Absolute value - Wikipedia

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href="#Definition_and_properties"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Definition and properties</span> </div> </a> <button aria-controls="toc-Definition_and_properties-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 Definition and properties subsection</span> </button> <ul id="toc-Definition_and_properties-sublist" class="vector-toc-list"> <li id="toc-Real_numbers" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Real_numbers"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Real numbers</span> </div> </a> <ul id="toc-Real_numbers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Complex_numbers" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Complex_numbers"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Complex numbers</span> </div> </a> <ul id="toc-Complex_numbers-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Absolute_value_function" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Absolute_value_function"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Absolute value function</span> </div> </a> <button aria-controls="toc-Absolute_value_function-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 Absolute value function subsection</span> </button> <ul id="toc-Absolute_value_function-sublist" class="vector-toc-list"> <li id="toc-Relationship_to_the_sign_function" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Relationship_to_the_sign_function"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Relationship to the sign function</span> </div> </a> <ul id="toc-Relationship_to_the_sign_function-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relationship_to_the_max_and_min_functions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Relationship_to_the_max_and_min_functions"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Relationship to the max and min functions</span> </div> </a> <ul id="toc-Relationship_to_the_max_and_min_functions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Derivative" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Derivative"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Derivative</span> </div> </a> <ul id="toc-Derivative-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Antiderivative" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Antiderivative"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Antiderivative</span> </div> </a> <ul id="toc-Antiderivative-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Derivatives_of_compositions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Derivatives_of_compositions"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.5</span> <span>Derivatives of compositions</span> </div> </a> <ul id="toc-Derivatives_of_compositions-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Distance" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Distance"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Distance</span> </div> </a> <ul id="toc-Distance-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Generalizations" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Generalizations"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Generalizations</span> </div> </a> <button aria-controls="toc-Generalizations-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 Generalizations subsection</span> </button> <ul id="toc-Generalizations-sublist" class="vector-toc-list"> <li id="toc-Ordered_rings" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ordered_rings"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Ordered rings</span> </div> </a> <ul id="toc-Ordered_rings-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Fields" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Fields"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Fields</span> </div> </a> <ul id="toc-Fields-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Vector_spaces" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Vector_spaces"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3</span> <span>Vector spaces</span> </div> </a> <ul id="toc-Vector_spaces-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Composition_algebras" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Composition_algebras"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.4</span> <span>Composition algebras</span> </div> </a> <ul id="toc-Composition_algebras-sublist" class="vector-toc-list"> </ul> </li> </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">6</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-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">8</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">9</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" > <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">Absolute value</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" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 76 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-76" 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">76 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/Absolute_waarde" title="Absolute waarde – Afrikaans" lang="af" hreflang="af" data-title="Absolute waarde" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%8A%95%E1%8C%A5%E1%88%A8_%E1%8A%A5%E1%88%B4%E1%89%B5" title="ንጥረ እሴት – Amharic" lang="am" hreflang="am" data-title="ንጥረ እሴት" data-language-autonym="አማርኛ" data-language-local-name="Amharic" class="interlanguage-link-target"><span>አማርኛ</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%82%D9%8A%D9%85%D8%A9_%D9%85%D8%B7%D9%84%D9%82%D8%A9" title="قيمة مطلقة – Arabic" lang="ar" hreflang="ar" data-title="قيمة مطلقة" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/M%C3%BCtl%C9%99q_qiym%C9%99t" title="Mütləq qiymət – Azerbaijani" lang="az" hreflang="az" data-title="Mütləq qiymət" 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%AA%E0%A6%B0%E0%A6%AE_%E0%A6%AE%E0%A6%BE%E0%A6%A8" title="পরম মান – Bangla" lang="bn" hreflang="bn" data-title="পরম মান" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%90%D0%B1%D1%81%D0%B0%D0%BB%D1%8E%D1%82%D0%BD%D0%B0%D1%8F_%D0%B2%D0%B5%D0%BB%D1%96%D1%87%D1%8B%D0%BD%D1%8F" title="Абсалютная велічыня – Belarusian" lang="be" hreflang="be" data-title="Абсалютная велічыня" 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%90%D0%B1%D1%81%D0%B0%D0%BB%D1%8E%D1%82%D0%BD%D0%B0%D1%8F_%D0%B2%D0%B5%D0%BB%D1%96%D1%87%D1%8B%D0%BD%D1%8F" title="Абсалютная велічыня – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Абсалютная велічыня" 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-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%90%D0%B1%D1%81%D0%BE%D0%BB%D1%8E%D1%82%D0%BD%D0%B0_%D1%81%D1%82%D0%BE%D0%B9%D0%BD%D0%BE%D1%81%D1%82" title="Абсолютна стойност – Bulgarian" lang="bg" hreflang="bg" data-title="Абсолютна стойност" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bo mw-list-item"><a href="https://bo.wikipedia.org/wiki/%E0%BD%9A%E0%BD%B4%E0%BD%82%E0%BD%A6%E0%BC%8B%E0%BD%90%E0%BD%B4%E0%BD%96%E0%BC%8B%E0%BD%A2%E0%BD%B2%E0%BD%93%E0%BC%8B%E0%BD%90%E0%BD%84%E0%BC%8B%E0%BC%8D" title="ཚུགས་ཐུབ་རིན་ཐང་། – Tibetan" lang="bo" hreflang="bo" data-title="ཚུགས་ཐུབ་རིན་ཐང་།" data-language-autonym="བོད་ཡིག" data-language-local-name="Tibetan" class="interlanguage-link-target"><span>བོད་ཡིག</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Apsolutna_vrijednost" title="Apsolutna vrijednost – Bosnian" lang="bs" hreflang="bs" data-title="Apsolutna vrijednost" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Valor_absolut" title="Valor absolut – Catalan" lang="ca" hreflang="ca" data-title="Valor absolut" 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%90%D0%B1%D1%81%D0%BE%D0%BB%D1%8E%D1%82%D0%BB%C4%83_%D0%BA%D0%B0%D0%BF" title="Абсолютлă кап – Chuvash" lang="cv" hreflang="cv" data-title="Абсолютлă кап" 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/Absolutn%C3%AD_hodnota" title="Absolutní hodnota – Czech" lang="cs" hreflang="cs" data-title="Absolutní hodnota" 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/Gwerth_absoliwt" title="Gwerth absoliwt – Welsh" lang="cy" hreflang="cy" data-title="Gwerth absoliwt" 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/Numerisk_v%C3%A6rdi" title="Numerisk værdi – Danish" lang="da" hreflang="da" data-title="Numerisk værdi" 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/Betragsfunktion" title="Betragsfunktion – German" lang="de" hreflang="de" data-title="Betragsfunktion" 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/Absoluutv%C3%A4%C3%A4rtus" title="Absoluutväärtus – Estonian" lang="et" hreflang="et" data-title="Absoluutväärtus" 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%91%CF%80%CF%8C%CE%BB%CF%85%CF%84%CE%B7_%CF%84%CE%B9%CE%BC%CE%AE" title="Απόλυτη τιμή – Greek" lang="el" hreflang="el" data-title="Απόλυτη τιμή" 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/Valor_absoluto" title="Valor absoluto – Spanish" lang="es" hreflang="es" data-title="Valor absoluto" 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/Absoluta_valoro" title="Absoluta valoro – Esperanto" lang="eo" hreflang="eo" data-title="Absoluta valoro" 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/Balio_absolutu" title="Balio absolutu – Basque" lang="eu" hreflang="eu" data-title="Balio absolutu" 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/%D9%82%D8%AF%D8%B1_%D9%85%D8%B7%D9%84%D9%82_(%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C)" title="قدر مطلق (ریاضی) – Persian" lang="fa" hreflang="fa" data-title="قدر مطلق (ریاضی)" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Valeur_absolue" title="Valeur absolue – French" lang="fr" hreflang="fr" data-title="Valeur absolue" 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/Val%C3%B4r_assol%C3%BBt" title="Valôr assolût – Friulian" lang="fur" hreflang="fur" data-title="Valôr assolût" 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/Valor_absoluto" title="Valor absoluto – Galician" lang="gl" hreflang="gl" data-title="Valor absoluto" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%A0%88%EB%8C%93%EA%B0%92" title="절댓값 – Korean" lang="ko" hreflang="ko" data-title="절댓값" 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/%D4%B2%D5%A1%D6%81%D5%A1%D6%80%D5%B1%D5%A1%D5%AF_%D5%A1%D6%80%D5%AA%D5%A5%D6%84" title="Բացարձակ արժեք – Armenian" lang="hy" hreflang="hy" data-title="Բացարձակ արժեք" 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%A8%E0%A4%BF%E0%A4%B0%E0%A4%AA%E0%A5%87%E0%A4%95%E0%A5%8D%E0%A4%B7_%E0%A4%AE%E0%A4%BE%E0%A4%A8" title="निरपेक्ष मान – Hindi" lang="hi" hreflang="hi" data-title="निरपेक्ष मान" 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/Apsolutna_vrijednost_broja" title="Apsolutna vrijednost broja – Croatian" lang="hr" hreflang="hr" data-title="Apsolutna vrijednost broja" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Nilai_absolut" title="Nilai absolut – Indonesian" lang="id" hreflang="id" data-title="Nilai absolut" 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-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Valor_absolute" title="Valor absolute – Interlingua" lang="ia" hreflang="ia" data-title="Valor absolute" data-language-autonym="Interlingua" data-language-local-name="Interlingua" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Algildi" title="Algildi – Icelandic" lang="is" hreflang="is" data-title="Algildi" 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/Valore_assoluto" title="Valore assoluto – Italian" lang="it" hreflang="it" data-title="Valore assoluto" 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%A2%D7%A8%D7%9A_%D7%9E%D7%95%D7%97%D7%9C%D7%98" title="ערך מוחלט – Hebrew" lang="he" hreflang="he" data-title="ערך מוחלט" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%90%E1%83%91%E1%83%A1%E1%83%9D%E1%83%9A%E1%83%A3%E1%83%A2%E1%83%A3%E1%83%A0%E1%83%98_%E1%83%A1%E1%83%98%E1%83%93%E1%83%98%E1%83%93%E1%83%94" title="აბსოლუტური სიდიდე – Georgian" lang="ka" hreflang="ka" data-title="აბსოლუტური სიდიდე" 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%90%D0%B1%D1%81%D0%BE%D0%BB%D1%8E%D1%82%D1%82%D1%96_%D1%88%D0%B0%D0%BC%D0%B0" title="Абсолютті шама – Kazakh" lang="kk" hreflang="kk" data-title="Абсолютті шама" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%90%D0%B1%D1%81%D0%BE%D0%BB%D1%8E%D1%82%D1%82%D1%83%D0%BA_%D1%87%D0%BE%D2%A3%D0%B4%D1%83%D0%BA" title="Абсолюттук чоңдук – Kyrgyz" lang="ky" hreflang="ky" data-title="Абсолюттук чоңдук" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Magnitudo_absoluta" title="Magnitudo absoluta – Latin" lang="la" hreflang="la" data-title="Magnitudo absoluta" 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/Absol%C5%ABt%C4%81_v%C4%93rt%C4%ABba" title="Absolūtā vērtība – Latvian" lang="lv" hreflang="lv" data-title="Absolūtā vērtība" 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/Modulis" title="Modulis – Lithuanian" lang="lt" hreflang="lt" data-title="Modulis" 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/Valua_asoluta" title="Valua asoluta – Lingua Franca Nova" lang="lfn" hreflang="lfn" data-title="Valua asoluta" 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-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Abszol%C3%BAt%C3%A9rt%C3%A9k-f%C3%BCggv%C3%A9ny" title="Abszolútérték-függvény – Hungarian" lang="hu" hreflang="hu" data-title="Abszolútérték-függvény" 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%90%D0%BF%D1%81%D0%BE%D0%BB%D1%83%D1%82%D0%BD%D0%B0_%D0%B2%D1%80%D0%B5%D0%B4%D0%BD%D0%BE%D1%81%D1%82" title="Апсолутна вредност – Macedonian" lang="mk" hreflang="mk" data-title="Апсолутна вредност" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Nilai_mutlak" title="Nilai mutlak – Malay" lang="ms" hreflang="ms" data-title="Nilai mutlak" 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%90%E1%80%99%E1%80%BA%E1%80%B8%E1%80%95%E1%80%9C%E1%80%AD%E1%80%90%E1%80%BA:Absolute_value" title="တမ်းပလိတ်:Absolute value – Burmese" lang="my" hreflang="my" data-title="တမ်းပလိတ်:Absolute value" 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/Absolute_waarde" title="Absolute waarde – Dutch" lang="nl" hreflang="nl" data-title="Absolute waarde" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E7%B5%B6%E5%AF%BE%E5%80%A4" title="絶対値 – Japanese" lang="ja" hreflang="ja" data-title="絶対値" 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/Absoluttverdi" title="Absoluttverdi – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Absoluttverdi" 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/Absoluttverdi" title="Absoluttverdi – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Absoluttverdi" 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-km mw-list-item"><a href="https://km.wikipedia.org/wiki/%E1%9E%8F%E1%9E%98%E1%9F%92%E1%9E%9B%E1%9F%83%E1%9E%8A%E1%9E%B6%E1%9E%85%E1%9F%8B%E1%9E%81%E1%9E%B6%E1%9E%8F" title="តម្លៃដាច់ខាត – Khmer" lang="km" hreflang="km" data-title="តម្លៃដាច់ខាត" data-language-autonym="ភាសាខ្មែរ" data-language-local-name="Khmer" class="interlanguage-link-target"><span>ភាសាខ្មែរ</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Valor_assol%C3%B9" title="Valor assolù – Piedmontese" lang="pms" hreflang="pms" data-title="Valor assolù" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Warto%C5%9B%C4%87_bezwzgl%C4%99dna" title="Wartość bezwzględna – Polish" lang="pl" hreflang="pl" data-title="Wartość bezwzględna" 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/Fun%C3%A7%C3%A3o_modular" title="Função modular – Portuguese" lang="pt" hreflang="pt" data-title="Função modular" 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/Modul" title="Modul – Romanian" lang="ro" hreflang="ro" data-title="Modul" 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%90%D0%B1%D1%81%D0%BE%D0%BB%D1%8E%D1%82%D0%BD%D0%B0%D1%8F_%D0%B2%D0%B5%D0%BB%D0%B8%D1%87%D0%B8%D0%BD%D0%B0" title="Абсолютная величина – Russian" lang="ru" hreflang="ru" data-title="Абсолютная величина" 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/Vlera_absolute" title="Vlera absolute – Albanian" lang="sq" hreflang="sq" data-title="Vlera absolute" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Absolute_value" title="Absolute value – Simple English" lang="en-simple" hreflang="en-simple" data-title="Absolute value" 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-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Absol%C3%BAtna_hodnota_(re%C3%A1lne_a_komplexn%C3%A9_%C4%8D%C3%ADslo)" title="Absolútna hodnota (reálne a komplexné číslo) – Slovak" lang="sk" hreflang="sk" data-title="Absolútna hodnota (reálne a komplexné číslo)" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Absolutna_vrednost" title="Absolutna vrednost – Slovenian" lang="sl" hreflang="sl" data-title="Absolutna vrednost" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-so mw-list-item"><a href="https://so.wikipedia.org/wiki/Qiime_sugan" title="Qiime sugan – Somali" lang="so" hreflang="so" data-title="Qiime sugan" data-language-autonym="Soomaaliga" data-language-local-name="Somali" class="interlanguage-link-target"><span>Soomaaliga</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D9%86%D8%B1%D8%AE%DB%8C_%DA%95%DB%95%DA%BE%D8%A7" title="نرخی