Square root - Wikipedia

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class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>As decimal expansions</span> </div> </a> <ul id="toc-As_decimal_expansions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-As_expansions_in_other_numeral_systems" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#As_expansions_in_other_numeral_systems"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>As expansions in other numeral systems</span> </div> </a> <ul id="toc-As_expansions_in_other_numeral_systems-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-As_periodic_continued_fractions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#As_periodic_continued_fractions"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>As periodic continued fractions</span> </div> </a> <ul id="toc-As_periodic_continued_fractions-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Computation" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Computation"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Computation</span> </div> </a> <ul id="toc-Computation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Square_roots_of_negative_and_complex_numbers" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Square_roots_of_negative_and_complex_numbers"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Square roots of negative and complex numbers</span> </div> </a> <button aria-controls="toc-Square_roots_of_negative_and_complex_numbers-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Square roots of negative and complex numbers subsection</span> </button> <ul id="toc-Square_roots_of_negative_and_complex_numbers-sublist" class="vector-toc-list"> <li id="toc-Principal_square_root_of_a_complex_number" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Principal_square_root_of_a_complex_number"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Principal square root of a complex number</span> </div> </a> <ul id="toc-Principal_square_root_of_a_complex_number-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Algebraic_formula" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Algebraic_formula"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Algebraic formula</span> </div> </a> <ul id="toc-Algebraic_formula-sublist" class="vector-toc-list"> </ul> </li> <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"> <span class="vector-toc-numb">5.3</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-nth_roots_and_polynomial_roots" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#nth_roots_and_polynomial_roots"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span><span>n</span>th roots and polynomial roots</span> </div> </a> <ul id="toc-nth_roots_and_polynomial_roots-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Square_roots_of_matrices_and_operators" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Square_roots_of_matrices_and_operators"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Square roots of matrices and operators</span> </div> </a> <ul id="toc-Square_roots_of_matrices_and_operators-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-In_integral_domains,_including_fields" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#In_integral_domains,_including_fields"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>In integral domains, including fields</span> </div> </a> <ul id="toc-In_integral_domains,_including_fields-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-In_rings_in_general" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#In_rings_in_general"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>In rings in general</span> </div> </a> <ul id="toc-In_rings_in_general-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Geometric_construction_of_the_square_root" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Geometric_construction_of_the_square_root"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Geometric construction of the square root</span> </div> </a> <ul id="toc-Geometric_construction_of_the_square_root-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"> <span class="vector-toc-numb">11</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes_2" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>Notes</span> </div> </a> <ul id="toc-Notes_2-sublist" class="vector-toc-list"> </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"> <span class="vector-toc-numb">13</span> <span>References</span> </div> </a> <ul id="toc-References-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"> <span class="vector-toc-numb">14</span> <span>External links</span> </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" title="Table of Contents" > <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" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Square root</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" 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Available in 98 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-98" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">98 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Vierkantswortel" title="Vierkantswortel – Afrikaans" lang="af" hreflang="af" data-title="Vierkantswortel" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AC%D8%B0%D8%B1_%D8%AA%D8%B1%D8%A8%D9%8A%D8%B9%D9%8A" title="جذر تربيعي – Arabic" lang="ar" hreflang="ar" data-title="جذر تربيعي" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Ra%C3%ADz_cuadrada" title="Raíz cuadrada – Asturian" lang="ast" hreflang="ast" data-title="Raíz cuadrada" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Kvadrat_k%C3%B6kl%C9%99r" title="Kvadrat köklər – Azerbaijani" lang="az" hreflang="az" data-title="Kvadrat köklər" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AC%E0%A6%B0%E0%A7%8D%E0%A6%97%E0%A6%AE%E0%A7%82%E0%A6%B2" title="বর্গমূল – Bangla" lang="bn" hreflang="bn" data-title="বর্গমূল" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/P%C3%AAng-hong-kin" title="Pêng-hong-kin – Minnan" lang="nan" hreflang="nan" data-title="Pêng-hong-kin" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%B4%D1%80%D0%B0%D1%82_%D1%82%D0%B0%D0%BC%D1%8B%D1%80" title="Квадрат тамыр – Bashkir" lang="ba" hreflang="ba" data-title="Квадрат тамыр" data-language-autonym="Башҡортса" data-language-local-name="Bashkir" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%B4%D1%80%D0%B0%D1%82%D0%BD%D1%8B_%D0%BA%D0%BE%D1%80%D0%B0%D0%BD%D1%8C" title="Квадратны корань – Belarusian" lang="be" hreflang="be" data-title="Квадратны корань" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%B4%D1%80%D0%B0%D1%82%D0%BD%D1%8B_%D0%BA%D0%BE%D1%80%D0%B0%D0%BD%D1%8C" title="Квадратны корань – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Квадратны корань" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bcl mw-list-item"><a href="https://bcl.wikipedia.org/wiki/Kwadradong_gamot" title="Kwadradong gamot – Central Bikol" lang="bcl" hreflang="bcl" data-title="Kwadradong gamot" data-language-autonym="Bikol Central" data-language-local-name="Central Bikol" class="interlanguage-link-target"><span>Bikol Central</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%B4%D1%80%D0%B0%D1%82%D0%B5%D0%BD_%D0%BA%D0%BE%D1%80%D0%B5%D0%BD" title="Квадратен корен – Bulgarian" lang="bg" hreflang="bg" data-title="Квадратен корен" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Kvadratni_korijen" title="Kvadratni korijen – Bosnian" lang="bs" hreflang="bs" data-title="Kvadratni korijen" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-br mw-list-item"><a href="https://br.wikipedia.org/wiki/Gwrizienn_garrez" title="Gwrizienn garrez – Breton" lang="br" hreflang="br" data-title="Gwrizienn garrez" data-language-autonym="Brezhoneg" data-language-local-name="Breton" class="interlanguage-link-target"><span>Brezhoneg</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Arrel_quadrada" title="Arrel quadrada – Catalan" lang="ca" hreflang="ca" data-title="Arrel quadrada" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A2%C4%83%D0%B2%D0%B0%D1%82%D0%BA%D0%B0%D0%BB%D0%BB%D0%B0_%D1%82%D1%8B%D0%BC%D0%B0%D1%80" title="Тăваткалла тымар – Chuvash" lang="cv" hreflang="cv" data-title="Тăваткалла тымар" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Druh%C3%A1_odmocnina" title="Druhá odmocnina – Czech" lang="cs" hreflang="cs" data-title="Druhá odmocnina" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Ail_isradd" title="Ail isradd – Welsh" lang="cy" hreflang="cy" data-title="Ail isradd" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Kvadratrod" title="Kvadratrod – Danish" lang="da" hreflang="da" data-title="Kvadratrod" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Quadratwurzel" title="Quadratwurzel – German" lang="de" hreflang="de" data-title="Quadratwurzel" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Ruutjuur" title="Ruutjuur – Estonian" lang="et" hreflang="et" data-title="Ruutjuur" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A4%CE%B5%CF%84%CF%81%CE%B1%CE%B3%CF%89%CE%BD%CE%B9%CE%BA%CE%AE_%CF%81%CE%AF%CE%B6%CE%B1" title="Τετραγωνική ρίζα – Greek" lang="el" hreflang="el" data-title="Τετραγωνική ρίζα" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Ra%C3%ADz_cuadrada" title="Raíz cuadrada – Spanish" lang="es" hreflang="es" data-title="Raíz cuadrada" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Kvadrata_radiko" title="Kvadrata radiko – Esperanto" lang="eo" hreflang="eo" data-title="Kvadrata radiko" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Erro_karratu" title="Erro karratu – Basque" lang="eu" hreflang="eu" data-title="Erro karratu" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%B1%DB%8C%D8%B4%D9%87_%D8%AF%D9%88%D9%85" title="ریشه دوم – Persian" lang="fa" hreflang="fa" data-title="ریشه دوم" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fo mw-list-item"><a href="https://fo.wikipedia.org/wiki/Kvadratr%C3%B3t" title="Kvadratrót – Faroese" lang="fo" hreflang="fo" data-title="Kvadratrót" data-language-autonym="Føroyskt" data-language-local-name="Faroese" class="interlanguage-link-target"><span>Føroyskt</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Racine_carr%C3%A9e" title="Racine carrée – French" lang="fr" hreflang="fr" data-title="Racine carrée" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-fur mw-list-item"><a href="https://fur.wikipedia.org/wiki/Lidr%C3%AEs_cuadrade" title="Lidrîs cuadrade – Friulian" lang="fur" hreflang="fur" data-title="Lidrîs cuadrade" data-language-autonym="Furlan" data-language-local-name="Friulian" class="interlanguage-link-target"><span>Furlan</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Ra%C3%ADz_cadrada" title="Raíz cadrada – Galician" lang="gl" hreflang="gl" data-title="Raíz cadrada" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-gan mw-list-item"><a href="https://gan.wikipedia.org/wiki/%E5%B9%B3%E6%96%B9%E6%A0%B9" title="平方根 – Gan" lang="gan" hreflang="gan" data-title="平方根" data-language-autonym="贛語" data-language-local-name="Gan" class="interlanguage-link-target"><span>贛語</span></a></li><li class="interlanguage-link interwiki-gu mw-list-item"><a href="https://gu.wikipedia.org/wiki/%E0%AA%B5%E0%AA%B0%E0%AB%8D%E0%AA%97%E0%AA%AE%E0%AB%82%E0%AA%B3" title="વર્ગમૂળ – Gujarati" lang="gu" hreflang="gu" data-title="વર્ગમૂળ" data-language-autonym="ગુજરાતી" data-language-local-name="Gujarati" class="interlanguage-link-target"><span>ગુજરાતી</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%A0%9C%EA%B3%B1%EA%B7%BC" title="제곱근 – Korean" lang="ko" hreflang="ko" data-title="제곱근" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%94%D5%A1%D5%BC%D5%A1%D5%AF%D5%B8%D6%82%D5%BD%D5%AB_%D5%A1%D6%80%D5%B4%D5%A1%D5%BF" title="Քառակուսի արմատ – Armenian" lang="hy" hreflang="hy" data-title="Քառակուսի արմատ" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B5%E0%A4%B0%E0%A5%8D%E0%A4%97%E0%A4%AE%E0%A5%82%E0%A4%B2" title="वर्गमूल – Hindi" lang="hi" hreflang="hi" data-title="वर्गमूल" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Kvadratni_korijen" title="Kvadratni korijen – Croatian" lang="hr" hreflang="hr" data-title="Kvadratni korijen" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Quadratala_radiko" title="Quadratala radiko – Ido" lang="io" hreflang="io" data-title="Quadratala radiko" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Akar_kuadrat" title="Akar kuadrat – Indonesian" lang="id" hreflang="id" data-title="Akar kuadrat" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Ferningsr%C3%B3t" title="Ferningsrót – Icelandic" lang="is" hreflang="is" data-title="Ferningsrót" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Radice_quadrata" title="Radice quadrata – Italian" lang="it" hreflang="it" data-title="Radice quadrata" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A9%D7%95%D7%A8%D7%A9_%D7%A8%D7%99%D7%91%D7%95%D7%A2%D7%99" title="שורש ריבועי – Hebrew" lang="he" hreflang="he" data-title="שורש ריבועי" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%B5%E0%B2%B0%E0%B3%8D%E0%B2%97%E0%B2%AE%E0%B3%82%E0%B2%B2" title="ವರ್ಗಮೂಲ – Kannada" lang="kn" hreflang="kn" data-title="ವರ್ಗಮೂಲ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="Kannada" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%99%E1%83%95%E1%83%90%E1%83%93%E1%83%A0%E1%83%90%E1%83%A2%E1%83%A3%E1%83%9A%E1%83%98_%E1%83%A4%E1%83%94%E1%83%A1%E1%83%95%E1%83%98" title="კვადრატული ფესვი – Georgian" lang="ka" hreflang="ka" data-title="კვადრატული ფესვი" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%B4%D1%80%D0%B0%D1%82%D1%82%D1%8B%D2%9B_%D1%82%D2%AF%D0%B1%D1%96%D1%80" title="Квадраттық түбір – Kazakh" lang="kk" hreflang="kk" data-title="Квадраттық түбір" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Kipeuo_cha_pili" title="Kipeuo cha pili – Swahili" lang="sw" hreflang="sw" data-title="Kipeuo cha pili" data-language-autonym="Kiswahili" data-language-local-name="Swahili" class="interlanguage-link-target"><span>Kiswahili</span></a></li><li class="interlanguage-link interwiki-gcr mw-list-item"><a href="https://gcr.wikipedia.org/wiki/Rasin_di_roun_nonm" title="Rasin di roun nonm – Guianan Creole" lang="gcr" hreflang="gcr" data-title="Rasin di roun nonm" data-language-autonym="Kriyòl gwiyannen" data-language-local-name="Guianan Creole" class="interlanguage-link-target"><span>Kriyòl gwiyannen</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%90%D1%80%D0%B8%D1%84%D0%BC%D0%B5%D1%82%D0%B8%D0%BA%D0%B0%D0%BB%D1%8B%D0%BA_%D0%BA%D0%B2%D0%B0%D0%B4%D1%80%D0%B0%D1%82_%D1%82%D0%B0%D0%BC%D1%8B%D1%80" title="Арифметикалык квадрат тамыр – Kyrgyz" lang="ky" hreflang="ky" data-title="Арифметикалык квадрат тамыр" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-lo mw-list-item"><a href="https://lo.wikipedia.org/wiki/%E0%BA%AE%E0%BA%B2%E0%BA%81%E0%BA%AA%E0%BA%B5%E0%BB%88%E0%BA%AB%E0%BA%A5%E0%BB%88%E0%BA%BD%E0%BA%A1" title="ຮາກສີ່ຫລ່ຽມ – Lao" lang="lo" hreflang="lo" data-title="ຮາກສີ່ຫລ່ຽມ" data-language-autonym="ລາວ" data-language-local-name="Lao" class="interlanguage-link-target"><span>ລາວ</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Radix_quadrata" title="Radix quadrata – Latin" lang="la" hreflang="la" data-title="Radix quadrata" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Kvadr%C4%81tsakne" title="Kvadrātsakne – Latvian" lang="lv" hreflang="lv" data-title="Kvadrātsakne" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Kvadratin%C4%97_%C5%A1aknis" title="Kvadratinė šaknis – Lithuanian" lang="lt" hreflang="lt" data-title="Kvadratinė šaknis" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lfn mw-list-item"><a href="https://lfn.wikipedia.org/wiki/Radis_cuadral" title="Radis cuadral – Lingua Franca Nova" lang="lfn" hreflang="lfn" data-title="Radis cuadral" data-language-autonym="Lingua Franca Nova" data-language-local-name="Lingua Franca Nova" class="interlanguage-link-target"><span>Lingua Franca Nova</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Radis_quadrada" title="Radis quadrada – Lombard" lang="lmo" hreflang="lmo" data-title="Radis quadrada" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/N%C3%A9gyzetgy%C3%B6k" title="Négyzetgyök – Hungarian" lang="hu" hreflang="hu" data-title="Négyzetgyök" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%B4%D1%80%D0%B0%D1%82%D0%B5%D0%BD_%D0%BA%D0%BE%D1%80%D0%B5%D0%BD" title="Квадратен корен – Macedonian" lang="mk" hreflang="mk" data-title="Квадратен корен" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%B5%E0%B5%BC%E0%B4%97%E0%B5%8D%E0%B4%97%E0%B4%AE%E0%B5%82%E0%B4%B2%E0%B4%82" title="വർഗ്ഗമൂലം – Malayalam" lang="ml" hreflang="ml" data-title="വർഗ്ഗമൂലം" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mt mw-list-item"><a href="https://mt.wikipedia.org/wiki/G%C4%A7erq_kwadru" title="Għerq kwadru – Maltese" lang="mt" hreflang="mt" data-title="Għerq kwadru" data-language-autonym="Malti" data-language-local-name="Maltese" class="interlanguage-link-target"><span>Malti</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%B5%E0%A4%B0%E0%A5%8D%E0%A4%97%E0%A4%AE%E0%A5%82%E0%A4%B3" title="वर्गमूळ – Marathi" lang="mr" hreflang="mr" data-title="वर्गमूळ" data-language-autonym="मराठी" data-language-local-name="Marathi" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Punca_kuasa_dua" title="Punca kuasa dua – Malay" lang="ms" hreflang="ms" data-title="Punca kuasa dua" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%94%E1%80%BE%E1%80%85%E1%80%BA%E1%80%91%E1%80%95%E1%80%BA%E1%80%80%E1%80%AD%E1%80%94%E1%80%BA%E1%80%B8%E1%80%9B%E1%80%84%E1%80%BA%E1%80%B8" title="နှစ်ထပ်ကိန်းရင်း – Burmese" lang="my" hreflang="my" data-title="နှစ်ထပ်ကိန်းရင်း" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="Burmese" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Vierkantswortel" title="Vierkantswortel – Dutch" lang="nl" hreflang="nl" data-title="Vierkantswortel" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ne mw-list-item"><a href="https://ne.wikipedia.org/wiki/%E0%A4%B5%E0%A4%B0%E0%A5%8D%E0%A4%97%E0%A4%AE%E0%A5%82%E0%A4%B2" title="वर्गमूल – Nepali" lang="ne" hreflang="ne" data-title="वर्गमूल" data-language-autonym="नेपाली" data-language-local-name="Nepali" class="interlanguage-link-target"><span>नेपाली</span></a></li><li class="interlanguage-link interwiki-new mw-list-item"><a href="https://new.wikipedia.org/wiki/%E0%A4%B5%E0%A4%B0%E0%A5%8D%E0%A4%97%E0%A4%AE%E0%A5%82%E0%A4%B2" title="वर्गमूल – Newari" lang="new" hreflang="new" data-title="वर्गमूल" data-language-autonym="नेपाल भाषा" data-language-local-name="Newari" class="interlanguage-link-target"><span>नेपाल भाषा</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%B9%B3%E6%96%B9%E6%A0%B9" title="平方根 – Japanese" lang="ja" hreflang="ja" data-title="平方根" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Kvadratrot" title="Kvadratrot – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Kvadratrot" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Kvadratrot" title="Kvadratrot – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Kvadratrot" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Rai%C3%A7_carrada" title="Raiç carrada – Occitan" lang="oc" hreflang="oc" data-title="Raiç carrada" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-om mw-list-item"><a href="https://om.wikipedia.org/wiki/Caaroo_Kaaslamee" title="Caaroo Kaaslamee – Oromo" lang="om" hreflang="om" data-title="Caaroo Kaaslamee" data-language-autonym="Oromoo" data-language-local-name="Oromo" class="interlanguage-link-target"><span>Oromoo</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Kvadrat_ildiz" title="Kvadrat ildiz – Uzbek" lang="uz" hreflang="uz" data-title="Kvadrat ildiz" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%B5%E0%A8%B0%E0%A8%97_%E0%A8%AE%E0%A9%82%E0%A8%B2" title="ਵਰਗ ਮੂਲ – Punjabi" lang="pa" hreflang="pa" data-title="ਵਰਗ ਮੂਲ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-jam mw-list-item"><a href="https://jam.wikipedia.org/wiki/Skwier_ruut" title="Skwier ruut – Jamaican Creole English" lang="jam" hreflang="jam" data-title="Skwier ruut" data-language-autonym="Patois" data-language-local-name="Jamaican Creole English" class="interlanguage-link-target"><span>Patois</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Pierwiastek_kwadratowy" title="Pierwiastek kwadratowy – Polish" lang="pl" hreflang="pl" data-title="Pierwiastek kwadratowy" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Raiz_quadrada" title="Raiz quadrada – Portuguese" lang="pt" hreflang="pt" data-title="Raiz quadrada" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/R%C4%83d%C4%83cin%C4%83_p%C4%83trat%C4%83" title="Rădăcină pătrată – Romanian" lang="ro" hreflang="ro" data-title="Rădăcină pătrată" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%B4%D1%80%D0%B0%D1%82%D0%BD%D1%8B%D0%B9_%D0%BA%D0%BE%D1%80%D0%B5%D0%BD%D1%8C" title="Квадратный корень – Russian" lang="ru" hreflang="ru" data-title="Квадратный корень" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Rr%C3%ABnja_katrore" title="Rrënja katrore – Albanian" lang="sq" hreflang="sq" data-title="Rrënja katrore" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Radici_quatrata" title="Radici quatrata – Sicilian" lang="scn" hreflang="scn" data-title="Radici quatrata" data-language-autonym="Sicilianu" data-language-local-name="Sicilian" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Square_root" title="Square root – Simple English" lang="en-simple" hreflang="en-simple" data-title="Square root" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Kvadratni_koren" title="Kvadratni koren – Slovenian" lang="sl" hreflang="sl" data-title="Kvadratni koren" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%DA%95%DB%95%DA%AF%DB%8C_%D8%AF%D9%88%D9%88%D8%AC%D8%A7" title="ڕەگی دووجا – Central Kurdish" lang="ckb" hreflang="ckb" data-title="ڕەگی دووجا" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%B4%D1%80%D0%B0%D1%82%D0%BD%D0%B8_%D0%BA%D0%BE%D1%80%D0%B5%D0%BD" title="Квадратни корен – Serbian" lang="sr" hreflang="sr" data-title="Квадратни корен" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Kvadratni_korijen" title="Kvadratni korijen – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Kvadratni korijen" data-language-autonym="Srpskohrvatski / српскохрватски" 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encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></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 whose square is a given number</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">"Square roots" redirects here. For other uses, see <a href="/wiki/Square_Roots_(disambiguation)" class="mw-disambig" title="Square Roots (disambiguation)">Square Roots (disambiguation)</a>.