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<title>Bijective numeration</title>

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</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="Positional_notation" title="Positional notation">Place-value notation</a></div></div><div class="sidebar-list-content mw-collapsible-content"><table class="sidebar-subgroup"><tbody><tr><td class="sidebar-content hlist">

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

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

<li><a href="Mixed_radix" title="Mixed radix">Mixed</a><span class="nowrap"> </span>(<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><span class="nowrap"> </span>(<a href="Quater-imaginary_base" title="Quater-imaginary base">2<i>i</i></a>)</li>

<li><a href="Non-integer_base_of_numeration" title="Non-integer base of numeration">Non-integer</a><span class="nowrap"> </span>(<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">

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<p><b>Bijective numeration</b> is any <a href="Numeral_system" title="Numeral system">numeral system</a> in which every non-negative <a href="Integer" title="Integer">integer</a> can be represented in exactly one way using a finite <a href="String_of_digits" class="mw-redirect" title="String of digits">string of digits</a>. The name refers to the <a href="Bijection" title="Bijection">bijection</a> (i.e. one-to-one correspondence) that exists in this case between the set of non-negative integers and the set of finite strings using a finite set of symbols (the "digits").

</p><p>Most ordinary numeral systems, such as the common <a href="Decimal" title="Decimal">decimal</a> system, are not bijective because more than one string of digits can represent the same positive integer. In particular, adding <a href="Leading_zero" title="Leading zero">leading zeroes</a> does not change the value represented, so "1", "01" and "001" all represent the number <a href="1_(number)" class="mw-redirect" title="1 (number)">one</a>. Even though only the first is usual, the fact that the others are possible means that the decimal system is not bijective. However, the <a href="Unary_numeral_system" title="Unary numeral system">unary numeral system</a>, with only one digit, <i>is</i> bijective.

</p><p>A <b>bijective <a href="Radix" title="Radix">base</a>-<i>k</i> numeration</b> is a bijective <a href="Positional_notation" title="Positional notation">positional notation</a>. It uses a string of digits from the set {1, 2, ..., <i>k</i>} (where <i>k</i> ≥ 1) to encode each positive integer; a digit's position in the string defines its value as a multiple of a power of <i>k</i>. <a href="#CITEREFSmullyan1961">Smullyan (1961)</a> calls this notation <i>k</i>-adic, but it should not be confused with the <a href="P-adic_number" title="P-adic number"><i>p</i>-adic numbers</a>: bijective numerals are a system for representing ordinary <a href="Integer" title="Integer">integers</a> by finite strings of nonzero digits, whereas the <i>p</i>-adic numbers are a system of mathematical values that contain the integers as a subset and may need infinite sequences of digits in any numerical representation.

</p>

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

<p>The <a href="Radix" title="Radix">base-<i>k</i></a> bijective numeration system uses the digit-set {1, 2, ..., <i>k</i>} (<i>k</i> ≥ 1) to uniquely represent every non-negative integer, as follows:

</p>

<ul><li>The integer zero is represented by the <i><a href="Empty_string" title="Empty string">empty string</a></i>.</li>

<li>The integer represented by the nonempty digit-string</li></ul>

<dd>is

<ul><li>The digit-string representing the integer <i>m</i> > 0 is</li></ul>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}a_{0}&=m-q_{0}k,&q_{0}&=f\left({\frac {m}{k}}\right)&\\a_{1}&=q_{0}-q_{1}k,&q_{1}&=f\left({\frac {q_{0}}{k}}\right)&\\a_{2}&=q_{1}-q_{2}k,&q_{2}&=f\left({\frac {q_{1}}{k}}\right)&\\&\,\,\,\vdots &&\,\,\,\vdots \\a_{n}&=q_{n-1}-0k,&q_{n}&=f\left({\frac {q_{n-1}}{k}}\right)=0\end{aligned}}}</annotation>

