p-adic number - Wikipedia

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class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#p-adic_integers"> <div class="vector-toc-text"> 5 p-adic integers </div> </a> <ul id="toc-p-adic_integers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Topological_properties" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Topological_properties"> <div class="vector-toc-text"> 6 Topological properties </div> </a> <ul id="toc-Topological_properties-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-p-adic_expansion_of_rational_numbers" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#p-adic_expansion_of_rational_numbers"> <div class="vector-toc-text"> 7 p-adic expansion of rational numbers </div> </a> <button aria-controls="toc-p-adic_expansion_of_rational_numbers-sublist" class="cdx-button cdx-button--weight-quiet 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class="vector-toc-list"> </ul> </li> <li id="toc-Notation" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notation"> <div class="vector-toc-text"> 9 Notation </div> </a> <ul id="toc-Notation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Cardinality" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Cardinality"> <div class="vector-toc-text"> 10 Cardinality </div> </a> <ul id="toc-Cardinality-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Algebraic_closure" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Algebraic_closure"> <div class="vector-toc-text"> 11 Algebraic closure </div> </a> <ul id="toc-Algebraic_closure-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Multiplicative_group" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Multiplicative_group"> <div class="vector-toc-text"> 12 Multiplicative group </div> </a> <ul id="toc-Multiplicative_group-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Local–global_principle" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Local–global_principle"> <div class="vector-toc-text"> 13 Local–global principle </div> </a> <ul id="toc-Local–global_principle-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Rational_arithmetic_with_Hensel_lifting" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Rational_arithmetic_with_Hensel_lifting"> <div class="vector-toc-text"> 14 Rational arithmetic with Hensel lifting </div> </a> <ul id="toc-Rational_arithmetic_with_Hensel_lifting-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Generalizations_and_related_concepts" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Generalizations_and_related_concepts"> <div class="vector-toc-text"> 15 Generalizations and related concepts </div> </a> <ul id="toc-Generalizations_and_related_concepts-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> 16 See also </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Footnotes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Footnotes"> <div class="vector-toc-text"> 17 Footnotes </div> </a> <button aria-controls="toc-Footnotes-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Footnotes subsection </button> <ul id="toc-Footnotes-sublist" class="vector-toc-list"> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> 17.1 Notes </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Citations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Citations"> <div class="vector-toc-text"> 17.2 Citations </div> </a> <ul id="toc-Citations-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> 18 References </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> 19 Further reading </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> 20 External links </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" 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Available in 28 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-28" aria-hidden="true" > 28 languages </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B9%D8%AF%D8%AF_%D8%AA%D9%82%D8%A7%D8%B1%D8%A8%D9%8A_%D8%A8%D8%AA%D8%B1%D8%AF%D8%AF_p" title="عدد تقاربي بتردد p – Arabic" lang="ar" hreflang="ar" data-title="عدد تقاربي بتردد p" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/P-%D0%B0%D0%B4%D1%8B%D1%87%D0%BD%D1%8B_%D0%BB%D1%96%D0%BA" title="P-адычны лік – Belarusian" lang="be" hreflang="be" data-title="P-адычны лік" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target">Беларуская</a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/P-%D0%B0%D0%B4%D0%B8%D1%87%D0%BD%D0%BE_%D1%87%D0%B8%D1%81%D0%BB%D0%BE" title="P-адично число – Bulgarian" lang="bg" hreflang="bg" data-title="P-адично число" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target">Български</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Nombre_p-%C3%A0dic" title="Nombre p-àdic – Catalan" lang="ca" hreflang="ca" data-title="Nombre p-àdic" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/P-adick%C3%A9_%C4%8D%C3%ADslo" title="P-adické číslo – Czech" lang="cs" hreflang="cs" data-title="P-adické číslo" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/P-adische_Zahl" title="P-adische Zahl – German" lang="de" hreflang="de" data-title="P-adische Zahl" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/N%C3%BAmero_p-%C3%A1dico" title="Número p-ádico – Spanish" lang="es" hreflang="es" data-title="Número p-ádico" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%B9%D8%AF%D8%AF_%D9%BE%DB%8C-%D8%A7%D8%AF%DB%8C%DA%A9" title="عدد پی-ادیک – Persian" lang="fa" hreflang="fa" data-title="عدد پی-ادیک" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Nombre_p-adique" title="Nombre p-adique – French" lang="fr" hreflang="fr" data-title="Nombre p-adique" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/N%C3%BAmero_p-%C3%A1dico" title="Número p-ádico – Galician" lang="gl" hreflang="gl" data-title="Número p-ádico" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target">Galego</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/P%EC%A7%84%EC%88%98" title="P진수 – Korean" lang="ko" hreflang="ko" data-title="P진수" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Numero_p-adico" title="Numero p-adico – Italian" lang="it" hreflang="it" data-title="Numero p-adico" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A1%D7%A4%D7%A8_p-%D7%90%D7%93%D7%99" title="מספר p-אדי – Hebrew" lang="he" hreflang="he" data-title="מספר p-אדי" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/P-adikus_sz%C3%A1mok" title="P-adikus számok – Hungarian" lang="hu" hreflang="hu" data-title="P-adikus számok" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target">Magyar</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/P-adisch_getal" title="P-adisch getal – Dutch" lang="nl" hreflang="nl" data-title="P-adisch getal" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/P%E9%80%B2%E6%95%B0" title="P進数 – Japanese" lang="ja" hreflang="ja" data-title="P進数" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/P-adisk_tall" title="P-adisk tall – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="P-adisk tall" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target">Norsk bokmål</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Liczby_p-adyczne" title="Liczby p-adyczne – Polish" lang="pl" hreflang="pl" data-title="Liczby p-adyczne" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/N%C3%BAmero_p-%C3%A1dico" title="Número p-ádico – Portuguese" lang="pt" hreflang="pt" data-title="Número p-ádico" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/P-%D0%B0%D0%B4%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%BE%D0%B5_%D1%87%D0%B8%D1%81%D0%BB%D0%BE" title="P-адическое число – Russian" lang="ru" hreflang="ru" data-title="P-адическое число" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/P-adic_number" title="P-adic number – Simple English" lang="en-simple" hreflang="en-simple" data-title="P-adic number" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target">Simple English</a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/P-adiska_tal" title="P-adiska tal – Swedish" lang="sv" hreflang="sv" data-title="P-adiska tal" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target">Svenska</a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/P-sel_say%C4%B1lar" title="P-sel sayılar – Turkish" lang="tr" hreflang="tr" data-title="P-sel sayılar" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target">Türkçe</a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/P-%D0%B0%D0%B4%D0%B8%D1%87%D0%BD%D0%B5_%D1%87%D0%B8%D1%81%D0%BB%D0%BE" title="P-адичне число – Ukrainian" lang="uk" hreflang="uk" data-title="P-адичне число" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target">Українська</a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/S%E1%BB%91_p-adic" title="Số p-adic – Vietnamese" lang="vi" hreflang="vi" data-title="Số p-adic" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target">Tiếng Việt</a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E9%80%B2%E6%95%B8" title="進數 – Literary Chinese" lang="lzh" hreflang="lzh" data-title="進數" data-language-autonym="文言" data-language-local-name="Literary Chinese" class="interlanguage-link-target">文言</a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/P%E9%80%B2%E6%95%B8" title="P進數 – Cantonese" lang="yue" hreflang="yue" data-title="P進數" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target">粵語</a></li><li class="interlanguage-link interwiki-zh badge-Q17437798 badge-goodarticle mw-list-item" title="good article badge"><a href="https://zh.wikipedia.org/wiki/P%E9%80%B2%E6%95%B8" title="P進數 – Chinese" lang="zh" hreflang="zh" data-title="P進數" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target">中文</a></li> </ul> <div class="after-portlet after-portlet-lang"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q311627#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit 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</div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle">(Redirected from <a href="/w/index.php?title=P-adic_field&redirect=no" class="mw-redirect" title="P-adic field">P-adic field</a>)</div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Number system extending the rational numbers</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">In this article, unless otherwise stated, p denotes a prime number that is fixed once for all.</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:3-adic_integers_with_dual_colorings.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/3-adic_integers_with_dual_colorings.svg/220px-3-adic_integers_with_dual_colorings.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/3-adic_integers_with_dual_colorings.svg/330px-3-adic_integers_with_dual_colorings.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/ce/3-adic_integers_with_dual_colorings.svg/440px-3-adic_integers_with_dual_colorings.svg.png 2x" data-file-width="600" data-file-height="600" /></a><figcaption>The 3-adic integers, with selected corresponding characters on their <a href="/wiki/Pontryagin_dual" class="mw-redirect" title="Pontryagin dual">Pontryagin dual</a> group</figcaption></figure> In <a href="/wiki/Number_theory" title="Number theory">number theory</a>, given a <a href="/wiki/Prime_number" title="Prime number">prime number</a> p, the p-adic numbers form an extension of the <a href="/wiki/Rational_number" title="Rational number">rational numbers</a> which is distinct from the <a href="/wiki/Real_number" title="Real number">real numbers</a>, though with some similar properties; p-adic numbers can be written in a form similar to (possibly <a href="/wiki/Infinity_(mathematics)" class="mw-redirect" title="Infinity (mathematics)">infinite</a>) <a href="/wiki/Decimal_representation" title="Decimal representation">decimals</a>, but with digits based on a <a href="/wiki/Prime_number" title="Prime number">prime number</a> p rather than ten, and extending to the left rather than to the right. For example, comparing the expansion of the rational number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{5}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>5</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{5}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7cba10df075ad5061b732c02e1101b8f9237558e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:1.658ex; height:3.676ex;" alt="{\displaystyle {\tfrac {1}{5}}}"> in <a href="/wiki/Ternary_numeral_system" title="Ternary numeral system">base 3</a> vs. the 3-adic expansion, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{alignedat}{3}{\tfrac {1}{5}}&{}=0.01210121\ldots \ ({\text{base }}3)&&{}=0\cdot 3^{0}+0\cdot 3^{-1}+1\cdot 3^{-2}+2\cdot 3^{-3}+\cdots \\[5mu]{\tfrac {1}{5}}&{}=\dots 121012102\ \ ({\text{3-adic}})&&{}=\cdots +2\cdot 3^{3}+1\cdot 3^{2}+0\cdot 3^{1}+2\cdot 3^{0}.\end{alignedat}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left" rowspacing="0.578em 0.3em" columnspacing="0em 0em 0em 0em 0em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>5</mn> </mfrac> </mstyle> </mrow> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mn>0.01210121</mn> <mo>…</mo> <mtext> </mtext> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>base </mtext> </mrow> <mn>3</mn> <mo stretchy="false">)</mo> </mtd> <mtd /> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mn>0</mn> <mo>⋅</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo>+</mo> <mn>0</mn> <mo>⋅</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>⋅</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mo>⋅</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>5</mn> </mfrac> </mstyle> </mrow> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mo>…</mo> <mn>121012102</mn> <mtext> </mtext> <mtext> </mtext> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>3-adic</mtext> </mrow> <mo stretchy="false">)</mo> </mtd> <mtd /> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mo>⋯</mo> <mo>+</mo> <mn>2</mn> <mo>⋅</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>⋅</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>0</mn> <mo>⋅</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mo>⋅</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{alignedat}{3}{\tfrac {1}{5}}&{}=0.01210121\ldots \ ({\text{base }}3)&&{}=0\cdot 3^{0}+0\cdot 3^{-1}+1\cdot 3^{-2}+2\cdot 3^{-3}+\cdots \\[5mu]{\tfrac {1}{5}}&{}=\dots 121012102\ \ ({\text{3-adic}})&&{}=\cdots +2\cdot 3^{3}+1\cdot 3^{2}+0\cdot 3^{1}+2\cdot 3^{0}.\end{alignedat}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a91337f3d3043e9236041f8f7e6bec74389665dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:69.903ex; height:8.176ex;" alt="{\displaystyle {\begin{alignedat}{3}{\tfrac {1}{5}}&{}=0.01210121\ldots \ ({\text{base }}3)&&{}=0\cdot 3^{0}+0\cdot 3^{-1}+1\cdot 3^{-2}+2\cdot 3^{-3}+\cdots \\[5mu]{\tfrac {1}{5}}&{}=\dots 121012102\ \ ({\text{3-adic}})&&{}=\cdots +2\cdot 3^{3}+1\cdot 3^{2}+0\cdot 3^{1}+2\cdot 3^{0}.\end{alignedat}}}"></dd></dl> Formally, given a prime number p, a p-adic number can be defined as a <a href="/wiki/Series_(mathematics)" title="Series (mathematics)">series</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s=\sum _{i=k}^{\infty }a_{i}p^{i}=a_{k}p^{k}+a_{k+1}p^{k+1}+a_{k+2}p^{k+2}+\cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>2</mn> </mrow> </msub> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s=\sum _{i=k}^{\infty }a_{i}p^{i}=a_{k}p^{k}+a_{k+1}p^{k+1}+a_{k+2}p^{k+2}+\cdots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6ad992b963bfa46cf597d00daf64ba798a83fd0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:48.405ex; height:6.843ex;" alt="{\displaystyle s=\sum _{i=k}^{\infty }a_{i}p^{i}=a_{k}p^{k}+a_{k+1}p^{k+1}+a_{k+2}p^{k+2}+\cdots }"></dd></dl> where k is an <a href="/wiki/Integer" title="Integer">integer</a> (possibly negative), and each <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bc77764b2e74e64a63341054fa90f3e07db275f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.029ex; height:2.009ex;" alt="{\displaystyle a_{i}}"> is an integer such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq a_{i}<p.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>≤</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo><</mo> <mi>p</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\leq a_{i}<p.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3f17182b9eaa86a5121a307f7784b01140ea0f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.205ex; height:2.509ex;" alt="{\displaystyle 0\leq a_{i}<p.}"> A p-adic integer is a p-adic number such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\geq 0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>≥</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\geq 0.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6a21e868e400e73512aa47ff205ebe6c2421919" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.119ex; height:2.343ex;" alt="{\displaystyle k\geq 0.}"> In general the series that represents a p-adic number is not <a href="/wiki/Convergent_series" title="Convergent series">convergent</a> in the usual sense, but it is convergent for the <a href="/wiki/P-adic_absolute_value" class="mw-redirect" title="P-adic absolute value">p-adic absolute value</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |s|_{p}=p^{-k},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>s</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>=</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>k</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |s|_{p}=p^{-k},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e46fc684774c8b4f9b7129085d64ec1074c16ba1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:10.725ex; height:3.509ex;" alt="{\displaystyle |s|_{p}=p^{-k},}"> where k is the least integer i such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{i}\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{i}\neq 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d66efc850e79e220d4f51bc2a2333af3d7325180" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.29ex; height:2.676ex;" alt="{\displaystyle a_{i}\neq 0}"> (if all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bc77764b2e74e64a63341054fa90f3e07db275f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.029ex; height:2.009ex;" alt="{\displaystyle a_{i}}"> are zero, one has the zero p-adic number, which has 0 as its p-adic absolute value). Every rational number can be uniquely expressed as the sum of a series as above, with respect to the p-adic absolute value. This allows considering rational numbers as special p-adic numbers, and alternatively defining the p-adic numbers as the <a href="/wiki/Completion_(metric_space)" class="mw-redirect" title="Completion (metric space)">completion</a> of the rational numbers for the p-adic absolute value, exactly as the real numbers are the completion of the rational numbers for the usual absolute value. p-adic numbers were first described by <a href="/wiki/Kurt_Hensel" title="Kurt Hensel">Kurt Hensel</a> in 1897,<a href="#cite_note-1">[1]</a> though, with hindsight, some of <a href="/wiki/Ernst_Kummer" title="Ernst Kummer">Ernst Kummer's</a> earlier work can be interpreted as implicitly using p-adic numbers.<a href="#cite_note-2">[note 1]</a> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Motivation">Motivation</h2>[<a href="/w/index.php?title=P-adic_number&action=edit&section=1" title="Edit section: Motivation">edit</a>]</div> Roughly speaking, <a href="/wiki/Modular_arithmetic" title="Modular arithmetic">modular arithmetic</a> modulo a positive integer n consists of "approximating" every integer by the remainder of its <a href="/wiki/Euclidean_division" title="Euclidean division">division</a> by n, called its residue modulo n. The main property of modular arithmetic is that the residue modulo n of the result of a succession of operations on integers is the same as the result of the same succession of operations on residues modulo n. If one knows that the absolute value of the result is less than n/2, this allows a computation of the result which does not involve any integer larger than n. For larger results, an old method, still in common use, consists of using several small moduli that are pairwise coprime, and applying the <a href="/wiki/Chinese_remainder_theorem" title="Chinese remainder theorem">Chinese remainder theorem</a> for recovering the result modulo the product of the moduli. Another method discovered by <a href="/wiki/Kurt_Hensel" title="Kurt Hensel">Kurt Hensel</a> consists of using a prime modulus p, and applying <a href="/wiki/Hensel%27s_lemma" title="Hensel's lemma">Hensel's lemma</a> for recovering iteratively the result modulo <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p^{2},p^{3},\ldots ,p^{n},\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p^{2},p^{3},\ldots ,p^{n},\ldots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5afa45c920651c8ae0f16177fdf4c42809b49649" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:16.894ex; height:3.009ex;" alt="{\displaystyle p^{2},p^{3},\ldots ,p^{n},\ldots }"> If the process is continued infinitely, this provides eventually a result which is a p-adic number. <div class="mw-heading mw-heading2"><h2 id="Basic_lemmas">Basic lemmas</h2>[<a href="/w/index.php?title=P-adic_number&action=edit&section=2" title="Edit section: Basic lemmas">edit</a>]</div> The theory of p-adic numbers is fundamentally based on the two following lemmas Every nonzero rational number can be written <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle p^{v}{\frac {m}{n}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>m</mi> <mi>n</mi> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle p^{v}{\frac {m}{n}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/421d6bca84cf67d97b1a385225d807cdba555a03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.089ex; width:5.214ex; height:3.009ex;" alt="{\textstyle p^{v}{\frac {m}{n}},}"> where v, m, and n are integers and neither m nor n is divisible by p. The exponent v is uniquely determined by the rational number and is called its p-adic valuation (this definition is a particular case of a more general definition, given below). The proof of the lemma results directly from the <a href="/wiki/Fundamental_theorem_of_arithmetic" title="Fundamental theorem of arithmetic">fundamental theorem of arithmetic</a>. Every nonzero rational number r of valuation v can be uniquely written <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=ap^{v}+s,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mi>a</mi> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msup> <mo>+</mo> <mi>s</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=ap^{v}+s,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4b33c7a9b5fc02bd139e057de580b53c9fb55ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.154ex; height:2.676ex;" alt="{\displaystyle r=ap^{v}+s,}"> where s is a rational number of valuation greater than v, and a is an integer such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<a<p.