</div> <p class="mw-empty-elt"> </p> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Nuvola_apps_edu_mathematics_blue-p.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/168px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="168" height="168" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/252px-Nuvola_apps_edu_mathematics_blue-p.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/336px-Nuvola_apps_edu_mathematics_blue-p.svg.png 2x" data-file-width="128" data-file-height="128" /></a><figcaption>Notation for the (principal) square root of <span class="texhtml mvar" style="font-style:italic;">x</span>.</figcaption></figure> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Five_Squared.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/Five_Squared.svg/168px-Five_Squared.svg.png" decoding="async" width="168" height="216" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/Five_Squared.svg/252px-Five_Squared.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/60/Five_Squared.svg/336px-Five_Squared.svg.png 2x" data-file-width="600" data-file-height="770" /></a><figcaption>For example, <span class="texhtml"><span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">25</span></span> = 5</span>, since <span class="texhtml">25 = 5 ⋅ 5</span>, or <span class="texhtml">5<sup>2</sup></span> (5 squared).</figcaption></figure> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, a <b>square root</b> of a number <span class="texhtml mvar" style="font-style:italic;">x</span> is a number <span class="texhtml mvar" style="font-style:italic;">y</span> such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y^{2}=x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y^{2}=x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb1cfda1f3310a5c649d6847b1b2325968850889" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.643ex; height:3.009ex;" alt="{\displaystyle y^{2}=x}" /></span>; in other words, a number <span class="texhtml mvar" style="font-style:italic;">y</span> whose <i><a href="/wiki/Square_(algebra)" title="Square (algebra)">square</a></i> (the result of multiplying the number by itself, or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\cdot y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>⋅</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y\cdot y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d08caf988ca4edd35b9dbb75955937bc1b61c02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.99ex; height:2.009ex;" alt="{\displaystyle y\cdot y}" /></span>) is <span class="texhtml mvar" style="font-style:italic;">x</span>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> For example, 4 and −4 are square roots of 16 because <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4^{2}=(-4)^{2}=16}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>4</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>16</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4^{2}=(-4)^{2}=16}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4fe90cd19335908e3320902af69ed20c7bc0396" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.573ex; height:3.176ex;" alt="{\displaystyle 4^{2}=(-4)^{2}=16}" /></span>. </p><p>Every <a href="/wiki/Nonnegative" class="mw-redirect" title="Nonnegative">nonnegative</a> <a href="/wiki/Real_number" title="Real number">real number</a> <span class="texhtml mvar" style="font-style:italic;">x</span> has a unique nonnegative square root, called the <i>principal square root</i> or simply <i>the square root</i> (with a definite article, see below), which is denoted by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {x}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>x</mi> </msqrt> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {x}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9763d9ff8638b882608213f6dc8e9fc8ddc79d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.912ex; height:3.009ex;" alt="{\displaystyle {\sqrt {x}},}" /></span> where the symbol "<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {~^{~}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mtext> </mtext> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {~^{~}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f289ecbf34503791aa2a837dd0dfbbab15cb0f93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.329ex; height:3.009ex;" alt="{\displaystyle {\sqrt {~^{~}}}}" /></span>" is called the <i><a href="/wiki/Radical_sign" class="mw-redirect" title="Radical sign">radical sign</a></i><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> or <i>radix</i>. For example, to express the fact that the principal square root of 9 is 3, we write <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {9}}=3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>9</mn> </msqrt> </mrow> <mo>=</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {9}}=3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44820d293d56e49db705b37ad756363767077121" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.359ex; height:2.843ex;" alt="{\displaystyle {\sqrt {9}}=3}" /></span>. The term (or number) whose square root is being considered is known as the <i>radicand</i>. The radicand is the number or expression underneath the radical sign, in this case, 9. For non-negative <span class="texhtml mvar" style="font-style:italic;">x</span>, the principal square root can also be written in <a href="/wiki/Exponentiation" title="Exponentiation">exponent</a> notation, as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{1/2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{1/2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32077e2f0f843fd11f5440a9818e6b2d353fb6c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.028ex; height:2.843ex;" alt="{\displaystyle x^{1/2}}" /></span>. </p><p>Every <a href="/wiki/Positive_number" class="mw-redirect" title="Positive number">positive number</a> <span class="texhtml mvar" style="font-style:italic;">x</span> has two square roots: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>x</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d62b24be305beff66cba9bfbcc01a362ba390f44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.266ex; height:3.009ex;" alt="{\displaystyle {\sqrt {x}}}" /></span> (which is positive) and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\sqrt {x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>x</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\sqrt {x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b2eaf3e2777b21ab1c891db88d1db286bd167d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.074ex; height:3.009ex;" alt="{\displaystyle -{\sqrt {x}}}" /></span> (which is negative). The two roots can be written more concisely using the <a href="/wiki/Plus%E2%80%93minus_sign" title="Plus–minus sign">± sign</a> as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm {\sqrt {x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>x</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pm {\sqrt {x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/015b9bb6aff25be9b75393afd06d3976adeb11ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.074ex; height:3.009ex;" alt="{\displaystyle \pm {\sqrt {x}}}" /></span>. Although the principal square root of a positive number is only one of its two square roots, the designation "<i>the</i> square root" is often used to refer to the principal square root.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>Square roots of negative numbers can be discussed within the framework of <a href="/wiki/Complex_number" title="Complex number">complex numbers</a>. More generally, square roots can be considered in any context in which a notion of the "<a href="/wiki/Square_(algebra)" title="Square (algebra)">square</a>" of a mathematical object is defined. These include <a href="/wiki/Function_space" title="Function space">function spaces</a> and <a href="/wiki/Square_matrices" class="mw-redirect" title="Square matrices">square matrices</a>, among other <a href="/wiki/Mathematical_structure" title="Mathematical structure">mathematical structures</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History">History</h2></div> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Ybc7289-bw.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Ybc7289-bw.jpg/220px-Ybc7289-bw.jpg" decoding="async" width="220" height="205" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Ybc7289-bw.jpg/330px-Ybc7289-bw.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/0/0b/Ybc7289-bw.jpg 2x" data-file-width="338" data-file-height="315" /></a><figcaption>YBC 7289 clay tablet</figcaption></figure> <p>The <a href="/wiki/Yale_Babylonian_Collection" title="Yale Babylonian Collection">Yale Babylonian Collection</a> clay tablet <a href="/wiki/YBC_7289" title="YBC 7289">YBC 7289</a> was created between 1800 BC and 1600 BC, showing <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4afc1e27d418021bf10898eb44a7f5f315735ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}}" /></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {\sqrt {2}}{2}}={\frac {1}{\sqrt {2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>2</mn> </msqrt> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\frac {\sqrt {2}}{2}}={\frac {1}{\sqrt {2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb14d1a927b83225a24543b3a5fd764359e60b10" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.152ex; height:4.843ex;" alt="{\textstyle {\frac {\sqrt {2}}{2}}={\frac {1}{\sqrt {2}}}}" /></span> respectively as 1;24,51,10 and 0;42,25,35 <a href="/wiki/Sexagesimal" title="Sexagesimal">base 60</a> numbers on a square crossed by two diagonals.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> (1;24,51,10) base 60 corresponds to 1.41421296, which is correct to 5 decimal places (1.41421356...). </p><p>The <a href="/wiki/Rhind_Mathematical_Papyrus" title="Rhind Mathematical Papyrus">Rhind Mathematical Papyrus</a> is a copy from 1650 BC of an earlier <a href="/wiki/Berlin_Papyrus_6619" title="Berlin Papyrus 6619">Berlin Papyrus</a> and other texts – possibly the <a href="/wiki/Kahun_Papyrus" class="mw-redirect" title="Kahun Papyrus">Kahun Papyrus</a> – that shows how the Egyptians extracted square roots by an inverse proportion method.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>In <a href="/wiki/History_of_India" title="History of India">Ancient India</a>, the knowledge of theoretical and applied aspects of square and square root was at least as old as the <i><a href="/wiki/Sulba_Sutras" class="mw-redirect" title="Sulba Sutras">Sulba Sutras</a></i>, dated around 800–500 BC (possibly much earlier).<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> A method for finding very good approximations to the square roots of 2 and 3 are given in the <i><a href="/wiki/Baudhayana_Sulba_Sutra" class="mw-redirect" title="Baudhayana Sulba Sutra">Baudhayana Sulba Sutra</a></i>.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Apastamba" class="mw-redirect" title="Apastamba">Apastamba</a> who was dated around 600 BCE has given a strikingly accurate value for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4afc1e27d418021bf10898eb44a7f5f315735ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}}" /></span> which is correct up to five decimal places as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle 1+{\frac {1}{3}}+{\frac {1}{3\times 4}}-{\frac {1}{3\times 4\times 34}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>3</mn> <mo>×</mo> <mn>4</mn> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>3</mn> <mo>×</mo> <mn>4</mn> <mo>×</mo> <mn>34</mn> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle 1+{\frac {1}{3}}+{\frac {1}{3\times 4}}-{\frac {1}{3\times 4\times 34}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/228dc099dde171fe76e9bc1a35fa52597a7993d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:21.782ex; height:3.676ex;" alt="{\textstyle 1+{\frac {1}{3}}+{\frac {1}{3\times 4}}-{\frac {1}{3\times 4\times 34}}}" /></span>.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> <sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Aryabhata" title="Aryabhata">Aryabhata</a>, in the <i><a href="/wiki/Aryabhatiya" title="Aryabhatiya">Aryabhatiya</a></i> (section 2.4), has given a method for finding the square root of numbers having many digits. </p><p>It was known to the ancient Greeks that square roots of <a href="/wiki/Natural_number" title="Natural number">positive integers</a> that are not <a href="/wiki/Square_number" title="Square number">perfect squares</a> are always <a href="/wiki/Irrational_number" title="Irrational number">irrational numbers</a>: numbers not expressible as a <a href="/wiki/Ratio" title="Ratio">ratio</a> of two integers (that is, they cannot be written exactly as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {m}{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>m</mi> <mi>n</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {m}{n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d48d87468620ad6c70385ddd0d024577ccb559e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:2.877ex; height:4.676ex;" alt="{\displaystyle {\frac {m}{n}}}" /></span>, where <span class="texhtml mvar" style="font-style:italic;">m</span> and <span class="texhtml mvar" style="font-style:italic;">n</span> are integers). This is the theorem <a href="/wiki/Euclid%27s_Elements" title="Euclid's Elements"><i>Euclid X, 9</i></a>, almost certainly due to <a href="/wiki/Theaetetus_(mathematician)" title="Theaetetus (mathematician)">Theaetetus</a> dating back to <abbr title="circa">c.</abbr><span style="white-space:nowrap;"> 380 BC</span>.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> The discovery of irrational numbers, including the particular case of the <a href="/wiki/Square_root_of_2" title="Square root of 2">square root of 2</a>, is widely associated with the Pythagorean school.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> Although some accounts attribute the discovery to <a href="/wiki/Hippasus" title="Hippasus">Hippasus</a>, the specific contributor remains uncertain due to the scarcity of primary sources and the secretive nature of the brotherhood.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> It is exactly the length of the <a href="/wiki/Diagonal" title="Diagonal">diagonal</a> of a <a href="/wiki/Unit_square" title="Unit square">square with side length 1</a>. </p><p>In the Chinese mathematical work <i><a href="/wiki/Writings_on_Reckoning" class="mw-redirect" title="Writings on Reckoning">Writings on Reckoning</a></i>, written between 202 BC and 186 BC during the early <a href="/wiki/Han_dynasty" title="Han dynasty">Han dynasty</a>, the square root is approximated by using an "excess and deficiency" method, which says to "...combine the excess and deficiency as the divisor; (taking) the deficiency numerator multiplied by the excess denominator and the excess numerator times the deficiency denominator, combine them as the dividend."<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p><p>A symbol for square roots, written as an elaborate R, was invented by <a href="/wiki/Regiomontanus" title="Regiomontanus">Regiomontanus</a> (1436–1476). An R was also used for radix to indicate square roots in <a href="/wiki/Gerolamo_Cardano" title="Gerolamo Cardano">Gerolamo Cardano</a>'s <i><a href="/wiki/Ars_Magna_(Gerolamo_Cardano)" class="mw-redirect" title="Ars Magna (Gerolamo Cardano)">Ars Magna</a></i>.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </p><p>According to historian of mathematics <a href="/wiki/David_Eugene_Smith" title="David Eugene Smith">D.E. Smith</a>, Aryabhata's method for finding the square root was first introduced in Europe by <a href="/wiki/Pietro_di_Giacomo_Cataneo" title="Pietro di Giacomo Cataneo">Cataneo</a>—in 1546. </p><p>According to Jeffrey A. Oaks, Arabs used the letter <i><a href="/wiki/Gimel#Arabic_ĝīm" title="Gimel">jīm/ĝīm</a></i> (<span title="Arabic-language text"><span lang="ar" dir="rtl">ج</span></span>), the first letter of the word "<span title="Arabic-language text"><span lang="ar" dir="rtl">جذر</span></span>" (variously transliterated as <i>jaḏr</i>, <i>jiḏr</i>, <i>ǧaḏr</i> or <i>ǧiḏr</i>, "root"), placed in its initial form (<span title="Arabic-language text"><span lang="ar" dir="rtl">ﺟ</span></span>) over a number to indicate its square root. The letter <i>jīm</i> resembles the present square root shape. Its usage goes as far as the end of the twelfth century in the works of the Moroccan mathematician <a href="/wiki/Ibn_al-Yasamin" title="Ibn al-Yasamin">Ibn al-Yasamin</a>.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> </p><p>The symbol "√" for the square root was first used in print in 1525, in <a href="/wiki/Christoph_Rudolff" title="Christoph Rudolff">Christoph Rudolff</a>'s <i>Coss</i>.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Properties_and_uses">Properties and uses</h2></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Square_root_0_25.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Square_root_0_25.svg/500px-Square_root_0_25.svg.png" decoding="async" width="400" height="208" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Square_root_0_25.svg/600px-Square_root_0_25.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Square_root_0_25.svg/800px-Square_root_0_25.svg.png 2x" data-file-width="535" data-file-height="278" /></a><figcaption>The graph of the function <span class="texhtml"><i>f</i>(<i>x</i>) = √<i>x</i></span>, made up of half a <a href="/wiki/Parabola" title="Parabola">parabola</a> with a vertical <a href="/wiki/Directrix_(conic_section)#Eccentricity,_focus_and_directrix" class="mw-redirect" title="Directrix (conic section)">directrix</a> </figcaption></figure> <p>The principal square root function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)={\sqrt {x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>x</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)={\sqrt {x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f4850536e7a37db22aacbc552b03f195a3eceaa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.782ex; height:3.009ex;" alt="{\displaystyle f(x)={\sqrt {x}}}" /></span> (usually just referred to as the "square root function") is a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> that maps the <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> of nonnegative real numbers onto itself. In <a href="/wiki/Geometry" title="Geometry">geometrical</a> terms, the square root function maps the <a href="/wiki/Area" title="Area">area</a> of a square to its side length. </p><p>The square root of <span class="texhtml mvar" style="font-style:italic;">x</span> is rational if and only if <span class="texhtml mvar" style="font-style:italic;">x</span> is a <a href="/wiki/Rational_number" title="Rational number">rational number</a> that can be represented as a ratio of two perfect squares. (See <a href="/wiki/Square_root_of_2" title="Square root of 2">square root of 2</a> for proofs that this is an irrational number, and <a href="/wiki/Quadratic_irrational" class="mw-redirect" title="Quadratic irrational">quadratic irrational</a> for a proof for all non-square natural numbers.) The square root function maps rational numbers into <a href="/wiki/Algebraic_number" title="Algebraic number">algebraic numbers</a>, the latter being a <a href="/wiki/Superset" class="mw-redirect" title="Superset">superset</a> of the rational numbers). </p><p>For all real numbers <span class="texhtml mvar" style="font-style:italic;">x</span>,<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {x^{2}}}=\left|x\right|={\begin{cases}x,&{\text{if }}x\geq 0\\-x,&{\text{if }}x<0.