</semantics>

</math></span><img src="./0a36bd81384d4da57626f26a5d15acdcf56645c4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -11.505ex; width:45.054ex; height:24.176ex;" alt="{\displaystyle {\begin{aligned}a_{0}&=m-q_{0}k,&q_{0}&=f\left({\frac {m}{k}}\right)&\\a_{1}&=q_{0}-q_{1}k,&q_{1}&=f\left({\frac {q_{0}}{k}}\right)&\\a_{2}&=q_{1}-q_{2}k,&q_{2}&=f\left({\frac {q_{1}}{k}}\right)&\\&\,\,\,\vdots &&\,\,\,\vdots \\a_{n}&=q_{n-1}-0k,&q_{n}&=f\left({\frac {q_{n-1}}{k}}\right)=0\end{aligned}}}" loading="lazy"></span></dd></dl></dd>

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<annotation encoding="application/x-tex">{\displaystyle f(x)=\lceil x\rceil -1,}</annotation>

</semantics>

</math></span><img src="./02bc2b94ca8b0c6e3588bf808b04e7cdbaecc5be.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.56ex; height:2.843ex;" alt="{\displaystyle f(x)=\lceil x\rceil -1,}" loading="lazy"></span></dd></dl></dd></dl>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./5ac7f37c8288700904b4a22a2f7c94d45ba917de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.394ex; height:2.843ex;" alt="{\displaystyle \lceil x\rceil }" loading="lazy"></span> being the least integer not less than <i>x</i> (the <a href="Ceiling_function" class="mw-redirect" title="Ceiling function">ceiling function</a>).</dd></dl>

<p>In contrast, standard <a href="Positional_notation" title="Positional notation">positional notation</a> can be defined with a similar recursive algorithm where

</p>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)=\lfloor x\rfloor ,}</annotation>

</semantics>

</math></span><img src="./190c830a762b949dfa2739040a4a55551338d2ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.557ex; height:2.843ex;" alt="{\displaystyle f(x)=\lfloor x\rfloor ,}" loading="lazy"></span></dd></dl></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Extension_to_integers">Extension to integers</h3></div>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./5cda43bd4034dc2d04cd562005d0af81d3d2dbc6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.472ex; height:2.176ex;" alt="{\displaystyle k>1}" loading="lazy"></span>, the bijective base-<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"></span> numeration system could be extended to negative integers <a href="Radix_complement" class="mw-redirect" title="Radix complement">in the same way as the standard base-<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}">

</mstyle>

</mrow>

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

</semantics>

</math></span><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"></span> numeral system</a> by use of an infinite number of the digit <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{k-1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./74e726983141293b94078583acbb7440815a4c8b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.398ex; height:2.509ex;" alt="{\displaystyle d_{k-1}}" loading="lazy"></span>, where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(d_{k-1})=k-1}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(d_{k-1})=k-1}</annotation>

</semantics>

</math></span><img src="./3afa0162f33798b64661f7a6404115a56f258096.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.799ex; height:2.843ex;" alt="{\displaystyle f(d_{k-1})=k-1}" loading="lazy"></span>, represented as a left-infinite sequence of digits <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ldots d_{k-1}d_{k-1}d_{k-1}={\overline {d_{k-1}}}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mover>

<msub>

</mrow>

</msub>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ldots d_{k-1}d_{k-1}d_{k-1}={\overline {d_{k-1}}}}</annotation>

</semantics>

</math></span><img src="./cb96df2b9e2525c2be01d536c2ae68d0ed0b3e94.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:23.916ex; height:3.343ex;" alt="{\displaystyle \ldots d_{k-1}d_{k-1}d_{k-1}={\overline {d_{k-1}}}}" loading="lazy"></span>. This is because the <a href="Euler_summation" title="Euler summation">Euler summation</a>

</p>

<mover>

<msub>

</mrow>

</msub>

</mover>

</mrow>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g({\overline {d_{k-1}}})=\sum _{i=0}^{\infty }f(d_{k-1})k^{i}=-{\frac {k-1}{k-1}}=-1}</annotation>