ڕەھا – Central Kurdish" lang="ckb" hreflang="ckb" data-title="نرخی ڕەھا" 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%90%D0%BF%D1%81%D0%BE%D0%BB%D1%83%D1%82%D0%BD%D0%B0_%D0%B2%D1%80%D0%B5%D0%B4%D0%BD%D0%BE%D1%81%D1%82" title="Апсолутна вредност – Serbian" lang="sr" hreflang="sr" data-title="Апсолутна вредност" 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/Apsolutna_vrijednost" title="Apsolutna vrijednost – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Apsolutna vrijednost" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Itseisarvo" title="Itseisarvo – Finnish" lang="fi" hreflang="fi" data-title="Itseisarvo" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Absolutbelopp" title="Absolutbelopp – Swedish" lang="sv" hreflang="sv" data-title="Absolutbelopp" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Ganap_na_halaga" title="Ganap na halaga – Tagalog" lang="tl" hreflang="tl" data-title="Ganap na halaga" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%A4%E0%AE%A9%E0%AE%BF_%E0%AE%AE%E0%AE%A4%E0%AE%BF%E0%AE%AA%E0%AF%8D%E0%AE%AA%E0%AF%81" title="தனி மதிப்பு – Tamil" lang="ta" hreflang="ta" data-title="தனி மதிப்பு" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%84%E0%B9%88%E0%B8%B2%E0%B8%AA%E0%B8%B1%E0%B8%A1%E0%B8%9A%E0%B8%B9%E0%B8%A3%E0%B8%93%E0%B9%8C" title="ค่าสัมบูรณ์ – Thai" lang="th" hreflang="th" data-title="ค่าสัมบูรณ์" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Mutlak_de%C4%9Fer" title="Mutlak değer – Turkish" lang="tr" hreflang="tr" data-title="Mutlak değer" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-tk mw-list-item"><a href="https://tk.wikipedia.org/wiki/Absol%C3%BDut_ululyk" title="Absolýut ululyk – Turkmen" lang="tk" hreflang="tk" data-title="Absolýut ululyk" data-language-autonym="Türkmençe" data-language-local-name="Turkmen" class="interlanguage-link-target"><span>Türkmençe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9C%D0%BE%D0%B4%D1%83%D0%BB%D1%8C_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Модуль (математика) – Ukrainian" lang="uk" hreflang="uk" data-title="Модуль (математика)" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Gi%C3%A1_tr%E1%BB%8B_tuy%E1%BB%87t_%C4%91%E1%BB%91i" title="Giá trị tuyệt đối – Vietnamese" lang="vi" hreflang="vi" data-title="Giá trị tuyệt đối" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E7%B5%95%E5%B0%8D%E5%80%BC" title="絕對值 – Literary Chinese" lang="lzh" hreflang="lzh" data-title="絕對值" data-language-autonym="文言" data-language-local-name="Literary Chinese" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E7%BB%9D%E5%AF%B9%E5%80%BC" title="绝对值 – Wu" lang="wuu" hreflang="wuu" data-title="绝对值" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E7%B5%95%E5%B0%8D%E5%80%BC" title="絕對值 – Cantonese" lang="yue" hreflang="yue" data-title="絕對值" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E7%BB%9D%E5%AF%B9%E5%80%BC" title="绝对值 – Chinese" lang="zh" hreflang="zh" data-title="绝对值" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q120812#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit 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class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Distance from zero to a 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">This article is about the absolute value of real and complex numbers. For other absolute values in mathematics, see <a href="/wiki/Absolute_value_(algebra)" title="Absolute value (algebra)">Absolute value (algebra)</a>. For other uses, see <a href="/wiki/Absolute_value_(disambiguation)" class="mw-disambig" title="Absolute value (disambiguation)">Absolute value (disambiguation)</a>.</div> <p class="mw-empty-elt"> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Absolute_value.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6b/Absolute_value.svg/220px-Absolute_value.svg.png" decoding="async" width="220" height="147" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6b/Absolute_value.svg/330px-Absolute_value.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6b/Absolute_value.svg/440px-Absolute_value.svg.png 2x" data-file-width="600" data-file-height="400" /></a><figcaption>The <a href="/wiki/Graph_of_a_function" title="Graph of a function">graph</a> of the absolute value function for real numbers</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:AbsoluteValueDiagram.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f1/AbsoluteValueDiagram.svg/220px-AbsoluteValueDiagram.svg.png" decoding="async" width="220" height="88" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f1/AbsoluteValueDiagram.svg/330px-AbsoluteValueDiagram.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f1/AbsoluteValueDiagram.svg/440px-AbsoluteValueDiagram.svg.png 2x" data-file-width="720" data-file-height="288" /></a><figcaption>The absolute value of a number may be thought of as its distance from zero.</figcaption></figure> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, the <b>absolute value</b> or <b>modulus</b> of a <a href="/wiki/Real_number" title="Real number">real number</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 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>, <span class="nowrap">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 |x|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </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/4eb41e5fd5dc37eaa1718dfbf4bc082edb991936" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.623ex; height:2.843ex;" alt="{\displaystyle |x|}"></span>,</span> is the <a href="/wiki/Non-negative" class="mw-redirect" title="Non-negative">non-negative</a> value <span class="nowrap">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></span> without regard to its <a href="/wiki/Sign_(mathematics)" title="Sign (mathematics)">sign</a>. Namely, <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|=x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |x|=x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55d2c1e557cc3eedc06f8cc736efca5add7066dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.052ex; height:2.843ex;" alt="{\displaystyle |x|=x}"></span> 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 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 <a href="/wiki/Positive_number" class="mw-redirect" title="Positive number">positive number</a>, 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 |x|=-x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mo>−</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |x|=-x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42dcc4b88f38f60c8a96a5b2f8d76f5e4ed8ea2c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.86ex; height:2.843ex;" alt="{\displaystyle |x|=-x}"></span> 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 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 href="/wiki/Negative_number" title="Negative number">negative</a> (in which case negating <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> makes <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"> <mo>−</mo> <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/ae55e66aeffc525917eed885b4b753ba5a7f8b3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.138ex; height:2.176ex;" alt="{\displaystyle -x}"></span> positive), and <span class="nowrap"><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 |0|=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |0|=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/550214d956a5bde2719eccd73a8e90cb72a4f1fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.717ex; height:2.843ex;" alt="{\displaystyle |0|=0}"></span>.</span> For example, the absolute value of 3 <span class="nowrap">is 3,</span> and the absolute value of −3 is <span class="nowrap">also 3.</span> The absolute value of a number may be thought of as its <a href="/wiki/Distance" title="Distance">distance</a> from zero. </p><p>Generalisations of the absolute value for real numbers occur in a wide variety of mathematical settings. For example, an absolute value is also defined for the <a href="/wiki/Complex_number" title="Complex number">complex numbers</a>, the <a href="/wiki/Quaternion" title="Quaternion">quaternions</a>, <a href="/wiki/Ordered_ring" title="Ordered ring">ordered rings</a>, <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">fields</a> and <a href="/wiki/Vector_space" title="Vector space">vector spaces</a>. The absolute value is closely related to the notions of <a href="/wiki/Magnitude_(mathematics)" title="Magnitude (mathematics)">magnitude</a>, <a href="/wiki/Distance" title="Distance">distance</a>, and <a href="/wiki/Norm_(mathematics)" title="Norm (mathematics)">norm</a> in various mathematical and physical contexts. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Terminology_and_notation">Terminology and notation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Absolute_value&action=edit&section=1" title="Edit section: Terminology and notation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In 1806, <a href="/wiki/Jean-Robert_Argand" title="Jean-Robert Argand">Jean-Robert Argand</a> introduced the term <i>module</i>, meaning <i>unit of measure</i> in French, specifically for the <i>complex</i> absolute value,<sup id="cite_ref-oed_1-0" class="reference"><a href="#cite_note-oed-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><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> and it was borrowed into English in 1866 as the Latin equivalent <i>modulus</i>.<sup id="cite_ref-oed_1-1" class="reference"><a href="#cite_note-oed-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> The term <i>absolute value</i> has been used in this sense from at least 1806 in French<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> and 1857 in English.<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> The notation <span class="texhtml">|<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;"><span class="texhtml mvar" style="font-style:italic;">x</span></span>|</span>, with a <a href="/wiki/Vertical_bar" title="Vertical bar">vertical bar</a> on each side, was introduced by <a href="/wiki/Karl_Weierstrass" title="Karl Weierstrass">Karl Weierstrass</a> in 1841.<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> Other names for <i>absolute value</i> include <i>numerical value</i><sup id="cite_ref-oed_1-2" class="reference"><a href="#cite_note-oed-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> and <i>magnitude</i>.<sup id="cite_ref-oed_1-3" class="reference"><a href="#cite_note-oed-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> In programming languages and computational software packages, the absolute value 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="{\textstyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d951e0f3b54b6a3d73bb9a0a005749046cbce781" 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="{\textstyle x}"></span> is generally represented by <code>abs(<i>x</i>)</code>, or a similar expression. </p><p>The vertical bar notation also appears in a number of other mathematical contexts: for example, when applied to a set, it denotes its <a href="/wiki/Cardinality" title="Cardinality">cardinality</a>; when applied to a <a href="/wiki/Matrix_(math)" class="mw-redirect" title="Matrix (math)">matrix</a>, it denotes its <a href="/wiki/Determinant" title="Determinant">determinant</a>. Vertical bars denote the absolute value only for algebraic objects for which the notion of an absolute value is defined, notably an <a href="/wiki/Element_(mathematics)" title="Element (mathematics)">element</a> of a <a href="/wiki/Normed_division_algebra" class="mw-redirect" title="Normed division algebra">normed division algebra</a>, for example a real number, a complex number, or a quaternion. A closely related but distinct notation is the use of vertical bars for either the <a href="/wiki/Euclidean_norm" class="mw-redirect" title="Euclidean norm">Euclidean norm</a><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> or <a href="/wiki/Sup_norm" class="mw-redirect" title="Sup norm">sup norm</a><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> of a vector <span class="nowrap">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 {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"></span>,</span> although double vertical bars with subscripts <span class="nowrap">(<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 \|\cdot \|_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mo>⋅</mo> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|\cdot \|_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3a8e44a2eb980f856968a6357e3d0a7c22c905f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.058ex; height:2.843ex;" alt="{\displaystyle \|\cdot \|_{2}}"></span></span> <span class="nowrap">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 \|\cdot \|_{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mo>⋅</mo> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|\cdot \|_{\infty }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b723232adf7317abb1fc1c1326e1e4f79616a7e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.879ex; height:2.843ex;" alt="{\displaystyle \|\cdot \|_{\infty }}"></span>,</span> respectively) are a more common and less ambiguous notation. </p> <div class="mw-heading mw-heading2"><h2 id="Definition_and_properties">Definition and properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Absolute_value&action=edit&section=2" title="Edit section: Definition and properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Real_numbers">Real numbers</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Absolute_value&action=edit&section=3" title="Edit section: Real numbers"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For any <span class="nowrap"><a href="/wiki/Real_number" title="Real number">real number</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 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>,</span> the <b>absolute value</b> or <b>modulus</b> <span class="nowrap">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></span> is denoted <span class="nowrap">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 |x|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </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/4eb41e5fd5dc37eaa1718dfbf4bc082edb991936" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.623ex; height:2.843ex;" alt="{\displaystyle |x|}"></span></span>, with a <a href="/wiki/Vertical_bar" title="Vertical bar">vertical bar</a> on each side of the quantity, and is defined as<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> <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|={\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"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</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 |x|={\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/2081a5ca887ae441236a175ae4a7f451e4632920" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:22.5ex; height:6.176ex;" alt="{\displaystyle |x|={\begin{cases}x,&{\text{if }}x\geq 0\\-x,&{\text{if }}x<0.\end{cases}}}"></span> </p><p>The absolute value <span class="nowrap">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></span> is thus always either a <a href="/wiki/Positive_number" class="mw-redirect" title="Positive number">positive number</a> or <a href="/wiki/0" title="0">zero</a>, but never <a href="/wiki/Negative_number" title="Negative number">negative</a>. 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 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> itself is negative <span class="nowrap">(<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<0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo><</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x<0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a4dbbf970b2d2863dcab589eafe006f08e727d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x<0}"></span>),</span> then its absolute value is necessarily positive <span class="nowrap">(<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|=-x>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mo>−</mo> <mi>x</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |x|=-x>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f32438005ede795236c4b3c33ba07b7573031a1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.121ex; height:2.843ex;" alt="{\displaystyle |x|=-x>0}"></span>).</span> </p><p>From an <a href="/wiki/Analytic_geometry" title="Analytic geometry">analytic geometry</a> point of view, the absolute value of a real number is that number's <a href="/wiki/Distance" title="Distance">distance</a> from zero along the <a href="/wiki/Real_number_line" class="mw-redirect" title="Real number line">real number line</a>, and more generally the absolute value of the difference of two real numbers (their <a href="/wiki/Absolute_difference" title="Absolute difference">absolute difference</a>) is the distance between them.