanumerics">alphanumerics</a> we could ultimately serve a base-62 numeral system, but we remove two digits, uppercase "I" and uppercase "O", to reduce confusion with digits "1" and "0".<a href="#cite_note-15">[15]</a>

We are left with a base-60, or sexagesimal numeral system utilizing 60 of the 62 standard alphanumerics. (But see <a href="#Sexagesimal_system">Sexagesimal system</a> below.) In general, the number of possible values that can be represented by a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d}</annotation>

</semantics>

</math><img src="./e85ff03cbe0c7341af6b982e47e9f90d235c66ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}" loading="lazy"> digit number in base <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r}</annotation>

</semantics>

</math><img src="./0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" loading="lazy"> is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r^{d}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r^{d}}</annotation>

</semantics>

</math><img src="./0d263be4cfbd2021700fbe1b8950ee76fc5f4da9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.141ex; height:2.676ex;" alt="{\displaystyle r^{d}}" loading="lazy">.

The common numeral systems in computer science are binary (radix 2), octal (radix 8), and hexadecimal (radix 16). In <a href="Binary_numeral_system" class="mw-redirect" title="Binary numeral system">binary</a> only digits "0" and "1" are in the numerals. In the <a href="Octal" title="Octal">octal</a> numerals, are the eight digits 0–7. <a href="Hexadecimal" title="Hexadecimal">Hex</a> is 0–9 A–F, where the ten numerics retain their usual meaning, and the alphabetics correspond to values 10–15, for a total of sixteen digits. The numeral "10" is binary numeral "2", octal numeral "8", or hexadecimal numeral "16".

<div class="mw-heading mw-heading3"><h3 id="Radix_point">Radix point</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Radix_point" class="mw-redirect" title="Radix point">Radix point</a></div>

The notation can be extended into the negative exponents of the base b. Thereby the so-called radix point, mostly ».«, is used as separator of the positions with non-negative from those with negative exponent.

Numbers that are not <a href="Integer" title="Integer">integers</a> use places beyond the <a href="Radix_point" class="mw-redirect" title="Radix point">radix point</a>. For every position behind this point (and thus after the units digit), the exponent n of the power bn decreases by 1 and the power approaches 0. For example, the number 2.35 is equal to:

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2\times 10^{0}+3\times 10^{-1}+5\times 10^{-2}}</annotation>

</semantics>

</math><img src="./1adbbdb77ef8fac9fbc78f404a393a9cd5b03d14.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:30.384ex; height:2.843ex;" alt="{\displaystyle 2\times 10^{0}+3\times 10^{-1}+5\times 10^{-2}}" loading="lazy"></dd></dl>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Sign_(mathematics)" title="Sign (mathematics)">Sign (mathematics)</a></div>

If the base and all the digits in the set of digits are non-negative, negative numbers cannot be expressed. To overcome this, a <a href="Negative_number" title="Negative number">minus sign</a>, here −, is added to the numeral system. In the usual notation it is prepended to the string of digits representing the otherwise non-negative number.

<div class="mw-heading mw-heading3"><h3 id="Base_conversion">Base conversion</h3></div>

The conversion to a base <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b_{2}}</annotation>

</semantics>

</math><img src="./2530a260ad35bf21ee61f1f4d6493ae0474f6068.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.052ex; height:2.509ex;" alt="{\displaystyle b_{2}}" loading="lazy"> of an integer n represented in base <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b_{1}}</annotation>

</semantics>

</math><img src="./9af2720c91be489f57ecde4bb651b95e113d0144.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.052ex; height:2.509ex;" alt="{\displaystyle b_{1}}" loading="lazy"> can be done by a succession of <a href="Euclidean_division" title="Euclidean division">Euclidean divisions</a> by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{2}:}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b_{2}:}</annotation>

</semantics>

</math><img src="./3c560364c83db1e1cae58ce7794c01555aeb2e1c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.344ex; height:2.509ex;" alt="{\displaystyle b_{2}:}" loading="lazy"> the right-most digit in base <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b_{2}}</annotation>

</semantics>

</math><img src="./2530a260ad35bf21ee61f1f4d6493ae0474f6068.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.052ex; height:2.509ex;" alt="{\displaystyle b_{2}}" loading="lazy"> is the remainder of the division of n by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{2};}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b_{2};}</annotation>

</semantics>

</math><img src="./3d8e457b3021926a4e447d6e28a95bbb80cae466.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.699ex; height:2.509ex;" alt="{\displaystyle b_{2};}" loading="lazy"> the second right-most digit is the remainder of the division of the quotient by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{2},}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b_{2},}</annotation>

</semantics>

</math><img src="./f0b02259cb1884d74ba5b44b6b7fe9f785547f4c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.699ex; height:2.509ex;" alt="{\displaystyle b_{2},}" loading="lazy"> and so on. The left-most digit is the last quotient. In general, the kth digit from the right is the remainder of the division by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b_{2}}</annotation>

</semantics>

</math><img src="./2530a260ad35bf21ee61f1f4d6493ae0474f6068.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.052ex; height:2.509ex;" alt="{\displaystyle b_{2}}" loading="lazy"> of the (k−1)th quotient.

For example: converting A10BHex to decimal (41227):

<pre>0xA10B/10 = Q: 0x101A R: 7 (ones place)

0x101A/10 = Q: 0x19C R: 2 (tens place)

0x19C/10 = Q: 0x29 R: 2 (hundreds place)

0x29/10 = Q: 0x4 R: 1 ...

4

</pre>

When converting to a larger base (such as from binary to decimal), the remainder represents <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b_{2}}</annotation>

</semantics>

</math><img src="./2530a260ad35bf21ee61f1f4d6493ae0474f6068.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.052ex; height:2.509ex;" alt="{\displaystyle b_{2}}" loading="lazy"> as a single digit, using digits from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b_{1}}</annotation>

</semantics>

</math><img src="./9af2720c91be489f57ecde4bb651b95e113d0144.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.052ex; height:2.509ex;" alt="{\displaystyle b_{1}}" loading="lazy">. For example: converting 0b11111001 (binary) to 249 (decimal):

<pre>0b11111001/10 = Q: 0b11000 R: 0b1001 (0b1001 = "9" for ones place)

0b11000/10 = Q: 0b10 R: 0b100 (0b100 = "4" for tens)

0b10/10 = Q: 0b0 R: 0b10 (0b10 = "2" for hundreds)

</pre>

For the <a href="Fraction_(mathematics)" class="mw-redirect" title="Fraction (mathematics)">fractional</a> part, conversion can be done by taking digits after the radix point (the numerator), and <a href="Long_division" title="Long division">dividing</a> it by the <a href="Fraction_(mathematics)" class="mw-redirect" title="Fraction (mathematics)">implied denominator</a> in the target radix. Approximation may be needed due to a possibility of <a href="Repeating_decimal#Extension_to_other_bases" title="Repeating decimal">non-terminating digits</a> if the <a href="Irreducible_fraction" title="Irreducible fraction">reduced</a> fraction's denominator has a prime factor other than any of the base's prime factor(s) to convert to. For example, 0.1 in decimal (1/10) is 0b1/0b1010 in binary, by dividing this in that radix, the result is 0b0.00011 (because one of the prime factors of 10 is 5). For more general fractions and bases see the <a href="Repeating_decimal#Algorithm_for_positive_bases" title="Repeating decimal">algorithm for positive bases</a>.