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo><</mo> <mi>a</mi> <mo><</mo> <mi>p</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0<a<p.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2cb1f170450e5661dec91f296b5536d510c8edd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.405ex; height:2.509ex;" alt="{\displaystyle 0<a<p.}"> The proof of this lemma results from <a href="/wiki/Modular_arithmetic" title="Modular arithmetic">modular arithmetic</a>: By the above lemma, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle r=p^{v}{\frac {m}{n}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>m</mi> <mi>n</mi> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle r=p^{v}{\frac {m}{n}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/caf4be1c0e345b3206baa5ab7d97ea94fa2927e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.272ex; height:3.009ex;" alt="{\textstyle r=p^{v}{\frac {m}{n}},}"> where m and n are integers <a href="/wiki/Coprime" class="mw-redirect" title="Coprime">coprime</a> with p. The <a href="/wiki/Modular_inverse" class="mw-redirect" title="Modular inverse">modular inverse</a> of n is an integer q such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle nq=1+ph}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mi>q</mi> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mi>p</mi> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle nq=1+ph}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29a1688a560f08cb0e72873b1b0a4451e0ecc1d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.074ex; height:2.509ex;" alt="{\displaystyle nq=1+ph}"> for some integer h. Therefore, one has <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {1}{n}}=q-p{\frac {h}{n}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <mo>=</mo> <mi>q</mi> <mo>−</mo> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>h</mi> <mi>n</mi> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\frac {1}{n}}=q-p{\frac {h}{n}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/224787dc3577af1121adfd361a4ae41364e68eb7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.469ex; height:3.509ex;" alt="{\textstyle {\frac {1}{n}}=q-p{\frac {h}{n}},}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle r=p^{v}mq-p^{v+1}{\frac {hm}{n}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msup> <mi>m</mi> <mi>q</mi> <mo>−</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>h</mi> <mi>m</mi> </mrow> <mi>n</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle r=p^{v}mq-p^{v+1}{\frac {hm}{n}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3ece2f96dd1ce05bfa3eaff3afc482791f00b38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.469ex; height:3.509ex;" alt="{\textstyle r=p^{v}mq-p^{v+1}{\frac {hm}{n}}.}"> The <a href="/wiki/Euclidean_division" title="Euclidean division">Euclidean division</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle mq}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle mq}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc22827374c8d773584aebfb96348ac4854320e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.11ex; height:2.009ex;" alt="{\displaystyle mq}"> by p gives <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle mq=pk+a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mi>q</mi> <mo>=</mo> <mi>p</mi> <mi>k</mi> <mo>+</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle mq=pk+a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/540331dd34c175effe355d1232ce403e984abbb9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.659ex; height:2.509ex;" alt="{\displaystyle mq=pk+a}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<a<p,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo><</mo> <mi>a</mi> <mo><</mo> <mi>p</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0<a<p,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1e854843aaf6baac40c51d47efe10cab21fbfcd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.405ex; height:2.509ex;" alt="{\displaystyle 0<a<p,}"> since mq is not divisible by p. So, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=ap^{v}+p^{v+1}{\frac {kn-hm}{n}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mi>a</mi> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msup> <mo>+</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>k</mi> <mi>n</mi> <mo>−</mo> <mi>h</mi> <mi>m</mi> </mrow> <mi>n</mi> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=ap^{v}+p^{v+1}{\frac {kn-hm}{n}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3f9f41a20d6b8e99df4dad6eb04dcc2749151a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:25.025ex; height:5.343ex;" alt="{\displaystyle r=ap^{v}+p^{v+1}{\frac {kn-hm}{n}},}"></dd></dl> which is the desired result. This can be iterated starting from s instead of r, giving the following. Given a nonzero rational number r of valuation v and a positive integer k, there are a rational number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04f159343172781e7666dbc88280c91f34117c30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.179ex; height:2.009ex;" alt="{\displaystyle s_{k}}"> of nonnegative valuation and k uniquely defined nonnegative integers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{0},\ldots ,a_{k-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{0},\ldots ,a_{k-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4db171b5bf04925f27efed545c02493caf65d08b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.881ex; height:2.009ex;" alt="{\displaystyle a_{0},\ldots ,a_{k-1}}"> less than p such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{0}>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{0}>0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1899d1c0713bcaa2fdb4c8bfea422c007fea38cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.545ex; height:2.509ex;" alt="{\displaystyle a_{0}>0}"> and <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=a_{0}p^{v}+a_{1}p^{v+1}+\cdots +a_{k-1}p^{v+k-1}+p^{v+k}s_{k}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msup> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> <mo>+</mo> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> <mo>+</mo> <mi>k</mi> </mrow> </msup> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=a_{0}p^{v}+a_{1}p^{v+1}+\cdots +a_{k-1}p^{v+k-1}+p^{v+k}s_{k}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64ae6b8c1d342654a911eb8ad3bdb6a59959009b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:47.312ex; height:3.009ex;" alt="{\displaystyle r=a_{0}p^{v}+a_{1}p^{v+1}+\cdots +a_{k-1}p^{v+k-1}+p^{v+k}s_{k}.}"></dd></dl> The p-adic numbers are essentially obtained by continuing this infinitely to produce an <a href="/wiki/Infinite_series" class="mw-redirect" title="Infinite series">infinite series</a>. <div class="mw-heading mw-heading2"><h2 id="p-adic_series">p-adic series</h2>[<a href="/w/index.php?title=P-adic_number&action=edit&section=3" title="Edit section: p-adic series">edit</a>]</div> The p-adic numbers are commonly defined by means of p-adic series. A p-adic series is a <a href="/wiki/Formal_power_series" title="Formal power series">formal power series</a> of the form <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i=v}^{\infty }r_{i}p^{i},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mi>v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i=v}^{\infty }r_{i}p^{i},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdb7b96b3be2d4484e36d6d6a8d6814e59318df0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:8.206ex; height:6.843ex;" alt="{\displaystyle \sum _{i=v}^{\infty }r_{i}p^{i},}"></dd></dl> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"> is an integer and the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0b6d651eaf432dbf1f106021c8bb499ae83fd1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.848ex; height:2.009ex;" alt="{\displaystyle r_{i}}"> are rational numbers that either are zero or have a nonnegative valuation (that is, the denominator of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0b6d651eaf432dbf1f106021c8bb499ae83fd1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.848ex; height:2.009ex;" alt="{\displaystyle r_{i}}"> is not divisible by p). Every rational number may be viewed as a p-adic series with a single nonzero term, consisting of its factorization of the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p^{k}{\tfrac {n}{d}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>n</mi> <mi>d</mi> </mfrac> </mstyle> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p^{k}{\tfrac {n}{d}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c27f7c0bc5ff9929c70a2d6e542284647aeee8e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; margin-left: -0.089ex; width:4.817ex; height:3.676ex;" alt="{\displaystyle p^{k}{\tfrac {n}{d}},}"> with n and d both coprime with p. Two p-adic series <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \sum _{i=v}^{\infty }r_{i}p^{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mi>v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \sum _{i=v}^{\infty }r_{i}p^{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37a10c607d4c1b667e05b0884c6310ad2341576f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.534ex; height:3.176ex;" alt="{\textstyle \sum _{i=v}^{\infty }r_{i}p^{i}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \sum _{i=w}^{\infty }s_{i}p^{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mi>w</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \sum _{i=w}^{\infty }s_{i}p^{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0e67c6a45f7246791f01e9dbb465c4c3379515b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.955ex; height:3.176ex;" alt="{\textstyle \sum _{i=w}^{\infty }s_{i}p^{i}}"> are equivalent if there is an integer N such that, for every integer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n>N,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>></mo> <mi>N</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n>N,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03d63161bd124b5a3709f827e364b6f8af291164" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.204ex; height:2.509ex;" alt="{\displaystyle n>N,}"> the rational number <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i=v}^{n}r_{i}p^{i}-\sum _{i=w}^{n}s_{i}p^{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mi>v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo>−</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mi>w</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i=v}^{n}r_{i}p^{i}-\sum _{i=w}^{n}s_{i}p^{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3420c57f084fb6aff8d659a533e1c4ffbe8dc1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:18.001ex; height:6.843ex;" alt="{\displaystyle \sum _{i=v}^{n}r_{i}p^{i}-\sum _{i=w}^{n}s_{i}p^{i}}"></dd></dl> is zero or has a p-adic valuation greater than n. A p-adic series <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \sum _{i=v}^{\infty }a_{i}p^{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mi>v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \sum _{i=v}^{\infty }a_{i}p^{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38a85d3f2881723135519504bfd5c3547510d6ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.715ex; height:3.176ex;" alt="{\textstyle \sum _{i=v}^{\infty }a_{i}p^{i}}"> is normalized if either all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bc77764b2e74e64a63341054fa90f3e07db275f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.029ex; height:2.009ex;" alt="{\displaystyle a_{i}}"> are integers such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq a_{i}<p,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>≤</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo><</mo> <mi>p</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\leq a_{i}<p,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf9e7e96581642f884dfe7efa018c0f4fcd7e142" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.205ex; height:2.509ex;" alt="{\displaystyle 0\leq a_{i}<p,}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{v}>0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> <mo>></mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{v}>0,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d91dc2348852dd682846429349ff3c926357a0a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.167ex; height:2.509ex;" alt="{\displaystyle a_{v}>0,}"> or all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bc77764b2e74e64a63341054fa90f3e07db275f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.029ex; height:2.009ex;" alt="{\displaystyle a_{i}}"> are zero. In the latter case, the series is called the zero series. Every p-adic series is equivalent to exactly one normalized series. This normalized series is obtained by a sequence of transformations, which are equivalences of series; see <a href="#Normalization_of_a_p-adic_series">§ Normalization of a p-adic series</a>, below. In other words, the equivalence of p-adic series is an <a href="/wiki/Equivalence_relation" title="Equivalence relation">equivalence relation</a>, and each <a href="/wiki/Equivalence_class" title="Equivalence class">equivalence class</a> contains exactly one normalized p-adic series. The usual operations of series (addition, subtraction, multiplication, division) are compatible with equivalence of p-adic series. That is, denoting the equivalence with ~, if S, T and U are nonzero p-adic series such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\sim T,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>∼</mo> <mi>T</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S\sim T,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78c4b74f492b4947ca8b8dcaeb9d967d1d34ff5f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.881ex; height:2.509ex;" alt="{\displaystyle S\sim T,}"> one has <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}S\pm U&\sim T\pm U,\\SU&\sim TU,\\1/S&\sim 1/T.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>S</mi> <mo>±</mo> <mi>U</mi> </mtd> <mtd> <mi></mi> <mo>∼</mo> <mi>T</mi> <mo>±</mo> <mi>U</mi> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>S</mi> <mi>U</mi> </mtd> <mtd> <mi></mi> <mo>∼</mo> <mi>T</mi> <mi>U</mi> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>S</mi> </mtd> <mtd> <mi></mi> <mo>∼</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>T</mi> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}S\pm U&\sim T\pm U,\\SU&\sim TU,\\1/S&\sim 1/T.\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b94e2860d294a359731cbfe353266e3ad3f357f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:16.878ex; height:8.843ex;" alt="{\displaystyle {\begin{aligned}S\pm U&\sim T\pm U,\\SU&\sim TU,\\1/S&\sim 1/T.\end{aligned}}}"></dd></dl> The p-adic numbers are often defined as the equivalence classes of p-adic series, in a similar way as the definition of the real numbers as equivalence classes of <a href="/wiki/Cauchy_sequence" title="Cauchy sequence">Cauchy sequences</a>. The uniqueness property of normalization, allows uniquely representing any p-adic number by the corresponding normalized p-adic series. The compatibility of the series equivalence leads almost immediately to basic properties of p-adic numbers: <ul><li>Addition, multiplication and <a href="/wiki/Multiplicative_inverse" title="Multiplicative inverse">multiplicative inverse</a> of p-adic numbers are defined as for <a href="/wiki/Formal_power_series" title="Formal power series">formal power series</a>, followed by the normalization of the result.</li> <li>With these operations, the p-adic numbers form a <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a>, which is an <a href="/wiki/Extension_field" class="mw-redirect" title="Extension field">extension field</a> of the rational numbers.</li> <li>The valuation of a nonzero p-adic number x, commonly denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{p}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{p}(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/963b72dbd91a6cf55797bc40f800ff77e3db1e5e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.326ex; height:3.009ex;" alt="{\displaystyle v_{p}(x)}"> is the exponent of p in the first non zero term of the corresponding normalized series; the valuation of zero is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{p}(0)=+\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{p}(0)=+\infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/559652dcdc06736c4ee03206ae3675bf1fe3e2ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.389ex; height:3.009ex;" alt="{\displaystyle v_{p}(0)=+\infty }"></li> <li>The p-adic absolute value of a nonzero p-adic number x, is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |x|_{p}=p^{-v(x)};}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>=</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>v</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |x|_{p}=p^{-v(x)};}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b66ef38ecd3e8b40e63cea95c85738e7ef0a498" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:13.125ex; height:3.676ex;" alt="{\displaystyle |x|_{p}=p^{-v(x)};}"> for the zero p-adic number, one has <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |0|_{p}=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>0</mn> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |0|_{p}=0.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d1d5397ccca35ad2edd8646408aee241666695e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:8.423ex; height:3.176ex;" alt="{\displaystyle |0|_{p}=0.}"></li></ul> <div class="mw-heading mw-heading3"><h3 id="Normalization_of_a_p-adic_series">Normalization of a p-adic series</h3>[<a href="/w/index.php?title=P-adic_number&action=edit&section=4" title="Edit section: Normalization of a p-adic series">edit</a>]</div> Starting with the series <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \sum _{i=v}^{\infty }r_{i}p^{i},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mi>v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \sum _{i=v}^{\infty }r_{i}p^{i},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0efaa06e5862a974e0066c947846ac0bbe317711" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.181ex; height:3.176ex;" alt="{\textstyle \sum _{i=v}^{\infty }r_{i}p^{i},}"> the first above lemma allows getting an equivalent series such that the p-adic valuation of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{v}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{v}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6351a9c12f3cd743f37050ddb9a0d0adfc33f190" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.078ex; height:2.009ex;" alt="{\displaystyle r_{v}}"> is zero. For that, one considers the first nonzero <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{i}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{i}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19fffb8f2f64839b26aeca3bf085400e0a3a232a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.495ex; height:2.009ex;" alt="{\displaystyle r_{i}.}"> If its p-adic valuation is zero, it suffices to change v into i, that is to start the summation from v. Otherwise, the p-adic valuation of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0b6d651eaf432dbf1f106021c8bb499ae83fd1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.848ex; height:2.009ex;" alt="{\displaystyle r_{i}}"> is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j>0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> <mo>></mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j>0,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa8ed85b5bd7100d6040337bb7f30b148a22b2ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:5.893ex; height:2.509ex;" alt="{\displaystyle j>0,}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{i}=p^{j}s_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{i}=p^{j}s_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/552ebbc923ec2ff9c5a36bb0fd7662c6f8cc64b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.916ex; height:3.009ex;" alt="{\displaystyle r_{i}=p^{j}s_{i}}"> where the valuation of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cfda82668232cbdc0874ed28ab8b6079420d1ffe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.89ex; height:2.009ex;" alt="{\displaystyle s_{i}}"> is zero; so, one gets an equivalent series by changing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0b6d651eaf432dbf1f106021c8bb499ae83fd1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.848ex; height:2.009ex;" alt="{\displaystyle r_{i}}"> to 0 and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{i+j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>+</mo> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{i+j}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56f1b5b62f8044b231757843416d99de7aad21b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.804ex; height:2.343ex;" alt="{\displaystyle r_{i+j}}"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{i+j}+s_{i}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>+</mo> <mi>j</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{i+j}+s_{i}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22ac511ca35e377c4cc71477f2ccf24ace4a7afe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.182ex; height:2.676ex;" alt="{\displaystyle r_{i+j}+s_{i}.}"> Iterating this process, one gets eventually, possibly after infinitely many steps, an equivalent series that either is the zero series or is a series such that the valuation of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{v}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{v}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6351a9c12f3cd743f37050ddb9a0d0adfc33f190" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.078ex; height:2.009ex;" alt="{\displaystyle r_{v}}"> is zero. Then, if the series is not normalized, consider the first nonzero <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0b6d651eaf432dbf1f106021c8bb499ae83fd1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.848ex; height:2.009ex;" alt="{\displaystyle r_{i}}"> that is not an integer in the interval <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,p-1].