\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>=</mo> <mrow> <mo>|</mo> <mi>x</mi> <mo>|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mi>x</mi> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>x</mi> <mo>≥</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mi>x</mi> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>x</mi> <mo><</mo> <mn>0.</mn> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {x^{2}}}=\left|x\right|={\begin{cases}x,&{\text{if }}x\geq 0\\-x,&{\text{if }}x<0.\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91c88a8c99a20438de963fc25db69d45ccbe167d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.307ex; height:6.176ex;" alt="{\displaystyle {\sqrt {x^{2}}}=\left|x\right|={\begin{cases}x,&{\text{if }}x\geq 0\\-x,&{\text{if }}x<0.\end{cases}}}" /></span> (see <a href="/wiki/Absolute_value" title="Absolute value">absolute value</a>). </p><p>For all nonnegative real numbers <span class="texhtml mvar" style="font-style:italic;">x</span> and <span class="texhtml mvar" style="font-style:italic;">y</span>,<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {xy}}={\sqrt {x}}{\sqrt {y}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>x</mi> <mi>y</mi> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>x</mi> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>y</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {xy}}={\sqrt {x}}{\sqrt {y}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7d12042622d1c1bf3a107300e4c269556ae6b93" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:13.876ex; height:3.176ex;" alt="{\displaystyle {\sqrt {xy}}={\sqrt {x}}{\sqrt {y}}}" /></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {x}}=x^{1/2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>x</mi> </msqrt> </mrow> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {x}}=x^{1/2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ceaad50b7a0ae8ad64014319f138887ec5147f6c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.039ex; height:3.509ex;" alt="{\displaystyle {\sqrt {x}}=x^{1/2}.}" /></span> </p><p>The square root function is <a href="/wiki/Continuous_function" title="Continuous function">continuous</a> for all nonnegative <span class="texhtml mvar" style="font-style:italic;">x</span>, and <a href="/wiki/Derivative" title="Derivative">differentiable</a> for all positive <span class="texhtml mvar" style="font-style:italic;">x</span>. If <span class="texhtml"><i>f</i></span> denotes the square root function, whose derivative is given by:<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f'(x)={\frac {1}{2{\sqrt {x}}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>x</mi> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f'(x)={\frac {1}{2{\sqrt {x}}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7063b75cc5175b0651c6448129d04f679c1cc30f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:14.154ex; height:6.176ex;" alt="{\displaystyle f'(x)={\frac {1}{2{\sqrt {x}}}}.}" /></span> </p><p>The <a href="/wiki/Taylor_series" title="Taylor series">Taylor series</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {1+x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>+</mo> <mi>x</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {1+x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8b52f282420b9d2b6625eb1889455838334976f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.268ex; height:3.009ex;" alt="{\displaystyle {\sqrt {1+x}}}" /></span> about <span class="texhtml"><i>x</i> = 0</span> converges for <span class="texhtml">|<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;"><i>x</i></span>| ≤ 1</span>, and is given by </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {1+x}}=\sum _{n=0}^{\infty }{\frac {(-1)^{n}(2n)!}{(1-2n)(n!)^{2}(4^{n})}}x^{n}=1+{\frac {1}{2}}x-{\frac {1}{8}}x^{2}+{\frac {1}{16}}x^{3}-{\frac {5}{128}}x^{4}+\cdots ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>+</mo> <mi>x</mi> </msqrt> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mn>2</mn> <mi>n</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>!</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <msup> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>x</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>8</mn> </mfrac> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>16</mn> </mfrac> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>128</mn> </mfrac> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {1+x}}=\sum _{n=0}^{\infty }{\frac {(-1)^{n}(2n)!}{(1-2n)(n!)^{2}(4^{n})}}x^{n}=1+{\frac {1}{2}}x-{\frac {1}{8}}x^{2}+{\frac {1}{16}}x^{3}-{\frac {5}{128}}x^{4}+\cdots ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e4f533de7fa291ba3d5837e3db4b7cb61ca9cf4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:77.141ex; height:6.843ex;" alt="{\displaystyle {\sqrt {1+x}}=\sum _{n=0}^{\infty }{\frac {(-1)^{n}(2n)!}{(1-2n)(n!)^{2}(4^{n})}}x^{n}=1+{\frac {1}{2}}x-{\frac {1}{8}}x^{2}+{\frac {1}{16}}x^{3}-{\frac {5}{128}}x^{4}+\cdots ,}" /></span> </p><p>The square root of a nonnegative number is used in the definition of <a href="/wiki/Euclidean_norm" class="mw-redirect" title="Euclidean norm">Euclidean norm</a> (and <a href="/wiki/Euclidean_distance" title="Euclidean distance">distance</a>), as well as in generalizations such as <a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert spaces</a>. It defines an important concept of <a href="/wiki/Standard_deviation" title="Standard deviation">standard deviation</a> used in <a href="/wiki/Probability_theory" title="Probability theory">probability theory</a> and <a href="/wiki/Statistics" title="Statistics">statistics</a>. It has a major use in the formula for solutions of a <a href="/wiki/Quadratic_equation" title="Quadratic equation">quadratic equation</a>. <a href="/wiki/Quadratic_field" title="Quadratic field">Quadratic fields</a> and rings of <a href="/wiki/Quadratic_integer" title="Quadratic integer">quadratic integers</a>, which are based on square roots, are important in algebra and have uses in geometry. Square roots frequently appear in mathematical formulas elsewhere, as well as in many <a href="/wiki/Physics" title="Physics">physical</a> laws. </p> <div class="mw-heading mw-heading2"><h2 id="Square_roots_of_positive_integers">Square roots of positive integers</h2></div> <p>A positive number has two square roots, one positive, and one negative, which are <a href="/wiki/Opposite_(mathematics)" class="mw-redirect" title="Opposite (mathematics)">opposite</a> to each other. When talking of <i>the</i> square root of a positive integer, it is usually the positive square root that is meant. </p><p>The square roots of an integer are <a href="/wiki/Algebraic_integer" title="Algebraic integer">algebraic integers</a>—more specifically <a href="/wiki/Quadratic_integer" title="Quadratic integer">quadratic integers</a>. </p><p>The square root of a positive integer is the product of the roots of its <a href="/wiki/Prime_number" title="Prime number">prime</a> factors, because the square root of a product is the product of the square roots of the factors. Since <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\sqrt {p^{2k}}}=p^{k},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> </mrow> </msup> </msqrt> </mrow> <mo>=</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\sqrt {p^{2k}}}=p^{k},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96b50a1e68ef9fa747bb2c6994d917ea5f1bad9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.407ex; height:3.509ex;" alt="{\textstyle {\sqrt {p^{2k}}}=p^{k},}" /></span> only roots of those primes having an odd power in the <a href="/wiki/Integer_factorization" title="Integer factorization">factorization</a> are necessary. More precisely, the square root of a prime factorization is<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {p_{1}^{2e_{1}+1}\cdots p_{k}^{2e_{k}+1}p_{k+1}^{2e_{k+1}}\dots p_{n}^{2e_{n}}}}=p_{1}^{e_{1}}\dots p_{n}^{e_{n}}{\sqrt {p_{1}\dots p_{k}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mo>⋯</mo> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mrow> </msubsup> <mo>…</mo> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msubsup> </msqrt> </mrow> <mo>=</mo> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msubsup> <mo>…</mo> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {p_{1}^{2e_{1}+1}\cdots p_{k}^{2e_{k}+1}p_{k+1}^{2e_{k+1}}\dots p_{n}^{2e_{n}}}}=p_{1}^{e_{1}}\dots p_{n}^{e_{n}}{\sqrt {p_{1}\dots p_{k}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5b6847ee2b6e9ace9d3f9c700bcb9231ff57bc7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:53.99ex; height:4.843ex;" alt="{\displaystyle {\sqrt {p_{1}^{2e_{1}+1}\cdots p_{k}^{2e_{k}+1}p_{k+1}^{2e_{k+1}}\dots p_{n}^{2e_{n}}}}=p_{1}^{e_{1}}\dots p_{n}^{e_{n}}{\sqrt {p_{1}\dots p_{k}}}.}" /></span> </p> <div class="mw-heading mw-heading3"><h3 id="As_decimal_expansions">As decimal expansions</h3></div> <p>The square roots of the <a href="/wiki/Square_number" title="Square number">perfect squares</a> (e.g., 0, 1, 4, 9, 16) are <a href="/wiki/Integers" class="mw-redirect" title="Integers">integers</a>. In all other cases, the square roots of positive integers are <a href="/wiki/Irrational_number" title="Irrational number">irrational numbers</a>, and hence have non-<a href="/wiki/Repeating_decimal" title="Repeating decimal">repeating decimals</a> in their <a href="/wiki/Decimal_representation" title="Decimal representation">decimal representations</a>. Decimal approximations of the square roots of the first few natural numbers are given in the following table. </p> <table class="wikitable"> <tbody><tr> <th><span class="texhtml mvar" style="font-style:italic;">n</span></th> <th><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {n}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>n</mi> </msqrt> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {n}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7173717a40b3c0819656acd987cacbc2feaf533" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.977ex; height:3.009ex;" alt="{\displaystyle {\sqrt {n}},}" /></span> truncated to 50 decimal places </th></tr> <tr> <td align="right">0</td> <td>0 </td></tr> <tr> <td align="right">1</td> <td>1 </td></tr> <tr> <td align="right">2</td> <td><a href="/wiki/Square_root_of_2" title="Square root of 2"><span style="white-space:nowrap">1.4142135623<span style="margin-left:0.25em">7309504880</span><span style="margin-left:0.25em">1688724209</span><span style="margin-left:0.25em">6980785696</span><span style="margin-left:0.25em">7187537694</span></span></a> </td></tr> <tr> <td align="right">3</td> <td><a href="/wiki/Square_root_of_3" title="Square root of 3"><span style="white-space:nowrap">1.7320508075<span style="margin-left:0.25em">6887729352</span><span style="margin-left:0.25em">7446341505</span><span style="margin-left:0.25em">8723669428</span><span style="margin-left:0.25em">0525381038</span></span></a> </td></tr> <tr> <td align="right">4</td> <td>2 </td></tr> <tr> <td align="right">5</td> <td><a href="/wiki/Square_root_of_5" title="Square root of 5"><span style="white-space:nowrap">2.2360679774<span style="margin-left:0.25em">9978969640</span><span style="margin-left:0.25em">9173668731</span><span style="margin-left:0.25em">2762354406</span><span style="margin-left:0.25em">1835961152</span></span></a> </td></tr> <tr> <td align="right">6</td> <td><a href="/wiki/Square_root_of_6" title="Square root of 6"><span style="white-space:nowrap">2.4494897427<span style="margin-left:0.25em">8317809819</span><span style="margin-left:0.25em">7284074705</span><span style="margin-left:0.25em">8913919659</span><span style="margin-left:0.25em">4748065667</span></span></a> </td></tr> <tr> <td align="right">7</td> <td><a href="/wiki/Square_root_of_7" title="Square root of 7"><span style="white-space:nowrap">2.6457513110<span style="margin-left:0.25em">6459059050</span><span style="margin-left:0.25em">1615753639</span><span style="margin-left:0.25em">2604257102</span><span style="margin-left:0.25em">5918308245</span></span></a> </td></tr> <tr> <td align="right">8</td> <td><span style="white-space:nowrap">2.8284271247<span style="margin-left:0.25em">4619009760</span><span style="margin-left:0.25em">3377448419</span><span style="margin-left:0.25em">3961571393</span><span style="margin-left:0.25em">4375075389</span></span> </td></tr> <tr> <td align="right">9</td> <td>3 </td></tr> <tr> <td align="right">10</td> <td><span style="white-space:nowrap">3.1622776601<span style="margin-left:0.25em">6837933199</span><span style="margin-left:0.25em">8893544432</span><span style="margin-left:0.25em">7185337195</span><span style="margin-left:0.25em">5513932521</span></span> </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="As_expansions_in_other_numeral_systems">As expansions in other numeral systems</h3></div> <p>As with before, the square roots of the <a href="/wiki/Square_number" title="Square number">perfect squares</a> (e.g., 0, 1, 4, 9, 16) are integers. In all other cases, the square roots of positive integers are <a href="/wiki/Irrational_number" title="Irrational number">irrational numbers</a>, and therefore have non-repeating digits in any standard <a href="/wiki/Positional_notation" title="Positional notation">positional notation</a> system. </p><p>The square roots of small integers are used in both the <a href="/wiki/SHA-1" title="SHA-1">SHA-1</a> and <a href="/wiki/SHA-2" title="SHA-2">SHA-2</a> hash function designs to provide <a href="/wiki/Nothing_up_my_sleeve_number" class="mw-redirect" title="Nothing up my sleeve number">nothing up my sleeve numbers</a>. </p> <div class="mw-heading mw-heading3"><h3 id="As_periodic_continued_fractions">As periodic continued fractions</h3></div> <p>A result from the study of <a href="/wiki/Irrational_number" title="Irrational number">irrational numbers</a> as <a href="/wiki/Simple_continued_fraction" title="Simple continued fraction">simple continued fractions</a> was obtained by <a href="/wiki/Joseph_Louis_Lagrange" class="mw-redirect" title="Joseph Louis Lagrange">Joseph Louis Lagrange</a> <abbr title="circa">c.</abbr><span style="white-space:nowrap;"> 1780</span>. Lagrange found that the representation of the square root of any non-square positive integer as a continued fraction is <a href="/wiki/Periodic_continued_fraction" title="Periodic continued fraction">periodic</a>. That is, a certain pattern of partial denominators repeats indefinitely in the continued fraction. In a sense these square roots are the very simplest irrational numbers, because they can be represented with a simple repeating pattern of integers. </p> <table> <tbody><tr> <td align="right"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4afc1e27d418021bf10898eb44a7f5f315735ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}}" /></span></td> <td>= [1; 2, 2, ...] </td></tr> <tr> <td align="right"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b19c09494138b5082459afac7f9a8d99c546fcd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:2.843ex;" alt="{\displaystyle {\sqrt {3}}}" /></span></td> <td>= [1; 1, 2, 1, 2, ...] </td></tr> <tr> <td align="right"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>4</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {4}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ddc524346bd183be9f2c63f906d73910844ef1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:3.009ex;" alt="{\displaystyle {\sqrt {4}}}" /></span></td> <td>= [2] </td></tr> <tr> <td align="right"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {5}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {5}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b78ccdb7e18e02d4fc567c66aac99bf524acb5f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:2.843ex;" alt="{\displaystyle {\sqrt {5}}}" /></span></td> <td>= [2; 4, 4, ...] </td></tr> <tr> <td align="right"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {6}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>6</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {6}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a857de6bca2591cfad08e4378634825b6be66a01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:2.843ex;" alt="{\displaystyle {\sqrt {6}}}" /></span></td> <td>= [2; 2, 4, 2, 4, ...] </td></tr> <tr> <td align="right"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {7}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>7</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {7}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ca16e62fc47de4252d87457029895a954d91a42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:3.009ex;" alt="{\displaystyle {\sqrt {7}}}" /></span></td> <td>= [2; 1, 1, 1, 4, 1, 1, 1, 4, ...] </td></tr> <tr> <td align="right"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {8}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>8</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {8}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74655655dfdd370266c9238e7ba06ff9cc9d43f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:2.843ex;" alt="{\displaystyle {\sqrt {8}}}" /></span></td> <td>= [2; 1, 4, 1, 4, ...] </td></tr> <tr> <td align="right"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {9}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>9</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {9}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac20efa072b0e9082e65122a4104a02f4a746986" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:2.843ex;" alt="{\displaystyle {\sqrt {9}}}" /></span></td> <td>= [3] </td></tr> <tr> <td align="right"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {10}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>10</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {10}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd7409b0ddbc1f90280265e7bc95dd20626ebf1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.261ex; height:2.843ex;" alt="{\displaystyle {\sqrt {10}}}" /></span></td> <td>= [3; 6, 6, ...] </td></tr> <tr> <td align="right"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {11}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>11</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {11}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b713509221c99940e3bfc0eeb7ddafe6ec870ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.261ex; height:3.009ex;" alt="{\displaystyle {\sqrt {11}}}" /></span></td> <td>= [3; 3, 6, 3, 6, ...] </td></tr> <tr> <td align="right"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {12}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>12</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {12}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c878efdf227cf70df28aa7d43cea0069e6f515e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.261ex; height:3.009ex;" alt="{\displaystyle {\sqrt {12}}}" /></span></td> <td>= [3; 2, 6, 2, 6, ...] </td></tr> <tr> <td align="right"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {13}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>13</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {13}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a95228048246821171e1789114839cbd00978027" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.261ex; height:2.843ex;" alt="{\displaystyle {\sqrt {13}}}" /></span></td> <td>= [3; 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, ...] </td></tr> <tr> <td align="right"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {14}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>14</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {14}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19763bff915d190ddb1eb7b8dd9beb7ede194bff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.261ex; height:3.009ex;" alt="{\displaystyle {\sqrt {14}}}" /></span></td> <td>= [3; 1, 2, 1, 6, 1, 2, 1, 6, ...] </td></tr> <tr> <td align="right"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {15}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>15</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {15}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e343ba1fba1d0222e6d6b02e264aec5717548f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.261ex; height:2.843ex;" alt="{\displaystyle {\sqrt {15}}}" /></span></td> <td>= [3; 1, 6, 1, 6, ...] </td></tr> <tr> <td align="right"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {16}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>16</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {16}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/513d1ff9b2ee38ab35caf25b67c82f17a6f99a9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.261ex; height:2.843ex;" alt="{\displaystyle {\sqrt {16}}}" /></span></td> <td>= [4] </td></tr> <tr> <td