Alternatively, <a href="Horner's_method" title="Horner's method">Horner's method</a> can be used for base conversion using repeated multiplications, with the same computational complexity as repeated divisions.<a href="#cite_note-16">[16]</a> A number in positional notation can be thought of as a polynomial, where each digit is a coefficient. Coefficients can be larger than one digit, so an efficient way to convert bases is to convert each digit, then evaluate the polynomial via Horner's method within the target base. Converting each digit is a simple <a href="Lookup_table" title="Lookup table">lookup table</a>, removing the need for expensive division or modulus operations; and multiplication by x becomes right-shifting. However, other polynomial evaluation algorithms would work as well, like <a href="Exponentiation_by_squaring" title="Exponentiation by squaring">repeated squaring</a> for single or sparse digits. Example:

<pre>Convert 0xA10B to 41227

A10B = (10*16^3) + (1*16^2) + (0*16^1) + (11*16^0)

</pre>

<pre> Lookup table:

0x0 = 0

0x1 = 1

...

0x9 = 9

0xA = 10

0xB = 11

0xC = 12

0xD = 13

0xE = 14

0xF = 15

Therefore 0xA10B's decimal digits are 10, 1, 0, and 11.

</pre>

<pre> Lay out the digits out like this. The most significant digit (10) is "dropped":

10 1 0 11 <- Digits of 0xA10B

</pre>

<pre> ---------------

10

Then we multiply the bottom number from the source base (16), the product is placed under the next digit of the source value, and then add:

10 1 0 11

160

---------------

10 161

</pre>

<pre> Repeat until the final addition is performed:

10 1 0 11

160 2576 41216

---------------

10 161 2576 41227

</pre>

<pre> and that is 41227 in decimal.

</pre>

<pre>Convert 0b11111001 to 249

Lookup table:

0b0 = 0

0b1 = 1

</pre>

<pre>Result:

1 1 1 1 1 0 0 1 <- Digits of 0b11111001

2 6 14 30 62 124 248

-------------------------

1 3 7 15 31 62 124 249

</pre>

<div class="mw-heading mw-heading3"><h3 id="Terminating_fractions">Terminating fractions</h3></div>

The numbers which have a finite representation form the <a href="Semiring" title="Semiring">semiring</a>

<mfrac>

<msub>

</mrow>

</mrow>

</msub>

<msup>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</msup>

</mfrac>

</mrow>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\mathbb {N} _{0}}{b^{\mathbb {N} _{0}}}}:=\left\{mb^{-\nu }\mid m\in \mathbb {N} _{0}\wedge \nu \in \mathbb {N} _{0}\right\}.}</annotation>

</semantics>

</math><img src="./93576bc370e9fc7a7c9db689001beec98c817efe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:35.932ex; height:5.676ex;" alt="{\displaystyle {\frac {\mathbb {N} _{0}}{b^{\mathbb {N} _{0}}}}:=\left\{mb^{-\nu }\mid m\in \mathbb {N} _{0}\wedge \nu \in \mathbb {N} _{0}\right\}.}" loading="lazy"></dd></dl>

More explicitly, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{1}^{\nu _{1}}\cdot \ldots \cdot p_{n}^{\nu _{n}}:=b}">

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msubsup>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p_{1}^{\nu _{1}}\cdot \ldots \cdot p_{n}^{\nu _{n}}:=b}</annotation>

</semantics>

</math><img src="./f39d27bc8cab730c76d3f65abca3b3f5ecc41549.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.089ex; width:17.138ex; height:3.176ex;" alt="{\displaystyle p_{1}^{\nu _{1}}\cdot \ldots \cdot p_{n}^{\nu _{n}}:=b}" loading="lazy"> is a <a href="Factorization" title="Factorization">factorization</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b}</annotation>

</semantics>

</math><img src="./f11423fbb2e967f986e36804a8ae4271734917c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" loading="lazy"> into the primes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{1},\ldots ,p_{n}\in \mathbb {P} }">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p_{1},\ldots ,p_{n}\in \mathbb {P} }</annotation>

</semantics>

</math><img src="./ebddbdf58eaac77ec8a1fe86a430dd60cd0eb906.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:14.14ex; height:2.509ex;" alt="{\displaystyle p_{1},\ldots ,p_{n}\in \mathbb {P} }" loading="lazy"> with exponents <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu _{1},\ldots ,\nu _{n}\in \mathbb {N} }">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \nu _{1},\ldots ,\nu _{n}\in \mathbb {N} }</annotation>

</semantics>

</math><img src="./746602e826861d4480c6e230af7d54e6d0741263.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.267ex; height:2.509ex;" alt="{\displaystyle \nu _{1},\ldots ,\nu _{n}\in \mathbb {N} }" loading="lazy">,<a href="#cite_note-17">[17]</a> then with the non-empty set of denominators <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S:=\{p_{1},\ldots ,p_{n}\}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S:=\{p_{1},\ldots ,p_{n}\}}</annotation>

</semantics>

</math><img src="./4583c2d28c58882d11b3b20e6395c4ac3cb7286e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.359ex; height:2.843ex;" alt="{\displaystyle S:=\{p_{1},\ldots ,p_{n}\}}" loading="lazy">

we have

<msub>

</mrow>

</mrow>

</msub>

<mrow>

<mrow>

</mrow>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

</munderover>

<msup>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} _{S}:=\left\{x\in \mathbb {Q} \left|\,\exists \mu _{i}\in \mathbb {Z} :x\prod _{i=1}^{n}{p_{i}}^{\mu _{i}}\in \mathbb {Z} \right.\right\}=b^{\mathbb {Z} }\,\mathbb {Z} ={\langle S\rangle }^{-1}\mathbb {Z} }</annotation>

</semantics>

</math><img src="./3807e120a0dff6436e643f47b58e1a2c7088413a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:58.49ex; height:7.509ex;" alt="{\displaystyle \mathbb {Z} _{S}:=\left\{x\in \mathbb {Q} \left|\,\exists \mu _{i}\in \mathbb {Z} :x\prod _{i=1}^{n}{p_{i}}^{\mu _{i}}\in \mathbb {Z} \right.\right\}=b^{\mathbb {Z} }\,\mathbb {Z} ={\langle S\rangle }^{-1}\mathbb {Z} }" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle S\rangle }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \langle S\rangle }</annotation>

</semantics>

</math><img src="./43e09a95f17a6c1836a61f42b133a066fd2edd0e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.309ex; height:2.843ex;" alt="{\displaystyle \langle S\rangle }" loading="lazy"> is the group generated by the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\in S}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\in S}</annotation>

</semantics>

</math><img src="./9f96a607d844af38d7ed59c774c1d0607ec0b9ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.599ex; height:2.509ex;" alt="{\displaystyle p\in S}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\langle S\rangle }^{-1}\mathbb {Z} }">

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\langle S\rangle }^{-1}\mathbb {Z} }</annotation>

</semantics>

</math><img src="./488d5b196638bfed271baf82eaffc3c885d149b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.192ex; height:3.343ex;" alt="{\displaystyle {\langle S\rangle }^{-1}\mathbb {Z} }" loading="lazy"> is the so-called <a href="Localization_(algebra)" class="mw-redirect" title="Localization (algebra)">localization</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation>

</semantics>

</math><img src="./449494a083e0a1fda2b61c62b2f09b6bee4633dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }" loading="lazy"> with respect to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S}</annotation>

</semantics>

</math><img src="./4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy">.

The <a href="Fraction_(mathematics)" class="mw-redirect" title="Fraction (mathematics)">denominator</a> of an element of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} _{S}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} _{S}}</annotation>

</semantics>

</math><img src="./06accfafc512e72329e0a582e1dc513319730a74.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.843ex; height:2.509ex;" alt="{\displaystyle \mathbb {Z} _{S}}" loading="lazy"> contains if reduced to lowest terms only prime factors out of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S}</annotation>

</semantics>

</math><img src="./4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy">.