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>p</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">]</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [0,p-1].}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d784d3c09f6c16f2be810fc933b13c064cf70df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.309ex; height:2.843ex;" alt="{\displaystyle [0,p-1].}"> The second above lemma allows writing it <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{i}=a_{i}+ps_{i};}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <mi>p</mi> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{i}=a_{i}+ps_{i};}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42552ceca649e885430a7a853b80fca9c2928fa1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.523ex; height:2.343ex;" alt="{\displaystyle r_{i}=a_{i}+ps_{i};}"> one gets n equivalent series by replacing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0b6d651eaf432dbf1f106021c8bb499ae83fd1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.848ex; height:2.009ex;" alt="{\displaystyle r_{i}}"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{i},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{i},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae3818f8bf83ca0d4d4d1f218f6cd437da185282" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.676ex; height:2.009ex;" alt="{\displaystyle a_{i},}"> and adding <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cfda82668232cbdc0874ed28ab8b6079420d1ffe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.89ex; height:2.009ex;" alt="{\displaystyle s_{i}}"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{i+1}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{i+1}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d32fed9bf6462768ecb36b47745d413f902c8821" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.596ex; height:2.009ex;" alt="{\displaystyle r_{i+1}.}"> Iterating this process, possibly infinitely many times, provides eventually the desired normalized p-adic series. <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2>[<a href="/w/index.php?title=P-adic_number&action=edit&section=5" title="Edit section: Definition">edit</a>]</div> There are several equivalent definitions of p-adic numbers. The one that is given here is relatively elementary, since it does not involve any other mathematical concepts than those introduced in the preceding sections. Other equivalent definitions use <a href="/wiki/Completion_of_a_ring" title="Completion of a ring">completion</a> of a <a href="/wiki/Discrete_valuation_ring" title="Discrete valuation ring">discrete valuation ring</a> (see <a href="#p-adic_integers">§ p-adic integers</a>), <a href="/wiki/Completion_of_a_metric_space" class="mw-redirect" title="Completion of a metric space">completion of a metric space</a> (see <a href="#Topological_properties">§ Topological properties</a>), or <a href="/wiki/Inverse_limit" title="Inverse limit">inverse limits</a> (see <a href="#Modular_properties">§ Modular properties</a>). A p-adic number can be defined as a normalized p-adic series. Since there are other equivalent definitions that are commonly used, one says often that a normalized p-adic series represents a p-adic number, instead of saying that it is a p-adic number. One can say also that any p-adic series represents a p-adic number, since every p-adic series is equivalent to a unique normalized p-adic series. This is useful for defining operations (addition, subtraction, multiplication, division) of p-adic numbers: the result of such an operation is obtained by normalizing the result of the corresponding operation on series. This well defines operations on p-adic numbers, since the series operations are compatible with equivalence of p-adic series. With these operations, p-adic numbers form a <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a> called the field of p-adic numbers and denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} _{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35f44bc6894c682710705f3ea74f33042e0acc3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.867ex; height:2.843ex;" alt="{\displaystyle \mathbb {Q} _{p}}"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {Q} _{p}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {Q} _{p}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffb8ef2b1cbdfbb8c17f0d775571c5d995084b07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.714ex; height:3.009ex;" alt="{\displaystyle \mathbf {Q} _{p}.}"> There is a unique <a href="/wiki/Field_homomorphism" class="mw-redirect" title="Field homomorphism">field homomorphism</a> from the rational numbers into the p-adic numbers, which maps a rational number to its p-adic expansion. The <a href="/wiki/Image_(mathematics)" title="Image (mathematics)">image</a> of this homomorphism is commonly identified with the field of rational numbers. This allows considering the p-adic numbers as an <a href="/wiki/Extension_field" class="mw-redirect" title="Extension field">extension field</a> of the rational numbers, and the rational numbers as a <a href="/wiki/Subfield_(mathematics)" class="mw-redirect" title="Subfield (mathematics)">subfield</a> of the p-adic numbers. The valuation of a nonzero p-adic number x, commonly denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{p}(x),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{p}(x),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ecc2317dd5cf29a453f7aaee465e2e62c99c733a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.973ex; height:3.009ex;" alt="{\displaystyle v_{p}(x),}"> is the exponent of p in the first nonzero term of every p-adic series that represents x. By convention, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{p}(0)=\infty ;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">∞</mi> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{p}(0)=\infty ;}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fec82dc9be17c0b5a0c48494293f3dc787e177a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.228ex; height:3.009ex;" alt="{\displaystyle v_{p}(0)=\infty ;}"> that is, the valuation of zero is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \infty .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∞</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \infty .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f6ccaf28d90bcfce739aca4c5ff119a60e08031" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.971ex; height:1.676ex;" alt="{\displaystyle \infty .}"> This valuation is a <a href="/wiki/Discrete_valuation" title="Discrete valuation">discrete valuation</a>. The restriction of this valuation to the rational numbers is the p-adic valuation of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91185244fbdded6ea99a5e9e6603299128b10928" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.455ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} ,}"> that is, the exponent v in the factorization of a rational number as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {n}{d}}p^{v},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>n</mi> <mi>d</mi> </mfrac> </mstyle> </mrow> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {n}{d}}p^{v},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5b8be7bb79d535cc4953cf2071f82d7780f30e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:4.668ex; height:3.343ex;" alt="{\displaystyle {\tfrac {n}{d}}p^{v},}"> with both n and d <a href="/wiki/Coprime" class="mw-redirect" title="Coprime">coprime</a> with p. <div class="mw-heading mw-heading2"><h2 id="p-adic_integers">p-adic integers</h2>[<a href="/w/index.php?title=P-adic_number&action=edit&section=6" title="Edit section: p-adic integers">edit</a>]</div> The p-adic integers are the p-adic numbers with a nonnegative valuation. A p-adic integer can be represented as a sequence <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=(x_{1}\operatorname {mod} p,~x_{2}\operatorname {mod} p^{2},~x_{3}\operatorname {mod} p^{3},~\ldots )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>mod</mi> <mo>⁡</mo> <mi>p</mi> <mo>,</mo> <mtext> </mtext> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi>mod</mi> <mo>⁡</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> <mtext> </mtext> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mi>mod</mi> <mo>⁡</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>,</mo> <mtext> </mtext> <mo>…</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=(x_{1}\operatorname {mod} p,~x_{2}\operatorname {mod} p^{2},~x_{3}\operatorname {mod} p^{3},~\ldots )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd84307f195f4bfd37e82069ca10d304d587bd76" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:42.455ex; height:3.176ex;" alt="{\displaystyle x=(x_{1}\operatorname {mod} p,~x_{2}\operatorname {mod} p^{2},~x_{3}\operatorname {mod} p^{3},~\ldots )}"></dd></dl> of residues xe mod pe for each integer e, satisfying the compatibility relations <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}\equiv x_{j}~(\operatorname {mod} p^{i})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>≡</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mtext> </mtext> <mo stretchy="false">(</mo> <mi>mod</mi> <mo>⁡</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{i}\equiv x_{j}~(\operatorname {mod} p^{i})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f51c0a9bffcfb32cc07278bd66fc0e8bfd7570d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.604ex; height:3.343ex;" alt="{\displaystyle x_{i}\equiv x_{j}~(\operatorname {mod} p^{i})}"> for i < j. Every <a href="/wiki/Integer" title="Integer">integer</a> is a p-adic integer (including zero, since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo><</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0<\infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be89330211a0d428d00411ae0a9123ece45096cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.585ex; height:2.176ex;" alt="{\displaystyle 0<\infty }">). The rational numbers of the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\tfrac {n}{d}}p^{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>n</mi> <mi>d</mi> </mfrac> </mstyle> </mrow> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\tfrac {n}{d}}p^{k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22f880d59d16d31a87abad283f396747941425c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:4.08ex; height:3.509ex;" alt="{\textstyle {\tfrac {n}{d}}p^{k}}"> with d coprime with p and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\geq 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79214ef55efadfb1d9c9b02252eb8a71cf6f8b6b" 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 0}"> are also p-adic integers (for the reason that d has an inverse mod pe for every e). The p-adic integers form a <a href="/wiki/Commutative_ring" title="Commutative ring">commutative ring</a>, denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} _{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbc1df7227ef11fe88dccd2dae3adc7bbdeae5f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.609ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} _{p}}"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {Z} _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {Z} _{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d24ebd1a053ea692c0dc6be7333408a81ea39f11" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.693ex; height:2.843ex;" alt="{\displaystyle \mathbf {Z} _{p}}">, that has the following properties. <ul><li>It is an <a href="/wiki/Integral_domain" title="Integral domain">integral domain</a>, since it is a <a href="/wiki/Subring" title="Subring">subring</a> of a field, or since the first term of the series representation of the product of two non zero p-adic series is the product of their first terms.</li> <li>The <a href="/wiki/Unit_(ring_theory)" title="Unit (ring theory)">units</a> (invertible elements) of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} _{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbc1df7227ef11fe88dccd2dae3adc7bbdeae5f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.609ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} _{p}}"> are the p-adic numbers of valuation zero.</li> <li>It is a <a href="/wiki/Principal_ideal_domain" title="Principal ideal domain">principal ideal domain</a>, such that each <a href="/wiki/Ideal_(ring_theory)" title="Ideal (ring theory)">ideal</a> is generated by a power of p.</li> <li>It is a <a href="/wiki/Local_ring" title="Local ring">local ring</a> of <a href="/wiki/Krull_dimension" title="Krull dimension">Krull dimension</a> one, since its only <a href="/wiki/Prime_ideal" title="Prime ideal">prime ideals</a> are the <a href="/wiki/Zero_ideal" class="mw-redirect" title="Zero ideal">zero ideal</a> and the ideal generated by p, the unique <a href="/wiki/Maximal_ideal" title="Maximal ideal">maximal ideal</a>.</li> <li>It is a <a href="/wiki/Discrete_valuation_ring" title="Discrete valuation ring">discrete valuation ring</a>, since this results from the preceding properties.</li> <li>It is the <a href="/wiki/Completion_of_a_ring" title="Completion of a ring">completion</a> of the local ring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} _{(p)}=\{{\tfrac {n}{d}}\mid n,d\in \mathbb {Z} ,\,d\not \in p\mathbb {Z} \},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>n</mi> <mi>d</mi> </mfrac> </mstyle> </mrow> <mo>∣</mo> <mi>n</mi> <mo>,</mo> <mi>d</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mo>∉</mo> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} _{(p)}=\{{\tfrac {n}{d}}\mid n,d\in \mathbb {Z} ,\,d\not \in p\mathbb {Z} \},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1bdc845e7901f2b9fad2c07cecaf3c6c1130b516" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:29.951ex; height:3.343ex;" alt="{\displaystyle \mathbb {Z} _{(p)}=\{{\tfrac {n}{d}}\mid n,d\in \mathbb {Z} ,\,d\not \in p\mathbb {Z} \},}"> which is the <a href="/wiki/Localization_(commutative_algebra)" title="Localization (commutative algebra)">localization</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" 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} }"> at the prime ideal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\mathbb {Z} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\mathbb {Z} .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b46b98d11e4640e1f540c82e87e96d0dbfb2b1c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:3.456ex; height:2.509ex;" alt="{\displaystyle p\mathbb {Z} .}"></li></ul> The last property provides a definition of the p-adic numbers that is equivalent to the above one: the field of the p-adic numbers is the <a href="/wiki/Field_of_fractions" title="Field of fractions">field of fractions</a> of the completion of the localization of the integers at the prime ideal generated by p. <div class="mw-heading mw-heading2"><h2 id="Topological_properties">Topological properties</h2>[<a href="/w/index.php?title=P-adic_number&action=edit&section=7" title="Edit section: Topological properties">edit</a>]</div> The p-adic valuation allows defining an <a href="/wiki/Absolute_value_(algebra)" title="Absolute value (algebra)">absolute value</a> on p-adic numbers: the p-adic absolute value of a nonzero p-adic number x is <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |x|_{p}=p^{-v_{p}(x)},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>=</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |x|_{p}=p^{-v_{p}(x)},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7ca71c5a28376310a53da55a48475aae8540b10" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:13.961ex; height:3.676ex;" alt="{\displaystyle |x|_{p}=p^{-v_{p}(x)},}"></dd></dl> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{p}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{p}(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/963b72dbd91a6cf55797bc40f800ff77e3db1e5e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.326ex; height:3.009ex;" alt="{\displaystyle v_{p}(x)}"> is the p-adic valuation of x. The p-adic absolute value of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"> is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |0|_{p}=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>0</mn> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |0|_{p}=0.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d1d5397ccca35ad2edd8646408aee241666695e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:8.423ex; height:3.176ex;" alt="{\displaystyle |0|_{p}=0.}"> This is an absolute value that satisfies the <a href="/wiki/Strong_triangle_inequality" class="mw-redirect" title="Strong triangle inequality">strong triangle inequality</a> since, for every x and y one has <ul><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |x|_{p}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |x|_{p}=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24e0fe4140ad6740fa489518745e5d023591516b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:7.943ex; height:3.176ex;" alt="{\displaystyle |x|_{p}=0}"> if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=0;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>0</mn> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=0;}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a7217b7522dd29a6a4c9a2775f5c85c6c41ea11" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.237ex; height:2.509ex;" alt="{\displaystyle x=0;}"></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |x|_{p}\cdot |y|_{p}=|xy|_{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>y</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mi>y</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |x|_{p}\cdot |y|_{p}=|xy|_{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a81505d011ef7ebfe9790d4884cf4d01b077a61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:16.806ex; height:3.176ex;" alt="{\displaystyle |x|_{p}\cdot |y|_{p}=|xy|_{p}}"></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |x+y|_{p}\leq \max(|x|_{p},|y|_{p})\leq |x|_{p}+|y|_{p}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mo>+</mo> <mi>y</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>≤</mo> <mo movablelimits="true" form="prefix">max</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>y</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>y</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |x+y|_{p}\leq \max(|x|_{p},|y|_{p})\leq |x|_{p}+|y|_{p}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b9ff1963e96ebcec4ecca3b327b5de0488a7bcd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:38.913ex; height:3.176ex;" alt="{\displaystyle |x+y|_{p}\leq \max(|x|_{p},|y|_{p})\leq |x|_{p}+|y|_{p}.}"></li></ul> Moreover, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |x|_{p}\neq |y|_{p},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>≠</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>y</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |x|_{p}\neq |y|_{p},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc904a7659b3361f8869caad8b8aa6967633fc20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:10.936ex; height:3.176ex;" alt="{\displaystyle |x|_{p}\neq |y|_{p},}"> one has <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |x+y|_{p}=\max(|x|_{p},|y|_{p}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mo>+</mo> <mi>y</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>=</mo> <mo movablelimits="true" form="prefix">max</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>y</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |x+y|_{p}=\max(|x|_{p},|y|_{p}).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a634664d59273d59577e16c486829b7c69db86e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:25.784ex; height:3.176ex;" alt="{\displaystyle |x+y|_{p}=\max(|x|_{p},|y|_{p}).}"> This makes the p-adic numbers a <a href="/wiki/Metric_space" title="Metric space">metric space</a>, and even an <a href="/wiki/Ultrametric_space" title="Ultrametric space">ultrametric space</a>, with the p-adic distance defined by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{p}(x,y)=|x-y|_{p}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mo>−</mo> <mi>y</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{p}(x,y)=|x-y|_{p}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a6f06439fced98e86e140bebebb14c0392d781d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:19.02ex; height:3.176ex;" alt="{\displaystyle d_{p}(x,y)=|x-y|_{p}.}"> As a metric space, the p-adic numbers form the <a href="/wiki/Completion_(metric_space)" class="mw-redirect" title="Completion (metric space)">completion</a> of the rational numbers equipped with the p-adic absolute value. This provides another way for defining the p-adic numbers. However, the general construction of a completion can be simplified in this case, because the metric is defined by a discrete valuation (in short, one can extract from every <a href="/wiki/Cauchy_sequence" title="Cauchy sequence">Cauchy sequence</a> a subsequence such that the differences between two consecutive terms have strictly decreasing absolute values; such a subsequence is the sequence of the <a href="/wiki/Partial_sum" class="mw-redirect" title="Partial sum">partial sums</a> of a p-adic series, and thus a unique normalized p-adic series can be associated to every equivalence class of Cauchy sequences; so, for building the completion, it suffices to consider normalized p-adic series instead of equivalence classes of Cauchy sequences). As the metric is defined from a discrete valuation, every <a href="/wiki/Open_ball" class="mw-redirect" title="Open ball">open ball</a> is also <a href="/wiki/Closed_ball" class="mw-redirect" title="Closed ball">closed</a>. More precisely, the open ball <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{r}(x)=\{y\mid d_{p}(x,y)<r\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>y</mi> <mo>∣</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo><</mo> <mi>r</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{r}(x)=\{y\mid d_{p}(x,y)<r\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/849c8ecad67fb23cbb6974e3e4b9cfd217a33dcb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:26.136ex; height:3.009ex;" alt="{\displaystyle B_{r}(x)=\{y\mid d_{p}(x,y)<r\}}"> equals the closed ball <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{p^{-v}}[x]=\{y\mid d_{p}(x,y)\leq p^{-v}\},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>v</mi> </mrow> </msup> </mrow> </msub> <mo stretchy="false">[</mo> <mi>x</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>y</mi> <mo>∣</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>v</mi> </mrow> </msup> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{p^{-v}}[x]=\{y\mid d_{p}(x,y)\leq p^{-v}\},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ec235ca576f811368e1d965ac817ace63b0ff0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:30.631ex; height:3.176ex;" alt="{\displaystyle B_{p^{-v}}[x]=\{y\mid d_{p}(x,y)\leq p^{-v}\},}"> where v is the least integer such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p^{-v}<r.