</semantics>

</math></span><img src="./0e906a3a53a057ea5cfebb4e164ab2887fa1cca7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:40.802ex; height:6.843ex;" alt="{\displaystyle g({\overline {d_{k-1}}})=\sum _{i=0}^{\infty }f(d_{k-1})k^{i}=-{\frac {k-1}{k-1}}=-1}" loading="lazy"></span></dd></dl>

<p>meaning that

</p>

<mover>

<msub>

</mrow>

</msub>

</mover>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g({\overline {d_{k-1}}}d_{k})=f(d_{k})\sum _{i=1}^{\infty }f(d_{k-1})k^{i}=1+\sum _{i=0}^{\infty }f(d_{k-1})k^{i}=0}</annotation>

</semantics>

</math></span><img src="./2294b8ffa74a905e7d2b9846d4d57c33b4bb806b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:56.447ex; height:6.843ex;" alt="{\displaystyle g({\overline {d_{k-1}}}d_{k})=f(d_{k})\sum _{i=1}^{\infty }f(d_{k-1})k^{i}=1+\sum _{i=0}^{\infty }f(d_{k-1})k^{i}=0}" loading="lazy"></span></dd></dl>

<p>and for every positive number <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span> with bijective numeration digit representation <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}">

</mstyle>

</mrow>

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

</semantics>

</math></span><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"></span> is represented by <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {d_{k-1}}}d_{k}d}">

<mover>

<msub>

</mrow>

</msub>

</mover>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\overline {d_{k-1}}}d_{k}d}</annotation>

</semantics>

</math></span><img src="./b037be41cc7eb46fa99d776aef7be7266524feb6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.027ex; height:3.343ex;" alt="{\displaystyle {\overline {d_{k-1}}}d_{k}d}" loading="lazy"></span>. For base <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k>2}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./fa7a4bb0e17a911cb4b3f6c4e455be472a295299.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.472ex; height:2.176ex;" alt="{\displaystyle k>2}" loading="lazy"></span>, negative numbers <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n<-1}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./68a1619773e5c76afa9368ac32eca2dfd0a5b52a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.464ex; height:2.343ex;" alt="{\displaystyle n<-1}" loading="lazy"></span> are represented by <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {d_{k-1}}}d_{i}d}">

<mover>

<msub>

</mrow>

</msub>

</mover>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\overline {d_{k-1}}}d_{i}d}</annotation>

</semantics>

</math></span><img src="./36864e6421f9fa055d74869e3b18c4a882d2c4cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.738ex; height:3.343ex;" alt="{\displaystyle {\overline {d_{k-1}}}d_{i}d}" loading="lazy"></span> with <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i<k-1}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./3076fe55d2c7771614e4b1c1be4d865246383b63.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.115ex; height:2.343ex;" alt="{\displaystyle i<k-1}" loading="lazy"></span>, while for base <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=2}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./0bd301789e1f25a3da4be297ff637754ebee5f5d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.472ex; height:2.176ex;" alt="{\displaystyle k=2}" loading="lazy"></span>, negative numbers <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n<-1}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./68a1619773e5c76afa9368ac32eca2dfd0a5b52a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.464ex; height:2.343ex;" alt="{\displaystyle n<-1}" loading="lazy"></span> are represented by <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {d_{k}}}d}">

<mover>

<msub>

</mrow>

</msub>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./f652440bf1cf79c3476b0bf59b6616faca88022c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.628ex; height:3.343ex;" alt="{\displaystyle {\overline {d_{k}}}d}" loading="lazy"></span>. This is similar to how in <a href="Signed-digit_representation" title="Signed-digit representation">signed-digit representations</a>, all integers <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span> with digit representations <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}">

</mstyle>

</mrow>

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

</semantics>

</math></span><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"></span> are represented as <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {d_{0}}}d}">

<mover>

<msub>

</mrow>

</msub>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\overline {d_{0}}}d}</annotation>

</semantics>

</math></span><img src="./6f6c490ff9aa6ac2eabe1e7a1bf0247b0d1b8471.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.594ex; height:3.343ex;" alt="{\displaystyle {\overline {d_{0}}}d}" loading="lazy"></span> where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(d_{0})=0}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(d_{0})=0}</annotation>

</semantics>

</math></span><img src="./98ddd9468b388f31154c4b346d1035493de6357e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.612ex; height:2.843ex;" alt="{\displaystyle f(d_{0})=0}" loading="lazy"></span>. This representation is no longer bijective, as the entire set of left-infinite sequences of digits is used to represent the <a href="P-adic_integer" class="mw-redirect" title="P-adic integer"><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"></span>-adic integers</a>, of which the integers are only a subset.