align="right"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {17}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>17</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {17}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d090c5c91c92d2926ceeece2133403c09bdf4dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.261ex; height:3.009ex;" alt="{\displaystyle {\sqrt {17}}}" /></span></td> <td>= [4; 8, 8, ...] </td></tr> <tr> <td align="right"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {18}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>18</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {18}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a72a94abb379143c58c86b6db1eddbba27a9b5b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.261ex; height:2.843ex;" alt="{\displaystyle {\sqrt {18}}}" /></span></td> <td>= [4; 4, 8, 4, 8, ...] </td></tr> <tr> <td align="right"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {19}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>19</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {19}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16a05218b63c0a1f5fb64f9bdd886e04ec9d1f3c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.261ex; height:2.843ex;" alt="{\displaystyle {\sqrt {19}}}" /></span></td> <td>= [4; 2, 1, 3, 1, 2, 8, 2, 1, 3, 1, 2, 8, ...] </td></tr> <tr> <td align="right"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {20}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>20</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {20}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9d32ce4e3e5160ec9e305bedcdd1fbbdd775c9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.261ex; height:2.843ex;" alt="{\displaystyle {\sqrt {20}}}" /></span></td> <td>= [4; 2, 8, 2, 8, ...] </td></tr></tbody></table> <p>The <a href="/wiki/Square_bracket" class="mw-redirect" title="Square bracket">square bracket</a> notation used above is a short form for a continued fraction. Written in the more suggestive algebraic form, the simple continued fraction for the square root of 11, [3; 3, 6, 3, 6, ...], looks like this:<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {11}}=3+{\cfrac {1}{3+{\cfrac {1}{6+{\cfrac {1}{3+{\cfrac {1}{6+{\cfrac {1}{3+\ddots }}}}}}}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>11</mn> </msqrt> </mrow> <mo>=</mo> <mn>3</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow></mrow> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow></mrow> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow></mrow> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow></mrow> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow></mrow> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow></mrow> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow></mrow> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow></mrow> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow></mrow> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow></mrow> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mo>+</mo> <mo>⋱</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {11}}=3+{\cfrac {1}{3+{\cfrac {1}{6+{\cfrac {1}{3+{\cfrac {1}{6+{\cfrac {1}{3+\ddots }}}}}}}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f60a868f2ab13366547648c71052731f7e92ea0f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -18.838ex; width:38.536ex; height:23.009ex;" alt="{\displaystyle {\sqrt {11}}=3+{\cfrac {1}{3+{\cfrac {1}{6+{\cfrac {1}{3+{\cfrac {1}{6+{\cfrac {1}{3+\ddots }}}}}}}}}}}" /></span> </p><p>where the two-digit pattern {3, 6} repeats over and over again in the partial denominators. Since <span class="texhtml">11 = 3<sup>2</sup> + 2</span>, the above is also identical to the following <a href="/wiki/Generalized_continued_fraction#Roots_of_positive_numbers" class="mw-redirect" title="Generalized continued fraction">generalized continued fractions</a>: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {11}}=3+{\cfrac {2}{6+{\cfrac {2}{6+{\cfrac {2}{6+{\cfrac {2}{6+{\cfrac {2}{6+\ddots }}}}}}}}}}=3+{\cfrac {6}{20-1-{\cfrac {1}{20-{\cfrac {1}{20-{\cfrac {1}{20-{\cfrac {1}{20-\ddots }}}}}}}}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>11</mn> </msqrt> </mrow> <mo>=</mo> <mn>3</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow></mrow> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow></mrow> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow></mrow> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow></mrow> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow></mrow> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow></mrow> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow></mrow> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow></mrow> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow></mrow> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow></mrow> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> <mo>+</mo> <mo>⋱</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>3</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow></mrow> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow></mrow> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>20</mn> <mo>−</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow></mrow> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow></mrow> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>20</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow></mrow> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow></mrow> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>20</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow></mrow> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow></mrow> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>20</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow></mrow> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow></mrow> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>20</mn> <mo>−</mo> <mo>⋱</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {11}}=3+{\cfrac {2}{6+{\cfrac {2}{6+{\cfrac {2}{6+{\cfrac {2}{6+{\cfrac {2}{6+\ddots }}}}}}}}}}=3+{\cfrac {6}{20-1-{\cfrac {1}{20-{\cfrac {1}{20-{\cfrac {1}{20-{\cfrac {1}{20-\ddots }}}}}}}}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f30022140c83bd3315dab88806fbee3e5bdbc685" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -18.838ex; width:83.273ex; height:23.009ex;" alt="{\displaystyle {\sqrt {11}}=3+{\cfrac {2}{6+{\cfrac {2}{6+{\cfrac {2}{6+{\cfrac {2}{6+{\cfrac {2}{6+\ddots }}}}}}}}}}=3+{\cfrac {6}{20-1-{\cfrac {1}{20-{\cfrac {1}{20-{\cfrac {1}{20-{\cfrac {1}{20-\ddots }}}}}}}}}}.}" /></span> </p> <div class="mw-heading mw-heading2"><h2 id="Computation">Computation</h2></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/Methods_of_computing_square_roots" title="Methods of computing square roots">Methods of computing square roots</a></div> <p>Square roots of positive numbers are not in general <a href="/wiki/Rational_number" title="Rational number">rational numbers</a>, and so cannot be written as a terminating or recurring decimal expression. Therefore in general any attempt to compute a square root expressed in decimal form can only yield an approximation, though a sequence of increasingly accurate approximations can be obtained. </p><p>Most <a href="/wiki/Pocket_calculator" class="mw-redirect" title="Pocket calculator">pocket calculators</a> have a square root key. Computer <a href="/wiki/Spreadsheet" title="Spreadsheet">spreadsheets</a> and other <a href="/wiki/Software" title="Software">software</a> are also frequently used to calculate square roots. Pocket calculators typically implement efficient routines, such as the <a href="/wiki/Newton%27s_method" title="Newton's method">Newton's method</a> (frequently with an initial guess of 1), to compute the square root of a positive real number.<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> When computing square roots with <a href="/wiki/Common_logarithm" title="Common logarithm">logarithm tables</a> or <a href="/wiki/Slide_rule" title="Slide rule">slide rules</a>, one can exploit the identities<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {a}}=e^{(\ln a)/2}=10^{(\log _{10}a)/2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>a</mi> </msqrt> </mrow> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>ln</mi> <mo>⁡</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {a}}=e^{(\ln a)/2}=10^{(\log _{10}a)/2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0fc6f923e7eae218254b1a6f2d03a979d0f15364" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:27.214ex; height:3.509ex;" alt="{\displaystyle {\sqrt {a}}=e^{(\ln a)/2}=10^{(\log _{10}a)/2},}" /></span> where <span class="texhtml">ln</span> and <span class="texhtml">log<sub>10</sub></span> are the <a href="/wiki/Natural_logarithm" title="Natural logarithm">natural</a> and <a href="/wiki/Base-10_logarithm" class="mw-redirect" title="Base-10 logarithm">base-10 logarithms</a>. </p><p>By trial-and-error,<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> one can square an estimate for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>a</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afccc332c876539296df1a980127d86173e59ef0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.166ex; height:3.009ex;" alt="{\displaystyle {\sqrt {a}}}" /></span> and raise or lower the estimate until it agrees to sufficient accuracy. For this technique it is prudent to use the identity<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x+c)^{2}=x^{2}+2xc+c^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>c</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>x</mi> <mi>c</mi> <mo>+</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x+c)^{2}=x^{2}+2xc+c^{2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bab75cc8675e7a9ef99649eb59a7ee7c43e37f04" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.411ex; height:3.176ex;" alt="{\displaystyle (x+c)^{2}=x^{2}+2xc+c^{2},}" /></span> as it allows one to adjust the estimate <span class="texhtml mvar" style="font-style:italic;">x</span> by some amount <span class="texhtml mvar" style="font-style:italic;">c</span> and measure the square of the adjustment in terms of the original estimate and its square. </p><p>The most common <a href="/wiki/Iterative_method" title="Iterative method">iterative method</a> of square root calculation by hand is known as the "<a href="/wiki/Babylonian_method" class="mw-redirect" title="Babylonian method">Babylonian method</a>" or "Heron's method" after the first-century Greek philosopher <a href="/wiki/Hero_of_Alexandria" title="Hero of Alexandria">Heron of Alexandria</a>, who first described it.<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> The method uses the same iterative scheme as the <a href="/wiki/Newton%E2%80%93Raphson_method" class="mw-redirect" title="Newton–Raphson method">Newton–Raphson method</a> yields when applied to the function <span class="texhtml"><i>y</i> = <i>f</i>(<i>x</i>) = <i>x</i><sup>2</sup> − <i>a</i></span>, using the fact that its slope at any point is <span class="texhtml"><i>dy</i>/<i>dx</i> = <i>f<span class="nowrap" style="padding-left:0.15em;">′</span></i>(<i>x</i>) = 2<i>x</i></span>, but predates it by many centuries.<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> The algorithm is to repeat a simple calculation that results in a number closer to the actual square root each time it is repeated with its result as the new input. The motivation is that if <span class="texhtml mvar" style="font-style:italic;">x</span> is an overestimate to the square root of a nonnegative real number <span class="texhtml mvar" style="font-style:italic;">a</span> then <span class="texhtml"><i>a</i>/<i>x</i></span> will be an underestimate and so the average of these two numbers is a better approximation than either of them. However, the <a href="/wiki/Inequality_of_arithmetic_and_geometric_means" class="mw-redirect" title="Inequality of arithmetic and geometric means">inequality of arithmetic and geometric means</a> shows this average is always an overestimate of the square root (as noted <a class="mw-selflink-fragment" href="#Geometric_construction_of_the_square_root">below</a>), and so it can serve as a new overestimate with which to repeat the process, which <a href="/wiki/Limit_of_a_sequence" title="Limit of a sequence">converges</a> as a consequence of the successive overestimates and underestimates being closer to each other after each iteration. To find <span class="texhtml mvar" style="font-style:italic;">x</span>: </p> <ol><li>Start with an arbitrary positive start value <span class="texhtml mvar" style="font-style:italic;">x</span>. The closer to the square root of <span class="texhtml mvar" style="font-style:italic;">a</span>, the fewer the iterations that will be needed to achieve the desired precision.</li> <li>Replace <span class="texhtml mvar" style="font-style:italic;">x</span> by the average <span class="texhtml">(<i>x</i> + <i>a</i>/<i>x</i>) / 2</span> between <span class="texhtml"><i>x</i></span> and <span class="texhtml"><i>a</i>/<i>x</i></span>.</li> <li>Repeat from step 2, using this average as the new value of <span class="texhtml mvar" style="font-style:italic;">x</span>.</li></ol> <p>That is, if an arbitrary guess for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>a</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afccc332c876539296df1a980127d86173e59ef0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.166ex; height:3.009ex;" alt="{\displaystyle {\sqrt {a}}}" /></span> is <span class="texhtml"><i>x</i><sub>0</sub></span>, and <span class="texhtml"><i>x</i><sub><i>n</i> + 1</sub> = (<i>x<sub>n</sub></i> + <i>a</i>/<i>x<sub>n</sub></i>) / 2</span>, then each <span class="texhtml"><i>x</i><sub><i>n</i></sub></span> is an approximation of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>a</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afccc332c876539296df1a980127d86173e59ef0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.166ex; height:3.009ex;" alt="{\displaystyle {\sqrt {a}}}" /></span> which is better for large <span class="texhtml mvar" style="font-style:italic;">n</span> than for small <span class="texhtml mvar" style="font-style:italic;">n</span>. If <span class="texhtml mvar" style="font-style:italic;">a</span> is positive, the convergence is <a href="/wiki/Rate_of_convergence" title="Rate of convergence">quadratic</a>, which means that in approaching the limit, the number of correct digits roughly doubles in each next iteration. If <span class="texhtml"><i>a</i> = 0</span>, the convergence is only linear; however, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {0}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>0</mn> </msqrt> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {0}}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d765967f8f0a4aec6fac7e551f8c65dec48a12e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.359ex; height:2.843ex;" alt="{\displaystyle {\sqrt {0}}=0}" /></span> so in this case no iteration is needed. </p><p>Using the identity<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {a}}=2^{-n}{\sqrt {4^{n}a}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>a</mi> </msqrt> </mrow> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>n</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mi>a</mi> </msqrt> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {a}}=2^{-n}{\sqrt {4^{n}a}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d45a4a036fe41cf541aa1cb278058766f55aefe" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.117ex; height:3.343ex;" alt="{\displaystyle {\sqrt {a}}=2^{-n}{\sqrt {4^{n}a}},}" /></span> the computation of the square root of a positive number can be reduced to that of a number in the range <span class="texhtml">[1, 4)</span>. This simplifies finding a start value for the iterative method that is close to the square root, for which a <a href="/wiki/Polynomial_function" class="mw-redirect" title="Polynomial function">polynomial</a> or <a href="/wiki/Piecewise_linear_function" title="Piecewise linear function">piecewise-linear</a> <a href="/wiki/Approximation_theory" title="Approximation theory">approximation</a> can be used. </p><p>The <a href="/wiki/Computational_complexity_theory" title="Computational complexity theory">time complexity</a> for computing a square root with <span class="texhtml mvar" style="font-style:italic;">n</span> digits of precision is equivalent to that of multiplying two <span class="texhtml mvar" style="font-style:italic;">n</span>-digit numbers. </p><p>Another useful method for calculating the square root is the shifting nth root algorithm, applied for <span class="texhtml"><i>n</i> = 2</span>. </p><p>The name of the square root <a href="/wiki/Function_(programming)" class="mw-redirect" title="Function (programming)">function</a> varies from <a href="/wiki/Programming_language" title="Programming language">programming language</a> to programming language, with <code>sqrt</code><sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> (often pronounced "squirt"<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup>) being common, used in <a href="/wiki/C_(programming_language)" title="C (programming language)">C</a> and derived languages such as <a href="/wiki/C%2B%2B" title="C++">C++</a>, <a href="/wiki/JavaScript" title="JavaScript">JavaScript</a>, <a href="/wiki/PHP" title="PHP">PHP</a>, and <a href="/wiki/Python_(programming_language)" title="Python (programming language)">Python</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Square_roots_of_negative_and_complex_numbers">Square roots of negative and complex numbers</h2></div> <style data-mw-deduplicate="TemplateStyles:r1273380762/mw-parser-output/.tmulti">.mw-parser-output .tmulti .multiimageinner{display:flex;flex-direction:column}.mw-parser-output .tmulti .trow{display:flex;flex-direction:row;clear:left;flex-wrap:wrap;width:100%;box-sizing:border-box}.mw-parser-output .tmulti .tsingle{margin:1px;float:left}.mw-parser-output .tmulti .theader{clear:both;font-weight:bold;text-align:center;align-self:center;background-color:transparent;width:100%}.mw-parser-output .tmulti .thumbcaption{background-color:transparent}.mw-parser-output .tmulti .text-align-left{text-align:left}.mw-parser-output .tmulti .text-align-right{text-align:right}.mw-parser-output .tmulti .text-align-center{text-align:center}@media all and (max-width:720px){.mw-parser-output .tmulti .thumbinner{width:100%!important;box-sizing:border-box;max-width:none!important;align-items:center}.mw-parser-output .tmulti .trow{justify-content:center}.mw-parser-output .tmulti .tsingle{float:none!important;max-width:100%!important;box-sizing:border-box;text-align:center}.mw-parser-output .tmulti .tsingle .thumbcaption{text-align:left}.mw-parser-output .tmulti .trow>.thumbcaption{text-align:center}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .tmulti .multiimageinner span:not(.skin-invert-image):not(.skin-invert):not(.bg-transparent) img{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .tmulti .multiimageinner span:not(.skin-invert-image):not(.skin-invert):not(.bg-transparent) img{background-color:white}}</style><div class="thumb tmulti tleft"><div class="thumbinner multiimageinner" style="width:612px;max-width:612px"><div class="trow"><div class="tsingle" style="width:202px;max-width:202px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:Complex_sqrt_leaf1.jpg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e2/Complex_sqrt_leaf1.jpg/200px-Complex_sqrt_leaf1.jpg" decoding="async" width="200" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e2/Complex_sqrt_leaf1.jpg/300px-Complex_sqrt_leaf1.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e2/Complex_sqrt_leaf1.jpg/400px-Complex_sqrt_leaf1.jpg 2x" data-file-width="641" data-file-height="641" /></a></span></div><div class="thumbcaption">First leaf of the complex square root</div></div><div class="tsingle" style="width:202px;max-width:202px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:Complex_sqrt_leaf2.jpg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f6/Complex_sqrt_leaf2.jpg/200px-Complex_sqrt_leaf2.jpg" decoding="async" width="200" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f6/Complex_sqrt_leaf2.jpg/300px-Complex_sqrt_leaf2.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f6/Complex_sqrt_leaf2.jpg/400px-Complex_sqrt_leaf2.jpg 2x" data-file-width="649" data-file-height="649" /></a></span></div><div class="thumbcaption">Second leaf of the complex square root</div></div><div class="tsingle" style="width:202px;max-width:202px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:Riemann_surface_sqrt.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Riemann_surface_sqrt.svg/250px-Riemann_surface_sqrt.svg.png" decoding="async" width="200" height="210" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Riemann_surface_sqrt.svg/330px-Riemann_surface_sqrt.