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>v</mi> </mrow> </msup> <mo><</mo> <mi>r</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p^{-v}<r.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e209c4c22250d212c7d9655d2ca49b13d101d11" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:8.361ex; height:2.843ex;" alt="{\displaystyle p^{-v}<r.}"> Similarly, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{r}[x]=B_{p^{-w}}(x),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mi>x</mi> <mo stretchy="false">]</mo> <mo>=</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>w</mi> </mrow> </msup> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{r}[x]=B_{p^{-w}}(x),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/129ea320fea4edec556f7cac1fef3e24170488d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.226ex; height:3.009ex;" alt="{\displaystyle B_{r}[x]=B_{p^{-w}}(x),}"> where w is the greatest integer such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p^{-w}>r.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>w</mi> </mrow> </msup> <mo>></mo> <mi>r</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p^{-w}>r.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/601b1fd1589aa6e706e720a7946976032b3d3652" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:8.74ex; height:2.843ex;" alt="{\displaystyle p^{-w}>r.}"> This implies that the p-adic numbers form a <a href="/wiki/Locally_compact_space" title="Locally compact space">locally compact space</a>, and the p-adic integers—that is, the ball <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{1}[0]=B_{p}(0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">[</mo> <mn>0</mn> <mo stretchy="false">]</mo> <mo>=</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{1}[0]=B_{p}(0)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66e7159da7964d6b254684541eb5542dcfd42203" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.168ex; height:3.009ex;" alt="{\displaystyle B_{1}[0]=B_{p}(0)}">—form a <a href="/wiki/Compact_space" title="Compact space">compact space</a>. <div class="mw-heading mw-heading2"><h2 id="p-adic_expansion_of_rational_numbers">p-adic expansion of rational numbers</h2>[<a href="/w/index.php?title=P-adic_number&action=edit&section=8" title="Edit section: p-adic expansion of rational numbers">edit</a>]</div> The <a href="/wiki/Decimal_expansion" class="mw-redirect" title="Decimal expansion">decimal expansion</a> of a positive <a href="/wiki/Rational_number" title="Rational number">rational number</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" 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}"> is its representation as a <a href="/wiki/Series_(mathematics)" title="Series (mathematics)">series</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=\sum _{i=k}^{\infty }a_{i}10^{-i},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=\sum _{i=k}^{\infty }a_{i}10^{-i},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edc747f95d8233c0d304c3cca39e8a77d6c37e3a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:14.969ex; height:6.843ex;" alt="{\displaystyle r=\sum _{i=k}^{\infty }a_{i}10^{-i},}"></dd></dl> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" 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}"> is an integer and each <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bc77764b2e74e64a63341054fa90f3e07db275f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.029ex; height:2.009ex;" alt="{\displaystyle a_{i}}"> is also an <a href="/wiki/Integer" title="Integer">integer</a> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq a_{i}<10.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>≤</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo><</mo> <mn>10.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\leq a_{i}<10.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98c779d4674546dae96aac3916937ff62c518e4c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.361ex; height:2.509ex;" alt="{\displaystyle 0\leq a_{i}<10.}"> This expansion can be computed by <a href="/wiki/Long_division" title="Long division">long division</a> of the numerator by the denominator, which is itself based on the following theorem: If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r={\tfrac {n}{d}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>n</mi> <mi>d</mi> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r={\tfrac {n}{d}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1863604be784c2d35cc4489d80251cd442c1a9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:5.969ex; height:3.343ex;" alt="{\displaystyle r={\tfrac {n}{d}}}"> is a rational number such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 10^{k}\leq r<10^{k+1},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>≤</mo> <mi>r</mi> <mo><</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 10^{k}\leq r<10^{k+1},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f87a5dd931cc35d5cbe628331b54f064f43ed7d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.82ex; height:3.009ex;" alt="{\displaystyle 10^{k}\leq r<10^{k+1},}"> there is an integer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<a<10,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo><</mo> <mi>a</mi> <mo><</mo> <mn>10</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0<a<10,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f0ee1c2c7ad292160107845a881662d299840f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.561ex; height:2.509ex;" alt="{\displaystyle 0<a<10,}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=a\,10^{k}+r',}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mi>a</mi> <mspace width="thinmathspace" /> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>+</mo> <msup> <mi>r</mi> <mo>′</mo> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=a\,10^{k}+r',}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9d4549d39210f78fe67f12f0e74e4da460fc7d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.398ex; height:3.009ex;" alt="{\displaystyle r=a\,10^{k}+r',}"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r'<10^{k}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>r</mi> <mo>′</mo> </msup> <mo><</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r'<10^{k}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edb7f1e97a75c292686825c625f56c5094e14d7c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.892ex; height:2.676ex;" alt="{\displaystyle r'<10^{k}.}"> The decimal expansion is obtained by repeatedly applying this result to the remainder <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>r</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r'}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20f4d06078c3550f2cd0812005ba6301d12cc4c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.733ex; height:2.509ex;" alt="{\displaystyle r'}"> which in the iteration assumes the role of the original rational number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" 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}">. The p-adic expansion of a rational number is defined similarly, but with a different division step. More precisely, given a fixed <a href="/wiki/Prime_number" title="Prime number">prime number</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" 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}">, every nonzero rational number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" 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}"> can be uniquely written as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=p^{k}{\tfrac {n}{d}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>n</mi> <mi>d</mi> </mfrac> </mstyle> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=p^{k}{\tfrac {n}{d}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/969d5e596e6931f5bbc7e28c44ee577b40311b8c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:8.874ex; height:3.676ex;" alt="{\displaystyle r=p^{k}{\tfrac {n}{d}},}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" 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}"> is a (possibly negative) integer, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" 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}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" 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}"> are <a href="/wiki/Coprime_integer" class="mw-redirect" title="Coprime integer">coprime integers</a> both coprime with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" 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}">, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" 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}"> is positive. The integer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" 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}"> is the p-adic valuation of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" 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}">, denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{p}(r),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{p}(r),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a3be3c684f7d4c613757b584931f382337aeba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.692ex; height:3.009ex;" alt="{\displaystyle v_{p}(r),}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p^{-k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p^{-k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67bc5c8000950bdb633a8c02bd1cbb44f59b1ca4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:3.626ex; height:3.009ex;" alt="{\displaystyle p^{-k}}"> is its p-adic absolute value, denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |r|_{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>r</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |r|_{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57de38fbfe8aa69428fc6911ea367768dd7bb61d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.401ex; height:3.176ex;" alt="{\displaystyle |r|_{p}}"> (the absolute value is small when the valuation is large). The division step consists of writing <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=a\,p^{k}+r'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mi>a</mi> <mspace width="thinmathspace" /> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>+</mo> <msup> <mi>r</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=a\,p^{k}+r'}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ff7a40ba4425971d8089ba7168309bcfd90f76c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.596ex; height:3.009ex;" alt="{\displaystyle r=a\,p^{k}+r'}"></dd></dl> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"> is an integer such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq a<p,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>≤</mo> <mi>a</mi> <mo><</mo> <mi>p</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\leq a<p,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73409b94faf202729f520fac3c316c83d0666ba0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.405ex; height:2.509ex;" alt="{\displaystyle 0\leq a<p,}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>r</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r'}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20f4d06078c3550f2cd0812005ba6301d12cc4c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.733ex; height:2.509ex;" alt="{\displaystyle r'}"> is either zero, or a rational number such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |r'|_{p}<p^{-k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mi>r</mi> <mo>′</mo> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo><</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |r'|_{p}<p^{-k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9bcc107f6439db3fc2cb351aa785460e83a0ae48" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:10.721ex; height:3.509ex;" alt="{\displaystyle |r'|_{p}<p^{-k}}"> (that is, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{p}(r')>k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msup> <mi>r</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo>></mo> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{p}(r')>k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e4192817893b7c267e1120a5a5ac15a7741d290" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.039ex; height:3.176ex;" alt="{\displaystyle v_{p}(r')>k}">). The <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" 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}">-adic expansion of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" 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}"> is the <a href="/wiki/Formal_power_series" title="Formal power series">formal power series</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=\sum _{i=k}^{\infty }a_{i}p^{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=\sum _{i=k}^{\infty }a_{i}p^{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e36ca98dc5782096e3c41548b2e7472ab28d4b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:11.888ex; height:6.843ex;" alt="{\displaystyle r=\sum _{i=k}^{\infty }a_{i}p^{i}}"></dd></dl> obtained by repeating indefinitely the <a href="#division_step">above</a> division step on successive remainders. In a p-adic expansion, all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bc77764b2e74e64a63341054fa90f3e07db275f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.029ex; height:2.009ex;" alt="{\displaystyle a_{i}}"> are integers such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq a_{i}<p.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>≤</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo><</mo> <mi>p</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\leq a_{i}<p.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3f17182b9eaa86a5121a307f7784b01140ea0f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.205ex; height:2.509ex;" alt="{\displaystyle 0\leq a_{i}<p.}"> If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=p^{k}{\tfrac {n}{1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>n</mi> <mn>1</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=p^{k}{\tfrac {n}{1}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae47fbf430f68af6fc80938d3938f6ed19bde821" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:8.228ex; height:3.509ex;" alt="{\displaystyle r=p^{k}{\tfrac {n}{1}}}"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n>0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27a6a5d982d54202a14f111cb8a49210501b2c96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n>0}">, the process stops eventually with a zero remainder; in this case, the series is completed by trailing terms with a zero coefficient, and is the representation of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" 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}"> in <a href="/wiki/Base-N" class="mw-redirect" title="Base-N">base-p</a>. The existence and the computation of the p-adic expansion of a rational number results from <a href="/wiki/B%C3%A9zout%27s_identity" title="Bézout's identity">Bézout's identity</a> in the following way. If, as above, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=p^{k}{\tfrac {n}{d}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>n</mi> <mi>d</mi> </mfrac> </mstyle> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=p^{k}{\tfrac {n}{d}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/969d5e596e6931f5bbc7e28c44ee577b40311b8c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:8.874ex; height:3.676ex;" alt="{\displaystyle r=p^{k}{\tfrac {n}{d}},}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" 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}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" 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}"> are coprime, there exist integers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle td+up=1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mi>d</mi> <mo>+</mo> <mi>u</mi> <mi>p</mi> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle td+up=1.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/246b768b0810699606a56ed2930732dd12ddf93d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.303ex; height:2.509ex;" alt="{\displaystyle td+up=1.}"> So <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=p^{k}{\tfrac {n}{d}}(td+up)=p^{k}nt+p^{k+1}{\frac {un}{d}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>n</mi> <mi>d</mi> </mfrac> </mstyle> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mi>d</mi> <mo>+</mo> <mi>u</mi> <mi>p</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mi>n</mi> <mi>t</mi> <mo>+</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>u</mi> <mi>n</mi> </mrow> <mi>d</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=p^{k}{\tfrac {n}{d}}(td+up)=p^{k}nt+p^{k+1}{\frac {un}{d}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e140ec22d136867524839ba6d6e77d4d52170e20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:36.429ex; height:4.843ex;" alt="{\displaystyle r=p^{k}{\tfrac {n}{d}}(td+up)=p^{k}nt+p^{k+1}{\frac {un}{d}}.}"></dd></dl> Then, the <a href="/wiki/Euclidean_division" title="Euclidean division">Euclidean division</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle nt}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle nt}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82c03b2d7710c484147f365d7073fd65e2827d67" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.234ex; height:2.009ex;" alt="{\displaystyle nt}"> by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" 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}"> gives <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle nt=qp+a,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mi>t</mi> <mo>=</mo> <mi>q</mi> <mi>p</mi> <mo>+</mo> <mi>a</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle nt=qp+a,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0c9da70e224bfb6eafa76dd360f8213421169bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.289ex; height:2.343ex;" alt="{\displaystyle nt=qp+a,}"></dd></dl> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq a<p.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>≤</mo> <mi>a</mi> <mo><</mo> <mi>p</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\leq a<p.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22dcdda6b48c6574bd19b524a654d4e8f9c76b60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.405ex; height:2.509ex;" alt="{\displaystyle 0\leq a<p.}"> This gives the division step as <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{lcl}r&=&p^{k}(qp+a)+p^{k+1}{\frac {un}{d}}\\&=&ap^{k}+p^{k+1}\,{\frac {qd+un}{d}},\\\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="left center left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>r</mi> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>q</mi> <mi>p</mi> <mo>+</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>u</mi> <mi>n</mi> </mrow> <mi>d</mi> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mo>=</mo> </mtd> <mtd> <mi>a</mi> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>+</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>q</mi> <mi>d</mi> <mo>+</mo> <mi>u</mi> <mi>n</mi> </mrow> <mi>d</mi> </mfrac> </mrow> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{lcl}r&=&p^{k}(qp+a)+p^{k+1}{\frac {un}{d}}\\&=&ap^{k}+p^{k+1}\,{\frac {qd+un}{d}},\\\end{array}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd247fdb129331fac456233c4c9627093abf472c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:28.592ex; height:7.843ex;" alt="{\displaystyle {\begin{array}{lcl}r&=&p^{k}(qp+a)+p^{k+1}{\frac {un}{d}}\\&=&ap^{k}+p^{k+1}\,{\frac {qd+un}{d}},\\\end{array}}}"></dd></dl> so that in the iteration <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r'=p^{k+1}\,{\frac {qd+un}{d}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>r</mi> <mo>′</mo> </msup> <mo>=</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>q</mi> <mi>d</mi> <mo>+</mo> <mi>u</mi> <mi>n</mi> </mrow> <mi>d</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r'=p^{k+1}\,{\frac {qd+un}{d}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e6219653af94be9659ed25e92ff8a680dfa495c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:18.264ex; height:5.509ex;" alt="{\displaystyle r'=p^{k+1}\,{\frac {qd+un}{d}}}"></dd></dl> is the new rational number. The uniqueness of the division step and of the whole p-adic expansion is easy: if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p^{k}a_{1}+p^{k+1}s_{1}=p^{k}a_{2}+p^{k+1}s_{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p^{k}a_{1}+p^{k+1}s_{1}=p^{k}a_{2}+p^{k+1}s_{2},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43161d59cdb95482e717e5a705a87798c277f58e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:31.607ex; height:3.009ex;" alt="{\displaystyle p^{k}a_{1}+p^{k+1}s_{1}=p^{k}a_{2}+p^{k+1}s_{2},}"> one has <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1}-a_{2}=p(s_{2}-s_{1}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mi>p</mi> <mo stretchy="false">(</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1}-a_{2}=p(s_{2}-s_{1}).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db6e8b998220ad120918b9ae57bb3debdbe18ef0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.262ex; height:2.843ex;" alt="{\displaystyle a_{1}-a_{2}=p(s_{2}-s_{1}).}"> This means <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" 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}"> divides <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1}-a_{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1}-a_{2}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d1fae5138293a59f1cc440721512330498d3971" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.055ex; height:2.343ex;" alt="{\displaystyle a_{1}-a_{2}.