<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> The notion of an abstract <a href="/wiki/Distance_function" class="mw-redirect" title="Distance function">distance function</a> in mathematics can be seen to be a generalisation of the absolute value of the difference (see <a href="#Distance">"Distance"</a> below). </p><p>Since the <a href="/wiki/Radical_symbol" title="Radical symbol">square root symbol</a> represents the unique <i>positive</i> <a href="/wiki/Square_root" title="Square root">square root</a>, when applied to a positive number, it follows that <span class="mwe-math-element" data-qid="Q120645811"><a href="/w/index.php?title=Special:MathWikibase&qid=Q120645811" style="color:inherit;"><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|={\sqrt {x^{2}}}.}"> <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> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |x|={\sqrt {x^{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f540e273f9ebfef18a3c4458da2e6bfeea99edb1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.076ex; height:3.509ex;" alt="{\displaystyle |x|={\sqrt {x^{2}}}.}"></a></span> This is equivalent to the definition above, and may be used as an alternative definition of the absolute value of real numbers.<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> </p><p>The absolute value has the following four fundamental properties (<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 a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a503f107a7c104e40e484cee9e1f5993d28ffd8" 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="{\textstyle a}"></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="{\textstyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73a780b69dfc55238880ef18a134dc65260877e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\textstyle b}"></span> are real numbers), that are used for generalization of this notion to other domains: </p> <table style="margin-left:1.6em"> <tbody><tr> <td style="width: 250px"><span class="mwe-math-element" data-qid="Q120645720"><a href="/w/index.php?title=Special:MathWikibase&qid=Q120645720" style="color:inherit;"><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|\geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |a|\geq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/939352424bd8ad4eb5af77202976c6a0edb1f2de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.784ex; height:2.843ex;" alt="{\displaystyle |a|\geq 0}"></a></span> </td> <td>Non-negativity </td></tr> <tr> <td><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|=0\iff a=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mn>0</mn> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> <mi>a</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |a|=0\iff a=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd2ce5b7c72f4380c43c6960fec025b6b6625eac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.172ex; height:2.843ex;" alt="{\displaystyle |a|=0\iff a=0}"></span> </td> <td>Positive-definiteness </td></tr> <tr> <td><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 |ab|=\left|a\right|\left|b\right|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>a</mi> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow> <mo>|</mo> <mi>a</mi> <mo>|</mo> </mrow> <mrow> <mo>|</mo> <mi>b</mi> <mo>|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |ab|=\left|a\right|\left|b\right|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70eae5914ce7318e2536cdf0f7425f2637dec37a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.821ex; height:2.843ex;" alt="{\displaystyle |ab|=\left|a\right|\left|b\right|}"></span> </td> <td><a href="/wiki/Multiplicativeness" class="mw-redirect" title="Multiplicativeness">Multiplicativity</a> </td></tr> <tr> <td><span class="mwe-math-element" data-qid="Q120645947"><a href="/w/index.php?title=Special:MathWikibase&qid=Q120645947" style="color:inherit;"><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+b|\leq |a|+|b|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |a+b|\leq |a|+|b|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5686d5cc5c91eee0c276359372efd32f5da08bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.115ex; height:2.843ex;" alt="{\displaystyle |a+b|\leq |a|+|b|}"></a></span> </td> <td><a href="/wiki/Subadditivity" title="Subadditivity">Subadditivity</a>, specifically the <a href="/wiki/Triangle_inequality" title="Triangle inequality">triangle inequality</a> </td></tr></tbody></table> <p>Non-negativity, positive definiteness, and multiplicativity are readily apparent from the definition. To see that subadditivity holds, first note 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 |a+b|=s(a+b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mi>s</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |a+b|=s(a+b)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c7baf70944f30399cf718e0b5860fff9618b134" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.427ex; height:2.843ex;" alt="{\displaystyle |a+b|=s(a+b)}"></span> <span class="nowrap">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 s=\pm 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>=</mo> <mo>±</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s=\pm 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64de17d8811b8b810f5ef7573efd10016f7938c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.16ex; height:2.176ex;" alt="{\displaystyle s=\pm 1}"></span>,</span> with its sign chosen to make the result positive. Now, 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="{\displaystyle -1\cdot x\leq |x|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>1</mn> <mo>⋅</mo> <mi>x</mi> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -1\cdot x\leq |x|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54d6e1224cb498da00871000894cfb57348b7886" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.701ex; height:2.843ex;" alt="{\displaystyle -1\cdot x\leq |x|}"></span> <span class="nowrap">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 +1\cdot x\leq |x|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mn>1</mn> <mo>⋅</mo> <mi>x</mi> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +1\cdot x\leq |x|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8353ed78e125c88c77efcf2c9907e8ca20782c2c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.701ex; height:2.843ex;" alt="{\displaystyle +1\cdot x\leq |x|}"></span>,</span> it follows that, whichever 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 \pm 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>±</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pm 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bfeaa85da53ad1947d8000926cfea33827ef1e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.971ex; height:2.176ex;" alt="{\displaystyle \pm 1}"></span> is the value <span class="nowrap">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 s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"></span>,</span> one has <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 s\cdot x\leq |x|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>⋅</mo> <mi>x</mi> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s\cdot x\leq |x|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9bd84298ffb155a9078b8ce7ece81ec0b2022d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.821ex; height:2.843ex;" alt="{\displaystyle s\cdot x\leq |x|}"></span> for all <span class="nowrap">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>.