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Riemann_surface_sqrt.svg/500px-Riemann_surface_sqrt.svg.png 2x" data-file-width="1098" data-file-height="1153" /></a></span></div><div class="thumbcaption">Using the <a href="/wiki/Riemann_surface" title="Riemann surface">Riemann surface</a> of the square root, it is shown how the two leaves fit together</div></div></div></div></div> <div style="clear:both;" class=""></div> <p>The square of any positive or negative number is positive, and the square of 0 is 0. Therefore, no negative number can have a <a href="/wiki/Real_number" title="Real number">real</a> square root. However, it is possible to work with a more inclusive set of numbers, called the <a href="/wiki/Complex_number" title="Complex number">complex numbers</a>, that does contain solutions to the square root of a negative number. This is done by introducing a new number, denoted by <span class="texhtml"><i>i</i></span> (sometimes by <span class="texhtml"><i>j</i></span>, especially in the context of <a href="/wiki/Electric_current" title="Electric current">electricity</a> where <i>i</i> traditionally represents electric current) and called the <a href="/wiki/Imaginary_unit" title="Imaginary unit">imaginary unit</a>, which is <i>defined</i> such that <span class="texhtml"><i>i</i><sup>2</sup> = −1</span>. Using this notation, we can think of <span class="texhtml"><i>i</i></span> as the square root of −1, but we also have <span class="texhtml">(−<i>i</i>)<sup>2</sup> = <i>i</i><sup>2</sup> = −1</span> and so <span class="texhtml">−<i>i</i></span> is also a square root of −1. By convention, the principal square root of −1 is <span class="texhtml"><i>i</i></span>, or more generally, if <span class="texhtml"><i>x</i></span> is any nonnegative number, then the principal square root of <span class="texhtml">−<i>x</i></span> is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {-x}}=i{\sqrt {x}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mi>x</mi> </msqrt> </mrow> <mo>=</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>x</mi> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {-x}}=i{\sqrt {x}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48c21734a1d7b8a5d724ceed1106e1f4302e57ef" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.887ex; height:3.009ex;" alt="{\displaystyle {\sqrt {-x}}=i{\sqrt {x}}.}" /></span> </p><p>The right side (as well as its negative) is indeed a square root of <span class="texhtml">−<i>x</i></span>, since<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (i{\sqrt {x}})^{2}=i^{2}({\sqrt {x}})^{2}=(-1)x=-x.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>x</mi> </msqrt> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>x</mi> </msqrt> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>x</mi> <mo>=</mo> <mo>−</mo> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (i{\sqrt {x}})^{2}=i^{2}({\sqrt {x}})^{2}=(-1)x=-x.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4238d7bebc5832f0cb56d22037a084b9c71f7eed" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:34.107ex; height:3.343ex;" alt="{\displaystyle (i{\sqrt {x}})^{2}=i^{2}({\sqrt {x}})^{2}=(-1)x=-x.}" /></span> </p><p>For every non-zero complex number <span class="texhtml mvar" style="font-style:italic;">z</span> there exist precisely two numbers <span class="texhtml mvar" style="font-style:italic;">w</span> such that <span class="texhtml"><i>w</i><sup>2</sup> = <i>z</i></span>: the principal square root of <span class="texhtml mvar" style="font-style:italic;">z</span> (defined below), and its negative. </p> <div class="mw-heading mw-heading3"><h3 id="Principal_square_root_of_a_complex_number">Principal square root of a complex number</h3></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Visualisation_complex_number_roots.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Visualisation_complex_number_roots.svg/250px-Visualisation_complex_number_roots.svg.png" decoding="async" width="250" height="333" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Visualisation_complex_number_roots.svg/500px-Visualisation_complex_number_roots.svg.png 1.5x" data-file-width="512" data-file-height="683" /></a><figcaption>Geometric representation of the 2nd to 6th roots of a complex number <span class="texhtml mvar" style="font-style:italic;">z</span>, in polar form <span class="nowrap"><span class="texhtml"><i>re</i><sup><i>iφ</i></sup></span> </span> where <span class="nowrap"><span class="texhtml"><i>r</i> = |<i>z</i> |</span></span> and <span class="nowrap"><span class="texhtml"><i>φ</i> = arg <i>z</i></span></span>. If <span class="texhtml mvar" style="font-style:italic;">z</span> is real, <span class="nowrap"><span class="texhtml"><i>φ</i> = 0</span> or <span class="texhtml mvar" style="font-style:italic;">π</span></span>. Principal roots are shown in black.</figcaption></figure> <p>To find a definition for the square root that allows us to consistently choose a single value, called the <a href="/wiki/Principal_value" title="Principal value">principal value</a>, we start by observing that any complex number <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x+iy}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>+</mo> <mi>i</mi> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x+iy}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ceb1c6ce62a20dbfe9cb3d82dca889577b469703" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.128ex; height:2.509ex;" alt="{\displaystyle x+iy}" /></span> can be viewed as a point in the plane, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4facf4be4b190ba97a34e73f2f761df72ec13313" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.975ex; height:2.843ex;" alt="{\displaystyle (x,y),}" /></span> expressed using <a href="/wiki/Cartesian_coordinate_system" title="Cartesian coordinate system">Cartesian coordinates</a>. The same point may be reinterpreted using <a href="/wiki/Polar_coordinates" class="mw-redirect" title="Polar coordinates">polar coordinates</a> as the pair <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (r,\varphi ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>r</mi> <mo>,</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (r,\varphi ),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5f96086fe5185510d47808c3f9a3ee4ce2a3208" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.059ex; height:2.843ex;" alt="{\displaystyle (r,\varphi ),}" /></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r\geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r\geq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa96c19954fcda2695f988938ccf091d2bc2bbae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.31ex; height:2.343ex;" alt="{\displaystyle r\geq 0}" /></span> is the distance of the point from the origin, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }" /></span> is the angle that the line from the origin to the point makes with the positive real (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" 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 x}" /></span>) axis. In complex analysis, the location of this point is conventionally written <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle re^{i\varphi }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>φ</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle re^{i\varphi }.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4e3db48ba9cf6dac607fe95a0615e2caaf96333" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.654ex; height:2.676ex;" alt="{\displaystyle re^{i\varphi }.}" /></span> If<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=re^{i\varphi }{\text{ with }}-\pi <\varphi \leq \pi ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mi>r</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>φ</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mtext> with </mtext> </mrow> <mo>−</mo> <mi>π</mi> <mo><</mo> <mi>φ</mi> <mo>≤</mo> <mi>π</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=re^{i\varphi }{\text{ with }}-\pi <\varphi \leq \pi ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/880753468cb5d28421f035236555ef024d6711e8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.745ex; height:3.176ex;" alt="{\displaystyle z=re^{i\varphi }{\text{ with }}-\pi <\varphi \leq \pi ,}" /></span> then the <em><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><span class="vanchor"><span id="principal_square_root"></span><span class="vanchor-text">principal square root</span></span></em> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" /></span> is defined to be the following:<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {z}}={\sqrt {r}}e^{i\varphi /2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>z</mi> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>r</mi> </msqrt> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {z}}={\sqrt {r}}e^{i\varphi /2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1edff7851737fcd3646b9397a6ab44677ba42c2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.356ex; height:3.509ex;" alt="{\displaystyle {\sqrt {z}}={\sqrt {r}}e^{i\varphi /2}.}" /></span> The principal square root function is thus defined using the non-positive real axis as a <a href="/wiki/Branch_cut" class="mw-redirect" title="Branch cut">branch cut</a>. If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" /></span> is a non-negative real number (which happens if and only if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/192287b02f5764a18fe39f37b8199d72000aa220" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.781ex; height:2.676ex;" alt="{\displaystyle \varphi =0}" /></span>) then the principal square root of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" /></span> is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {r}}e^{i(0)/2}={\sqrt {r}};}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>r</mi> </msqrt> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>r</mi> </msqrt> </mrow> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {r}}e^{i(0)/2}={\sqrt {r}};}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7927dfa14d103ce168e11eebdb3c8510b2ec7c10" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.343ex; height:3.509ex;" alt="{\displaystyle {\sqrt {r}}e^{i(0)/2}={\sqrt {r}};}" /></span> in other words, the principal square root of a non-negative real number is just the usual non-negative square root. It is important that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\pi <\varphi \leq \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>π</mi> <mo><</mo> <mi>φ</mi> <mo>≤</mo> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\pi <\varphi \leq \pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b27eb4930922e649015e37b8d782b6b1ad7d55b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.189ex; height:2.509ex;" alt="{\displaystyle -\pi <\varphi \leq \pi }" /></span> because if, for example, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=-2i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mo>−</mo> <mn>2</mn> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=-2i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a95599ddc200f43a61e3441144343ca7d37fcaca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.96ex; height:2.343ex;" alt="{\displaystyle z=-2i}" /></span> (so <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi =-\pi /2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo>=</mo> <mo>−</mo> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi =-\pi /2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d200f54eb3c0a721104aa004043d03ee3e1837ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.084ex; height:2.843ex;" alt="{\displaystyle \varphi =-\pi /2}" /></span>) then the principal square root is<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {-2i}}={\sqrt {2e^{i\varphi }}}={\sqrt {2}}e^{i\varphi /2}={\sqrt {2}}e^{i(-\pi /4)}=1-i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>2</mn> <mi>i</mi> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>φ</mi> </mrow> </msup> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo stretchy="false">(</mo> <mo>−</mo> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {-2i}}={\sqrt {2e^{i\varphi }}}={\sqrt {2}}e^{i\varphi /2}={\sqrt {2}}e^{i(-\pi /4)}=1-i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/680bbdf3423e66e92b742b21b695a34cde3d7623" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:47.178ex; height:3.509ex;" alt="{\displaystyle {\sqrt {-2i}}={\sqrt {2e^{i\varphi }}}={\sqrt {2}}e^{i\varphi /2}={\sqrt {2}}e^{i(-\pi /4)}=1-i}" /></span> but using <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {\varphi }}:=\varphi +2\pi =3\pi /2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>φ</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo>:=</mo> <mi>φ</mi> <mo>+</mo> <mn>2</mn> <mi>π</mi> <mo>=</mo> <mn>3</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tilde {\varphi }}:=\varphi +2\pi =3\pi /2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d13bf85dc07dd6e91a2dd9406a035a0fd75f06e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.053ex; height:2.843ex;" alt="{\displaystyle {\tilde {\varphi }}:=\varphi +2\pi =3\pi /2}" /></span> would instead produce the other square root <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2}}e^{i{\tilde {\varphi }}/2}={\sqrt {2}}e^{i(3\pi /4)}=-1+i=-{\sqrt {-2i}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>φ</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo stretchy="false">(</mo> <mn>3</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo>=</mo> <mo>−</mo> <mn>1</mn> <mo>+</mo> <mi>i</mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>2</mn> <mi>i</mi> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}e^{i{\tilde {\varphi }}/2}={\sqrt {2}}e^{i(3\pi /4)}=-1+i=-{\sqrt {-2i}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65b35f4508f0d58c39a6de36bf16fc56ee492b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.509ex; height:3.509ex;" alt="{\displaystyle {\sqrt {2}}e^{i{\tilde {\varphi }}/2}={\sqrt {2}}e^{i(3\pi /4)}=-1+i=-{\sqrt {-2i}}.}" /></span> </p><p>The principal square root function is <a href="/wiki/Holomorphic_function" title="Holomorphic function">holomorphic</a> everywhere except on the set of non-positive real numbers (on strictly negative reals it is not even <a href="/wiki/Continuous_function" title="Continuous function">continuous</a>). The above Taylor series for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {1+x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>+</mo> <mi>x</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {1+x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8b52f282420b9d2b6625eb1889455838334976f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.268ex; height:3.009ex;" alt="{\displaystyle {\sqrt {1+x}}}" /></span> remains valid for complex numbers <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" 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 x}" /></span> with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |x|<1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |x|<1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4e657241d23e0514c31745c2d302fffa61a77ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.531ex; height:2.843ex;" alt="{\displaystyle |x|<1.}" /></span> </p><p>The above can also be expressed in terms of <a href="/wiki/Trigonometric_function" class="mw-redirect" title="Trigonometric function">trigonometric functions</a>:<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {r\left(\cos \varphi +i\sin \varphi \right)}}={\sqrt {r}}\left(\cos {\frac {\varphi }{2}}+i\sin {\frac {\varphi }{2}}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>r</mi> <mrow> <mo>(</mo> <mrow> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <mo>+</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> </mrow> <mo>)</mo> </mrow> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>r</mi> </msqrt> </mrow> <mrow> <mo>(</mo> <mrow> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>φ</mi> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>φ</mi> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {r\left(\cos \varphi +i\sin \varphi \right)}}={\sqrt {r}}\left(\cos {\frac {\varphi }{2}}+i\sin {\frac {\varphi }{2}}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b1c6547d2ba001ee223f8823934863ace7496b0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:45.143ex; height:4.843ex;" alt="{\displaystyle {\sqrt {r\left(\cos \varphi +i\sin \varphi \right)}}={\sqrt {r}}\left(\cos {\frac {\varphi }{2}}+i\sin {\frac {\varphi }{2}}\right).}" /></span> </p> <div class="mw-heading mw-heading3"><h3 id="Algebraic_formula">Algebraic formula</h3></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Imaginary2Root.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/32/Imaginary2Root.svg/220px-Imaginary2Root.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/32/Imaginary2Root.svg/330px-Imaginary2Root.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/32/Imaginary2Root.svg/440px-Imaginary2Root.svg.png 2x" data-file-width="480" data-file-height="480" /></a><figcaption>The square roots of <span class="texhtml mvar" style="font-style:italic;">i</span></figcaption></figure> <p>When the number is expressed using its real and imaginary parts, the following formula can be used for the principal square root:<sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {x+iy}}={\sqrt {{\tfrac {1}{2}}{\bigl (}{\sqrt {\textstyle x^{2}+y^{2}}}+x{\bigr )}}}+i\operatorname {sgn}(y){\sqrt {{\tfrac {1}{2}}{\bigl (}{\sqrt {\textstyle x^{2}+y^{2}}}-x{\bigr )}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>x</mi> <mo>+</mo> <mi>i</mi> <mi>y</mi> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mstyle displaystyle="false" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </msqrt> </mrow> <mo>+</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </msqrt> </mrow> <mo>+</mo> <mi>i</mi> <mi>sgn</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mstyle displaystyle="false" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </msqrt> </mrow> <mo>−</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </msqrt> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {x+iy}}={\sqrt {{\tfrac {1}{2}}{\bigl (}{\sqrt {\textstyle x^{2}+y^{2}}}+x{\bigr )}}}+i\operatorname {sgn}(y){\sqrt {{\tfrac {1}{2}}{\bigl (}{\sqrt {\textstyle x^{2}+y^{2}}}-x{\bigr )}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/027bef226c1a102d193ad1d2108163d407cfcd50" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:62.652ex; height:4.843ex;" alt="{\displaystyle {\sqrt {x+iy}}={\sqrt {{\tfrac {1}{2}}{\bigl (}{\sqrt {\textstyle x^{2}+y^{2}}}+x{\bigr )}}}+i\operatorname {sgn}(y){\sqrt {{\tfrac {1}{2}}{\bigl (}{\sqrt {\textstyle x^{2}+y^{2}}}-x{\bigr )}}},}" /></span> </p><p>where <span class="texhtml">sgn(<i>y</i>) = 1</span> if <span class="texhtml"><i>y</i> ≥ 0</span> and <span class="texhtml">sgn(<i>y</i>) = −1</span> otherwise.<sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> In particular, the imaginary parts of the original number and the principal value of its square root have the same sign. The real part of the principal value of the square root is always nonnegative. </p><p>For example, the principal square roots of <span class="texhtml">±<i>i</i></span> are given by: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {i}}={\frac {1+i}{\sqrt {2}}},\qquad {\sqrt {-i}}={\frac {1-i}{\sqrt {2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>i</mi> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <mi>i</mi> </mrow> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mrow> <mo>,</mo> <mspace width="2em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mi>i</mi> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>i</mi> </mrow> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {i}}={\frac {1+i}{\sqrt {2}}},\qquad {\sqrt {-i}}={\frac {1-i}{\sqrt {2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ec1c829100c331b08564d2998495e19c282331d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:31.091ex; height:6.176ex;" alt="{\displaystyle {\sqrt {i}}={\frac {1+i}{\sqrt {2}}},\qquad {\sqrt {-i}}={\frac {1-i}{\sqrt {2}}}.