}"> Since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq a_{1}<p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>≤</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo><</mo> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\leq a_{1}<p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e03e6d9915ff7f28bf51f2ee0610cb6b3f861157" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.813ex; height:2.509ex;" alt="{\displaystyle 0\leq a_{1}<p}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq a_{2}<p,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>≤</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo><</mo> <mi>p</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\leq a_{2}<p,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3351867292792eebedd594c26685e5d1a3ba0185" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.46ex; height:2.509ex;" alt="{\displaystyle 0\leq a_{2}<p,}"> the following must be true: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq a_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>≤</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\leq a_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd6c4852b2230851269f30d4ab0473ac158d441a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.545ex; height:2.509ex;" alt="{\displaystyle 0\leq a_{1}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{2}<p.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo><</mo> <mi>p</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{2}<p.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e4b665fe8da46eb1c9f58c241c6cc41a46edc19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.199ex; height:2.176ex;" alt="{\displaystyle a_{2}<p.}"> Thus, one gets <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -p<a_{1}-a_{2}<p,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>p</mi> <mo><</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo><</mo> <mi>p</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -p<a_{1}-a_{2}<p,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9369438864e51432c27e9cb6190b8f7cd86a43f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.399ex; height:2.343ex;" alt="{\displaystyle -p<a_{1}-a_{2}<p,}"> and since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" 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}"> divides <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1}-a_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1}-a_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/297ac08b38ecf44d707b6595acfa9fad474f5648" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.408ex; height:2.343ex;" alt="{\displaystyle a_{1}-a_{2}}"> it must be that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1}=a_{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1}=a_{2}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c71b24ac3abd6950cd493942d1ccdc70c48205c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.313ex; height:2.009ex;" alt="{\displaystyle a_{1}=a_{2}.}"> The p-adic expansion of a rational number is a series that converges to the rational number, if one applies the definition of a <a href="/wiki/Convergent_series" title="Convergent series">convergent series</a> with the p-adic absolute value. In the standard p-adic notation, the digits are written in the same order as in a <a href="/wiki/Positional_notation#Base_of_the_numeral_system" title="Positional notation">standard base-p system</a>, namely with the powers of the base increasing to the left. This means that the production of the digits is reversed and the limit happens on the left hand side. The p-adic expansion of a rational number is eventually <a href="/wiki/Periodic_function" title="Periodic function">periodic</a>. <a href="/wiki/Converse_(logic)" title="Converse (logic)">Conversely</a>, a series <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \sum _{i=k}^{\infty }a_{i}p^{i},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \sum _{i=k}^{\infty }a_{i}p^{i},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9cb4f62d5aea3480af7ffb6232bb22767f345207" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.421ex; height:3.176ex;" alt="{\textstyle \sum _{i=k}^{\infty }a_{i}p^{i},}"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq a_{i}<p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>≤</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo><</mo> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\leq a_{i}<p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f575e443f9bd3ed528e6134963643b49bff8e3a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.558ex; height:2.509ex;" alt="{\displaystyle 0\leq a_{i}<p}"> converges (for the p-adic absolute value) to a rational number <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> it is eventually periodic; in this case, the series is the p-adic expansion of that rational number. The <a href="/wiki/Mathematical_proof" title="Mathematical proof">proof</a> is similar to that of the similar result for <a href="/wiki/Repeating_decimal" title="Repeating decimal">repeating decimals</a>. <div class="mw-heading mw-heading3"><h3 id="Example">Example</h3>[<a href="/w/index.php?title=P-adic_number&action=edit&section=9" title="Edit section: Example">edit</a>]</div> Let us compute the 5-adic expansion of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{3}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{3}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a372c26fd0fc7fe00bec4ec23f3b4ae7df729c07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:2.305ex; height:3.676ex;" alt="{\displaystyle {\tfrac {1}{3}}.}"> Bézout's identity for 5 and the denominator 3 is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\cdot 3+(-1)\cdot 5=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>⋅</mo> <mn>3</mn> <mo>+</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>⋅</mo> <mn>5</mn> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\cdot 3+(-1)\cdot 5=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7bc0762b0738e932d7b3319fbb665f318edcf919" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.727ex; height:2.843ex;" alt="{\displaystyle 2\cdot 3+(-1)\cdot 5=1}"> (for larger examples, this can be computed with the <a href="/wiki/Extended_Euclidean_algorithm" title="Extended Euclidean algorithm">extended Euclidean algorithm</a>). Thus <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{3}}=2+5({\frac {-1}{3}}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mo>=</mo> <mn>2</mn> <mo>+</mo> <mn>5</mn> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mn>1</mn> </mrow> <mn>3</mn> </mfrac> </mrow> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{3}}=2+5({\frac {-1}{3}}).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13a294ba6f1ce38c91baf86b27004e72613caf00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:16.525ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{3}}=2+5({\frac {-1}{3}}).}"></dd></dl> For the next step, one has to expand <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -1/3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -1/3}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d26272f09f96204eaaaa4236a61b57e3fe9f1be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.296ex; height:2.843ex;" alt="{\displaystyle -1/3}"> (the factor 5 has to be viewed as a "<a href="/wiki/Arithmetic_shift" title="Arithmetic shift">shift</a>" of the p-adic valuation, similar to the basis of any number expansion, and thus it should not be itself expanded). To expand <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -1/3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -1/3}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d26272f09f96204eaaaa4236a61b57e3fe9f1be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.296ex; height:2.843ex;" alt="{\displaystyle -1/3}">, we start from the same Bézout's identity and multiply it by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/704fb0427140d054dd267925495e78164fee9aac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle -1}">, giving <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\frac {1}{3}}=-2+{\frac {5}{3}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mn>2</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>3</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\frac {1}{3}}=-2+{\frac {5}{3}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d55d123b24b6404fe35fe906fc5d30bf14c84453" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:15.362ex; height:5.176ex;" alt="{\displaystyle -{\frac {1}{3}}=-2+{\frac {5}{3}}.}"></dd></dl> The "integer part" <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46e5b5b462e546b1d3d7e5f9a23efece405b2e78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle -2}"> is not in the right interval. So, one has to use <a href="/wiki/Euclidean_division" title="Euclidean division">Euclidean division</a> by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29483407999b8763f0ea335cf715a6a5e809f44b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 5}"> for getting <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -2=3-1\cdot 5,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>2</mn> <mo>=</mo> <mn>3</mn> <mo>−</mo> <mn>1</mn> <mo>⋅</mo> <mn>5</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -2=3-1\cdot 5,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91b2712718554cb7919eaf2bffbd5d403afd9d9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.723ex; height:2.509ex;" alt="{\displaystyle -2=3-1\cdot 5,}"> giving <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\frac {1}{3}}=3-5+{\frac {5}{3}}=3-{\frac {10}{3}}=3+5({\frac {-2}{3}}),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mo>=</mo> <mn>3</mn> <mo>−</mo> <mn>5</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>3</mn> </mfrac> </mrow> <mo>=</mo> <mn>3</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>10</mn> <mn>3</mn> </mfrac> </mrow> <mo>=</mo> <mn>3</mn> <mo>+</mo> <mn>5</mn> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mn>2</mn> </mrow> <mn>3</mn> </mfrac> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\frac {1}{3}}=3-5+{\frac {5}{3}}=3-{\frac {10}{3}}=3+5({\frac {-2}{3}}),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/779f40c81494db4448ceb1423c6a5535e9a317a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:41.698ex; height:5.176ex;" alt="{\displaystyle -{\frac {1}{3}}=3-5+{\frac {5}{3}}=3-{\frac {10}{3}}=3+5({\frac {-2}{3}}),}"></dd></dl> and the expansion in the first step becomes <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{3}}=2+5\cdot (3+5\cdot ({\frac {-2}{3}}))=2+3\cdot 5+{\frac {-2}{3}}\cdot 5^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mo>=</mo> <mn>2</mn> <mo>+</mo> <mn>5</mn> <mo>⋅</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mo>+</mo> <mn>5</mn> <mo>⋅</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mn>2</mn> </mrow> <mn>3</mn> </mfrac> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <mo>+</mo> <mn>3</mn> <mo>⋅</mo> <mn>5</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mn>2</mn> </mrow> <mn>3</mn> </mfrac> </mrow> <mo>⋅</mo> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{3}}=2+5\cdot (3+5\cdot ({\frac {-2}{3}}))=2+3\cdot 5+{\frac {-2}{3}}\cdot 5^{2}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eac8d54a4f8dae4401d6143b9bbbec8c300acc75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:48.506ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{3}}=2+5\cdot (3+5\cdot ({\frac {-2}{3}}))=2+3\cdot 5+{\frac {-2}{3}}\cdot 5^{2}.}"></dd></dl> Similarly, one has <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\frac {2}{3}}=1-{\frac {5}{3}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>3</mn> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\frac {2}{3}}=1-{\frac {5}{3}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71b706877718904d93d3325bc554e2b43a6c9e66" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.553ex; height:5.176ex;" alt="{\displaystyle -{\frac {2}{3}}=1-{\frac {5}{3}},}"></dd></dl> and <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{3}}=2+3\cdot 5+1\cdot 5^{2}+{\frac {-1}{3}}\cdot 5^{3}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mo>=</mo> <mn>2</mn> <mo>+</mo> <mn>3</mn> <mo>⋅</mo> <mn>5</mn> <mo>+</mo> <mn>1</mn> <mo>⋅</mo> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mn>1</mn> </mrow> <mn>3</mn> </mfrac> </mrow> <mo>⋅</mo> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{3}}=2+3\cdot 5+1\cdot 5^{2}+{\frac {-1}{3}}\cdot 5^{3}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0853231a320ed08c0e1c9c75c4cdb86e9f807fbb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:32.192ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{3}}=2+3\cdot 5+1\cdot 5^{2}+{\frac {-1}{3}}\cdot 5^{3}.}"></dd></dl> As the "remainder" <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\tfrac {1}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\tfrac {1}{3}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edb3e88f738057777c96ebd0a7a7c7c1d1934696" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:3.466ex; height:3.676ex;" alt="{\displaystyle -{\tfrac {1}{3}}}"> has already been found, the process can be continued easily, giving coefficients <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/991e33c6e207b12546f15bdfee8b5726eafbbb2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 3}"> for <a href="/wiki/Parity_(mathematics)" title="Parity (mathematics)">odd</a> powers of five, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"> for <a href="/wiki/Parity_(mathematics)" title="Parity (mathematics)">even</a> powers. Or in the standard 5-adic notation <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{3}}=\ldots 1313132_{5}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mo>=</mo> <mo>…</mo> <msub> <mn>1313132</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{3}}=\ldots 1313132_{5}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77a64ed5a0d3a026876dbeb296ef3912faaae00f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:17.399ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{3}}=\ldots 1313132_{5}}"></dd></dl> with the <a href="/wiki/Ellipsis" title="Ellipsis">ellipsis</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ldots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b8619532e44ee1ccae3ab03405a6885260d09ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.723ex; height:0.843ex;" alt="{\displaystyle \ldots }"> on the left hand side. <div class="mw-heading mw-heading3"><h3 id="Positional_notation">Positional notation</h3>[<a href="/w/index.php?title=P-adic_number&action=edit&section=10" title="Edit section: Positional notation">edit</a>]</div> It is possible to use a <a href="/wiki/Positional_notation" title="Positional notation">positional notation</a> similar to that which is used to represent numbers in <a href="/wiki/Radix" title="Radix">base</a> p. Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \sum _{i=k}^{\infty }a_{i}p^{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \sum _{i=k}^{\infty }a_{i}p^{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbe62cb94fff59f2058172778bad0519c3d3e8b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.774ex; height:3.176ex;" alt="{\textstyle \sum _{i=k}^{\infty }a_{i}p^{i}}"> be a normalized p-adic series, i.e. each <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bc77764b2e74e64a63341054fa90f3e07db275f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.029ex; height:2.009ex;" alt="{\displaystyle a_{i}}"> is an integer in the interval <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,p-1].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>p</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">]</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [0,p-1].}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d784d3c09f6c16f2be810fc933b13c064cf70df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.309ex; height:2.843ex;" alt="{\displaystyle [0,p-1].}"> One can suppose that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\leq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>≤</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\leq 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4fa6cb13b5d8c1dfc91921ead9a8d89c347aad5" 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\leq 0}"> by setting <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{i}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{i}=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1d2e283d81c74e9c6742dcbbcad8c622ef0c3c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.29ex; height:2.509ex;" alt="{\displaystyle a_{i}=0}"> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq i<k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>≤</mo> <mi>i</mi> <mo><</mo> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\leq i<k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac8047b84849528b307835b4debef40d14a60ffb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.373ex; height:2.343ex;" alt="{\displaystyle 0\leq i<k}"> (if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k>0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27b3af208b148139eefc03f0f80fa94c38c5af45" 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>0}">), and adding the resulting zero terms to the series. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\geq 0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>≥</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\geq 0,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c10f12b66e1ef733d7d4a58e6ac0dc86550bf94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.119ex; height:2.509ex;" alt="{\displaystyle k\geq 0,}"> the positional notation consists of writing the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bc77764b2e74e64a63341054fa90f3e07db275f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.029ex; height:2.009ex;" alt="{\displaystyle a_{i}}"> consecutively, ordered by decreasing values of i, often with p appearing on the right as an index: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ldots a_{n}\ldots a_{1}{a_{0}}_{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>…</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>…</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ldots a_{n}\ldots a_{1}{a_{0}}_{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ce9cb73197e93a12804d9cf61a7328f61d0874c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.683ex; height:2.343ex;" alt="{\displaystyle \ldots a_{n}\ldots a_{1}{a_{0}}_{p}}"></dd></dl> So, the computation of the <a href="#Example">example above</a> shows that <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{3}}=\ldots 1313132_{5},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mo>=</mo> <mo>…</mo> <msub> <mn>1313132</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{3}}=\ldots 1313132_{5},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4e226e48cdcc81d34c913826dc44b5d799e05c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:18.046ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{3}}=\ldots 1313132_{5},}"></dd></dl> and <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {25}{3}}=\ldots 131313200_{5}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>25</mn> <mn>3</mn> </mfrac> </mrow> <mo>=</mo> <mo>…</mo> <msub> <mn>131313200</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {25}{3}}=\ldots 131313200_{5}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8143440235096ce4728f5b2e89b99c95439e0743" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:21.533ex; height:5.176ex;" alt="{\displaystyle {\frac {25}{3}}=\ldots 131313200_{5}.}"></dd></dl> When <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k<0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo><</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k<0,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/360bdeec6d5ccbc863a0e29e8108555711a98ac2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.119ex; height:2.509ex;" alt="{\displaystyle k<0,}"> a separating dot is added before the digits with negative index, and, if the index p is present, it appears just after the separating dot. For example, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{15}}=\ldots 3131313._{5}2,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>15</mn> </mfrac> </mrow> <mo>=</mo> <mo>…</mo> <msub> <mn>3131313.</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mn>2</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{15}}=\ldots 3131313._{5}2,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab3e558cbd675445e3014f9f80d129faf2e04a8f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:21.017ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{15}}=\ldots 3131313._{5}2,}"></dd></dl> and <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{75}}=\ldots 1313131._{5}32.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>75</mn> </mfrac> </mrow> <mo>=</mo> <mo>…</mo> <msub> <mn>1313131.</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mn>32.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{75}}=\ldots 1313131._{5}32.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a6dca33d2cf1c26dd1c398b570eefdd4a0c91f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:22.18ex; height:5.343ex;" alt="{\displaystyle {\frac {1}{75}}=\ldots 1313131._{5}32.