</p>

<div class="mw-heading mw-heading2"><h2 id="Properties_of_bijective_base-k_numerals">Properties of bijective base-<i>k</i> numerals</h2></div>

<p>For a given base <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\geq 2}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./c797a67c0a51167d373c013a9a020f4568a11754.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.472ex; height:2.343ex;" alt="{\displaystyle k\geq 2}" loading="lazy"></span>,

</p>

<ul><li>the number of digits in the bijective base-<i>k</i> numeral representing a nonnegative integer <i>n</i> is

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lfloor \log _{k}((n+1)(k-1))\rfloor }</annotation>

</semantics>

</math></span><img src="./889fa4606cc8d6a53ea1fc785760002f42fc47d2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.165ex; height:2.843ex;" alt="{\displaystyle \lfloor \log _{k}((n+1)(k-1))\rfloor }" loading="lazy"></span></a>,<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> in contrast to <a href="Floor_and_ceiling_functions#Number_of_digits" title="Floor and ceiling functions"><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lceil \log _{k}(n+1)\rceil }">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lceil \log _{k}(n+1)\rceil }</annotation>

</semantics>

</math></span><img src="./492c6e9c6b24b962e0701259316ccaa1a111c811.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.332ex; height:2.843ex;" alt="{\displaystyle \lceil \log _{k}(n+1)\rceil }" loading="lazy"></span></a> for ordinary base-<i>k</i> numerals;<br>if <i>k</i> = 1 (i.e., unary), then the number of digits is just <i>n</i>;</dd></dl></li>

<li>the smallest nonnegative integer, representable in a bijective base-<i>k</i> numeral of length <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell \geq 0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ell \geq 0}</annotation>

</semantics>

</math></span><img src="./3ea34eac88d806a616239ef6bb0be2f8b6e3ce2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.231ex; height:2.343ex;" alt="{\displaystyle \ell \geq 0}" loading="lazy"></span>, is

</mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {min} (\ell )={\frac {k^{\ell }-1}{k-1}}}</annotation>

</semantics>

</math></span><img src="./c8cda91fd4548832eccf4a5c327ea9fde6b4d89b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:16.721ex; height:6.009ex;" alt="{\displaystyle \mathrm {min} (\ell )={\frac {k^{\ell }-1}{k-1}}}" loading="lazy"></span>;</dd></dl></li>

<li>the largest nonnegative integer, representable in a bijective base-<i>k</i> numeral of length <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell \geq 0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ell \geq 0}</annotation>

</semantics>

</math></span><img src="./3ea34eac88d806a616239ef6bb0be2f8b6e3ce2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.231ex; height:2.343ex;" alt="{\displaystyle \ell \geq 0}" loading="lazy"></span>, is

</mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {max} (\ell )={\frac {k^{\ell +1}-k}{k-1}}}</annotation>

</semantics>

</math></span><img src="./ec3e5992fdd1461f331ee9b16290dca011b269b8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:19.321ex; height:6.009ex;" alt="{\displaystyle \mathrm {max} (\ell )={\frac {k^{\ell +1}-k}{k-1}}}" loading="lazy"></span>, equivalent to <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {max} (\ell )=k\times \mathrm {min} (\ell )}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {max} (\ell )=k\times \mathrm {min} (\ell )}</annotation>

</semantics>

</math></span><img src="./d4cd3dd9e5a89eaf2e68d130316b4ecba2e0c661.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.909ex; height:2.843ex;" alt="{\displaystyle \mathrm {max} (\ell )=k\times \mathrm {min} (\ell )}" loading="lazy"></span>, or <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {max} (\ell )=\mathrm {min} (\ell +1)-1}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {max} (\ell )=\mathrm {min} (\ell +1)-1}</annotation>