</span> Consequently, <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+b|=s\cdot (a+b)=s\cdot a+s\cdot b\leq |a|+|b|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mi>s</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>s</mi> <mo>⋅</mo> <mi>a</mi> <mo>+</mo> <mi>s</mi> <mo>⋅</mo> <mi>b</mi> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |a+b|=s\cdot (a+b)=s\cdot a+s\cdot b\leq |a|+|b|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cec10a2f66f01024056c66db69400262958adfdf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.565ex; height:2.843ex;" alt="{\displaystyle |a+b|=s\cdot (a+b)=s\cdot a+s\cdot b\leq |a|+|b|}"></span>, as desired. </p><p>Some additional useful properties are given below. These are either immediate consequences of the definition or implied by the four fundamental properties above. </p> <table style="margin-left:1.6em"> <tbody><tr> <td style="width:250px"><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 {\bigl |}\left|a\right|{\bigr |}=|a|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">|</mo> </mrow> </mrow> <mrow> <mo>|</mo> <mi>a</mi> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">|</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bigl |}\left|a\right|{\bigr |}=|a|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7041c0e5448125f04ef95d12b2a05f22a950086a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.213ex; height:3.176ex;" alt="{\displaystyle {\bigl |}\left|a\right|{\bigr |}=|a|}"></span> </td> <td><a href="/wiki/Idempotence" title="Idempotence">Idempotence</a> (the absolute value of the absolute value is the absolute value) </td></tr> <tr> <td style="width:250px"><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 \left|-a\right|=|a|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <mrow> <mo>−</mo> <mi>a</mi> </mrow> <mo>|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|-a\right|=|a|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7e71ae5b11e947d9d87e4deb2763e2998b6812b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.954ex; height:2.843ex;" alt="{\displaystyle \left|-a\right|=|a|}"></span> </td> <td><a href="/wiki/Even_function" class="mw-redirect" title="Even function">Evenness</a> (<a href="/wiki/Reflection_symmetry" title="Reflection symmetry">reflection symmetry</a> of the graph) </td></tr> <tr> <td><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-b|=0\iff a=b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>a</mi> <mo>−</mo> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mn>0</mn> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> <mi>a</mi> <mo>=</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |a-b|=0\iff a=b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6137de7a60350b20e0a469a348c6c0d74315bfe1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.845ex; height:2.843ex;" alt="{\displaystyle |a-b|=0\iff a=b}"></span> </td> <td><a href="/wiki/Identity_of_indiscernibles" title="Identity of indiscernibles">Identity of indiscernibles</a> (equivalent to positive-definiteness) </td></tr> <tr> <td><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-b|\leq |a-c|+|c-b|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>a</mi> <mo>−</mo> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>a</mi> <mo>−</mo> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>c</mi> <mo>−</mo> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |a-b|\leq |a-c|+|c-b|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a62b70acc54310f42eca0e1fd56766531ac49b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.81ex; height:2.843ex;" alt="{\displaystyle |a-b|\leq |a-c|+|c-b|}"></span> </td> <td><a href="/wiki/Triangle_inequality#Example_norms" title="Triangle inequality">Triangle inequality</a> (equivalent to subadditivity) </td></tr> <tr> <td><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 \left|{\frac {a}{b}}\right|={\frac {|a|}{|b|}}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mrow> <mo>|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|{\frac {a}{b}}\right|={\frac {|a|}{|b|}}\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/503662d3c96f72ed3eea0d902d6964b780565903" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:10.398ex; height:6.509ex;" alt="{\displaystyle \left|{\frac {a}{b}}\right|={\frac {|a|}{|b|}}\ }"></span> (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 b\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b\neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad073253b4c817f2ec7e3dd7517b7f89a8e581dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.258ex; height:2.676ex;" alt="{\displaystyle b\neq 0}"></span>) </td> <td>Preservation of division (equivalent to multiplicativity) </td></tr> <tr> <td><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-b|\geq {\bigl |}\left|a\right|-\left|b\right|{\bigr |}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>a</mi> <mo>−</mo> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≥</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">|</mo> </mrow> </mrow> <mrow> <mo>|</mo> <mi>a</mi> <mo>|</mo> </mrow> <mo>−</mo> <mrow> <mo>|</mo> <mi>b</mi> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">|</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |a-b|\geq {\bigl |}\left|a\right|-\left|b\right|{\bigr |}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca8d6b26e1c4027b5c03e92505abcb8b55c2cea8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.183ex; height:3.176ex;" alt="{\displaystyle |a-b|\geq {\bigl |}\left|a\right|-\left|b\right|{\bigr |}}"></span> </td> <td><a href="/wiki/Reverse_triangle_inequality" class="mw-redirect" title="Reverse triangle inequality">Reverse triangle inequality</a> (equivalent to subadditivity) </td></tr></tbody></table> <p>Two other useful properties concerning inequalities are: </p> <table style="margin-left:1.6em"> <tbody><tr> <td><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|\leq b\iff -b\leq a\leq b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <mi>b</mi> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> <mo>−</mo> <mi>b</mi> <mo>≤</mo> <mi>a</mi> <mo>≤</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |a|\leq b\iff -b\leq a\leq b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b73ea4b68d3a14d3859f90d823cbf3c49e5932a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.747ex; height:2.843ex;" alt="{\displaystyle |a|\leq b\iff -b\leq a\leq b}"></span> </td></tr> <tr> <td><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|\geq b\iff a\leq -b\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≥</mo> <mi>b</mi> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> <mi>a</mi> <mo>≤</mo> <mo>−</mo> <mi>b</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |a|\geq b\iff a\leq -b\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b19870c1c49c9f51ad1f27e251a778f3bd91014" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.231ex; height:2.843ex;" alt="{\displaystyle |a|\geq b\iff a\leq -b\ }"></span> 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 a\geq b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>≥</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\geq b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed5d3957d5f94566507526017e4ebb67c02efe81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.326ex; height:2.343ex;" alt="{\displaystyle a\geq b}"></span> </td></tr></tbody></table> <p>These relations may be used to solve inequalities involving absolute values. For example: </p> <table style="margin-left:1.6em"> <tbody><tr> <td><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-3|\leq 9}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mo>−</mo> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <mn>9</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |x-3|\leq 9}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6f301a657f9730baa59d2d8d2f4bbc8681b5bac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.887ex; height:2.843ex;" alt="{\displaystyle |x-3|\leq 9}"></span> </td> <td><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 \iff -9\leq x-3\leq 9}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> <mo>−</mo> <mn>9</mn> <mo>≤</mo> <mi>x</mi> <mo>−</mo> <mn>3</mn> <mo>≤</mo> <mn>9</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \iff -9\leq x-3\leq 9}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b30cac8b6465d29dff2ff8ab198b64112ff60691" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:21.915ex; height:2.343ex;" alt="{\displaystyle \iff -9\leq x-3\leq 9}"></span> </td></tr> <tr> <td> </td> <td><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 \iff -6\leq x\leq 12}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> <mo>−</mo> <mn>6</mn> <mo>≤</mo> <mi>x</mi> <mo>≤</mo> <mn>12</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \iff -6\leq x\leq 12}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38057478967bda0a9df934e3f24f256188e1c81a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:19.074ex; height:2.343ex;" alt="{\displaystyle \iff -6\leq x\leq 12}"></span> </td></tr></tbody></table> <p>The absolute value, as "distance from zero", is used to define the absolute difference between arbitrary real numbers, the standard <a href="/wiki/Metric_(mathematics)" class="mw-redirect" title="Metric (mathematics)">metric</a> on the real numbers. </p> <div class="mw-heading mw-heading3"><h3 id="Complex_numbers">Complex numbers</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Absolute_value&action=edit&section=4" title="Edit section: Complex numbers"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><span class="anchor" id="complex_modulus"></span></p><figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Complex_conjugate_picture.