This <a href="Ring_(mathematics)" title="Ring (mathematics)">ring</a> of all terminating fractions to base <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b}</annotation>

</semantics>

</math><img src="./f11423fbb2e967f986e36804a8ae4271734917c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" loading="lazy"> is <a href="Dense_set" title="Dense set">dense</a> in the field of <a href="Rational_number" title="Rational number">rational numbers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation>

</semantics>

</math><img src="./c5909f0b54e4718fa24d5fd34d54189d24a66e9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }" loading="lazy">. Its <a href="Complete_metric_space" title="Complete metric space">completion</a> for the usual (Archimedean) metric is the same as for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation>

</semantics>

</math><img src="./c5909f0b54e4718fa24d5fd34d54189d24a66e9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }" loading="lazy">, namely the real numbers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>

</semantics>

</math><img src="./786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy">. So, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=\{p\}}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S=\{p\}}</annotation>

</semantics>

</math><img src="./db7cda8a792f6e87b6ad4c7a16f898b2d08df280.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.092ex; height:2.843ex;" alt="{\displaystyle S=\{p\}}" loading="lazy"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} _{\{p\}}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} _{\{p\}}}</annotation>

</semantics>

</math><img src="./a78bc5c4689e84d65b8cda275d25ea2cd667e7ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:4.253ex; height:3.009ex;" alt="{\displaystyle \mathbb {Z} _{\{p\}}}" loading="lazy"> has not to be confused with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} _{(p)}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} _{(p)}}</annotation>

</semantics>

</math><img src="./31ece65379ee3430084adc10c17f0b4d0a0fad16.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.889ex; height:3.009ex;" alt="{\displaystyle \mathbb {Z} _{(p)}}" loading="lazy">, the <a href="Discrete_valuation_ring" title="Discrete valuation ring">discrete valuation ring</a> for the <a href="Prime_number" title="Prime number">prime</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy">, which is equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} _{T}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} _{T}}</annotation>

</semantics>

</math><img src="./05573c2f699963a1738af4451e3381d3b6b57b06.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.94ex; height:2.509ex;" alt="{\displaystyle \mathbb {Z} _{T}}" loading="lazy"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T=\mathbb {P} \setminus \{p\}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T=\mathbb {P} \setminus \{p\}}</annotation>

</semantics>

</math><img src="./3ef6e278e1745d644a6a66c83387ce16719a8772.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.844ex; height:2.843ex;" alt="{\displaystyle T=\mathbb {P} \setminus \{p\}}" loading="lazy">.

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b}</annotation>

</semantics>

</math><img src="./f11423fbb2e967f986e36804a8ae4271734917c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" loading="lazy"> divides <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c}</annotation>

</semantics>

</math><img src="./86a67b81c2de995bd608d5b2df50cd8cd7d92455.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" loading="lazy">, we have <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b^{\mathbb {Z} }\,\mathbb {Z} \subseteq c^{\mathbb {Z} }\,\mathbb {Z} .}">

<msup>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b^{\mathbb {Z} }\,\mathbb {Z} \subseteq c^{\mathbb {Z} }\,\mathbb {Z} .}</annotation>

</semantics>

</math><img src="./280a657033fce9790a610d97ae908f2ef2013f5f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.282ex; height:2.843ex;" alt="{\displaystyle b^{\mathbb {Z} }\,\mathbb {Z} \subseteq c^{\mathbb {Z} }\,\mathbb {Z} .}" loading="lazy">

<div class="mw-heading mw-heading3"><h3 id="Infinite_representations">Infinite representations</h3></div>

<div class="mw-heading mw-heading4"><h4 id="Rational_numbers">Rational numbers</h4></div>

The representation of non-integers can be extended to allow an infinite string of digits beyond the point. For example, 1.12112111211112 ... base-3 represents the sum of the infinite <a href="Series_(mathematics)" title="Series (mathematics)">series</a>:

<mtr>

<mtd>

<msup>

</mrow>

</msup>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{array}{l}1\times 3^{0\,\,\,}+{}\\1\times 3^{-1\,\,}+2\times 3^{-2\,\,\,}+{}\\1\times 3^{-3\,\,}+1\times 3^{-4\,\,\,}+2\times 3^{-5\,\,\,}+{}\\1\times 3^{-6\,\,}+1\times 3^{-7\,\,\,}+1\times 3^{-8\,\,\,}+2\times 3^{-9\,\,\,}+{}\\1\times 3^{-10}+1\times 3^{-11}+1\times 3^{-12}+1\times 3^{-13}+2\times 3^{-14}+\cdots \end{array}}}</annotation>

</semantics>

</math><img src="./ed37d8111edb82f7105f5b2c7ce1f5a368f5d362.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.671ex; width:59.277ex; height:16.509ex;" alt="{\displaystyle {\begin{array}{l}1\times 3^{0\,\,\,}+{}\\1\times 3^{-1\,\,}+2\times 3^{-2\,\,\,}+{}\\1\times 3^{-3\,\,}+1\times 3^{-4\,\,\,}+2\times 3^{-5\,\,\,}+{}\\1\times 3^{-6\,\,}+1\times 3^{-7\,\,\,}+1\times 3^{-8\,\,\,}+2\times 3^{-9\,\,\,}+{}\\1\times 3^{-10}+1\times 3^{-11}+1\times 3^{-12}+1\times 3^{-13}+2\times 3^{-14}+\cdots \end{array}}}" loading="lazy"></dd></dl>

Since a complete infinite string of digits cannot be explicitly written, the trailing ellipsis (...) designates the omitted digits, which may or may not follow a pattern of some kind. One common pattern is when a finite sequence of digits repeats infinitely. This is designated by drawing a <a href="Vinculum_(symbol)" title="Vinculum (symbol)">vinculum</a> across the repeating block:<a href="#cite_note-18">[18]</a>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2.42{\overline {314}}_{5}=2.42314314314314314\dots _{5}}</annotation>

</semantics>

</math><img src="./cf297a21ff387febbeed42ea1bc6a3ff2069b645.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:38.012ex; height:3.343ex;" alt="{\displaystyle 2.42{\overline {314}}_{5}=2.42314314314314314\dots _{5}}" loading="lazy"></dd></dl>

This is the <a href="Repeating_decimal#Notation" title="Repeating decimal">repeating decimal notation</a> (to which there does not exist a single universally accepted notation or phrasing).

For base 10 it is called a repeating decimal or recurring decimal.

An <a href="Irrational_number" title="Irrational number">irrational number</a> has an infinite non-repeating representation in all integer bases. Whether a <a href="Rational_number" title="Rational number">rational number</a> has a finite representation or requires an infinite repeating representation depends on the base. For example, one third can be represented by:

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0.1_{3}}</annotation>

</semantics>

</math><img src="./4dd84ca46ed40eb520b3064e2352ee52ce9986b8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.026ex; height:2.509ex;" alt="{\displaystyle 0.1_{3}}" loading="lazy"></dd>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0.{\overline {3}}_{10}=0.3333333\dots _{10}}</annotation>

</semantics>

</math><img src="./661667fd7e84f4db29a060504d25935c8ae91de0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:23.382ex; height:3.176ex;" alt="{\displaystyle 0.{\overline {3}}_{10}=0.3333333\dots _{10}}" loading="lazy">

<dl><dd>or, with the base implied:</dd>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0.{\overline {3}}=0.3333333\dots }</annotation>