</semantics>

</math></span><img src="./9cfdd61473e22e9700d4c5d26c2e061623829422.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.863ex; height:2.843ex;" alt="{\displaystyle \mathrm {max} (\ell )=\mathrm {min} (\ell +1)-1}" loading="lazy"></span>;</dd></dl></li>

<li>the bijective base-<i>k</i> and ordinary base-<i>k</i> numerals for a nonnegative integer <i>n</i> <i>are identical</i> if and only if the ordinary numeral does not contain the digit <i>0</i> (or, equivalently, the bijective numeral is neither the empty string nor contains the digit <i>k</i>).</li></ul>

<p>For a given base <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\geq 1}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./d30d7dcf305b7bce39d36df72fe3985b47aa9961.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.472ex; height:2.343ex;" alt="{\displaystyle k\geq 1}" loading="lazy"></span>,

</p>

<ul><li>there are exactly <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k^{\ell }}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./5efba7e2dfb1fb31ce737c776ce2b8b7d71440b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.129ex; height:2.676ex;" alt="{\displaystyle k^{\ell }}" loading="lazy"></span> bijective base-<i>k</i> numerals of length <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell \geq 0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ell \geq 0}</annotation>

</semantics>

</math></span><img src="./3ea34eac88d806a616239ef6bb0be2f8b6e3ce2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.231ex; height:2.343ex;" alt="{\displaystyle \ell \geq 0}" loading="lazy"></span>;<sup id="cite_ref-FOOTNOTEForslund1995_2-0" class="reference"><a href="#cite_note-FOOTNOTEForslund1995-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup></li>

<li>a list of bijective base-<i>k</i> numerals, in natural order of the integers represented, is automatically in <a href="Shortlex_order" title="Shortlex order">shortlex order</a> (shortest first, lexicographical within each length). Thus, using λ to denote the <a href="Empty_string" title="Empty string">empty string</a>, the base 1, 2, 3, 8, 10, 12, and 16 numerals are as follows (where the ordinary representations are listed for comparison):</li></ul>

<th>bijective base 1:

</th>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

<td colspan="11" align="left">(<a href="Unary_numeral_system" title="Unary numeral system">unary numeral system</a>)

</td></tr>

<th>bijective base 2:

</th>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td></tr>

<th>binary:

</th>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td></tr>

<th>bijective base 3:

</th>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td></tr>

<th>ternary:

</th>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td></tr>

<th>bijective base 8:

</th>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td></tr>

<th>octal:

</th>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td></tr>

<th>bijective base 10:

</th>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td></tr>

<th>decimal:

</th>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td></tr>

<th>bijective base 12:

</th>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td></tr>

<th>duodecimal:

</th>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td></tr>

<th>bijective base 16:

</th>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td></tr>

<th>hexadecimal:

</th>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

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

</div>

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

<dl><dd>34152 (in bijective base-5) = 3×5<sup>4</sup> + 4×5<sup>3</sup> + 1×5<sup>2</sup> + 5×5<sup>1</sup> + 2×1 = 2427 (in decimal).</dd></dl>

<dl><dd>119A (in bijective base-10, with "A" representing the digit value ten) = 1×10<sup>3</sup> + 1×10<sup>2</sup> + 9×10<sup>1</sup> + 10×1 = 1200 (in decimal).</dd></dl>

<dl><dd>A typical alphabetic list with more than 26 elements is bijective, using the order of A, B, C...X, Y, Z, AA, AB, AC...ZX, ZY, ZZ, AAA, AAB, AAC...</dd></dl>

<div class="mw-heading mw-heading2"><h2 id="The_bijective_base-10_system">The bijective base-10 system</h2></div>

<p>The bijective base-10 system is a base <a href="10_(number)" class="mw-redirect" title="10 (number)">ten</a> positional <a href="Numeral_system" title="Numeral system">numeral system</a> that does not use a digit to represent <a href="0_(number)" class="mw-redirect" title="0 (number)">zero</a>. It instead has a digit to represent ten, such as <i>A</i>.