}" /></span> </p> <div class="mw-heading mw-heading3"><h3 id="Notes">Notes</h3></div> <p>In the following, the complex <span class="texhtml mvar" style="font-style:italic;">z</span> and <span class="texhtml mvar" style="font-style:italic;">w</span> may be expressed as: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=|z|e^{i\theta _{z}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=|z|e^{i\theta _{z}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b832790e811c06d55e109324616b343c76e7b7b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.012ex; height:3.176ex;" alt="{\displaystyle z=|z|e^{i\theta _{z}}}" /></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w=|w|e^{i\theta _{w}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>w</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> </mrow> </msub> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w=|w|e^{i\theta _{w}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b643d6c33bca0267488d117798790d96855c093e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.494ex; height:3.176ex;" alt="{\displaystyle w=|w|e^{i\theta _{w}}}" /></span></li></ul> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\pi <\theta _{z}\leq \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>π</mi> <mo><</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>≤</mo> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\pi <\theta _{z}\leq \pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3aeabfa69185eedc1363619bb534927d46eb8aff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.761ex; height:2.509ex;" alt="{\displaystyle -\pi <\theta _{z}\leq \pi }" /></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\pi <\theta _{w}\leq \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>π</mi> <mo><</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> </mrow> </msub> <mo>≤</mo> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\pi <\theta _{w}\leq \pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/679df4a1684cbd40b9ff7ac9bbed56bffe6a2903" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.169ex; height:2.509ex;" alt="{\displaystyle -\pi <\theta _{w}\leq \pi }" /></span>. </p><p>Because of the discontinuous nature of the square root function in the complex plane, the following laws are <b>not true</b> in general. </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {zw}}={\sqrt {z}}{\sqrt {w}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>z</mi> <mi>w</mi> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>z</mi> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>w</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {zw}}={\sqrt {z}}{\sqrt {w}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69736e0fca5366225d71b03cbe0ae84ef8d7d9bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.411ex; height:3.009ex;" alt="{\displaystyle {\sqrt {zw}}={\sqrt {z}}{\sqrt {w}}}" /></span> <div class="paragraphbreak" style="margin-top:0.5em"></div> Counterexample for the principal square root: <span class="texhtml"><i>z</i> = −1</span> and <span class="texhtml"><i>w</i> = −1</span> <div class="paragraphbreak" style="margin-top:0.5em"></div> This equality is valid only when <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\pi <\theta _{z}+\theta _{w}\leq \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>π</mi> <mo><</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> </mrow> </msub> <mo>≤</mo> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\pi <\theta _{z}+\theta _{w}\leq \pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/616dd651f429490e09f3398a52a279be4c00a279" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.101ex; height:2.509ex;" alt="{\displaystyle -\pi <\theta _{z}+\theta _{w}\leq \pi }" /></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\sqrt {w}}{\sqrt {z}}}={\sqrt {\frac {w}{z}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mi>w</mi> </msqrt> <msqrt> <mi>z</mi> </msqrt> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mi>w</mi> <mi>z</mi> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\sqrt {w}}{\sqrt {z}}}={\sqrt {\frac {w}{z}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32905bc5f10ca0ce6afeea72186f2ac85d9d91ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:12.359ex; height:6.843ex;" alt="{\displaystyle {\frac {\sqrt {w}}{\sqrt {z}}}={\sqrt {\frac {w}{z}}}}" /></span> <div class="paragraphbreak" style="margin-top:0.5em"></div> Counterexample for the principal square root: <span class="texhtml"><i>w</i> = 1</span> and <span class="texhtml"><i>z</i> = −1</span> <div class="paragraphbreak" style="margin-top:0.5em"></div> This equality is valid only when <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\pi <\theta _{w}-\theta _{z}\leq \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>π</mi> <mo><</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>≤</mo> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\pi <\theta _{w}-\theta _{z}\leq \pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd8739a5027cfcf839f6d2f8105a1e94a115622" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.101ex; height:2.509ex;" alt="{\displaystyle -\pi <\theta _{w}-\theta _{z}\leq \pi }" /></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {z^{*}}}=\left({\sqrt {z}}\right)^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </msqrt> </mrow> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>z</mi> </msqrt> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {z^{*}}}=\left({\sqrt {z}}\right)^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7de0bb8f7c38b7810de19ebe31b599569de31b15" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.066ex; height:3.343ex;" alt="{\displaystyle {\sqrt {z^{*}}}=\left({\sqrt {z}}\right)^{*}}" /></span> <div class="paragraphbreak" style="margin-top:0.5em"></div> Counterexample for the principal square root: <span class="texhtml"><i>z</i> = −1</span>) <div class="paragraphbreak" style="margin-top:0.5em"></div> This equality is valid only when <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta _{z}\neq \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>≠</mo> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta _{z}\neq \pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/034dc5bfb7a900ff7984ee8d6205d4a77711daf5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.523ex; height:2.676ex;" alt="{\displaystyle \theta _{z}\neq \pi }" /></span></li></ul> <p>A similar problem appears with other complex functions with branch cuts, e.g., the <a href="/wiki/Complex_logarithm" title="Complex logarithm">complex logarithm</a> and the relations <span class="texhtml">log<i>z</i> + log<i>w</i> = log(<i>zw</i>)</span> or <span class="texhtml">log(<i>z</i><sup>*</sup>) = log(<i>z</i>)<sup>*</sup></span> which are not true in general. </p><p>Wrongly assuming one of these laws underlies several faulty "proofs", for instance the following one showing that <span class="texhtml">−1 = 1</span>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}-1&=i\cdot i\\&={\sqrt {-1}}\cdot {\sqrt {-1}}\\&={\sqrt {\left(-1\right)\cdot \left(-1\right)}}\\&={\sqrt {1}}\\&=1.\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> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>i</mi> <mo>⋅</mo> <mi>i</mi> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>⋅</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </msqrt> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> </msqrt> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <mn>1.</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}-1&=i\cdot i\\&={\sqrt {-1}}\cdot {\sqrt {-1}}\\&={\sqrt {\left(-1\right)\cdot \left(-1\right)}}\\&={\sqrt {1}}\\&=1.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c9f2dfa4d8a723116461a4761a3e2b92d875653" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.005ex; width:20.383ex; height:17.176ex;" alt="{\displaystyle {\begin{aligned}-1&=i\cdot i\\&={\sqrt {-1}}\cdot {\sqrt {-1}}\\&={\sqrt {\left(-1\right)\cdot \left(-1\right)}}\\&={\sqrt {1}}\\&=1.\end{aligned}}}" /></span> </p><p>The third equality cannot be justified (see <a href="/wiki/Invalid_proof" class="mw-redirect" title="Invalid proof">invalid proof</a>).<sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Location: Chapter VI, Section I, Subsection 2 The fallacy that +1 = -1">: Chapter VI, Section I, Subsection 2 <i>The fallacy that +1 = -1</i> </span></sup> It can be made to hold by changing the meaning of √ so that this no longer represents the principal square root (see above) but selects a branch for the square root that contains <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {1}}\cdot {\sqrt {-1}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> </msqrt> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {1}}\cdot {\sqrt {-1}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2e44299c601dae6dec16d70e4e91e5bae063d0a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.331ex; height:3.176ex;" alt="{\displaystyle {\sqrt {1}}\cdot {\sqrt {-1}}.}" /></span> The left-hand side becomes either<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {-1}}\cdot {\sqrt {-1}}=i\cdot i=-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> <mo>=</mo> <mi>i</mi> <mo>⋅</mo> <mi>i</mi> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {-1}}\cdot {\sqrt {-1}}=i\cdot i=-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8e26e9d17bd17d6512366a57b45dbdceafd41cb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.944ex; height:3.009ex;" alt="{\displaystyle {\sqrt {-1}}\cdot {\sqrt {-1}}=i\cdot i=-1}" /></span> if the branch includes <span class="texhtml">+<i>i</i></span> or<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {-1}}\cdot {\sqrt {-1}}=(-i)\cdot (-i)=-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>i</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>i</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {-1}}\cdot {\sqrt {-1}}=(-i)\cdot (-i)=-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b3acf7c67af449365b332ef5f5df925a7240e18" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.178ex; height:3.009ex;" alt="{\displaystyle {\sqrt {-1}}\cdot {\sqrt {-1}}=(-i)\cdot (-i)=-1}" /></span> if the branch includes <span class="texhtml">−<i>i</i></span>, while the right-hand side becomes<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {\left(-1\right)\cdot \left(-1\right)}}={\sqrt {1}}=-1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>⋅</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> </msqrt> </mrow> <mo>=</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {\left(-1\right)\cdot \left(-1\right)}}={\sqrt {1}}=-1,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8ad32cf0f034e23845cf5c93cf9cce914cefe8a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:26.475ex; height:4.843ex;" alt="{\displaystyle {\sqrt {\left(-1\right)\cdot \left(-1\right)}}={\sqrt {1}}=-1,}" /></span> where the last equality, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {1}}=-1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> </msqrt> </mrow> <mo>=</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {1}}=-1,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d575e5316ea60baefbad924bea7ed1117258e52" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.814ex; height:3.009ex;" alt="{\displaystyle {\sqrt {1}}=-1,}" /></span> is a consequence of the choice of branch in the redefinition of <span class="texhtml">√</span>. </p> <div class="mw-heading mw-heading2"><h2 id="nth_roots_and_polynomial_roots"><span class="texhtml mvar" style="font-style:italic;">n</span>th roots and polynomial roots</h2></div> <p>The definition of a square root of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" 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 x}" /></span> as a number <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" /></span> such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y^{2}=x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y^{2}=x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb1cfda1f3310a5c649d6847b1b2325968850889" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.643ex; height:3.009ex;" alt="{\displaystyle y^{2}=x}" /></span> has been generalized in the following way. </p><p>A <a href="/wiki/Cube_root" title="Cube root">cube root</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" 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 x}" /></span> is a number <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" /></span> such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y^{3}=x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y^{3}=x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1da9d6d6c56d24492a6c80d82aa531bb4ef8ed3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.643ex; height:3.009ex;" alt="{\displaystyle y^{3}=x}" /></span>; it is denoted <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt[{3}]{x}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </mroot> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{3}]{x}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d19f445fd1e8ab7046f090279ee7cf3506f0cf50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.912ex; height:3.009ex;" alt="{\displaystyle {\sqrt[{3}]{x}}.}" /></span> </p><p>If <span class="texhtml mvar" style="font-style:italic;">n</span> is an integer greater than two, a <a href="/wiki/Nth_root" title="Nth root"><span class="texhtml mvar" style="font-style:italic;">n</span>-th root</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" 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 x}" /></span> is a number <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" /></span> such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y^{n}=x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y^{n}=x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/287a41964d6e61f0251e3e13a9fd032c558e8a41" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.807ex; height:2.676ex;" alt="{\displaystyle y^{n}=x}" /></span>; it is denoted <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt[{n}]{x}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mroot> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{n}]{x}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8562e64a6bc6e408ddf67f055682c4dc9c9f957f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.912ex; height:3.009ex;" alt="{\displaystyle {\sqrt[{n}]{x}}.}" /></span> </p><p>Given any <a href="/wiki/Polynomial" title="Polynomial">polynomial</a> <span class="texhtml"><i>p</i></span>, a <a href="/wiki/Polynomial_root" class="mw-redirect" title="Polynomial root">root</a> of <span class="texhtml"><i>p</i></span> is a number <span class="texhtml mvar" style="font-style:italic;">y</span> such that <span class="texhtml"><i>p</i>(<i>y</i>) = 0</span>. For example, the <span class="texhtml mvar" style="font-style:italic;">n</span>th roots of <span class="texhtml mvar" style="font-style:italic;">x</span> are the roots of the polynomial (in <span class="texhtml mvar" style="font-style:italic;">y</span>) <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y^{n}-x.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y^{n}-x.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d0fe48dec343ce88b30171f0bbc7d7ef5f398f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.196ex; height:2.676ex;" alt="{\displaystyle y^{n}-x.}" /></span> </p><p><a href="/wiki/Abel%E2%80%93Ruffini_theorem" title="Abel–Ruffini theorem">Abel–Ruffini theorem</a> states that, in general, the roots of a polynomial of degree five or higher cannot be expressed in terms of <span class="texhtml mvar" style="font-style:italic;">n</span>th roots. </p> <div class="mw-heading mw-heading2"><h2 id="Square_roots_of_matrices_and_operators">Square roots of matrices and operators</h2></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/Square_root_of_a_matrix" title="Square root of a matrix">Square root of a matrix</a></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Square_root_of_a_2_by_2_matrix" title="Square root of a 2 by 2 matrix">Square root of a 2 by 2 matrix</a></div> <p>If <i>A</i> is a <a href="/wiki/Positive-definite_matrix" class="mw-redirect" title="Positive-definite matrix">positive-definite matrix</a> or operator, then there exists precisely one positive definite matrix or operator <i>B</i> with <span class="texhtml"><i>B</i><sup>2</sup> = <i>A</i></span>; we then define <span class="texhtml"><i>A</i><sup>1/2</sup> = <i>B</i></span>. In general matrices may have multiple square roots or even an infinitude of them. For example, the <span class="nowrap">2 × 2</span> <a href="/wiki/Identity_matrix" title="Identity matrix">identity matrix</a> has an infinity of square roots,<sup id="cite_ref-32" class="reference"><a href="#cite_note-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> though only one of them is positive definite. </p> <div class="mw-heading mw-heading2"><h2 id="In_integral_domains,_including_fields"><span id="In_integral_domains.2C_including_fields"></span>In integral domains, including fields</h2></div> <p>Each element of an <a href="/wiki/Integral_domain" title="Integral domain">integral domain</a> has no more than 2 square roots. The <a href="/wiki/Difference_of_two_squares" title="Difference of two squares">difference of two squares</a> identity <span class="texhtml"><i>u</i><sup>2</sup> − <i>v</i><sup>2</sup> = (<i>u</i> − <i>v</i>)(<i>u</i> + <i>v</i>)</span> is proved using the <a href="/wiki/Commutative_ring" title="Commutative ring">commutativity of multiplication</a>. If <span class="texhtml mvar" style="font-style:italic;">u</span> and <span class="texhtml mvar" style="font-style:italic;">v</span> are square roots of the same element, then <span class="texhtml"><i>u</i><sup>2</sup> − <i>v</i><sup>2</sup> = 0</span>. Because there are no <a href="/wiki/Zero_divisors" class="mw-redirect" title="Zero divisors">zero divisors</a> this implies <span class="texhtml"><i>u</i> = <i>v</i></span> or <span class="texhtml"><i>u</i> + <i>v</i> = 0</span>, where the latter means that two roots are <a href="/wiki/Additive_inverse" title="Additive inverse">additive inverses</a> of each other. In other words if an element a square root <span class="texhtml mvar" style="font-style:italic;">u</span> of an element <span class="texhtml mvar" style="font-style:italic;">a</span> exists, then the only square roots of <span class="texhtml mvar" style="font-style:italic;">a</span> are <span class="texhtml mvar" style="font-style:italic;">u</span> and <span class="texhtml mvar" style="font-style:italic;">−u</span>. The only square root of 0 in an integral domain is 0 itself. </p><p>In a field of <a href="/wiki/Characteristic_(algebra)" title="Characteristic (algebra)">characteristic</a> 2, an element either has one square root or does not have any at all, because each element is its own additive inverse, so that <span class="texhtml">−<i>u</i> = <i>u</i></span>. If the field is <a href="/wiki/Finite_field" title="Finite field">finite</a> of characteristic 2 then every element has a unique square root. In a <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a> of any other characteristic, any non-zero element either has two square roots, as explained above, or does not have any. </p><p>Given an odd <a href="/wiki/Prime_number" title="Prime number">prime number</a> <span class="texhtml mvar" style="font-style:italic;">p</span>, let <span class="texhtml"><i>q</i> = <i>p</i><sup><i>e</i></sup></span> for some positive integer <span class="texhtml mvar" style="font-style:italic;">e</span>. A non-zero element of the field <span class="texhtml"><a href="/wiki/Finite_field" title="Finite field"><b>F</b><sub><i>q</i></sub></a></span> with <span class="texhtml mvar" style="font-style:italic;">q</span> elements is a <a href="/wiki/Quadratic_residue" title="Quadratic residue">quadratic residue</a> if it has a square root in <span class="texhtml"><b>F</b><sub><i>q</i></sub></span>. Otherwise, it is a quadratic non-residue. There are <span class="texhtml">(<i>q</i> − 1)/2</span> quadratic residues and <span class="texhtml">(<i>q</i> − 1)/2</span> quadratic non-residues; zero is not counted in either class. The quadratic residues form a <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a> under multiplication. The properties of quadratic residues are widely used in <a href="/wiki/Number_theory" title="Number theory">number theory</a>. </p> <div class="mw-heading mw-heading2"><h2 id="In_rings_in_general">In rings in general</h2></div> <p>Unlike in an integral domain, a square root in an arbitrary (unital) ring need not be unique up to sign. For example, in the ring <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} /8\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> <mn>8</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} /8\mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34e4b343f070c08c1500541ce768b60f9b37bf9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.426ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} /8\mathbb {Z} }" /></span> of integers <a href="/wiki/Modular_arithmetic" title="Modular arithmetic">modulo 8</a> (which is commutative, but has zero divisors), the element 1 has four distinct square roots: ±1 and ±3. </p><p>Another example is provided by the ring of <a href="/wiki/Quaternion" title="Quaternion">quaternions</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {H} ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d178e5ac94e706fdb8d8733d567b7c087b23195" 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 {H} ,}" /></span> which has no zero divisors, but is not commutative. Here, the element −1 has <a href="/wiki/Quaternion#Square_roots_of_−1" title="Quaternion">infinitely many square roots</a>, including <span class="texhtml">±<i>i</i></span>, <span class="texhtml">±<i>j</i></span>, and <span class="texhtml">±<i>k</i></span>. In fact, the set of square roots of <span class="texhtml">−1</span> is exactly<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{ai+bj+ck\mid a^{2}+b^{2}+c^{2}=1\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>a</mi> <mi>i</mi> <mo>+</mo> <mi>b</mi> <mi>j</mi> <mo>+</mo> <mi>c</mi> <mi>k</mi> <mo>∣</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{ai+bj+ck\mid a^{2}+b^{2}+c^{2}=1\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0dc159d3e291b844ca2393f6c32af2dc8a71e98" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.134ex; height:3.176ex;" alt="{\displaystyle \{ai+bj+ck\mid a^{2}+b^{2}+c^{2}=1\}.