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/69/Complex_conjugate_picture.svg/220px-Complex_conjugate_picture.svg.png" decoding="async" width="220" height="309" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/69/Complex_conjugate_picture.svg/330px-Complex_conjugate_picture.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/69/Complex_conjugate_picture.svg/440px-Complex_conjugate_picture.svg.png 2x" data-file-width="300" data-file-height="422" /></a><figcaption>The absolute value of a <span class="nowrap"><a href="/wiki/Complex_number" title="Complex number">complex number</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 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></span> is the <span class="nowrap">distance <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span></span> <span class="nowrap">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></span> from the origin. It is also seen in the picture 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 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> and its <span class="nowrap"><a href="/wiki/Complex_conjugate" title="Complex conjugate">complex conjugate</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 {\bar {z}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>z</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {z}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52dd0599595d539f7d757ec21da6c6e6ac3ad427" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.296ex; height:2.009ex;" alt="{\displaystyle {\bar {z}}}"></span></span> have the same absolute value.</figcaption></figure> <p>Since the <a href="/wiki/Complex_number" title="Complex number">complex numbers</a> are not <a href="/wiki/Totally_ordered_set" class="mw-redirect" title="Totally ordered set">ordered</a>, the definition given at the top for the real absolute value cannot be directly applied to complex numbers. However, the geometric interpretation of the absolute value of a real number as its distance from 0 can be generalised. The absolute value of a complex number is defined by the Euclidean distance of its corresponding point in the <a href="/wiki/Complex_plane" title="Complex plane">complex plane</a> from the <a href="/wiki/Origin_(mathematics)" title="Origin (mathematics)">origin</a>. This can be computed using the <a href="/wiki/Pythagorean_theorem" title="Pythagorean theorem">Pythagorean theorem</a>: for any complex number <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=x+iy,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mi>x</mi> <mo>+</mo> <mi>i</mi> <mi>y</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=x+iy,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ee21a775fda65273b6e1a6786b7f3d51a85de65" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.961ex; height:2.509ex;" alt="{\displaystyle z=x+iy,}"></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 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> 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 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> are real numbers, the <b>absolute value</b> or <b>modulus</b> <span class="nowrap">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></span> is <span class="nowrap">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 |z|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </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/28fd4d7dcabf618d707c21bd08306c7b3aa8b68e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.382ex; height:2.843ex;" alt="{\displaystyle |z|}"></span></span> and is defined by<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> <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|={\sqrt {\operatorname {Re} (z)^{2}+\operatorname {Im} (z)^{2}}}={\sqrt {x^{2}+y^{2}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>Re</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>Im</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <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> </msqrt> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |z|={\sqrt {\operatorname {Re} (z)^{2}+\operatorname {Im} (z)^{2}}}={\sqrt {x^{2}+y^{2}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67e409c351dc2e2bad8572f1f0ed266695c89fd2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:37.574ex; height:4.843ex;" alt="{\displaystyle |z|={\sqrt {\operatorname {Re} (z)^{2}+\operatorname {Im} (z)^{2}}}={\sqrt {x^{2}+y^{2}}},}"></span> the <a href="/wiki/Pythagorean_addition" title="Pythagorean addition">Pythagorean addition</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> 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 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>, 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 \operatorname {Re} (z)=x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Re</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Re} (z)=x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7bb2bd3edb038f55aaeef527e6e4ed5d0b448a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.069ex; height:2.843ex;" alt="{\displaystyle \operatorname {Re} (z)=x}"></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 \operatorname {Im} (z)=y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Im</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Im} (z)=y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2870a1a8895bfa307c08befe34843bfbecefde65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.927ex; height:2.843ex;" alt="{\displaystyle \operatorname {Im} (z)=y}"></span> denote the real and imaginary parts <span class="nowrap">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>,</span> respectively. When the <span class="nowrap">imaginary part <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></span> is zero, this coincides with the definition of the absolute value of the <span class="nowrap">real 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}"> <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>.</span> </p><p>When a 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 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 expressed in its <a href="/wiki/Complex_number#Polar_form" title="Complex number">polar form</a> <span class="nowrap">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 z=re^{i\theta },}"> <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>θ</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=re^{i\theta },}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbc951a75dd090066c10e578c7d1fc24482beee1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.536ex; height:3.009ex;" alt="{\displaystyle z=re^{i\theta },}"></span></span> its absolute value <span class="nowrap">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 |z|=r.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mi>r</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |z|=r.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60ef4a30c33776c4013ad32b70f67c87efe028c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.176ex; height:2.843ex;" alt="{\displaystyle |z|=r.}"></span></span> </p><p>Since the product of 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 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> and its <span class="nowrap"><a href="/wiki/Complex_conjugate" title="Complex conjugate">complex conjugate</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 {\bar {z}}=x-iy}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>z</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi>x</mi> <mo>−</mo> <mi>i</mi> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {z}}=x-iy}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f3fcfb169c13babe58e60f75cbc06c185193dfa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.523ex; height:2.509ex;" alt="{\displaystyle {\bar {z}}=x-iy}"></span>,</span> with the same absolute value, is always the non-negative real number <span class="nowrap"><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 \left(x^{2}+y^{2}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <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> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(x^{2}+y^{2}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8264e5255fff4ebecc4405c9ea1962650bf41b12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.569ex; height:3.343ex;" alt="{\displaystyle \left(x^{2}+y^{2}\right)}"></span>,</span> the absolute value of a 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 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 the square root <span class="nowrap">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\cdot {\overline {z}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>z</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z\cdot {\overline {z}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb44fb593e1cc480fdd9d1927e5455867cd1361f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.622ex; height:2.676ex;" alt="{\displaystyle z\cdot {\overline {z}},}"></span></span> which is therefore called the <a href="/wiki/Absolute_square" class="mw-redirect" title="Absolute square">absolute square</a> or <i>squared modulus</i> <span class="nowrap">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>:</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 |z|={\sqrt {z\cdot {\overline {z}}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>z</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>z</mi> <mo accent="false">¯</mo> </mover> </mrow> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |z|={\sqrt {z\cdot {\overline {z}}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/618bb1288d3937c5a0323293903581a127d3d373" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.426ex; height:3.509ex;" alt="{\displaystyle |z|={\sqrt {z\cdot {\overline {z}}}}.