}"></dd></dl> If a p-adic representation is finite on the left (that is, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{i}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{i}=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1d2e283d81c74e9c6742dcbbcad8c622ef0c3c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.29ex; height:2.509ex;" alt="{\displaystyle a_{i}=0}"> for large values of i), then it has the value of a nonnegative rational number of the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle np^{v},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle np^{v},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc7623a4405153c4adb524097423a2d1f996695e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.241ex; height:2.676ex;" alt="{\displaystyle np^{v},}"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n,v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>,</mo> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n,v}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d95601a6c27c62237f4480a4e4f875ef0c324f1c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.556ex; height:2.009ex;" alt="{\displaystyle n,v}"> integers. These rational numbers are exactly the nonnegative rational numbers that have a finite representation in <a href="/wiki/Radix" title="Radix">base</a> p. For these rational numbers, the two representations are the same. <div class="mw-heading mw-heading2"><h2 id="Modular_properties">Modular properties</h2>[<a href="/w/index.php?title=P-adic_number&action=edit&section=11" title="Edit section: Modular properties">edit</a>]</div> The <a href="/wiki/Quotient_ring" title="Quotient ring">quotient ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} _{p}/p^{n}\mathbb {Z} _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} _{p}/p^{n}\mathbb {Z} _{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d97b008b5d7ced177ca33fa8fddba98e3a396e42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.769ex; height:3.009ex;" alt="{\displaystyle \mathbb {Z} _{p}/p^{n}\mathbb {Z} _{p}}"> may be identified with the <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} /p^{n}\mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} /p^{n}\mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f568eec3d80f812c3d02a19a445f088e888c180" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.651ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} /p^{n}\mathbb {Z} }"> of the integers <a href="/wiki/Modular_arithmetic" title="Modular arithmetic">modulo</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p^{n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p^{n}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8ab248d6bd65126e21bd19002f669086dcf30c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:3.124ex; height:2.676ex;" alt="{\displaystyle p^{n}.}"> This can be shown by remarking that every p-adic integer, represented by its normalized p-adic series, is congruent modulo <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6a7a7e74ae90ab94f01e1629177758fb68b423b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.477ex; height:2.676ex;" alt="{\displaystyle p^{n}}"> with its <a href="/wiki/Partial_sum" class="mw-redirect" title="Partial sum">partial sum</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \sum _{i=0}^{n-1}a_{i}p^{i},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \sum _{i=0}^{n-1}a_{i}p^{i},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e1d606f955789c2fd797545f47d2323ab76717a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.805ex; height:3.509ex;" alt="{\textstyle \sum _{i=0}^{n-1}a_{i}p^{i},}"> whose value is an integer in the interval <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,p^{n}-1].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo stretchy="false">]</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [0,p^{n}-1].}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7bb48ca2618ca918f1b5c06f0fa6991b711ec163" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.528ex; height:2.843ex;" alt="{\displaystyle [0,p^{n}-1].}"> A straightforward verification shows that this defines a <a href="/wiki/Ring_isomorphism" class="mw-redirect" title="Ring isomorphism">ring isomorphism</a> from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} _{p}/p^{n}\mathbb {Z} _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} _{p}/p^{n}\mathbb {Z} _{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d97b008b5d7ced177ca33fa8fddba98e3a396e42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.769ex; height:3.009ex;" alt="{\displaystyle \mathbb {Z} _{p}/p^{n}\mathbb {Z} _{p}}"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} /p^{n}\mathbb {Z} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} /p^{n}\mathbb {Z} .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f6a705e0c4f6b3e0cbf77a60e0367f1e2ed1c56" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.298ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} /p^{n}\mathbb {Z} .}"> The <a href="/wiki/Inverse_limit" title="Inverse limit">inverse limit</a> of the rings <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} _{p}/p^{n}\mathbb {Z} _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} _{p}/p^{n}\mathbb {Z} _{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d97b008b5d7ced177ca33fa8fddba98e3a396e42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.769ex; height:3.009ex;" alt="{\displaystyle \mathbb {Z} _{p}/p^{n}\mathbb {Z} _{p}}"> is defined as the ring formed by the sequences <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{0},a_{1},\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{0},a_{1},\ldots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee0930c2573669657257678785c2c0e97ea38afc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.359ex; height:2.009ex;" alt="{\displaystyle a_{0},a_{1},\ldots }"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{i}\in \mathbb {Z} /p^{i}\mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{i}\in \mathbb {Z} /p^{i}\mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/837d12f2f05c4ad54595d0956d9e4a2a0db32310" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.102ex; height:3.176ex;" alt="{\displaystyle a_{i}\in \mathbb {Z} /p^{i}\mathbb {Z} }"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle a_{i}\equiv a_{i+1}{\pmod {p^{i}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>≡</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0.444em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle a_{i}\equiv a_{i+1}{\pmod {p^{i}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d33ab9c63288e560b8e8a072eb6294184e106dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.621ex; height:3.009ex;" alt="{\textstyle a_{i}\equiv a_{i+1}{\pmod {p^{i}}}}"> for every i. The mapping that maps a normalized p-adic series to the sequence of its partial sums is a ring isomorphism from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} _{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbc1df7227ef11fe88dccd2dae3adc7bbdeae5f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.609ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} _{p}}"> to the inverse limit of the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} _{p}/p^{n}\mathbb {Z} _{p}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} _{p}/p^{n}\mathbb {Z} _{p}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a3a0186a7f358ebb15d8b3fcc00d93a048f3013" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.416ex; height:3.009ex;" alt="{\displaystyle \mathbb {Z} _{p}/p^{n}\mathbb {Z} _{p}.}"> This provides another way for defining p-adic integers (<a href="/wiki/Up_to" title="Up to">up to</a> an isomorphism). This definition of p-adic integers is specially useful for practical computations, as allowing building p-adic integers by successive approximations. For example, for computing the p-adic (multiplicative) inverse of an integer, one can use <a href="/wiki/Newton%27s_method" title="Newton's method">Newton's method</a>, starting from the inverse modulo p; then, each Newton step computes the inverse modulo <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle p^{n^{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle p^{n^{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11cb92ac7e57d07cc6f704b4397d0000642e3d7c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:3.309ex; height:3.176ex;" alt="{\textstyle p^{n^{2}}}"> from the inverse modulo <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle p^{n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle p^{n}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f93c398e50fdd5643e70b93424e56d27e72c4e4d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:3.124ex; height:2.509ex;" alt="{\textstyle p^{n}.}"> The same method can be used for computing the p-adic <a href="/wiki/Square_root" title="Square root">square root</a> of an integer that is a <a href="/wiki/Quadratic_residue" title="Quadratic residue">quadratic residue</a> modulo p. This seems to be the fastest known method for testing whether a large integer is a square: it suffices to test whether the given integer is the square of the value found in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} _{p}/p^{n}\mathbb {Z} _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} _{p}/p^{n}\mathbb {Z} _{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d97b008b5d7ced177ca33fa8fddba98e3a396e42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.769ex; height:3.009ex;" alt="{\displaystyle \mathbb {Z} _{p}/p^{n}\mathbb {Z} _{p}}">. Applying Newton's method to find the square root requires <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle p^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle p^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/482c2e2e7b4f46628a60fb529d7effaa867e915e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.477ex; height:2.509ex;" alt="{\textstyle p^{n}}"> to be larger than twice the given integer, which is quickly satisfied. <a href="/wiki/Hensel_lifting" class="mw-redirect" title="Hensel lifting">Hensel lifting</a> is a similar method that allows to "lift" the factorization modulo p of a polynomial with integer coefficients to a factorization modulo <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle p^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle p^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/482c2e2e7b4f46628a60fb529d7effaa867e915e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.477ex; height:2.509ex;" alt="{\textstyle p^{n}}"> for large values of n. This is commonly used by <a href="/wiki/Polynomial_factorization" class="mw-redirect" title="Polynomial factorization">polynomial factorization</a> algorithms. <div class="mw-heading mw-heading2"><h2 id="Notation">Notation</h2>[<a href="/w/index.php?title=P-adic_number&action=edit&section=12" title="Edit section: Notation">edit</a>]</div> There are several different conventions for writing p-adic expansions. So far this article has used a notation for p-adic expansions in which <a href="/wiki/Exponentiation" title="Exponentiation">powers</a> of p increase from right to left. With this right-to-left notation the 3-adic expansion of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{5}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>5</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{5}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2bee75ce3139189cbe7440ed3c627f618fbf0ebf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:2.305ex; height:3.676ex;" alt="{\displaystyle {\tfrac {1}{5}},}"> for example, is written as <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{5}}=\dots 121012102_{3}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>5</mn> </mfrac> </mrow> <mo>=</mo> <mo>…</mo> <msub> <mn>121012102</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{5}}=\dots 121012102_{3}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1665cf94da1daee3f987c72fbb4c35f3ba29d48e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:20.371ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{5}}=\dots 121012102_{3}.}"></dd></dl> When performing arithmetic in this notation, digits are <a href="/wiki/Carry_(arithmetic)" title="Carry (arithmetic)">carried</a> to the left. It is also possible to write p-adic expansions so that the powers of p increase from left to right, and digits are carried to the right. With this left-to-right notation the 3-adic expansion of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{5}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>5</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{5}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7cba10df075ad5061b732c02e1101b8f9237558e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:1.658ex; height:3.676ex;" alt="{\displaystyle {\tfrac {1}{5}}}"> is <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{5}}=2.01210121\dots _{3}{\mbox{ or }}{\frac {1}{15}}=20.1210121\dots _{3}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>5</mn> </mfrac> </mrow> <mo>=</mo> <mn>2.01210121</mn> <mo>…</mo> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext> or </mtext> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>15</mn> </mfrac> </mrow> <mo>=</mo> <mn>20.1210121</mn> <mo>…</mo> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{5}}=2.01210121\dots _{3}{\mbox{ or }}{\frac {1}{15}}=20.1210121\dots _{3}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/551d140a00ed22743bdd4f30aa283bf8d885aac4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:46.56ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{5}}=2.01210121\dots _{3}{\mbox{ or }}{\frac {1}{15}}=20.1210121\dots _{3}.}"></dd></dl> p-adic expansions may be written with <a href="/wiki/Signed-digit_representation" title="Signed-digit representation">other sets of digits</a> instead of {0, 1, ..., p − 1}. For example, the 3-adic expansion of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{5}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>5</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{5}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7cba10df075ad5061b732c02e1101b8f9237558e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:1.658ex; height:3.676ex;" alt="{\displaystyle {\tfrac {1}{5}}}"> can be written using <a href="/wiki/Balanced_ternary" title="Balanced ternary">balanced ternary</a> digits {1, 0, 1}, with 1 representing negative one, as <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{5}}=\dots {\underline {1}}11{\underline {11}}11{\underline {11}}11{\underline {1}}_{\text{3}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>5</mn> </mfrac> </mrow> <mo>=</mo> <mo>…</mo> <mrow class="MJX-TeXAtom-ORD"> <munder> <mn>1</mn> <mo>_</mo> </munder> </mrow> <mn>11</mn> <mrow class="MJX-TeXAtom-ORD"> <munder> <mn>11</mn> <mo>_</mo> </munder> </mrow> <mn>11</mn> <mrow class="MJX-TeXAtom-ORD"> <munder> <mn>11</mn> <mo>_</mo> </munder> </mrow> <mn>11</mn> <msub> <mrow class="MJX-TeXAtom-ORD"> <munder> <mn>1</mn> <mo>_</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>3</mtext> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{5}}=\dots {\underline {1}}11{\underline {11}}11{\underline {11}}11{\underline {1}}_{\text{3}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c5b5f56730a3b1662a6e2ba31349374c04b30dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:23.863ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{5}}=\dots {\underline {1}}11{\underline {11}}11{\underline {11}}11{\underline {1}}_{\text{3}}.}"></dd></dl> In fact any set of p integers which are in distinct <a href="/wiki/Residue_class" class="mw-redirect" title="Residue class">residue classes</a> modulo p may be used as p-adic digits. In number theory, <a href="/wiki/Witt_vector#Motivation" title="Witt vector">Teichmüller representatives</a> are sometimes used as digits.<a href="#cite_note-3">[2]</a> <style data-mw-deduplicate="TemplateStyles:r1238216509">.mw-parser-output .vanchor>:target~.vanchor-text{background-color:#b1d2ff}@media screen{html.skin-theme-clientpref-night .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}</style>Quote notation is a variant of the p-adic representation of <a href="/wiki/Rational_number" title="Rational number">rational numbers</a> that was proposed in 1979 by <a href="/wiki/Eric_Hehner" title="Eric Hehner">Eric Hehner</a> and <a href="/wiki/Nigel_Horspool" title="Nigel Horspool">Nigel Horspool</a> for implementing on computers the (exact) arithmetic with these numbers.<a href="#cite_note-4">[3]</a> <div class="mw-heading mw-heading2"><h2 id="Cardinality">Cardinality</h2>[<a href="/w/index.php?title=P-adic_number&action=edit&section=13" title="Edit section: Cardinality">edit</a>]</div> Both <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} _{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbc1df7227ef11fe88dccd2dae3adc7bbdeae5f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.609ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} _{p}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} _{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35f44bc6894c682710705f3ea74f33042e0acc3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.867ex; height:2.843ex;" alt="{\displaystyle \mathbb {Q} _{p}}"> are <a href="/wiki/Uncountable_set" title="Uncountable set">uncountable</a> and have the <a href="/wiki/Cardinality_of_the_continuum" title="Cardinality of the continuum">cardinality of the continuum</a>.<a href="#cite_note-5">[4]</a> For <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} _{p},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} _{p},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7967f468b6a942f77dd96ada0815be530fcda626" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.256ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} _{p},}"> this results from the p-adic representation, which defines a <a href="/wiki/Bijection" title="Bijection">bijection</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} _{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbc1df7227ef11fe88dccd2dae3adc7bbdeae5f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.609ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} _{p}}"> on the <a href="/wiki/Power_set" title="Power set">power set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{0,\ldots ,p-1\}^{\mathbb {N} }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>p</mi> <mo>−</mo> <mn>1</mn> <msup> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{0,\ldots ,p-1\}^{\mathbb {N} }.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6befd2c851845de5af62150f59fd2aa232074ee1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.904ex; height:3.176ex;" alt="{\displaystyle \{0,\ldots ,p-1\}^{\mathbb {N} }.}"> For <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} _{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35f44bc6894c682710705f3ea74f33042e0acc3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.867ex; height:2.843ex;" alt="{\displaystyle \mathbb {Q} _{p}}"> this results from its expression as a <a href="/wiki/Countably_infinite" class="mw-redirect" title="Countably infinite">countably infinite</a> <a href="/wiki/Union_(set_theory)" title="Union (set theory)">union</a> of copies of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} _{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbc1df7227ef11fe88dccd2dae3adc7bbdeae5f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.609ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} _{p}}">: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} _{p}=\bigcup _{i=0}^{\infty }{\frac {1}{p^{i}}}\mathbb {Z} _{p}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>⋃</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} _{p}=\bigcup _{i=0}^{\infty }{\frac {1}{p^{i}}}\mathbb {Z} _{p}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/762936a5b1fb27213fc9b2c0edaf559cc20c36ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:15.082ex; height:6.843ex;" alt="{\displaystyle \mathbb {Q} _{p}=\bigcup _{i=0}^{\infty }{\frac {1}{p^{i}}}\mathbb {Z} _{p}.}"></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Algebraic_closure">Algebraic closure</h2>[<a href="/w/index.php?title=P-adic_number&action=edit&section=14" title="Edit section: Algebraic closure">edit</a>]</div> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} _{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35f44bc6894c682710705f3ea74f33042e0acc3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.867ex; height:2.843ex;" alt="{\displaystyle \mathbb {Q} _{p}}"> contains <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" 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} }"> and is a field of <a href="/wiki/Characteristic_(algebra)" title="Characteristic (algebra)">characteristic</a> 0. Because 0 can be written as sum of squares,<a href="#cite_note-6">[5]</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} _{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35f44bc6894c682710705f3ea74f33042e0acc3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.867ex; height:2.843ex;" alt="{\displaystyle \mathbb {Q} _{p}}"> cannot be turned into an <a href="/wiki/Ordered_field#Orderability_of_fields" title="Ordered field">ordered field</a>. The field of <a href="/wiki/Real_numbers" class="mw-redirect" title="Real numbers">real numbers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" 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} }"> has only a single proper <a href="/wiki/Algebraic_extension" title="Algebraic extension">algebraic extension</a>: the <a href="/wiki/Complex_numbers" class="mw-redirect" title="Complex numbers">complex numbers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" 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 {C} }">. In other words, this <a href="/wiki/Quadratic_extension" class="mw-redirect" title="Quadratic extension">quadratic extension</a> is already <a href="/wiki/Algebraically_closed_field" title="Algebraically closed field">algebraically closed</a>. By contrast, the <a href="/wiki/Algebraic_closure" title="Algebraic closure">algebraic closure</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} _{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35f44bc6894c682710705f3ea74f33042e0acc3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.867ex; height:2.843ex;" alt="{\displaystyle \mathbb {Q} _{p}}">, denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {\mathbb {Q} _{p}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo accent="false">¯</mo> </mover> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {\mathbb {Q} _{p}}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d123b037d0a0948fb45875aabfd95807918624" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.629ex; height:3.676ex;" alt="{\displaystyle {\overline {\mathbb {Q} _{p}}},}"> has infinite degree,<a href="#cite_note-7">[6]</a> that is, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} _{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35f44bc6894c682710705f3ea74f33042e0acc3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.867ex; height:2.843ex;" alt="{\displaystyle \mathbb {Q} _{p}}"> has infinitely many inequivalent algebraic extensions. Also contrasting the case of real numbers, although there is a unique extension of the p-adic valuation to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {\mathbb {Q} _{p}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo accent="false">¯</mo> </mover> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {\mathbb {Q} _{p}}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d123b037d0a0948fb45875aabfd95807918624" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.629ex; height:3.676ex;" alt="{\displaystyle {\overline {\mathbb {Q} _{p}}},}"> the latter is not (metrically) complete.