</semantics>

</math><img src="./b9254d625a10f2c80bbdc5156cb1e1a3e3d9b5a0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:19.242ex; height:2.843ex;" alt="{\displaystyle 0.{\overline {3}}=0.3333333\dots }" loading="lazy"> (see also <a href="0.999..." title="0.999...">0.999...</a>)</dd></dl></dd>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0.{\overline {01}}_{2}=0.010101\dots _{2}}</annotation>

</semantics>

</math><img src="./d2715f3a61f3fc2ff76575c039c1a8f28f563037.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:21.738ex; height:3.176ex;" alt="{\displaystyle 0.{\overline {01}}_{2}=0.010101\dots _{2}}" loading="lazy"></dd>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0.2_{6}}</annotation>

</semantics>

</math><img src="./c2f28ff9b3614d9edbfef94b3567a35b9b51f639.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.026ex; height:2.509ex;" alt="{\displaystyle 0.2_{6}}" loading="lazy"></dd></dl>

For integers p and q with <a href="Greatest_common_divisor" title="Greatest common divisor">gcd</a> (p, q) = 1, the <a href="Fraction_(mathematics)" class="mw-redirect" title="Fraction (mathematics)">fraction</a> p/q has a finite representation in base b if and only if each <a href="Prime_factor" class="mw-redirect" title="Prime factor">prime factor</a> of q is also a prime factor of b.

For a given base, any number that can be represented by a finite number of digits (without using the bar notation) will have multiple representations, including one or two infinite representations:

<ol><li>A finite or infinite number of zeroes can be appended:

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 3.46_{7}=3.460_{7}=3.460000_{7}=3.46{\overline {0}}_{7}}</annotation>

</semantics>

</math><img src="./eeca95a46de429e2a7ae3b43a77f212dc3f3254b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:37.139ex; height:3.176ex;" alt="{\displaystyle 3.46_{7}=3.460_{7}=3.460000_{7}=3.46{\overline {0}}_{7}}" loading="lazy"></dd></dl></li>

<li>The last non-zero digit can be reduced by one and an infinite string of digits, each corresponding to one less than the base, are appended (or replace any following zero digits):

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 3.46_{7}=3.45{\overline {6}}_{7}}</annotation>

</semantics>

</math><img src="./a086fed8191fbbaca78c257e53c5faeb66412083.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.753ex; height:3.176ex;" alt="{\displaystyle 3.46_{7}=3.45{\overline {6}}_{7}}" loading="lazy"></dd>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1_{10}=0.{\overline {9}}_{10}\qquad }</annotation>

</semantics>

</math><img src="./d22aadb5408e0bb589e7430fc6f5dcb19921154a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.745ex; height:3.176ex;" alt="{\displaystyle 1_{10}=0.{\overline {9}}_{10}\qquad }" loading="lazy"> (see also <a href="0.999..." title="0.999...">0.999...</a>)</dd>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 220_{5}=214.{\overline {4}}_{5}}</annotation>

</semantics>

</math><img src="./242adcda6a8ed40c24a68a0b238a7af986ef0f53.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.106ex; height:3.343ex;" alt="{\displaystyle 220_{5}=214.{\overline {4}}_{5}}" loading="lazy"></dd></dl></li></ol>

<div class="mw-heading mw-heading4"><h4 id="Irrational_numbers">Irrational numbers</h4></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Irrational_number" title="Irrational number">irrational number</a></div>

A (real) irrational number has an infinite non-repeating representation in all integer bases.<a href="#cite_note-19">[19]</a>

Examples are the non-solvable <a href="Nth_root" title="Nth root">nth roots</a>

<mroot>

</mrow>

</mroot>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y={\sqrt[{n}]{x}}}</annotation>

</semantics>

</math><img src="./70c948f50917ced2d3036c8c5edc92ee14a6b43a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.52ex; height:3.009ex;" alt="{\displaystyle y={\sqrt[{n}]{x}}}" loading="lazy"></dd></dl>

with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y^{n}=x}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y^{n}=x}</annotation>

</semantics>

</math><img src="./287a41964d6e61f0251e3e13a9fd032c558e8a41.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.807ex; height:2.676ex;" alt="{\displaystyle y^{n}=x}" loading="lazy"> and y ∉ Q, numbers which are called <a href="Algebraic_number" title="Algebraic number">algebraic</a>, or numbers like

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi ,e}</annotation>

</semantics>

</math><img src="./e023bb9bc4361bfc11ab9fdfb4559f682c1ad700.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.449ex; height:2.009ex;" alt="{\displaystyle \pi ,e}" loading="lazy"></dd></dl>

which are <a href="Transcendental_number" title="Transcendental number">transcendental</a>. The number of transcendentals is <a href="Uncountable" class="mw-redirect" title="Uncountable">uncountable</a> and the sole way to write them down with a finite number of symbols is to give them a symbol or a finite sequence of symbols.

<div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Decimal_system">Decimal system</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Decimal_representation" title="Decimal representation">Decimal representation</a></div>

In the <a href="Decimal" title="Decimal">decimal</a> (base-10) <a href="Hindu%E2%80%93Arabic_numeral_system" title="Hindu–Arabic numeral system">Hindu–Arabic numeral system</a>, each position starting from the right is a higher power of 10. The first position represents <a href="1_E0" class="mw-redirect" title="1 E0">100</a> (1), the second position <a href="1_E1" class="mw-redirect" title="1 E1">101</a> (10), the third position <a href="1_E2" class="mw-redirect" title="1 E2">102</a> (10 × 10 or 100), the fourth position <a href="1000_(number)" title="1000 (number)">103</a> (10 × 10 × 10 or 1000), and so on.

<a href="Decimal" title="Decimal">Fractional</a> values are indicated by a <a href="Decimal_separator" title="Decimal separator">separator</a>, which can vary in different locations. Usually this separator is a period or <a href="Full_stop" title="Full stop">full stop</a>, or a <a href="Comma_(punctuation)" class="mw-redirect" title="Comma (punctuation)">comma</a>. Digits to the right of it are multiplied by 10 raised to a negative power or exponent. The first position to the right of the separator indicates <a href="1_E-1" class="mw-redirect" title="1 E-1">10−1</a> (0.1), the second position <a href="1_E-2" class="mw-redirect" title="1 E-2">10−2</a> (0.01), and so on for each successive position.

As an example, the number 2674 in a base-10 numeral system is:

or

<div class="mw-heading mw-heading3"><h3 id="Sexagesimal_system">Sexagesimal system</h3></div>

The <a href="Sexagesimal" title="Sexagesimal">sexagesimal</a> or base-60 system was used for the integral and fractional portions of <a href="Babylonian_numerals" class="mw-redirect" title="Babylonian numerals">Babylonian numerals</a> and other Mesopotamian systems, by <a href="Hellenistic" class="mw-redirect" title="Hellenistic">Hellenistic</a> astronomers using <a href="Greek_numerals" title="Greek numerals">Greek numerals</a> for the fractional portion only, and is still used for modern time and angles, but only for minutes and seconds. However, not all of these uses were positional.