</p><p>As with conventional <a href="Decimal" title="Decimal">decimal</a>, each digit position represents a power of ten, so for example 123 is "one hundred, plus two tens, plus three units." All <a href="Positive_integer" class="mw-redirect" title="Positive integer">positive integers</a> that are represented solely with non-zero digits in conventional decimal (such as 123) have the same representation in the bijective base-10 system. Those that use a zero must be rewritten, so for example 10 becomes A, conventional 20 becomes 1A, conventional 100 becomes 9A, conventional 101 becomes A1, conventional 302 becomes 2A2, conventional 1000 becomes 99A, conventional 1110 becomes AAA, conventional 2010 becomes 19AA, and so on.

</p><p><a href="Addition" title="Addition">Addition</a> and <a href="Multiplication" title="Multiplication">multiplication</a> in this system are essentially the same as with conventional decimal, except that carries occur when a position exceeds ten, rather than when it exceeds nine. So to calculate 643 + 759, there are twelve units (write 2 at the right and carry 1 to the tens), ten tens (write A with no need to carry to the hundreds), thirteen hundreds (write 3 and carry 1 to the thousands), and one thousand (write 1), to give the result 13A2 rather than the conventional 1402.

</p>

<div class="mw-heading mw-heading2"><h2 id="The_bijective_base-26_system">The bijective base-26 system</h2></div>

<p>In the bijective base-26 system one may use the Latin alphabet letters "A" to "Z" to represent the 26 digit values <a href="1_(number)" class="mw-redirect" title="1 (number)">one</a> to <a href="26_(number)" title="26 (number)">twenty-six</a>. (A=1, B=2, C=3, ..., Z=26)

</p><p>With this choice of notation, the number sequence (starting from 1) begins A, B, C, ..., X, Y, Z, AA, AB, AC, ..., AX, AY, AZ, BA, BB, BC, ...

</p><p>Each digit position represents a power of twenty-six, so for example, the numeral WI represents the value <span class="nowrap">23 × 26<sup>1</sup> + 9 × 26<sup>0</sup></span> = 607 in base 10.

</p><p>Many <a href="Spreadsheet" title="Spreadsheet">spreadsheets</a> including <a href="Microsoft_Excel" title="Microsoft Excel">Microsoft Excel</a> use this system to assign labels to the columns of a spreadsheet, starting A, B, C, ..., Z, AA, AB, ..., AZ, BA, ..., ZZ, AAA, etc. For instance, in Excel 2013, there can be up to 16384 columns (2<sup>14</sup> in binary code), labeled from A to XFD.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> <a href="Malware" title="Malware">Malware</a> variants are also named using this system: for example, the first widespread Microsoft Word macro virus, Concept, is formally named WM/Concept.A, its 26th variant WM/Concept.Z, the 27th variant WM/Concept.AA, et seq. A variant of this system is used to name <a href="Variable_star_designation" class="mw-redirect" title="Variable star designation">variable stars</a>.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> It can be applied to any problem where a systematic naming using letters is desired, while using the shortest possible strings.

</p>

<div class="mw-heading mw-heading2"><h2 id="Historical_notes">Historical notes</h2></div>

<p>The fact that every non-negative integer has a unique representation in bijective base-<i>k</i> (<i>k</i> ≥ 1) is a "<a href="Mathematical_folklore" title="Mathematical folklore">folk theorem</a>" that has been rediscovered many times. Early instances are <a href="#CITEREFFoster1947">Foster (1947)</a> for the case <i>k</i> = 10, and <a href="#CITEREFSmullyan1961">Smullyan (1961)</a> and <a href="#CITEREFBöhm1964">Böhm (1964)</a> for all <i>k</i> ≥ 1. Smullyan uses this system to provide a <a href="G%C3%B6del_numbering" title="Gödel numbering">Gödel numbering</a> of the strings of symbols in a logical system; Böhm uses these representations to perform computations in the programming language <a href="P%E2%80%B2%E2%80%B2" title="P′′">P′′</a>. <a href="#CITEREFKnuth1969">Knuth (1969)</a> mentions the special case of <i>k</i> = 10, and <a href="#CITEREFSalomaa1973">Salomaa (1973)</a> discusses the cases <i>k</i> ≥ 2. <a href="#CITEREFForslund1995">Forslund (1995)</a> appears to be another rediscovery, and hypothesizes that if ancient numeration systems used bijective base-<i>k</i>, they might not be recognized as such in archaeological documents, due to general unfamiliarity with this system.