<a href="#cite_note-8">[7]</a><a href="#cite_note-C149-9">[8]</a> Its (metric) completion is called <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} _{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6f9e7692267c8a29ed4d848c3421eee929c23c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.737ex; height:2.843ex;" alt="{\displaystyle \mathbb {C} _{p}}"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega _{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/733a4950dd5b62722f046ecb52222c07dcaed57d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.737ex; height:2.843ex;" alt="{\displaystyle \Omega _{p}}">.<a href="#cite_note-C149-9">[8]</a><a href="#cite_note-K13-10">[9]</a> Here an end is reached, as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} _{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6f9e7692267c8a29ed4d848c3421eee929c23c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.737ex; height:2.843ex;" alt="{\displaystyle \mathbb {C} _{p}}"> is algebraically closed.<a href="#cite_note-C149-9">[8]</a><a href="#cite_note-11">[10]</a> However unlike <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" 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 {C} }"> this field is not <a href="/wiki/Locally_compact" class="mw-redirect" title="Locally compact">locally compact</a>.<a href="#cite_note-K13-10">[9]</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} _{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6f9e7692267c8a29ed4d848c3421eee929c23c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.737ex; height:2.843ex;" alt="{\displaystyle \mathbb {C} _{p}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" 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 {C} }"> are isomorphic as rings,<a href="#cite_note-12">[11]</a> so we may regard <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} _{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6f9e7692267c8a29ed4d848c3421eee929c23c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.737ex; height:2.843ex;" alt="{\displaystyle \mathbb {C} _{p}}"> as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" 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 {C} }"> endowed with an exotic metric. The proof of existence of such a field isomorphism relies on the <a href="/wiki/Axiom_of_choice" title="Axiom of choice">axiom of choice</a>, and does not provide an explicit example of such an isomorphism (that is, it is not <a href="/wiki/Constructive_proof" title="Constructive proof">constructive</a>). If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"> is any finite <a href="/wiki/Galois_extension" title="Galois extension">Galois extension</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} _{p},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} _{p},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34d194e3e8fce9335ed524db967666b4f02fb523" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.514ex; height:2.843ex;" alt="{\displaystyle \mathbb {Q} _{p},}">, the <a href="/wiki/Galois_group" title="Galois group">Galois group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Gal} \left(K/\mathbb {Q} _{p}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Gal</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Gal} \left(K/\mathbb {Q} _{p}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ee941bf5a7cd1b93f3f5486431980ebaaf5daac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.859ex; height:3.176ex;" alt="{\displaystyle \operatorname {Gal} \left(K/\mathbb {Q} _{p}\right)}"> is <a href="/wiki/Solvable_group" title="Solvable group">solvable</a>. Thus, the Galois group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Gal} \left({\overline {\mathbb {Q} _{p}}}/\mathbb {Q} _{p}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Gal</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Gal} \left({\overline {\mathbb {Q} _{p}}}/\mathbb {Q} _{p}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a08dccfbc4ae9d31e59870d0731df390b41634c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.421ex; height:4.843ex;" alt="{\displaystyle \operatorname {Gal} \left({\overline {\mathbb {Q} _{p}}}/\mathbb {Q} _{p}\right)}"> is <a href="/wiki/Prosolvable" class="mw-redirect" title="Prosolvable">prosolvable</a>. <div class="mw-heading mw-heading2"><h2 id="Multiplicative_group">Multiplicative group</h2>[<a href="/w/index.php?title=P-adic_number&action=edit&section=15" title="Edit section: Multiplicative group">edit</a>]</div> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} _{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35f44bc6894c682710705f3ea74f33042e0acc3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.867ex; height:2.843ex;" alt="{\displaystyle \mathbb {Q} _{p}}"> contains the n-th <a href="/wiki/Cyclotomic_field" title="Cyclotomic field">cyclotomic field</a> (n > 2) if and only if n | p − 1.<a href="#cite_note-13">[12]</a> For instance, the n-th cyclotomic field is a subfield of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} _{13}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} _{13}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9daa0400bff1269e8b00dc9a3e4bb8232ae8ddff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.684ex; height:2.676ex;" alt="{\displaystyle \mathbb {Q} _{13}}"> if and only if n = 1, 2, 3, 4, 6, or 12. In particular, there is no multiplicative p-<a href="/wiki/Torsion_(algebra)" title="Torsion (algebra)">torsion</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} _{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35f44bc6894c682710705f3ea74f33042e0acc3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.867ex; height:2.843ex;" alt="{\displaystyle \mathbb {Q} _{p}}"> if p > 2. Also, −1 is the only non-trivial torsion element in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} _{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d6705440d223bf7db3c6a49932b2583e6d59b59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.862ex; height:2.676ex;" alt="{\displaystyle \mathbb {Q} _{2}}">. Given a <a href="/wiki/Natural_number" title="Natural number">natural number</a> k, the <a href="/wiki/Index_(group_theory)" class="mw-redirect" title="Index (group theory)">index</a> of the multiplicative group of the k-th powers of the non-zero elements of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} _{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35f44bc6894c682710705f3ea74f33042e0acc3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.867ex; height:2.843ex;" alt="{\displaystyle \mathbb {Q} _{p}}"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} _{p}^{\times }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>×</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} _{p}^{\times }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11bc65ad8b3b0169f3558eb7ff5b90587314603c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.319ex; height:3.176ex;" alt="{\displaystyle \mathbb {Q} _{p}^{\times }}"> is finite. The number <a href="/wiki/E_(mathematical_constant)" title="E (mathematical constant)">e</a>, defined as the sum of <a href="/wiki/Reciprocal_(mathematics)" class="mw-redirect" title="Reciprocal (mathematics)">reciprocals</a> of <a href="/wiki/Factorial" title="Factorial">factorials</a>, is not a member of any p-adic field; but <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{p}\in \mathbb {Q} _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>∈</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{p}\in \mathbb {Q} _{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3fc9e51b97d70b296764f3d9f5ae93044330fb32" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.851ex; height:3.009ex;" alt="{\displaystyle e^{p}\in \mathbb {Q} _{p}}"> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\neq 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>≠</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\neq 2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/884d62fa2ca44dc88b2e2020d0921bd7bdcf1d2e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:5.52ex; height:2.676ex;" alt="{\displaystyle p\neq 2}">. For p = 2 one must take at least the fourth power.<a href="#cite_note-14">[13]</a> (Thus a number with similar properties as e — namely a p-th root of ep — is a member of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} _{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35f44bc6894c682710705f3ea74f33042e0acc3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.867ex; height:2.843ex;" alt="{\displaystyle \mathbb {Q} _{p}}"> for all p.) <div class="mw-heading mw-heading2"><h2 id="Local–global_principle">Local–global principle</h2>[<a href="/w/index.php?title=P-adic_number&action=edit&section=16" title="Edit section: Local–global principle">edit</a>]</div> <a href="/wiki/Helmut_Hasse" title="Helmut Hasse">Helmut Hasse</a>'s <a href="/wiki/Hasse_principle" title="Hasse principle">local–global principle</a> is said to hold for an equation if it can be solved over the rational numbers <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> it can be solved over the real numbers and over the p-adic numbers for every prime p. This principle holds, for example, for equations given by <a href="/wiki/Quadratic_form" title="Quadratic form">quadratic forms</a>, but fails for higher polynomials in several indeterminates. <div class="mw-heading mw-heading2"><h2 id="Rational_arithmetic_with_Hensel_lifting">Rational arithmetic with Hensel lifting</h2>[<a href="/w/index.php?title=P-adic_number&action=edit&section=17" title="Edit section: Rational arithmetic with Hensel lifting">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Hensel_lifting" class="mw-redirect" title="Hensel lifting">Hensel lifting</a></div> <div class="mw-heading mw-heading2"><h2 id="Generalizations_and_related_concepts">Generalizations and related concepts</h2>[<a href="/w/index.php?title=P-adic_number&action=edit&section=18" title="Edit section: Generalizations and related concepts">edit</a>]</div> The reals and the p-adic numbers are the completions of the rationals; it is also possible to complete other fields, for instance general <a href="/wiki/Algebraic_number_field" title="Algebraic number field">algebraic number fields</a>, in an analogous way. This will be described now. Suppose D is a <a href="/wiki/Dedekind_domain" title="Dedekind domain">Dedekind domain</a> and E is its <a href="/wiki/Field_of_fractions" title="Field of fractions">field of fractions</a>. Pick a non-zero <a href="/wiki/Prime_ideal" title="Prime ideal">prime ideal</a> P of D. If x is a non-zero element of E, then xD is a <a href="/wiki/Fractional_ideal" title="Fractional ideal">fractional ideal</a> and can be uniquely factored as a product of positive and negative powers of non-zero prime ideals of D. We write ordP(x) for the exponent of P in this factorization, and for any choice of number c greater than 1 we can set <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |x|_{P}=c^{-\!\operatorname {ord} _{P}(x)}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo>=</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mspace width="negativethinmathspace" /> <msub> <mi>ord</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |x|_{P}=c^{-\!\operatorname {ord} _{P}(x)}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55b332e0ef062cc927bbdd81180ad66fc327203f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.119ex; height:3.343ex;" alt="{\displaystyle |x|_{P}=c^{-\!\operatorname {ord} _{P}(x)}.}"></dd></dl> Completing with respect to this absolute value |⋅|P yields a field EP, the proper generalization of the field of p-adic numbers to this setting. The choice of c does not change the completion (different choices yield the same concept of Cauchy sequence, so the same completion). It is convenient, when the <a href="/wiki/Residue_field" title="Residue field">residue field</a> D/P is finite, to take for c the size of D/P. For example, when E is a <a href="/wiki/Number_field" class="mw-redirect" title="Number field">number field</a>, <a href="/wiki/Ostrowski%27s_theorem" title="Ostrowski's theorem">Ostrowski's theorem</a> says that every non-trivial <a href="/wiki/Absolute_value_(algebra)" title="Absolute value (algebra)">non-Archimedean absolute value</a> on E arises as some |⋅|P. The remaining non-trivial absolute values on E arise from the different embeddings of E into the real or complex numbers. (In fact, the non-Archimedean absolute values can be considered as simply the different embeddings of E into the fields Cp, thus putting the description of all the non-trivial absolute values of a number field on a common footing.) Often, one needs to simultaneously keep track of all the above-mentioned completions when E is a number field (or more generally a <a href="/wiki/Global_field" title="Global field">global field</a>), which are seen as encoding "local" information. This is accomplished by <a href="/wiki/Adele_ring" title="Adele ring">adele rings</a> and <a href="/wiki/Idele_group" class="mw-redirect" title="Idele group">idele groups</a>. p-adic integers can be extended to <a href="/wiki/Solenoid_(mathematics)#p-adic_solenoids" title="Solenoid (mathematics)">p-adic solenoids</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {T} _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">T</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {T} _{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ffef9c58fb5d7e73913e28f1438a7d969f4f993" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.609ex; height:2.843ex;" alt="{\displaystyle \mathbb {T} _{p}}">. There is a map from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {T} _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">T</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {T} _{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ffef9c58fb5d7e73913e28f1438a7d969f4f993" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.609ex; height:2.843ex;" alt="{\displaystyle \mathbb {T} _{p}}"> to the <a href="/wiki/Circle_group" title="Circle group">circle group</a> whose fibers are the p-adic integers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} _{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbc1df7227ef11fe88dccd2dae3adc7bbdeae5f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.609ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} _{p}}">, in analogy to how there is a map from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" 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} }"> to the circle whose fibers are <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" 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} }">. <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=P-adic_number&action=edit&section=19" title="Edit section: See also">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col"> <ul><li><a href="/wiki/Non-Archimedean_(disambiguation)" class="mw-redirect mw-disambig" title="Non-Archimedean (disambiguation)">Non-Archimedean</a></li> <li><a href="/wiki/P-adic_quantum_mechanics" title="P-adic quantum mechanics">p-adic quantum mechanics</a></li> <li><a href="/wiki/P-adic_Hodge_theory" title="P-adic Hodge theory">p-adic Hodge theory</a></li> <li><a href="/wiki/P-adic_Teichmuller_theory" class="mw-redirect" title="P-adic Teichmuller theory">p-adic Teichmuller theory</a></li> <li><a href="/wiki/P-adic_analysis" title="P-adic analysis">p-adic analysis</a></li> <li><a href="/wiki/P-adic_valuation" title="P-adic valuation">p-adic valuation</a></li> <li><a href="/wiki/1_%2B_2_%2B_4_%2B_8_%2B_..." class="mw-redirect" title="1 + 2 + 4 + 8 + ...">1 + 2 + 4 + 8 + ...</a></li> <li><a href="/wiki/Bijective_numeration" title="Bijective numeration">k-adic notation</a></li> <li><a href="/wiki/C-minimal_theory" title="C-minimal theory">C-minimal theory</a></li> <li><a href="/wiki/Hensel%27s_lemma" title="Hensel's lemma">Hensel's lemma</a></li> <li><a href="/wiki/Locally_compact_field" title="Locally compact field">Locally compact field</a></li> <li><a href="/wiki/Mahler%27s_theorem" title="Mahler's theorem">Mahler's theorem</a></li> <li><a href="/wiki/Profinite_integer" title="Profinite integer">Profinite integer</a></li> <li><a href="/wiki/Volkenborn_integral" title="Volkenborn integral">Volkenborn integral</a></li> <li><a href="/wiki/Two%27s_complement" title="Two's complement">Two's complement</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Footnotes">Footnotes</h2>[<a href="/w/index.php?title=P-adic_number&action=edit&section=20" title="Edit section: Footnotes">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Notes">Notes</h3>[<a href="/w/index.php?title=P-adic_number&action=edit&section=21" title="Edit section: Notes">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-2"><a href="#cite_ref-2">^</a> Translator's introduction, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Qxte2mhlEOYC&pg=PA35">page 35</a>: "Indeed, with hindsight it becomes apparent that a <a href="/wiki/Discrete_valuation" title="Discrete valuation">discrete valuation</a> is behind Kummer's concept of ideal numbers."(<a href="#CITEREFDedekindWeber2012">Dedekind & Weber 2012</a>, p. 35) </li> </ol></div></div> <div class="mw-heading mw-heading3"><h3 id="Citations">Citations</h3>[<a href="/w/index.php?title=P-adic_number&action=edit&section=22" title="Edit section: Citations">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1">^</a> (<a href="#CITEREFHensel1897">Hensel 1897</a>) </li> <li id="cite_note-3"><a href="#cite_ref-3">^</a> (<a href="#CITEREFHazewinkel2009">Hazewinkel 2009</a>, p. 342) </li> <li id="cite_note-4"><a href="#cite_ref-4">^</a> (<a href="#CITEREFHehnerHorspool1979">Hehner & Horspool 1979</a>, pp. 124–134) </li> <li id="cite_note-5"><a href="#cite_ref-5">^</a> (<a href="#CITEREFRobert2000">Robert 2000</a>, Chapter 1 Section 1.1) </li> <li id="cite_note-6"><a href="#cite_ref-6">^</a> According to <a href="/wiki/Hensel%27s_lemma#Examples" title="Hensel's lemma">Hensel's lemma</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} _{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d6705440d223bf7db3c6a49932b2583e6d59b59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.862ex; height:2.676ex;" alt="{\displaystyle \mathbb {Q} _{2}}"> contains a square root of −7, so that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{2}+1^{2}+1^{2}+1^{2}+\left({\sqrt {-7}}\right)^{2}=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>7</mn> </msqrt> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{2}+1^{2}+1^{2}+1^{2}+\left({\sqrt {-7}}\right)^{2}=0,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e0caaa012312f14c8bd25a1be420fdfed273470" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:33.227ex; height:3.676ex;" alt="{\displaystyle 2^{2}+1^{2}+1^{2}+1^{2}+\left({\sqrt {-7}}\right)^{2}=0,}"> and if p > 2 then also by Hensel's lemma <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} _{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35f44bc6894c682710705f3ea74f33042e0acc3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.867ex; height:2.843ex;" alt="{\displaystyle \mathbb {Q} _{p}}"> contains a square root of 1 − p, thus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (p-1)\times 1^{2}+\left({\sqrt {1-p}}\right)^{2}=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>p</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>×</mo> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <mi>p</mi> </msqrt> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (p-1)\times 1^{2}+\left({\sqrt {1-p}}\right)^{2}=0.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a87b5268511bc2e999bcaa5f212174dc6ce416e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:30.467ex; height:3.843ex;" alt="{\displaystyle (p-1)\times 1^{2}+\left({\sqrt {1-p}}\right)^{2}=0.}"> </li> <li id="cite_note-7"><a href="#cite_ref-7">^</a> (<a href="#CITEREFGouvêa1997">Gouvêa 1997</a>, Corollary 5.3.10) </li> <li id="cite_note-8"><a href="#cite_ref-8">^</a> (<a href="#CITEREFGouvêa1997">Gouvêa 1997</a>, Theorem 5.7.4) </li> <li id="cite_note-C149-9">^ <a href="#cite_ref-C149_9-0">a</a> <a href="#cite_ref-C149_9-1">b</a> <a href="#cite_ref-C149_9-2">c</a> (<a href="#CITEREFCassels1986">Cassels 1986</a>, p. 149) </li> <li id="cite_note-K13-10">^ <a href="#cite_ref-K13_10-0">a</a> <a href="#cite_ref-K13_10-1">b</a> (<a href="#CITEREFKoblitz1980">Koblitz 1980</a>, p. 13) </li> <li id="cite_note-11"><a href="#cite_ref-11">^</a> (<a href="#CITEREFGouvêa1997">Gouvêa 1997</a>, Proposition 5.7.8) </li> <li id="cite_note-12"><a href="#cite_ref-12">^</a> Two algebraically closed fields are isomorphic if and only if they have the same characteristic and transcendence degree (see, for example Lang’s Algebra X §1), and both <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} _{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6f9e7692267c8a29ed4d848c3421eee929c23c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.737ex; height:2.843ex;" alt="{\displaystyle \mathbb {C} _{p}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" 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 {C} }"> have characteristic zero and the cardinality of the continuum. </li> <li id="cite_note-13"><a href="#cite_ref-13">^</a> (<a href="#CITEREFGouvêa1997">Gouvêa 1997</a>, Proposition 3.4.2) </li> <li id="cite_note-14"><a href="#cite_ref-14">^</a> (<a href="#CITEREFRobert2000">Robert 2000</a>, Section 4.1) </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2>[<a href="/w/index.php?title=P-adic_number&action=edit&section=23" title="Edit section: References">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin" style=""> <ul><li><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFCassels1986" class="citation cs2"><a href="/wiki/J._W._S._Cassels" title="J. W. S. Cassels">Cassels, J. W. S.</a> (1986), Local Fields, London Mathematical Society Student Texts, vol. 3, <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-31525-5" title="Special:BookSources/0-521-31525-5"><bdi>0-521-31525-5</bdi></a>, <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0595.12006">0595.12006</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Local+Fields&rft.series=London+Mathematical+Society+Student+Texts&rft.pub=Cambridge+University+Press&rft.date=1986&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0595.12006%23id-name%3DZbl&rft.isbn=0-521-31525-5&rft.aulast=Cassels&rft.aufirst=J.+W.