Modern time separates each position by a colon or a <a href="Prime_(symbol)" title="Prime (symbol)">prime symbol</a>. For example, the time might be 10:25:59 (10 hours 25 minutes 59 seconds). Angles use similar notation. For example, an angle might be 10°‍25′‍59″ (10 <a href="Degree_(angle)" title="Degree (angle)">degrees</a> 25 <a href="Minute_(angle)" class="mw-redirect" title="Minute (angle)">minutes</a> 59 <a href="Second_(angle)" class="mw-redirect" title="Second (angle)">seconds</a>). In both cases, only minutes and seconds use sexagesimal notation—angular degrees can be larger than 59 (one rotation around a circle is 360°, two rotations are 720°, etc.), and both time and angles use decimal fractions of a second. This contrasts with the numbers used by Hellenistic and <a href="Renaissance" title="Renaissance">Renaissance</a> astronomers, who used <a href="Third_(angle)" class="mw-redirect" title="Third (angle)">thirds</a>, <a href="Fourth_(angle)" class="mw-redirect" title="Fourth (angle)">fourths</a>, etc. for finer increments. Where we might write 10°‍25′‍59.392″, they would have written 10°‍25′‍59′′‍23′′′‍31′′′′‍12′′′′′ or 10°‍25<style data-mw-deduplicate="TemplateStyles:r920966791">

/* start https://en.wikipedia.org/ */

.mw-parser-output span.smallcaps{font-variant:small-caps}.mw-parser-output span.smallcaps-smaller{font-size:85%}

/* end https://en.wikipedia.org/ */

</style>i‍59ii‍23iii‍31iv‍12v.

Using a digit set of digits with upper and lowercase letters allows short notation for sexagesimal numbers, e.g. 10:25:59 becomes 'ARz' (by omitting I and O, but not i and o), which is useful for use in URLs, etc., but it is not very intelligible to humans.

In the 1930s, <a href="Otto_Neugebauer" class="mw-redirect" title="Otto Neugebauer">Otto Neugebauer</a> introduced a modern notational system for Babylonian and Hellenistic numbers that substitutes modern decimal notation from 0 to 59 in each position, while using a semicolon (;) to separate the integral and fractional portions of the number and using a comma (,) to separate the positions within each portion.<a href="#cite_note-20">[20]</a> For example, the mean <a href="Synodic_month" class="mw-redirect" title="Synodic month">synodic month</a> used by both Babylonian and Hellenistic astronomers and still used in the <a href="Hebrew_calendar" title="Hebrew calendar">Hebrew calendar</a> is 29;31,50,8,20 days, and the angle used in the example above would be written 10;25,59,23,31,12 degrees.

<div class="mw-heading mw-heading3"><h3 id="Computing">Computing</h3></div>

In <a href="Computing" title="Computing">computing</a>, the <a href="Binary_numeral_system" class="mw-redirect" title="Binary numeral system">binary</a> (base-2), octal (base-8) and <a href="Hexadecimal" title="Hexadecimal">hexadecimal</a> (base-16) bases are most commonly used. Computers, at the most basic level, deal only with sequences of conventional zeroes and ones, thus it is easier in this sense to deal with powers of two. The hexadecimal system is used as "shorthand" for binary—every 4 binary digits (bits) relate to one and only one hexadecimal digit. In hexadecimal, the six digits after 9 are denoted by A, B, C, D, E, and F (and sometimes a, b, c, d, e, and f).

The <a href="Octal" title="Octal">octal</a> numbering system is also used as another way to represent binary numbers. In this case the base is 8 and therefore only digits 0, 1, 2, 3, 4, 5, 6, and 7 are used. When converting from binary to octal every 3 bits relate to one and only one octal digit.

Hexadecimal, decimal, octal, and a wide variety of other bases have been used for <a href="Binary-to-text_encoding" title="Binary-to-text encoding">binary-to-text encoding</a>, implementations of <a href="Arbitrary-precision_arithmetic" title="Arbitrary-precision arithmetic">arbitrary-precision arithmetic</a>, and other applications.

For a list of bases and their applications, see <a href="List_of_numeral_systems" title="List of numeral systems">list of numeral systems</a>.

<div class="mw-heading mw-heading3"><h3 id="Other_bases_in_human_language">Other bases in human language</h3></div>

Base-12 systems (<a href="Duodecimal" title="Duodecimal">duodecimal</a> or dozenal) have been popular because multiplication and division are easier than in base-10, with addition and subtraction being just as easy. Twelve is a useful base because it has many <a href="Divisor" title="Divisor">factors</a>. It is the smallest common multiple of one, two, three, four and six. There is still a special word for "dozen" in English, and by analogy with the word for 102, hundred, commerce developed a word for 122, gross. The standard 12-hour clock and common use of 12 in English units emphasize the utility of the base. In addition, prior to its conversion to decimal, the old British currency <a href="Pound_Sterling" class="mw-redirect" title="Pound Sterling">Pound Sterling</a> (GBP) partially used base-12; there were 12 pence (d) in a shilling (s), 20 shillings in a pound (£), and therefore 240 pence in a pound. Hence the term LSD or, more properly, <a href="%C2%A3sd" title="£sd">£sd</a>.

The <a href="Maya_numerals" title="Maya numerals">Maya civilization</a> and other civilizations of <a href="Pre-Columbian" class="mw-redirect" title="Pre-Columbian">pre-Columbian</a> <a href="Mesoamerica" title="Mesoamerica">Mesoamerica</a> used base-20 (<a href="Vigesimal" title="Vigesimal">vigesimal</a>), as did several North American tribes (two being in southern California). Evidence of base-20 counting systems is also found in the languages of central and western <a href="Africa" title="Africa">Africa</a>.

Remnants of a <a href="Gaulish_language" class="mw-redirect" title="Gaulish language">Gaulish</a> base-20 system also exist in French, as seen today in the names of the numbers from 60 through 99. For example, sixty-five is soixante-cinq (literally, "sixty [and] five"), while seventy-five is soixante-quinze (literally, "sixty [and] fifteen"). Furthermore, for any number between 80 and 99, the "tens-column" number is expressed as a multiple of twenty. For example, eighty-two is quatre-vingt-deux (literally, four twenty[s] [and] two), while ninety-two is quatre-vingt-douze (literally, four twenty[s] [and] twelve). In Old French, forty was expressed as two twenties and sixty was three twenties, so that fifty-three was expressed as two twenties [and] thirteen, and so on.

In English the same base-20 counting appears in the use of "<a href="20_(number)" title="20 (number)">scores</a>". Although mostly historical, it is occasionally used colloquially. Verse 10 of Psalm 90 in the King James Version of the Bible starts: "The days of our years are threescore years and ten; and if by reason of strength they be fourscore years, yet is their strength labour and sorrow". The Gettysburg Address starts: "Four score and seven years ago".

The <a href="Irish_language" title="Irish language">Irish language</a> also used base-20 in the past, twenty being fichid, forty dhá fhichid, sixty trí fhichid and eighty ceithre fhichid. A remnant of this system may be seen in the modern word for 40, daoichead.

The <a href="Welsh_language" title="Welsh language">Welsh language</a> continues to use a <a href="Vigesimal" title="Vigesimal">base-20</a> <a href="Welsh_language#Counting_system" title="Welsh language">counting system</a>, particularly for the age of people, dates and in common phrases. 15 is also important, with 16–19 being "one on 15", "two on 15" etc. 18 is normally "two nines". A decimal system is commonly used.

The <a href="Inuit_languages" title="Inuit languages">Inuit languages</a> use a <a href="Base-20" class="mw-redirect" title="Base-20">base-20</a> counting system. Students from <a href="Kaktovik%2C_Alaska" title="Kaktovik, Alaska">Kaktovik, Alaska</a> invented a <a href="Kaktovik_numerals" title="Kaktovik numerals">base-20 numeral system</a> in 1994<a href="#cite_note-kakt-21">[21]</a>

<a href="Danish_language#Numerals" title="Danish language">Danish numerals</a> display a similar <a href="Vigesimal" title="Vigesimal">base-20</a> structure.

The <a href="M%C4%81ori_language" title="Māori language">Māori language</a> of New Zealand also has evidence of an underlying base-20 system as seen in the terms Te Hokowhitu a Tu referring to a war party (literally "the seven 20s of Tu") and Tama-hokotahi, referring to a great warrior ("the one man equal to 20").