}" /></span> </p><p>A square root of 0 is either 0 or a zero divisor. Thus in rings where zero divisors do not exist, it is uniquely 0. However, rings with zero divisors may have multiple square roots of 0. For example, in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} /n^{2}\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>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </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} /n^{2}\mathbb {Z} ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c51eb14e6d2fb8a3c317c7d7e8f67e35a52a681" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.359ex; height:3.176ex;" alt="{\displaystyle \mathbb {Z} /n^{2}\mathbb {Z} ,}" /></span> any multiple of <span class="texhtml mvar" style="font-style:italic;">n</span> is a square root of 0. </p> <div class="mw-heading mw-heading2"><h2 id="Geometric_construction_of_the_square_root">Geometric construction of the square root</h2></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:SqrtGeom.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/07/SqrtGeom.gif/220px-SqrtGeom.gif" decoding="async" width="220" height="131" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/07/SqrtGeom.gif/330px-SqrtGeom.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/07/SqrtGeom.gif/440px-SqrtGeom.gif 2x" data-file-width="524" data-file-height="312" /></a><figcaption><a href="/wiki/Straightedge_and_compass_construction" title="Straightedge and compass construction">Constructing</a> the length <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x={\sqrt {a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>a</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x={\sqrt {a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/949c9b60025e426b8128dfdd4ae98cb07bd52969" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.594ex; height:3.009ex;" alt="{\displaystyle x={\sqrt {a}}}" /></span>, given the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}" /></span> and the unit length</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Spiral_of_Theodorus.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9f/Spiral_of_Theodorus.svg/220px-Spiral_of_Theodorus.svg.png" decoding="async" width="220" height="179" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9f/Spiral_of_Theodorus.svg/330px-Spiral_of_Theodorus.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9f/Spiral_of_Theodorus.svg/440px-Spiral_of_Theodorus.svg.png 2x" data-file-width="700" data-file-height="570" /></a><figcaption>The <a href="/wiki/Spiral_of_Theodorus" title="Spiral of Theodorus">Spiral of Theodorus</a> up to the triangle with a hypotenuse of <span class="texhtml"><span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">17</span></span></span></figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Root_rectangles_Hambidge_1920.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c6/Root_rectangles_Hambidge_1920.png/250px-Root_rectangles_Hambidge_1920.png" decoding="async" width="220" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c6/Root_rectangles_Hambidge_1920.png/330px-Root_rectangles_Hambidge_1920.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c6/Root_rectangles_Hambidge_1920.png/500px-Root_rectangles_Hambidge_1920.png 2x" data-file-width="500" data-file-height="341" /></a><figcaption><a href="/wiki/Jay_Hambidge" title="Jay Hambidge">Jay Hambidge</a>'s construction of successive square roots using <a href="/wiki/Root_rectangle" class="mw-redirect" title="Root rectangle">root rectangles</a></figcaption></figure> <p>The square root of a positive number is usually defined as the side length of a <a href="/wiki/Square" title="Square">square</a> with the <a href="/wiki/Area" title="Area">area</a> equal to the given number. But the square shape is not necessary for it: if one of two <a href="/wiki/Similarity_(geometry)" title="Similarity (geometry)">similar</a> <a href="/wiki/Euclidean_plane" title="Euclidean plane">planar Euclidean</a> objects has the area <i>a</i> times greater than another, then the ratio of their linear sizes is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>a</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afccc332c876539296df1a980127d86173e59ef0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.166ex; height:3.009ex;" alt="{\displaystyle {\sqrt {a}}}" /></span>. </p><p>A square root can be constructed with a compass and straightedge. In his <a href="/wiki/Euclid%27s_Elements" title="Euclid's Elements">Elements</a>, <a href="/wiki/Euclid" title="Euclid">Euclid</a> (<a href="/wiki/Floruit" title="Floruit">fl.</a> 300 BC) gave the construction of the <a href="/wiki/Geometric_mean" title="Geometric mean">geometric mean</a> of two quantities in two different places: <a rel="nofollow" class="external text" href="http://aleph0.clarku.edu/~djoyce/java/elements/bookII/propII14.html">Proposition II.14</a> and <a rel="nofollow" class="external text" href="http://aleph0.clarku.edu/~djoyce/java/elements/bookVI/propVI13.html">Proposition VI.13</a>. Since the geometric mean of <i>a</i> and <i>b</i> is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {ab}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>a</mi> <mi>b</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {ab}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8814066fc656817115551d0cd8ec8db74bb048aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.163ex; height:3.009ex;" alt="{\displaystyle {\sqrt {ab}}}" /></span>, one can construct <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>a</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afccc332c876539296df1a980127d86173e59ef0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.166ex; height:3.009ex;" alt="{\displaystyle {\sqrt {a}}}" /></span> simply by taking <span class="texhtml"><i>b</i> = 1</span>. </p><p>The construction is also given by <a href="/wiki/Descartes" class="mw-redirect" title="Descartes">Descartes</a> in his <i><a href="/wiki/La_G%C3%A9om%C3%A9trie" title="La Géométrie">La Géométrie</a></i>, see figure 2 on <a rel="nofollow" class="external text" href="http://historical.library.cornell.edu/cgi-bin/cul.math/docviewer?did=00570001&seq=12&frames=0&view=50">page 2</a>. However, Descartes made no claim to originality and his audience would have been quite familiar with Euclid. </p><p>Euclid's second proof in Book VI depends on the theory of <a href="/wiki/Similar_triangles#Similar_triangles" class="mw-redirect" title="Similar triangles">similar triangles</a>. Let AHB be a line segment of length <span class="texhtml"><i>a</i> + <i>b</i></span> with <span class="texhtml">AH = <i>a</i></span> and <span class="texhtml">HB = <i>b</i></span>. Construct the circle with AB as diameter and let C be one of the two intersections of the perpendicular chord at H with the circle and denote the length CH as <i>h</i>. Then, using <a href="/wiki/Thales%27_theorem" class="mw-redirect" title="Thales' theorem">Thales' theorem</a> and, as in the <a href="/wiki/Pythagorean_theorem#Proof_using_similar_triangles" title="Pythagorean theorem">proof of Pythagoras' theorem by similar triangles</a>, triangle AHC is similar to triangle CHB (as indeed both are to triangle ACB, though we don't need that, but it is the essence of the proof of Pythagoras' theorem) so that AH:CH is as HC:HB, i.e. <span class="texhtml"><i>a</i>/<i>h</i> = <i>h</i>/<i>b</i></span>, from which we conclude by cross-multiplication that <span class="texhtml"><i>h</i><sup>2</sup> = <i>ab</i></span>, and finally that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h={\sqrt {ab}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>a</mi> <mi>b</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h={\sqrt {ab}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7e64149049f753399863dcae96f5e79ca004c87" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.601ex; height:3.009ex;" alt="{\displaystyle h={\sqrt {ab}}}" /></span>. When marking the midpoint O of the line segment AB and drawing the radius OC of length <span class="texhtml">(<i>a</i> + <i>b</i>)/2</span>, then clearly OC > CH, i.e. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {a+b}{2}}\geq {\sqrt {ab}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mo>+</mo> <mi>b</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>≥</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>a</mi> <mi>b</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\frac {a+b}{2}}\geq {\sqrt {ab}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/943fd2010c0f1d2819ffbaaa68f2f8b40c02326c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:10.951ex; height:3.676ex;" alt="{\textstyle {\frac {a+b}{2}}\geq {\sqrt {ab}}}" /></span> (with equality if and only if <span class="texhtml"><i>a</i> = <i>b</i></span>), which is the <a href="/wiki/Inequality_of_arithmetic_and_geometric_means" class="mw-redirect" title="Inequality of arithmetic and geometric means">arithmetic–geometric mean inequality for two variables</a> and, as noted <a class="mw-selflink-fragment" href="#Computation">above</a>, is the basis of the <a href="/wiki/Greek_Mathematics" class="mw-redirect" title="Greek Mathematics">Ancient Greek</a> understanding of "Heron's method". </p><p>Another method of geometric construction uses <a href="/wiki/Right_triangle" title="Right triangle">right triangles</a> and <a href="/wiki/Mathematical_induction" title="Mathematical induction">induction</a>: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7bf2189fbb14a70abb4d7e3f2aedec1f3e787e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:3.009ex;" alt="{\displaystyle {\sqrt {1}}}" /></span> can be constructed, and once <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>x</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d62b24be305beff66cba9bfbcc01a362ba390f44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.266ex; height:3.009ex;" alt="{\displaystyle {\sqrt {x}}}" /></span> has been constructed, the right triangle with legs 1 and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>x</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d62b24be305beff66cba9bfbcc01a362ba390f44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.266ex; height:3.009ex;" alt="{\displaystyle {\sqrt {x}}}" /></span> has a <a href="/wiki/Hypotenuse" title="Hypotenuse">hypotenuse</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {x+1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>x</mi> <mo>+</mo> <mn>1</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {x+1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2c9507b348d8635f87afb425d80e65a0ee1d931" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.268ex; height:3.009ex;" alt="{\displaystyle {\sqrt {x+1}}}" /></span>. Constructing successive square roots in this manner yields the <a href="/wiki/Spiral_of_Theodorus" title="Spiral of Theodorus">Spiral of Theodorus</a> depicted above. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></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/Apotome_(mathematics)" title="Apotome (mathematics)">Apotome (mathematics)</a></li> <li><a href="/wiki/Cube_root" title="Cube root">Cube root</a></li> <li><a href="/wiki/Functional_square_root" title="Functional square root">Functional square root</a></li> <li><a href="/wiki/Integer_square_root" title="Integer square root">Integer square root</a></li> <li><a href="/wiki/Nested_radical" title="Nested radical">Nested radical</a></li> <li><a href="/wiki/Nth_root" title="Nth root">Nth root</a></li> <li><a href="/wiki/Root_of_unity" title="Root of unity">Root of unity</a></li> <li><a href="/wiki/Solving_quadratic_equations_with_continued_fractions" title="Solving quadratic equations with continued fractions">Solving quadratic equations with continued fractions</a></li> <li><a href="/wiki/Square-root_sum_problem" title="Square-root sum problem">Square-root sum problem</a></li> <li><a href="/wiki/Square_root_principle" class="mw-redirect" title="Square root principle">Square root principle</a></li> <li><a href="/wiki/Quantum_gate#Square_root_of_NOT_gate_(√NOT)" class="mw-redirect" title="Quantum gate">Quantum gate § Square root of NOT gate (√NOT)</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Notes_2">Notes</h2></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">Gel'fand, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Z9z7iliyFD0C&pg=PA120">p. 120</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160902151740/https://books.google.com/books?id=Z9z7iliyFD0C&pg=PA120">Archived</a> 2016-09-02 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><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 class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.mathsisfun.com/square-root.html">"Squares and Square Roots"</a>. <i>www.mathsisfun.com</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2020-08-28</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.mathsisfun.com&rft.atitle=Squares+and+Square+Roots&rft_id=https%3A%2F%2Fwww.mathsisfun.com%2Fsquare-root.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare+root" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFZillShanahan2008" class="citation book cs1">Zill, Dennis G.; Shanahan, Patrick (2008). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=YKZqY8PCNo0C"><i>A First Course in Complex Analysis With Applications</i></a> (2nd ed.). Jones & Bartlett Learning. p. 78. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7637-5772-4" title="Special:BookSources/978-0-7637-5772-4"><bdi>978-0-7637-5772-4</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160901081936/https://books.google.com/books?id=YKZqY8PCNo0C">Archived</a> from the original on 2016-09-01.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+First+Course+in+Complex+Analysis+With+Applications&rft.pages=78&rft.edition=2nd&rft.pub=Jones+%26+Bartlett+Learning&rft.date=2008&rft.isbn=978-0-7637-5772-4&rft.aulast=Zill&rft.aufirst=Dennis+G.&rft.au=Shanahan%2C+Patrick&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DYKZqY8PCNo0C&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare+root" class="Z3988"></span> <a rel="nofollow" class="external text" href="https://books.google.com/books?id=YKZqY8PCNo0C&pg=PA78">Extract of page 78</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160901091148/https://books.google.com/books?id=YKZqY8PCNo0C&pg=PA78">Archived</a> 2016-09-01 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFWeisstein" class="citation web cs1">Weisstein, Eric W. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/SquareRoot.html">"Square Root"</a>. <i>mathworld.wolfram.com</i><span class="reference-accessdate">. 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(1961). <a rel="nofollow" class="external text" href="https://dx.doi.org/10.1007/bf00327767">"The ritual origin of geometry"</a>. <i>Archive for History of Exact Sciences</i>. <b>1</b> (5): <span class="nowrap">488–</span>527. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fbf00327767">10.1007/bf00327767</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0003-9519">0003-9519</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:119992603">119992603</a>. <q>Seidenberg (pp. 501-505) proposes: "It is the distinction between use and origin." [By analogy] "KEPLER needed the ellipse to describe the paths of the planets around the sun; he did not, however invent the ellipse, but made use of a curve that had been lying around for nearly 2000 years". In this manner Seidenberg argues: "Although the date of a manuscript or text cannot give us the age of the practices it discloses, nonetheless the evidence is contained in manuscripts." Seidenberg quotes Thibaut from 1875: "Regarding the time in which the Sulvasutras may have been composed, it is impossible to give more accurate information than we are able to give about the date of the Kalpasutras. But whatever the period may have been during which Kalpasutras and Sulvasutras were composed in the form now before us, we must keep in view that they only give a systematically arranged description of sacrificial rites, which had been practiced during long preceding ages." Lastly, Seidenberg summarizes: "In 1899, THIBAUT ventured to assign the fourth or the third centuries B.C. as the latest possible date for the composition of the Sulvasutras (it being understood that this refers to a codification of far older material)."<span class="cs1-kern-right"></span></q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Archive+for+History+of+Exact+Sciences&rft.atitle=The+ritual+origin+of+geometry&rft.volume=1&rft.issue=5&rft.pages=%3Cspan+class%3D%22nowrap%22%3E488-%3C%2Fspan%3E527&rft.date=1961&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A119992603%23id-name%3DS2CID&rft.issn=0003-9519&rft_id=info%3Adoi%2F10.1007%2Fbf00327767&rft.aulast=Seidenberg&rft.aufirst=A.&rft_id=http%3A%2F%2Fdx.doi.org%2F10.1007%2Fbf00327767&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare+root" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text">Joseph, ch.8.</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFDutta1931" class="citation journal cs1">Dutta, Bibhutibhusan (1931). <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2300909">"On the Origin of the Hindu Terms for "Root"<span class="cs1-kern-right"></span>"</a>. <i>The American Mathematical Monthly</i>. <b>38</b> (7): <span class="nowrap">371–</span>376. <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%2F2300909">10.2307/2300909</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/2300909">2300909</a><span class="reference-accessdate">. Retrieved <span class="nowrap">30 March</span> 2024</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+American+Mathematical+Monthly&rft.atitle=On+the+Origin+of+the+Hindu+Terms+for+%22Root%22&rft.volume=38&rft.issue=7&rft.pages=%3Cspan+class%3D%22nowrap%22%3E371-%3C%2Fspan%3E376&rft.date=1931&rft_id=info%3Adoi%2F10.2307%2F2300909&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2300909%23id-name%3DJSTOR&rft.aulast=Dutta&rft.aufirst=Bibhutibhusan&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2300909&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare+root" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFCynthia_J._HuffmanScott_V._Thuong2015" class="citation web cs1">Cynthia J. Huffman; Scott V. Thuong (2015). <a rel="nofollow" class="external text" href="https://maa.org/press/periodicals/convergence/ancient-indian-rope-geometry-in-the-classroom-approximating-the-square-root-of-2#:~:text=The%20Śulba-sūtras%20of%20Āpastamba,is%20less%20than%200.0003%25!">"Ancient Indian Rope Geometry in the Classroom - Approximating the Square Root of 2"</a>. <i>www.maa.org</i><span class="reference-accessdate">. Retrieved <span class="nowrap">30 March</span> 2024</span>. <q>Increase the measure by its third and this third by its own fourth, less the thirty-fourth part of that fourth. This is the value with a special quantity in excess.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.maa.org&rft.atitle=Ancient+Indian+Rope+Geometry+in+the+Classroom+-+Approximating+the+Square+Root+of+2&rft.date=2015&rft.au=Cynthia+J.+Huffman&rft.au=Scott+V.+Thuong&rft_id=https%3A%2F%2Fmaa.org%2Fpress%2Fperiodicals%2Fconvergence%2Fancient-indian-rope-geometry-in-the-classroom-approximating-the-square-root-of-2%23%3A~%3Atext%3DThe%2520%C5%9Aulba-s%C5%ABtras%2520of%2520%C4%80pastamba%2Cis%2520less%2520than%25200.0003%2525%21&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare+root" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFJ_J_O'ConnorE_F_Robertson2020" class="citation web cs1">J J O'Connor; E F Robertson (November 2020). <a rel="nofollow" class="external text" href="https://mathshistory.st-andrews.ac.uk/Biographies/Apastamba/">"Apastamba"</a>. <i>www.mathshistory.st-andrews.ac.uk</i>. School of Mathematics and Statistics, University of St Andrews, Scotland<span class="reference-accessdate">. Retrieved <span class="nowrap">30 March</span> 2024</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.mathshistory.st-andrews.ac.uk&rft.atitle=Apastamba&rft.date=2020-11&rft.au=J+J+O%27Connor&rft.au=E+F+Robertson&rft_id=https%3A%2F%2Fmathshistory.st-andrews.ac.uk%2FBiographies%2FApastamba%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare+root" class="Z3988"></span></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHeath1908" class="citation book cs1">Heath, Sir Thomas L. (1908). <a rel="nofollow" class="external text" href="https://archive.org/stream/thirteenbookseu03heibgoog#page/n14/mode/1up"><i>The Thirteen Books of The Elements, Vol. 3</i></a>. Cambridge University Press. p. 3.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Thirteen+Books+of+The+Elements%2C+Vol.