</p>

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

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</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://math.stackexchange.com/q/607856">"How many digits are in the bijective base-k numeral for n?"</a>. <i>Stackexchange</i><span class="reference-accessdate">. Retrieved <span class="nowrap">22 September</span> 2018</span>.</cite></span>

</li>

<li id="cite_note-FOOTNOTEForslund1995-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEForslund1995_2-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFForslund1995">Forslund (1995)</a>.</span>

</li>

<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite id="CITEREFHarvey2013" class="citation cs2">Harvey, Greg (2013), <i>Excel 2013 For Dummies</i>, John Wiley & Sons, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9781118550007</bdi></cite>.</span>

</li>

<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><cite id="CITEREFHellier2001" class="citation cs2">Hellier, Coel (2001), "Appendix D: Variable star nomenclature", <a rel="nofollow" class="external text" href="https://books.google.com/books?id=5I-yLZ3oYxIC&pg=PA197"><i>Cataclysmic Variable Stars - How and Why They Vary</i></a>, Praxis Books in Astronomy and Space, Springer, p. 197, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9781852332112</bdi></cite>.</span>

</li>

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

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

<ul><li><cite id="CITEREFBöhm1964" class="citation cs2"><a href="Corrado_B%C3%B6hm" title="Corrado Böhm">Böhm, C.</a> (July 1964), "On a family of Turing machines and the related programming language", <i>ICC Bulletin</i>, <b>3</b>: 191</cite>.</li>

<li><cite id="CITEREFForslund1995" class="citation cs2">Forslund, Robert R. (1995), "A logical alternative to the existing positional number system", <i>Southwest Journal of Pure and Applied Mathematics</i>, <b>1</b>: <span class="nowrap">27–</span>29, <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=1386376">1386376</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:19010664">19010664</a></cite>.</li>

<li><cite id="CITEREFFoster1947" class="citation cs2">Foster, J. E. (1947), "A number system without a zero symbol", <i><a href="Mathematics_Magazine" title="Mathematics Magazine">Mathematics Magazine</a></i>, <b>21</b> (1): <span class="nowrap">39–</span>41, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F3029479">10.2307/3029479</a>, <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/3029479">3029479</a></cite>.</li>

<li><cite id="CITEREFKnuth1969" class="citation cs2"><a href="Donald_Knuth" title="Donald Knuth">Knuth, D. E.</a> (1969), <i>The Art of Computer Programming, Vol. 2: Seminumerical Algorithms</i> (1st ed.), Addison-Wesley, Solution to Exercise 4.1-24, p. 195</cite>. (Discusses bijective base-10.)</li>

<li><cite id="CITEREFSalomaa1973" class="citation cs2"><a href="Arto_Salomaa" title="Arto Salomaa">Salomaa, A.</a> (1973), <i>Formal Languages</i>, Academic Press, Note 9.1, pp. 90–91</cite>. (Discusses bijective base-<i>k</i> for all <i>k</i> ≥ 2.)</li>

<li><cite id="CITEREFSmullyan1961" class="citation cs2"><a href="Raymond_Smullyan" title="Raymond Smullyan">Smullyan, R.</a> (1961), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=O4wZLfMnd74C&pg=PA34">"9. Lexicographical ordering; <i>n</i>-adic representation of integers"</a>, <i>Theory of Formal Systems</i>, Annals of Mathematics Studies, vol. 47, Princeton University Press, pp. <span class="nowrap">34–</span>36</cite>.</li></ul>

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