<a href="Binary_numeral_system" class="mw-redirect" title="Binary numeral system">The binary system</a> was used in the Egyptian Old Kingdom, 3000 BC to 2050 BC. It was cursive by rounding off rational numbers smaller than 1 to 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64, with a 1/64 term thrown away (the system was called the <a href="Eye_of_Horus#Mathematics" title="Eye of Horus">Eye of Horus</a>).

A number of <a href="Australian_Aboriginal_languages" title="Australian Aboriginal languages">Australian Aboriginal languages</a> employ binary or binary-like counting systems. For example, in <a href="Kala_Lagaw_Ya" class="mw-redirect" title="Kala Lagaw Ya">Kala Lagaw Ya</a>, the numbers one through six are urapon, ukasar, ukasar-urapon, ukasar-ukasar, ukasar-ukasar-urapon, ukasar-ukasar-ukasar.

North and Central American natives used base-4 (<a href="Quaternary_numeral_system" title="Quaternary numeral system">quaternary</a>) to represent the four cardinal directions. Mesoamericans tended to add a second base-5 system to create a modified base-20 system.

A base-5 system (<a href="Quinary" title="Quinary">quinary</a>) has been used in many cultures for counting. Plainly it is based on the number of digits on a human hand. It may also be regarded as a sub-base of other bases, such as base-10, base-20, and base-60.

A base-8 system (<a href="Octal" title="Octal">octal</a>) was devised by the <a href="Yuki_tribe" class="mw-redirect" title="Yuki tribe">Yuki tribe</a> of Northern California, who used the spaces between the fingers to count, corresponding to the digits one through eight.<a href="#cite_note-22">[22]</a> There is also linguistic evidence which suggests that the Bronze Age <a href="Proto-Indo_European" class="mw-redirect" title="Proto-Indo European">Proto-Indo Europeans</a> (from whom most European and Indic languages descend) might have replaced a base-8 system (or a system which could only count up to 8) with a base-10 system. The evidence is that the word for 9, newm, is suggested by some to derive from the word for "new", newo-, suggesting that the number 9 had been recently invented and called the "new number".<a href="#cite_note-23">[23]</a>

Many ancient counting systems use five as a primary base, almost surely coming from the number of fingers on a person's hand. Often these systems are supplemented with a secondary base, sometimes ten, sometimes twenty. In some <a href="African_languages" class="mw-redirect" title="African languages">African languages</a> the word for five is the same as "hand" or "fist" (<a href="Dyola_language" class="mw-redirect" title="Dyola language">Dyola language</a> of <a href="Guinea-Bissau" title="Guinea-Bissau">Guinea-Bissau</a>, <a href="Banda_languages" title="Banda languages">Banda language</a> of <a href="Central_Africa" title="Central Africa">Central Africa</a>). Counting continues by adding 1, 2, 3, or 4 to combinations of 5, until the secondary base is reached. In the case of twenty, this word often means "man complete". This system is referred to as quinquavigesimal. It is found in many languages of the <a href="Sudan" title="Sudan">Sudan</a> region.

The <a href="Telefol_language" title="Telefol language">Telefol language</a>, spoken in <a href="Papua_New_Guinea" title="Papua New Guinea">Papua New Guinea</a>, is notable for possessing a base-27 numeral system.

<div class="mw-heading mw-heading2"><h2 id="Non-standard_positional_numeral_systems">Non-standard positional numeral systems</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Non-standard_positional_numeral_systems" title="Non-standard positional numeral systems">Non-standard positional numeral systems</a></div>

Interesting properties exist when the base is not fixed or positive and when the digit symbol sets denote negative values. There are many more variations. These systems are of practical and theoretic value to computer scientists.

<a href="Balanced_ternary" title="Balanced ternary">Balanced ternary</a><a href="#cite_note-24">[24]</a> uses a base of 3 but the digit set is {1,0,1} instead of {0,1,2}. The "1" has an equivalent value of −1. The negation of a number is easily formed by switching the    on the 1s. This system can be used to solve the balance problem, which requires finding a minimal set of known counter-weights to determine an unknown weight. Weights of 1, 3, 9, ..., 3n known units can be used to determine any unknown weight up to 1 + 3 + ... + 3n units. A weight can be used on either side of the balance or not at all. Weights used on the balance pan with the unknown weight are designated with 1, with 1 if used on the empty pan, and with 0 if not used. If an unknown weight W is balanced with 3 (31) on its pan and 1 and 27 (30 and 33) on the other, then its weight in decimal is 25 or 1011 in balanced base-3.

The <a href="Factorial_number_system" title="Factorial number system">factorial number system</a> uses a varying radix, giving <a href="Factorial" title="Factorial">factorials</a> as place values; they are related to <a href="Chinese_remainder_theorem" title="Chinese remainder theorem">Chinese remainder theorem</a> and <a href="Residue_number_system" title="Residue number system">residue number system</a> enumerations. This system effectively enumerates permutations. A derivative of this uses the <a href="Towers_of_Hanoi" class="mw-redirect" title="Towers of Hanoi">Towers of Hanoi</a> puzzle configuration as a counting system. The configuration of the towers can be put into 1-to-1 correspondence with the decimal count of the step at which the configuration occurs and vice versa.

<th align="left">Decimal equivalents

</th>

<td width="6%">−3

</td>

<td width="6%">−2

</td>

<td width="6%">−1

</td>

<td width="6%">0

</td>

<td width="6%">1

</td>

<td width="6%">2

</td>

<td width="6%">3

</td>

<td width="6%">4

</td>

<td width="6%">5

</td>

<td width="8%">6

</td>

<td width="8%">7

</td>

<td width="8%">8

</td></tr>

<tr>

<th align="left">Balanced base 3

</th>

<td>10

</td>

<td>11

</td>

<td>1

</td>

<td>0

</td>

<td>1

</td>

<td>11

</td>

<td>10

</td>

<td>11

</td>

<td>111

</td>

<td>110

</td>

<td>111

</td>

<td>101

</td></tr>

<tr>

<th align="left">Base −2

</th>

<td>1101

</td>

<td>10

</td>

<td>11

</td>

<td>0

</td>

<td>1

</td>

<td>110

</td>

<td>111

</td>

<td>100

</td>

<td>101

</td>

<td>11010

</td>

<td>11011

</td>

<td>11000

</td></tr>

<tr>

<th align="left">Factoroid

</th>

<td>1100

</td></tr></tbody></table>

<div class="mw-heading mw-heading2"><h2 id="Non-positional_positions">Non-positional positions</h2></div>

Each position does not need to be positional itself. <a href="Babylonian_cuneiform_numerals" title="Babylonian cuneiform numerals">Babylonian sexagesimal numerals</a> were positional, but in each position were groups of two kinds of wedges representing ones and tens (a narrow vertical wedge | for the one and an open left pointing wedge ⟨ for the ten) — up to 5+9=14 symbols per position (i.e. 5 tens ⟨⟨⟨⟨⟨ and 9 ones ||||||||| grouped into one or two near squares containing up to three tiers of symbols, or a place holder (⑊) for the lack of a position).<a href="#cite_note-25">[25]</a> Hellenistic astronomers used one or two alphabetic Greek numerals for each position (one chosen from 5 letters representing 10–50 and/or one chosen from 9 letters representing 1–9, or a <a href="Greek_numerals#Hellenistic_zero" title="Greek numerals">zero symbol</a>).<a href="#cite_note-26">[26]</a>

Examples:

<ul><li><a href="List_of_numeral_systems" title="List of numeral systems">List of numeral systems</a></li>

<li>Category: Positional numeral systems</li></ul>

Related topics:

<ul><li><a href="Algorism" title="Algorism">Algorism</a></li>

<li><a href="Hindu%E2%80%93Arabic_numeral_system" title="Hindu–Arabic numeral system">Hindu–Arabic numeral system</a></li>

<li><a href="Mixed_radix" title="Mixed radix">Mixed radix</a></li>

<li><a href="Non-standard_positional_numeral_systems" title="Non-standard positional numeral systems">Non-standard positional numeral systems</a></li>

<li><a href="Scientific_notation" title="Scientific notation">Scientific notation</a></li></ul>

Other:

<ul><li><a href="Significant_figures" title="Significant figures">Significant figures</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

/* start https://en.wikipedia.org/ */

.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}

/* end https://en.wikipedia.org/ */

</style><div class="reflist">

/* start https://en.wikipedia.org/ */

.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("./mw/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("./mw/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("./mw/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("./mw/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}}

/* end https://en.wikipedia.org/ */

</style><cite id="CITEREFKaplan2000" class="citation book cs1">Kaplan, Robert (2000). <a rel="nofollow" class="external text" href="https://archive.org/details/nothingthatisnat00kapl">The Nothing That Is: A Natural History of Zero</a>. Oxford: Oxford University Press. pp. 11–12 – via archive.org.</cite>

</li>

<li id="cite_note-Greek_numerals-2"><a href="#cite_ref-Greek_numerals_2-0">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20161126013536/http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Greek_numbers.html">"Greek numerals"</a>. Archived from <a rel="nofollow" class="external text" href="http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Greek_numbers.html">the original</a> on 26 November 2016. Retrieved 31 May 2016.</cite>

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> <a href="Karl_Menninger_(mathematics)" title="Karl Menninger (mathematics)">Menninger, Karl</a>: Zahlwort und Ziffer. Eine Kulturgeschichte der Zahl, Vandenhoeck und Ruprecht, 3rd. ed., 1979, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>3-525-40725-4</bdi>, pp. 150–153

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> Ifrah, page 187

</li>

<li id="cite_note-5"><a href="#cite_ref-5">^</a> L. F. Menabrea.

Translated by Ada Augusta, Countess of Lovelace.

<a rel="nofollow" class="external text" href="http://www.fourmilab.ch/babbage/sketch.html">"Sketch of The Analytical Engine Invented by Charles Babbage"</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20080915134651/http://www.fourmilab.ch/babbage/sketch.html">Archived</a> 15 September 2008 at the <a href="Wayback_Machine" title="Wayback Machine">Wayback Machine</a>.

1842.

</li>

<li id="cite_note-6"><a href="#cite_ref-6">^</a> <a href="Lam_Lay_Yong" title="Lam Lay Yong">Lam Lay Yong</a>, "The Development of Hindu-Arabic and Traditional Chinese Arithmetic", Chinese Science, 1996 p38, Kurt Vogel notation

</li>

<li id="cite_note-jnfractn1-7"><a href="#cite_ref-jnfractn1_7-0">^</a> <cite id="CITEREFJoseph_Needham1959" class="citation book cs1"><a href="Joseph_Needham" title="Joseph Needham">Joseph Needham</a> (1959). "Decimal System". <a href="Science_and_Civilisation_in_China" title="Science and Civilisation in China">Science and Civilisation in China, Volume III, Mathematics and the Sciences of the Heavens and the Earth</a>. Cambridge University Press.</cite>

</li>

<li id="cite_note-Berggren-8">^ <a href="#cite_ref-Berggren_8-0">a</a> <a href="#cite_ref-Berggren_8-1">b</a> <cite id="CITEREFBerggren2007" class="citation book cs1">Berggren, J. Lennart (2007). "Mathematics in Medieval Islam". The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook. Princeton University Press. p. 518. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-691-11485-9</bdi>.</cite>

</li>

<li id="cite_note-9"><a href="#cite_ref-9">^</a> <a href="Solomon_Gandz" title="Solomon Gandz">Gandz, S.</a>: The invention of the decimal fractions and the application of the exponential calculus by Immanuel Bonfils of Tarascon (c. 1350), Isis 25 (1936), 16–45.

</li>

<li id="cite_note-Lam-10">^ <a href="#cite_ref-Lam_10-0">a</a> <a href="#cite_ref-Lam_10-1">b</a> <a href="Lam_Lay_Yong" title="Lam Lay Yong">Lam Lay Yong</a>, "The Development of Hindu-Arabic and Traditional Chinese Arithmetic", Chinese Science, 1996, p. 38, Kurt Vogel notation

</li>

<li id="cite_note-11"><a href="#cite_ref-11">^</a> <cite id="CITEREFLay_Yong" class="citation journal cs1"><a href="Lam_Lay_Yong" title="Lam Lay Yong">Lay Yong, Lam</a>. "A Chinese Genesis, Rewriting the history of our numeral system". Archive for History of Exact Sciences. 38: 101–108.</cite>

</li>

<li id="cite_note-van-12"><a href="#cite_ref-van_12-0">^</a> <cite id="CITEREFB._L._van_der_Waerden1985" class="citation book cs1"><a href="Bartel_Leendert_van_der_Waerden" title="Bartel Leendert van der Waerden">B. L. van der Waerden</a> (1985). <a rel="nofollow" class="external text" href="https://archive.org/details/historyofalgebra0000waer">A History of Algebra. From Khwarizmi to Emmy Noether</a>. Berlin: Springer-Verlag.</cite>

</li>

<li id="cite_note-EJD-13">^ <a href="#cite_ref-EJD_13-0">a</a> <a href="#cite_ref-EJD_13-1">b</a> <a href="E._J._Dijksterhuis" class="mw-redirect" title="E. J. Dijksterhuis">E. J. Dijksterhuis</a> (1970) Simon Stevin: Science in the Netherlands around 1600, <a href="Martinus_Nijhoff_Publishers" class="mw-redirect" title="Martinus Nijhoff Publishers">Martinus Nijhoff Publishers</a>, Dutch original 1943

</li>

<li id="cite_note-14"><a href="#cite_ref-14">^</a> The digit will retain its meaning in other number bases, in general, because a higher number base would normally be a notational extension of the lower number base in any systematic organization. In the <a href="Mathematical_science" class="mw-redirect" title="Mathematical science">mathematical sciences</a> there is virtually only one positional-notation numeral system for each base below 10, and this extends with few, if insignificant, variations on the choice of alphabetic digits for those bases above 10.

</li>

<li id="cite_note-15"><a href="#cite_ref-15">^</a> We do not usually remove the lowercase digits "l" and lowercase "o", for in most fonts they are discernible from the digits "1" and "0".

</li>

<cite id="CITEREFCollinsMignotteWinkler1983" class="citation book cs1">Collins, G. E.; Mignotte, M.; Winkler, F. (1983). <a rel="nofollow" class="external text" href="https://www3.risc.jku.at/publications/download/risc_229/paper_55.pdf">"Arithmetic in basic algebraic domains"</a> (PDF). In Buchberger, Bruno; Collins, George Edwin; Loos, Rüdiger; Albrecht, Rudolf (eds.). Computer Algebra: Symbolic and Algebraic Computation. Computing Supplementa. Vol. 4. Vienna: Springer. pp. 189–220. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-7091-7551-4_13">10.1007/978-3-7091-7551-4_13</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>3-211-81776-X</bdi>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0728973">0728973</a>.</cite>

</li>

<li id="cite_note-17"><a href="#cite_ref-17">^</a> The exact size of the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu _{1},\ldots ,\nu _{n}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \nu _{1},\ldots ,\nu _{n}}</annotation>

</semantics>

</math><img src="./3ae7d48dd2a85c83184498787c56b8a8c413e4fb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.748ex; height:2.009ex;" alt="{\displaystyle \nu _{1},\ldots ,\nu _{n}}" loading="lazy"> does not matter. They only have to be ≥ 1.