+3&rft.pages=3&rft.pub=Cambridge+University+Press&rft.date=1908&rft.aulast=Heath&rft.aufirst=Sir+Thomas+L.&rft_id=https%3A%2F%2Farchive.org%2Fstream%2Fthirteenbookseu03heibgoog%23page%2Fn14%2Fmode%2F1up&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare+root" class="Z3988"></span></span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFCraig_Smorynski2007" class="citation book cs1">Craig Smorynski (2007). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=_zliInaOM8UC"><i>History of Mathematics: A Supplement</i></a> (illustrated, annotated ed.). Springer Science & Business Media. p. 49. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-75480-2" title="Special:BookSources/978-0-387-75480-2"><bdi>978-0-387-75480-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=History+of+Mathematics%3A+A+Supplement&rft.pages=49&rft.edition=illustrated%2C+annotated&rft.pub=Springer+Science+%26+Business+Media&rft.date=2007&rft.isbn=978-0-387-75480-2&rft.au=Craig+Smorynski&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D_zliInaOM8UC&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare+root" class="Z3988"></span> <a rel="nofollow" class="external text" href="https://books.google.com/books?id=_zliInaOM8UC&pg=PA49">Extract of page 49</a></span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBrian_E._BlankSteven_George_Krantz2006" class="citation book cs1">Brian E. Blank; Steven George Krantz (2006). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=hMY8lbX87Y8C"><i>Calculus: Single Variable, Volume 1</i></a> (illustrated ed.). Springer Science & Business Media. p. 71. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-931914-59-8" title="Special:BookSources/978-1-931914-59-8"><bdi>978-1-931914-59-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Calculus%3A+Single+Variable%2C+Volume+1&rft.pages=71&rft.edition=illustrated&rft.pub=Springer+Science+%26+Business+Media&rft.date=2006&rft.isbn=978-1-931914-59-8&rft.au=Brian+E.+Blank&rft.au=Steven+George+Krantz&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DhMY8lbX87Y8C&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare+root" class="Z3988"></span> <a rel="nofollow" class="external text" href="https://books.google.com/books?id=hMY8lbX87Y8C&pg=PA71">Extract of page 71</a></span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text">Boyer, Carl B.; Merzbach, Uta C. (2011). A History of Mathematics (3rd ed.). Hoboken, NJ: John Wiley & Sons. pp. 51–53. ISBN 978-0470525487.</span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text">Stillwell, John (2010). <i>Mathematics and Its History</i> (3rd ed.). New York, NY: Springer. pp. 14–15. ISBN 978-1441960528.</span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text">Dauben (2007), p. 210.</span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://nrich.maths.org/6546">"The Development of Algebra - 2"</a>. <i>maths.org</i>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20141124102946/http://nrich.maths.org/6546">Archived</a> from the original on 24 November 2014<span class="reference-accessdate">. Retrieved <span class="nowrap">19 January</span> 2015</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=maths.org&rft.atitle=The+Development+of+Algebra+-+2&rft_id=http%3A%2F%2Fnrich.maths.org%2F6546&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare+root" class="Z3988"></span></span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFOaks2012" class="citation thesis cs1">Oaks, Jeffrey A. (2012). <a rel="nofollow" class="external text" href="http://logica.ugent.be/philosophica/fulltexts/87-2.pdf"><i>Algebraic Symbolism in Medieval Arabic Algebra</i></a> <span class="cs1-format">(PDF)</span> (Thesis). Philosophica. p. 36. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20161203134229/http://logica.ugent.be/philosophica/fulltexts/87-2.pdf">Archived</a> <span class="cs1-format">(PDF)</span> from the original on 2016-12-03.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adissertation&rft.title=Algebraic+Symbolism+in+Medieval+Arabic+Algebra&rft.inst=Philosophica&rft.date=2012&rft.aulast=Oaks&rft.aufirst=Jeffrey+A.&rft_id=http%3A%2F%2Flogica.ugent.be%2Fphilosophica%2Ffulltexts%2F87-2.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare+root" class="Z3988"></span></span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFManguel2006" class="citation book cs1">Manguel, Alberto (2006). "Done on paper: the dual nature of numbers and the page". <i>The Life of Numbers</i>. Taric, S.A. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/84-86882-14-1" title="Special:BookSources/84-86882-14-1"><bdi>84-86882-14-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Done+on+paper%3A+the+dual+nature+of+numbers+and+the+page&rft.btitle=The+Life+of+Numbers&rft.pub=Taric%2C+S.A.&rft.date=2006&rft.isbn=84-86882-14-1&rft.aulast=Manguel&rft.aufirst=Alberto&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare+root" class="Z3988"></span></span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFParkhurst2006" class="citation book cs1">Parkhurst, David F. (2006). <span class="id-lock-limited" title="Free access subject to limited trial, subscription normally required"><a rel="nofollow" class="external text" href="https://archive.org/details/introductiontoap00park_663"><i>Introduction to Applied Mathematics for Environmental Science</i></a></span>. Springer. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/introductiontoap00park_663/page/n249">241</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780387342283" title="Special:BookSources/9780387342283"><bdi>9780387342283</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+Applied+Mathematics+for+Environmental+Science&rft.pages=241&rft.pub=Springer&rft.date=2006&rft.isbn=9780387342283&rft.aulast=Parkhurst&rft.aufirst=David+F.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fintroductiontoap00park_663&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare+root" class="Z3988"></span></span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSolow1993" class="citation book cs1">Solow, Anita E. (1993). <a rel="nofollow" class="external text" href="https://archive.org/details/learningbydiscov0001unse/page/48"><i>Learning by Discovery: A Lab Manual for Calculus</i></a>. Cambridge University Press. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/learningbydiscov0001unse/page/48">48</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780883850831" title="Special:BookSources/9780883850831"><bdi>9780883850831</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Learning+by+Discovery%3A+A+Lab+Manual+for+Calculus&rft.pages=48&rft.pub=Cambridge+University+Press&rft.date=1993&rft.isbn=9780883850831&rft.aulast=Solow&rft.aufirst=Anita+E.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Flearningbydiscov0001unse%2Fpage%2F48&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare+root" class="Z3988"></span></span> </li> <li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFAitkenBroadhurstHladky2009" class="citation book cs1">Aitken, Mike; Broadhurst, Bill; Hladky, Stephen (2009). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=KywWBAAAQBAJ"><i>Mathematics for Biological Scientists</i></a>. Garland Science. p. 41. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-136-84393-8" title="Special:BookSources/978-1-136-84393-8"><bdi>978-1-136-84393-8</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170301101038/https://books.google.com/books?id=KywWBAAAQBAJ">Archived</a> from the original on 2017-03-01.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematics+for+Biological+Scientists&rft.pages=41&rft.pub=Garland+Science&rft.date=2009&rft.isbn=978-1-136-84393-8&rft.aulast=Aitken&rft.aufirst=Mike&rft.au=Broadhurst%2C+Bill&rft.au=Hladky%2C+Stephen&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DKywWBAAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare+root" class="Z3988"></span> <a rel="nofollow" class="external text" href="https://books.google.com/books?id=KywWBAAAQBAJ&pg=PA41">Extract of page 41</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170301100516/https://books.google.com/books?id=KywWBAAAQBAJ&pg=PA41">Archived</a> 2017-03-01 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></span> </li> <li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHeath1921" class="citation book cs1">Heath, Sir Thomas L. (1921). <a rel="nofollow" class="external text" href="https://archive.org/details/ahistorygreekma00heatgoog"><i>A History of Greek Mathematics, Vol. 2</i></a>. Oxford: Clarendon Press. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/ahistorygreekma00heatgoog/page/n340">323</a>–324.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+History+of+Greek+Mathematics%2C+Vol.+2&rft.place=Oxford&rft.pages=323-324&rft.pub=Clarendon+Press&rft.date=1921&rft.aulast=Heath&rft.aufirst=Sir+Thomas+L.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fahistorygreekma00heatgoog&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare+root" class="Z3988"></span></span> </li> <li id="cite_note-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-25">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMuller2006" class="citation book cs1">Muller, Jean-Mic (2006). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=g3AlWip4R38C"><i>Elementary functions: algorithms and implementation</i></a>. Springer. pp. <span class="nowrap">92–</span>93. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-8176-4372-9" title="Special:BookSources/0-8176-4372-9"><bdi>0-8176-4372-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elementary+functions%3A+algorithms+and+implementation&rft.pages=%3Cspan+class%3D%22nowrap%22%3E92-%3C%2Fspan%3E93&rft.pub=Springer&rft.date=2006&rft.isbn=0-8176-4372-9&rft.aulast=Muller&rft.aufirst=Jean-Mic&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dg3AlWip4R38C&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare+root" class="Z3988"></span>, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=g3AlWip4R38C&pg=PA92">Chapter 5, p 92</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160901091516/https://books.google.com/books?id=g3AlWip4R38C&pg=PA92">Archived</a> 2016-09-01 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></span> </li> <li id="cite_note-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-26">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.cplusplus.com/reference/clibrary/cmath/sqrt/">"Function sqrt"</a>. <i>CPlusPlus.com</i>. The C++ Resources Network. 2016. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20121122050619/http://www.cplusplus.com/reference/clibrary/cmath/sqrt/">Archived</a> from the original on November 22, 2012<span class="reference-accessdate">. Retrieved <span class="nowrap">June 24,</span> 2016</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=CPlusPlus.com&rft.atitle=Function+sqrt&rft.date=2016&rft_id=http%3A%2F%2Fwww.cplusplus.com%2Freference%2Fclibrary%2Fcmath%2Fsqrt%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare+root" class="Z3988"></span></span> </li> <li id="cite_note-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-27">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFOverland2013" class="citation book cs1">Overland, Brian (2013). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=eJFpV-_t4WkC&q=%22squirt%22+sqrt+C%2B%2B&pg=PA338"><i>C++ for the Impatient</i></a>. Addison-Wesley. p. 338. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780133257120" title="Special:BookSources/9780133257120"><bdi>9780133257120</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/850705706">850705706</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160901082021/https://books.google.com/books?id=eJFpV-_t4WkC&pg=PA338&dq=%22squirt%22+sqrt+C%2B%2B&hl=en&sa=X&ved=0ahUKEwjEwfj04sHNAhUY0GMKHatGDnsQ6AEIKDAC#v=onepage&q=%22squirt%22%20sqrt%20C%2B%2B&f=false">Archived</a> from the original on September 1, 2016<span class="reference-accessdate">. Retrieved <span class="nowrap">June 24,</span> 2016</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=C%2B%2B+for+the+Impatient&rft.pages=338&rft.pub=Addison-Wesley&rft.date=2013&rft_id=info%3Aoclcnum%2F850705706&rft.isbn=9780133257120&rft.aulast=Overland&rft.aufirst=Brian&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DeJFpV-_t4WkC%26q%3D%2522squirt%2522%2Bsqrt%2BC%252B%252B%26pg%3DPA338&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare+root" class="Z3988"></span></span> </li> <li id="cite_note-28"><span class="mw-cite-backlink"><b><a href="#cite_ref-28">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFAbramowitzStegun1964" class="citation book cs1">Abramowitz, Milton; Stegun, Irene A. (1964). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=MtU8uP7XMvoC"><i>Handbook of mathematical functions with formulas, graphs, and mathematical tables</i></a>. Courier Dover Publications. p. 17. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-61272-4" title="Special:BookSources/0-486-61272-4"><bdi>0-486-61272-4</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160423180235/https://books.google.com/books?id=MtU8uP7XMvoC">Archived</a> from the original on 2016-04-23.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Handbook+of+mathematical+functions+with+formulas%2C+graphs%2C+and+mathematical+tables&rft.pages=17&rft.pub=Courier+Dover+Publications&rft.date=1964&rft.isbn=0-486-61272-4&rft.aulast=Abramowitz&rft.aufirst=Milton&rft.au=Stegun%2C+Irene+A.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DMtU8uP7XMvoC&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare+root" class="Z3988"></span>, <a rel="nofollow" class="external text" href="http://www.math.sfu.ca/~cbm/aands/page_17.htm">Section 3.7.27, p. 17</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20090910094533/http://www.math.sfu.ca/~cbm/aands/page_17.htm">Archived</a> 2009-09-10 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></span> </li> <li id="cite_note-29"><span class="mw-cite-backlink"><b><a href="#cite_ref-29">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFCooke2008" class="citation book cs1">Cooke, Roger (2008). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=lUcTsYopfhkC&pg=PA59"><i>Classical algebra: its nature, origins, and uses</i></a>. John Wiley and Sons. p. 59. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-470-25952-8" title="Special:BookSources/978-0-470-25952-8"><bdi>978-0-470-25952-8</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160423183239/https://books.google.com/books?id=lUcTsYopfhkC&pg=PA59">Archived</a> from the original on 2016-04-23.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Classical+algebra%3A+its+nature%2C+origins%2C+and+uses&rft.pages=59&rft.pub=John+Wiley+and+Sons&rft.date=2008&rft.isbn=978-0-470-25952-8&rft.aulast=Cooke&rft.aufirst=Roger&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DlUcTsYopfhkC%26pg%3DPA59&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare+root" class="Z3988"></span></span> </li> <li id="cite_note-30"><span class="mw-cite-backlink"><b><a href="#cite_ref-30">^</a></b></span> <span class="reference-text">This sign function differs from the usual <a href="/wiki/Sign_function" title="Sign function">sign function</a> by its value at <span class="texhtml">0</span>.</span> </li> <li id="cite_note-31"><span class="mw-cite-backlink"><b><a href="#cite_ref-31">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMaxwell1959" class="citation book cs1">Maxwell, E. A. (1959). <a rel="nofollow" class="external text" href="https://archive.org/details/fallaciesinmathe0000maxw_z1t0/page/n5/mode/2up"><i>Fallacies in Mathematics</i></a>. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780511569739" title="Special:BookSources/9780511569739"><bdi>9780511569739</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fallacies+in+Mathematics&rft.pub=Cambridge+University+Press&rft.date=1959&rft.isbn=9780511569739&rft.aulast=Maxwell&rft.aufirst=E.+A.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Ffallaciesinmathe0000maxw_z1t0%2Fpage%2Fn5%2Fmode%2F2up&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare+root" class="Z3988"></span></span> </li> <li id="cite_note-32"><span class="mw-cite-backlink"><b><a href="#cite_ref-32">^</a></b></span> <span class="reference-text">Mitchell, Douglas W., "Using Pythagorean triples to generate square roots of I<sub>2</sub>", <i>Mathematical Gazette</i> 87, November 2003, 499–500.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFDauben2007" class="citation book cs1"><a href="/wiki/Joseph_Dauben" title="Joseph Dauben">Dauben, Joseph W.</a> (2007). "Chinese Mathematics I". In Katz, Victor J. (ed.). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=3ullzl036UEC"><i>The Mathematics of Egypt, Mesopotamia, China, India, and Islam</i></a>. Princeton: Princeton University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-11485-9" title="Special:BookSources/978-0-691-11485-9"><bdi>978-0-691-11485-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Chinese+Mathematics+I&rft.btitle=The+Mathematics+of+Egypt%2C+Mesopotamia%2C+China%2C+India%2C+and+Islam&rft.place=Princeton&rft.pub=Princeton+University+Press&rft.date=2007&rft.isbn=978-0-691-11485-9&rft.aulast=Dauben&rft.aufirst=Joseph+W.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D3ullzl036UEC&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare+root" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFGel'fandShen1993" class="citation book cs1"><a href="/wiki/Israel_Gelfand" title="Israel Gelfand">Gel'fand, Izrael M.</a>; Shen, Alexander (1993). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Z9z7iliyFD0C"><i>Algebra</i></a> (3rd ed.). Birkhäuser. p. 120. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-8176-3677-3" title="Special:BookSources/0-8176-3677-3"><bdi>0-8176-3677-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebra&rft.pages=120&rft.edition=3rd&rft.pub=Birkh%C3%A4user&rft.date=1993&rft.isbn=0-8176-3677-3&rft.aulast=Gel%27fand&rft.aufirst=Izrael+M.&rft.au=Shen%2C+Alexander&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DZ9z7iliyFD0C&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare+root" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFJoseph2000" class="citation book cs1">Joseph, George (2000). <a rel="nofollow" class="external text" href="https://archive.org/details/crestofpeacockno00jose"><i>The Crest of the Peacock</i></a>. Princeton: Princeton University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-691-00659-8" title="Special:BookSources/0-691-00659-8"><bdi>0-691-00659-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Crest+of+the+Peacock&rft.place=Princeton&rft.pub=Princeton+University+Press&rft.date=2000&rft.isbn=0-691-00659-8&rft.aulast=Joseph&rft.aufirst=George&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fcrestofpeacockno00jose&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare+root" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSmith1958" class="citation book cs1"><a href="/wiki/David_Eugene_Smith" title="David Eugene Smith">Smith, David</a> (1958). <i>History of Mathematics</i>. Vol. 2. New York: Dover Publications. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-20430-7" title="Special:BookSources/978-0-486-20430-7"><bdi>978-0-486-20430-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=History+of+Mathematics&rft.place=New+York&rft.pub=Dover+Publications&rft.date=1958&rft.isbn=978-0-486-20430-7&rft.aulast=Smith&rft.aufirst=David&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare+root" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSelin2008" class="citation cs2"><a href="/wiki/Helaine_Selin" title="Helaine Selin">Selin, Helaine</a> (2008), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=kt9DIY1g9HYC&pg=PA1268"><i>Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures</i></a>, Springer, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2008ehst.book.....S">2008ehst.book.....S</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4020-4559-2" title="Special:BookSources/978-1-4020-4559-2"><bdi>978-1-4020-4559-2</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Encyclopaedia+of+the+History+of+Science%2C+Technology%2C+and+Medicine+in+Non-Western+Cultures&rft.pub=Springer&rft.date=2008&rft_id=info%3Abibcode%2F2008ehst.book.....S&rft.isbn=978-1-4020-4559-2&rft.aulast=Selin&rft.aufirst=Helaine&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dkt9DIY1g9HYC%26pg%3DPA1268&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare+root" class="Z3988"></span>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></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"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Commons-logo.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/40px-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/60px-Commons-logo.svg.png 1.5x" data-file-width="1024" data-file-height="1376" /></a></span></div> <div class="side-box-text plainlist">Wikimedia Commons has media related to <span style="font-weight: bold; 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