+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AP-adic+number" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDedekindWeber2012" class="citation cs2"><a href="/wiki/Richard_Dedekind" title="Richard Dedekind">Dedekind, Richard</a>; <a href="/wiki/Heinrich_Martin_Weber" title="Heinrich Martin Weber">Weber, Heinrich</a> (2012), Theory of Algebraic Functions of One Variable, History of mathematics, vol. 39, American Mathematical Society, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-8330-3" title="Special:BookSources/978-0-8218-8330-3"><bdi>978-0-8218-8330-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Theory+of+Algebraic+Functions+of+One+Variable&rft.series=History+of+mathematics&rft.pub=American+Mathematical+Society&rft.date=2012&rft.isbn=978-0-8218-8330-3&rft.aulast=Dedekind&rft.aufirst=Richard&rft.au=Weber%2C+Heinrich&rfr_id=info%3Asid%2Fen.wikipedia.org%3AP-adic+number" class="Z3988">. — Translation into English by <a href="/wiki/John_Stillwell" title="John Stillwell">John Stillwell</a> of Theorie der algebraischen Functionen einer Veränderlichen (1882).</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGouvêa1994" class="citation cs2">Gouvêa, F. Q. (March 1994), "A Marvelous Proof", <a href="/wiki/American_Mathematical_Monthly" class="mw-redirect" title="American Mathematical Monthly">American Mathematical Monthly</a>, 101 (3): 203–222, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2975598">10.2307/2975598</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2975598">2975598</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Mathematical+Monthly&rft.atitle=A+Marvelous+Proof&rft.volume=101&rft.issue=3&rft.pages=203-222&rft.date=1994-03&rft_id=info%3Adoi%2F10.2307%2F2975598&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2975598%23id-name%3DJSTOR&rft.aulast=Gouv%C3%AAa&rft.aufirst=F.+Q.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AP-adic+number" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGouvêa1997" class="citation cs2">Gouvêa, Fernando Q. (1997), p-adic Numbers: An Introduction (2nd ed.), Springer, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3-540-62911-4" title="Special:BookSources/3-540-62911-4"><bdi>3-540-62911-4</bdi></a>, <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0874.11002">0874.11002</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=p-adic+Numbers%3A+An+Introduction&rft.edition=2nd&rft.pub=Springer&rft.date=1997&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0874.11002%23id-name%3DZbl&rft.isbn=3-540-62911-4&rft.aulast=Gouv%C3%AAa&rft.aufirst=Fernando+Q.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AP-adic+number" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHazewinkel2009" class="citation cs2">Hazewinkel, M., ed. (2009), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=yimXZ-7L9ZoC&pg=PA342">Handbook of Algebra</a>, vol. 6, North Holland, p. 342, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-444-53257-2" title="Special:BookSources/978-0-444-53257-2"><bdi>978-0-444-53257-2</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Handbook+of+Algebra&rft.pages=342&rft.pub=North+Holland&rft.date=2009&rft.isbn=978-0-444-53257-2&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DyimXZ-7L9ZoC%26pg%3DPA342&rfr_id=info%3Asid%2Fen.wikipedia.org%3AP-adic+number" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHehnerHorspool1979" class="citation cs2"><a href="/wiki/Eric_C._R._Hehner" class="mw-redirect" title="Eric C. R. Hehner">Hehner, Eric C. R.</a>; Horspool, R. Nigel (1979), <a rel="nofollow" class="external text" href="https://www.researchgate.net/publication/220617770">"A new representation of the rational numbers for fast easy arithmetic"</a>, <a href="/wiki/SIAM_Journal_on_Computing" title="SIAM Journal on Computing">SIAM Journal on Computing</a>, 8 (2): 124–134, <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.64.7714">10.1.1.64.7714</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1137%2F0208011">10.1137/0208011</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=SIAM+Journal+on+Computing&rft.atitle=A+new+representation+of+the+rational+numbers+for+fast+easy+arithmetic&rft.volume=8&rft.issue=2&rft.pages=124-134&rft.date=1979&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.64.7714%23id-name%3DCiteSeerX&rft_id=info%3Adoi%2F10.1137%2F0208011&rft.aulast=Hehner&rft.aufirst=Eric+C.+R.&rft.au=Horspool%2C+R.+Nigel&rft_id=https%3A%2F%2Fwww.researchgate.net%2Fpublication%2F220617770&rfr_id=info%3Asid%2Fen.wikipedia.org%3AP-adic+number" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHensel1897" class="citation cs2"><a href="/wiki/Kurt_Hensel" title="Kurt Hensel">Hensel, Kurt</a> (1897), <a rel="nofollow" class="external text" href="http://www.digizeitschriften.de/resolveppn/GDZPPN00211612X&L=2">"Über eine neue Begründung der Theorie der algebraischen Zahlen"</a>, Jahresbericht der Deutschen Mathematiker-Vereinigung, 6 (3): 83–88</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Jahresbericht+der+Deutschen+Mathematiker-Vereinigung&rft.atitle=%C3%9Cber+eine+neue+Begr%C3%BCndung+der+Theorie+der+algebraischen+Zahlen&rft.volume=6&rft.issue=3&rft.pages=83-88&rft.date=1897&rft.aulast=Hensel&rft.aufirst=Kurt&rft_id=http%3A%2F%2Fwww.digizeitschriften.de%2Fresolveppn%2FGDZPPN00211612X%26L%3D2&rfr_id=info%3Asid%2Fen.wikipedia.org%3AP-adic+number" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKelley2008" class="citation cs2"><a href="/wiki/John_Leroy_Kelley" class="mw-redirect" title="John Leroy Kelley">Kelley, John L.</a> (2008) [1955], General Topology, New York: Ishi Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-923891-55-8" title="Special:BookSources/978-0-923891-55-8"><bdi>978-0-923891-55-8</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=General+Topology&rft.place=New+York&rft.pub=Ishi+Press&rft.date=2008&rft.isbn=978-0-923891-55-8&rft.aulast=Kelley&rft.aufirst=John+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AP-adic+number" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKoblitz1980" class="citation cs2"><a href="/wiki/Neal_Koblitz" title="Neal Koblitz">Koblitz, Neal</a> (1980), p-adic analysis: a short course on recent work, London Mathematical Society Lecture Note Series, vol. 46, <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-28060-5" title="Special:BookSources/0-521-28060-5"><bdi>0-521-28060-5</bdi></a>, <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0439.12011">0439.12011</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=p-adic+analysis%3A+a+short+course+on+recent+work&rft.series=London+Mathematical+Society+Lecture+Note+Series&rft.pub=Cambridge+University+Press&rft.date=1980&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0439.12011%23id-name%3DZbl&rft.isbn=0-521-28060-5&rft.aulast=Koblitz&rft.aufirst=Neal&rfr_id=info%3Asid%2Fen.wikipedia.org%3AP-adic+number" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRobert2000" class="citation cs2">Robert, Alain M. (2000), A Course in p-adic Analysis, Springer, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-98669-3" title="Special:BookSources/0-387-98669-3"><bdi>0-387-98669-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Course+in+p-adic+Analysis&rft.pub=Springer&rft.date=2000&rft.isbn=0-387-98669-3&rft.aulast=Robert&rft.aufirst=Alain+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AP-adic+number" class="Z3988"></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2>[<a href="/w/index.php?title=P-adic_number&action=edit&section=24" title="Edit section: Further reading">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239549316"><div class="refbegin" style=""> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBachman1964" class="citation cs2">Bachman, George (1964), Introduction to p-adic Numbers and Valuation Theory, Academic Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-12-070268-1" title="Special:BookSources/0-12-070268-1"><bdi>0-12-070268-1</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+p-adic+Numbers+and+Valuation+Theory&rft.pub=Academic+Press&rft.date=1964&rft.isbn=0-12-070268-1&rft.aulast=Bachman&rft.aufirst=George&rfr_id=info%3Asid%2Fen.wikipedia.org%3AP-adic+number" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBorevichShafarevich1986" class="citation cs2"><a href="/wiki/Zenon_Ivanovich_Borevich" class="mw-redirect" title="Zenon Ivanovich Borevich">Borevich, Z. I.</a>; <a href="/wiki/Igor_Rostislavovich_Shafarevich" class="mw-redirect" title="Igor Rostislavovich Shafarevich">Shafarevich, I. R.</a> (1986), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=njgVUjjO-EAC">Number Theory</a>, Pure and Applied Mathematics, vol. 20, Boston, MA: Academic Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-12-117851-2" title="Special:BookSources/978-0-12-117851-2"><bdi>978-0-12-117851-2</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0195803">0195803</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Number+Theory&rft.place=Boston%2C+MA&rft.series=Pure+and+Applied+Mathematics&rft.pub=Academic+Press&rft.date=1986&rft.isbn=978-0-12-117851-2&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0195803%23id-name%3DMR&rft.aulast=Borevich&rft.aufirst=Z.+I.&rft.au=Shafarevich%2C+I.+R.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DnjgVUjjO-EAC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AP-adic+number" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKoblitz1984" class="citation cs2"><a href="/wiki/Neal_Koblitz" title="Neal Koblitz">Koblitz, Neal</a> (1984), p-adic Numbers, p-adic Analysis, and Zeta-Functions, <a href="/wiki/Graduate_Texts_in_Mathematics" title="Graduate Texts in Mathematics">Graduate Texts in Mathematics</a>, vol. 58 (2nd ed.), Springer, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-96017-1" title="Special:BookSources/0-387-96017-1"><bdi>0-387-96017-1</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=p-adic+Numbers%2C+p-adic+Analysis%2C+and+Zeta-Functions&rft.series=Graduate+Texts+in+Mathematics&rft.edition=2nd&rft.pub=Springer&rft.date=1984&rft.isbn=0-387-96017-1&rft.aulast=Koblitz&rft.aufirst=Neal&rfr_id=info%3Asid%2Fen.wikipedia.org%3AP-adic+number" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMahler1981" class="citation cs2"><a href="/wiki/Kurt_Mahler" title="Kurt Mahler">Mahler, Kurt</a> (1981), <a rel="nofollow" class="external text" href="https://archive.org/details/padicnumbersthei0000mahl">p-adic numbers and their functions</a>, Cambridge Tracts in Mathematics, vol. 76 (2nd ed.), Cambridge: <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-23102-7" title="Special:BookSources/0-521-23102-7"><bdi>0-521-23102-7</bdi></a>, <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0444.12013">0444.12013</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=p-adic+numbers+and+their+functions&rft.place=Cambridge&rft.series=Cambridge+Tracts+in+Mathematics&rft.edition=2nd&rft.pub=Cambridge+University+Press&rft.date=1981&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0444.12013%23id-name%3DZbl&rft.isbn=0-521-23102-7&rft.aulast=Mahler&rft.aufirst=Kurt&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fpadicnumbersthei0000mahl&rfr_id=info%3Asid%2Fen.wikipedia.org%3AP-adic+number" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSteen1978" class="citation cs2"><a href="/wiki/Lynn_Arthur_Steen" class="mw-redirect" title="Lynn Arthur Steen">Steen, Lynn Arthur</a> (1978), <a href="/wiki/Counterexamples_in_Topology" title="Counterexamples in Topology">Counterexamples in Topology</a>, Dover, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-68735-X" title="Special:BookSources/0-486-68735-X"><bdi>0-486-68735-X</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Counterexamples+in+Topology&rft.pub=Dover&rft.date=1978&rft.isbn=0-486-68735-X&rft.aulast=Steen&rft.aufirst=Lynn+Arthur&rfr_id=info%3Asid%2Fen.wikipedia.org%3AP-adic+number" class="Z3988"></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2>[<a href="/w/index.php?title=P-adic_number&action=edit&section=25" title="Edit section: External links">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@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}}</style><div class="side-box side-box-right plainlinks sistersitebox"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></div> <div class="side-box-text plainlist">Wikimedia Commons has media related to <a href="https://commons.wikimedia.org/wiki/Category:P-adic_numbers" class="extiw" title="commons:Category:P-adic numbers">P-adic numbers</a>.</div></div> </div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/p-adicNumber.html">"p-adic Number"</a>. <a href="/wiki/MathWorld" title="MathWorld">MathWorld</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=p-adic+Number&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2Fp-adicNumber.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AP-adic+number" class="Z3988"></li> <li><a rel="nofollow" class="external text" href="http://www.encyclopediaofmath.org/index.php/P-adic_number">p-adic number</a> at <a href="/wiki/Encyclopaedia_of_Mathematics" class="mw-redirect" title="Encyclopaedia of Mathematics">Springer On-line Encyclopaedia of Mathematics</a></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist 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.navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Number_systems" title="Template:Number systems"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Number_systems" title="Template talk:Number systems"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Number_systems" title="Special:EditPage/Template:Number systems"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Number_systems" style="font-size:114%;margin:0 4em"><a href="/wiki/Number" title="Number">Number</a> systems</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Sets of <a href="/wiki/Definable_number" class="mw-redirect" title="Definable number">definable numbers</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Natural_number" title="Natural number">Natural numbers</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed" 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 {N} }">)</li> <li><a href="/wiki/Integer" title="Integer">Integers</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" 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} }">)</li> <li><a href="/wiki/Rational_number" title="Rational number">Rational numbers</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" 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} }">)</li> <li><a href="/wiki/Constructible_number" title="Constructible number">Constructible numbers</a></li> <li><a href="/wiki/Algebraic_number" title="Algebraic number">Algebraic numbers</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {A} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3fb423c16a5f403edbaf66438b75e7a36e725af6" 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 {A} }">)</li> <li><a href="/wiki/Closed-form_expression#Closed-form_number" title="Closed-form expression">Closed-form numbers</a></li> <li><a href="/wiki/Period_(algebraic_geometry)" title="Period (algebraic geometry)">Periods</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10d6ec962de5797ba4f161c40e66dca74ae95cc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.704ex; height:2.176ex;" alt="{\displaystyle {\mathcal {P}}}">)</li> <li><a href="/wiki/Computable_number" title="Computable number">Computable numbers</a></li> <li><a href="/wiki/Definable_real_number#Definability_in_arithmetic" title="Definable real number">Arithmetical numbers</a></li> <li><a href="/wiki/Definable_real_number#Definability_in_models_of_ZFC" title="Definable real number">Set-theoretically definable numbers</a></li> <li><a href="/wiki/Gaussian_integer" title="Gaussian integer">Gaussian integers</a> <ul><li><a href="/wiki/Gaussian_rational" title="Gaussian rational">Gaussian rationals</a></li></ul></li> <li><a href="/wiki/Eisenstein_integer" title="Eisenstein integer">Eisenstein integers</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Composition_algebra" title="Composition algebra">Composition algebras</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Division_algebra" title="Division algebra">Division algebras</a>: <a href="/wiki/Real_number" title="Real number">Real numbers</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" 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} }">)</li> <li><a href="/wiki/Complex_number" title="Complex number">Complex numbers</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" 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 {C} }">)</li> <li><a href="/wiki/Quaternion" title="Quaternion">Quaternions</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {H} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">H</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {H} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e050965453c42bcc6bd544546703c836bdafeac9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \mathbb {H} }">)</li> <li><a href="/wiki/Octonion" title="Octonion">Octonions</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {O} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">O</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {O} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1ed2664a4fe515e6fbed25a7193ce663b82920c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \mathbb {O} }">)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Split types</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li>Over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" 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} }">:</li> <li><a href="/wiki/Split-complex_number" title="Split-complex number">Split-complex numbers</a></li> <li><a href="/wiki/Split-quaternion" title="Split-quaternion">Split-quaternions</a></li> <li><a href="/wiki/Split-octonion" title="Split-octonion">Split-octonions</a> Over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" 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 {C} }">:</li> <li><a href="/wiki/Bicomplex_number" title="Bicomplex number">Bicomplex numbers</a></li> <li><a href="/wiki/Biquaternion" title="Biquaternion">Biquaternions</a></li> <li><a href="/wiki/Bioctonion" title="Bioctonion">Bioctonions</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other <a href="/wiki/Hypercomplex_number" title="Hypercomplex number">hypercomplex</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Dual_number" title="Dual number">Dual numbers</a></li> <li><a href="/wiki/Dual_quaternion" title="Dual quaternion">Dual quaternions</a></li> <li><a href="/wiki/Dual-complex_number" class="mw-redirect" title="Dual-complex number">Dual-complex numbers</a></li> <li><a href="/wiki/Hyperbolic_quaternion" title="Hyperbolic quaternion">Hyperbolic quaternions</a></li> <li><a href="/wiki/Sedenion" title="Sedenion">Sedenions</a>  (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {S} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">S</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {S} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f9d5874c5d7f68eba1cec9da9ccbe53903303bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.293ex; height:2.176ex;" alt="{\displaystyle \mathbb {S} }">)</li> <li><a href="/wiki/Trigintaduonion" title="Trigintaduonion">Trigintaduonions</a>  (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {T} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">T</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {T} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c039979935c00b3b216cbb065999207872677f6" 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 {T} }">)</li> <li><a href="/wiki/Split-biquaternion" title="Split-biquaternion">Split-biquaternions</a></li> <li><a href="/wiki/Multicomplex_number" title="Multicomplex number">Multicomplex numbers</a></li> <li><a href="/wiki/Geometric_algebra" title="Geometric algebra">Geometric algebra</a>/<a href="/wiki/Clifford_algebra" title="Clifford algebra">Clifford algebra</a> <ul><li><a href="/wiki/Algebra_of_physical_space" title="Algebra of physical space">Algebra of physical space</a></li> <li><a href="/wiki/Spacetime_algebra" title="Spacetime algebra">Spacetime algebra</a></li> <li><a href="/wiki/Plane-based_geometric_algebra" title="Plane-based geometric algebra">Plane-based geometric algebra</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Infinity" title="Infinity">Infinities</a> and <a href="/wiki/Infinitesimal" title="Infinitesimal">infinitesimals</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cardinal_number" title="Cardinal number">Cardinal numbers</a></li> <li><a href="/wiki/Extended_natural_numbers" title="Extended natural numbers">Extended natural numbers</a></li> <li><a href="/wiki/Extended_real_number_line" title="Extended real number line">Extended real numbers</a> <ul><li><a href="/wiki/Projectively_extended_real_line" title="Projectively extended real line">Projective</a></li></ul></li> <li><a href="/wiki/Riemann_sphere" title="Riemann sphere">Extended complex numbers</a></li> <li><a href="/wiki/Hyperreal_number" title="Hyperreal number">Hyperreal numbers</a></li> <li><a href="/wiki/Levi-Civita_field" title="Levi-Civita field">Levi-Civita field</a></li> <li><a href="/wiki/Ordinal_number" title="Ordinal number">Ordinal numbers</a></li> <li><a href="/wiki/Supernatural_number" title="Supernatural number">Supernatural numbers</a></li> <li><a href="/wiki/Surreal_number" title="Surreal number">Surreal numbers</a></li> <li><a href="/wiki/Superreal_number" title="Superreal number">Superreal numbers</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other types</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Irrational_number" title="Irrational number">Irrational numbers</a></li> <li><a href="/wiki/Fuzzy_number" title="Fuzzy number">Fuzzy numbers</a></li> <li><a href="/wiki/Transcendental_number" title="Transcendental number">Transcendental numbers</a></li> <li><a class="mw-selflink selflink">p-adic numbers</a> (<a href="/wiki/Solenoid_(mathematics)#p-adic_solenoids" title="Solenoid (mathematics)">p-adic solenoids</a>)</li> <li><a href="/wiki/Profinite_integer" title="Profinite integer">Profinite integers</a></li> <li><a href="/wiki/Normal_number" title="Normal number">Normal numbers</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow hlist" colspan="2"><div> <ul><li><a href="/wiki/Number#Main_classification" title="Number">Classification</a></li> <li><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/16px-Symbol_list_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/23px-Symbol_list_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/31px-Symbol_list_class.svg.png 2x" data-file-width="180" data-file-height="185" 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p-adic number - Wikipedia