</li>

<li id="cite_note-18"><a href="#cite_ref-18">^</a> <cite id="CITEREFWeisstein" class="citation web cs1">Weisstein, Eric W. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Vinculum.html">"Vinculum"</a>. mathworld.wolfram.com. Retrieved 22 August 2024.</cite>

</li>

<li id="cite_note-19"><a href="#cite_ref-19">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://flamath.com/en/irrational-numbers#:~:text=The%20main%20characteristic%20of%20irrational%20numbers%20is%20that,number%20where%20a%20group%20of%20digits%20repeats%20constantly.">"Irrational Numbers: Definition, Examples and Properties"</a>. flamath.com. 10 April 2024. Retrieved 22 August 2024.</cite>

</li>

<li id="cite_note-20"><a href="#cite_ref-20">^</a> <cite id="CITEREFNeugebauerSachsGötze1945" class="citation cs2"><a href="Otto_Neugebauer" class="mw-redirect" title="Otto Neugebauer">Neugebauer, Otto</a>; <a href="Abraham_Sachs" title="Abraham Sachs">Sachs, Abraham Joseph</a>; Götze, Albrecht (1945), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=i-juAAAAMAAJ&q=%22sexagesimal+notation%22&pg=PA1">Mathematical Cuneiform Texts</a>, American Oriental Series, vol. 29, New Haven: American Oriental Society and the American Schools of Oriental Research, p. 2, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780940490291</bdi>, <a rel="nofollow" class="external text" href="https://web.archive.org/web/20161001222131/https://books.google.com/books?hl=en&lr=&id=i-juAAAAMAAJ&oi=fnd&pg=PA1&dq=%22sexagesimal+notation%22&ots=nkmW8KTnQ_&sig=U_K4iOoKy5Xf70UrbjoTyS3hN2A#v=onepage&q=%22sexagesimal%20notation%22&f=false">archived</a> from the original on 1 October 2016, retrieved 18 September 2019</cite> <code class="cs1-code">{{citation}}</code>: ISBN / Date incompatibility (help)

</li>

<li id="cite_note-kakt-21"><a href="#cite_ref-kakt_21-0">^</a> <cite id="CITEREFBartley1997" class="citation journal cs1">Bartley, Wm. Clark (January–February 1997). <a rel="nofollow" class="external text" href="http://www.ankn.uaf.edu/sop/SOPv2i1.pdf">"Making the Old Way Count"</a> (PDF). Sharing Our Pathways. 2 (1): 12–13. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20130625225547/http://ankn.uaf.edu/SOP/SOPv2i1.pdf">Archived</a> (PDF) from the original on 25 June 2013. Retrieved 27 February 2017.</cite>

</li>

<li id="cite_note-22"><a href="#cite_ref-22">^</a> <cite id="CITEREFBarrow1992" class="citation cs2">Barrow, John D. (1992), <a rel="nofollow" class="external text" href="https://archive.org/details/piinskycounting00barr">Pi in the sky: counting, thinking, and being</a>, Clarendon Press, p. 38, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780198539568</bdi></cite>.

</li>

<li id="cite_note-23"><a href="#cite_ref-23">^</a> (Mallory & Adams 1997) <a href="Encyclopedia_of_Indo-European_Culture" title="Encyclopedia of Indo-European Culture">Encyclopedia of Indo-European Culture</a>

</li>

<li id="cite_note-24"><a href="#cite_ref-24">^</a> <a href="#Knuth">Knuth</a>, pages 195–213

</li>

<li id="cite_note-25"><a href="#cite_ref-25">^</a> Ifrah, pages 326, 379

</li>

<li id="cite_note-26"><a href="#cite_ref-26">^</a> Ifrah, pages 261–264

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

<ul><li><cite id="CITEREFO'ConnorRobertson2000" class="citation web cs1">O'Connor, John; Robertson, Edmund (December 2000). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20140911192557/http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Babylonian_numerals.html">"Babylonian Numerals"</a>. Archived from <a rel="nofollow" class="external text" href="http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Babylonian_numerals.html">the original</a> on 11 September 2014. Retrieved 21 August 2010.</cite></li>

<li><cite id="CITEREFKadvany2007" class="citation journal cs1">Kadvany, John (December 2007). "Positional Value and Linguistic Recursion". Journal of Indian Philosophy. 35 (5–6): 487–520. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs10781-007-9025-5">10.1007/s10781-007-9025-5</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:52885600">52885600</a>.</cite></li>

<li><cite id="CITEREFKnuth1997" class="citation book cs1"><a href="Donald_Knuth" title="Donald Knuth">Knuth, Donald</a> (1997). The art of Computer Programming. Vol. 2. Addison-Wesley. pp. 195–213. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-201-89684-2</bdi>.</cite></li>

<li><cite id="CITEREFIfrah2000" class="citation book cs1">Ifrah, George (2000). <a rel="nofollow" class="external text" href="https://archive.org/details/unset0000unse_w3q2">The Universal History of Numbers: From Prehistory to the Invention of the Computer</a>. Wiley. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-471-37568-3</bdi>.</cite></li>

<li><cite id="CITEREFKroeber1976" class="citation book cs1"><a href="Alfred_Kroeber" title="Alfred Kroeber">Kroeber, Alfred</a> (1976) [1925]. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Plqf_OTz4ukC">Handbook of the Indians of California</a>. Courier Dover Publications. p. 176. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780486233680</bdi>.</cite></li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

/* start https://en.wikipedia.org/ */

.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 .infobox .side-box{font-size:100%}.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}}

/* end https://en.wikipedia.org/ */

</style><style data-mw-deduplicate="TemplateStyles:r1237033735">

/* start https://en.wikipedia.org/ */

@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-en-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-en-v2.svg"]{background-color:white}}

/* end https://en.wikipedia.org/ */

</style><div class="side-box side-box-right sistersitebox"><style data-mw-deduplicate="TemplateStyles:r1126788409">

/* start https://en.wikipedia.org/ */

.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}

/* end https://en.wikipedia.org/ */

</style>

<div class="side-box-text plainlist">Wikimedia Commons has media related to <a href="https://commons.wikimedia.org/wiki/Category:Positional_numeral_systems" class="extiw external" title="commons:Category:Positional numeral systems">Positional numeral systems</a>.</div></div>

</div>

<ul><li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20170204004954/http://ultrastudio.org/en/MechengburakalkanApplet-1.7.zip">Accurate Base Conversion</a></li>

<li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20160310032143/http://ibrarian.net/navon/paper/the_development_of_hindu_arabic_and_traditional_c.pdf?paperid=1247217">The Development of Hindu Arabic and Traditional Chinese Arithmetics</a></li>

<li><a rel="nofollow" class="external text" href="http://www.cut-the-knot.org/recurrence/conversion.shtml">Implementation of Base Conversion</a> at <a href="Cut-the-knot" class="mw-redirect" title="Cut-the-knot">cut-the-knot</a></li>

<li><a rel="nofollow" class="external text" href="http://www.intuitor.com/counting/">Learn to count other bases on your fingers</a></li>

<li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20161109022004/http://thedevtoolkit.com/tools/base_conversion">Online Arbitrary Precision Base Converter</a></li></ul>

</div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-08-01" href="https://en.wikipedia.org/wiki/?title=Positional_notation&oldid=1303697942">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

</div>

</div></div>

</div>

</main>

</div>

</body></html>