}"></span> This generalizes the alternative definition for reals: <span class="nowrap"><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 |x|={\sqrt {x\cdot x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" 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> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>x</mi> <mo>⋅</mo> <mi>x</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle |x|={\sqrt {x\cdot x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/560d7ffd4c0c7c4ea0ac1cb2a8f810277c903bce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.996ex; height:2.843ex;" alt="{\textstyle |x|={\sqrt {x\cdot x}}}"></span>.</span> </p><p>The complex absolute value shares the four fundamental properties given above for the real absolute value. The identity <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|^{2}=|z^{2}|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |z|^{2}=|z^{2}|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5903fa61269ad02bd45d4cf3a664eb6bd74f263" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.973ex; height:3.343ex;" alt="{\displaystyle |z|^{2}=|z^{2}|}"></span> is a special case of multiplicativity that is often useful by itself. </p> <div class="mw-heading mw-heading2"><h2 id="Absolute_value_function">Absolute value function</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Absolute_value&action=edit&section=5" title="Edit section: Absolute value function"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Absolute_value.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6b/Absolute_value.svg/360px-Absolute_value.svg.png" decoding="async" width="360" height="240" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6b/Absolute_value.svg/540px-Absolute_value.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6b/Absolute_value.svg/720px-Absolute_value.svg.png 2x" data-file-width="600" data-file-height="400" /></a><figcaption>The <a href="/wiki/Graph_of_a_function" title="Graph of a function">graph</a> of the absolute value function for real numbers</figcaption></figure> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Absolute_value_composition.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Absolute_value_composition.svg/256px-Absolute_value_composition.svg.png" decoding="async" width="256" height="332" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Absolute_value_composition.svg/384px-Absolute_value_composition.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Absolute_value_composition.svg/512px-Absolute_value_composition.svg.png 2x" data-file-width="768" data-file-height="996" /></a><figcaption><a href="/wiki/Composition_of_functions" class="mw-redirect" title="Composition of functions">Composition</a> of absolute value with a <a href="/wiki/Cubic_function" title="Cubic function">cubic function</a> in different orders</figcaption></figure> <p>The real absolute value function is <a href="/wiki/Continuous_function" title="Continuous function">continuous</a> everywhere. It is <a href="/wiki/Differentiable" class="mw-redirect" title="Differentiable">differentiable</a> everywhere except for <span class="texhtml"><i>x</i> = 0</span>. It is <a href="/wiki/Monotonic_function" title="Monotonic function">monotonically decreasing</a> on the <a href="/wiki/Interval_(mathematics)" title="Interval (mathematics)">interval</a> <span class="texhtml">(−∞, 0]</span> and monotonically increasing on the interval <span class="texhtml">[0, +∞)</span>. Since a real number and its <a href="/wiki/Additive_inverse" title="Additive inverse">opposite</a> have the same absolute value, it is an <a href="/wiki/Even_function" class="mw-redirect" title="Even function">even function</a>, and is hence not <a href="/wiki/Inverse_function" title="Inverse function">invertible</a>. The real absolute value function is a <a href="/wiki/Piecewise_linear_function" title="Piecewise linear function">piecewise linear</a>, <a href="/wiki/Convex_function" title="Convex function">convex function</a>. </p><p>For both real and complex numbers the absolute value function is <a href="/wiki/Idempotent" class="mw-redirect" title="Idempotent">idempotent</a> (meaning that the absolute value of any absolute value is itself). </p> <div class="mw-heading mw-heading3"><h3 id="Relationship_to_the_sign_function">Relationship to the sign function</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Absolute_value&action=edit&section=6" title="Edit section: Relationship to the sign function"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The absolute value function of a real number returns its value irrespective of its sign, whereas the <a href="/wiki/Sign_function" title="Sign function">sign (or signum) function</a> returns a number's sign irrespective of its value. The following equations show the relationship between these two functions: </p> <dl><dd><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|=x\operatorname {sgn}(x),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mi>x</mi> <mi>sgn</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |x|=x\operatorname {sgn}(x),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1efa376005baefd13f0cba4b004931ae507146ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.596ex; height:2.843ex;" alt="{\displaystyle |x|=x\operatorname {sgn}(x),}"></span></dd></dl> <p>or </p> <dl><dd><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|\operatorname {sgn}(x)=x,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>sgn</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |x|\operatorname {sgn}(x)=x,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fcc3778def6f332f748431753847993f24a5a62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.596ex; height:2.843ex;" alt="{\displaystyle |x|\operatorname {sgn}(x)=x,}"></span></dd></dl> <p>and for <span class="texhtml"><i>x</i> ≠ 0</span>, </p> <dl><dd><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 \operatorname {sgn}(x)={\frac {|x|}{x}}={\frac {x}{|x|}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sgn</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mi>x</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {sgn}(x)={\frac {|x|}{x}}={\frac {x}{|x|}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/870c78b93ea803c26286a3cfc697b71540d50c6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:20.273ex; height:6.509ex;" alt="{\displaystyle \operatorname {sgn}(x)={\frac {|x|}{x}}={\frac {x}{|x|}}.}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Relationship_to_the_max_and_min_functions">Relationship to the max and min functions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Absolute_value&action=edit&section=7" title="Edit section: Relationship to the max and min functions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <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 s,t\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>,</mo> <mi>t</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s,t\in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bae1d32304f0a658e915eadc6c9c983c51bb2692" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.483ex; height:2.509ex;" alt="{\displaystyle s,t\in \mathbb {R} }"></span>, then </p> <dl><dd><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 |t-s|=-2\min(s,t)+s+t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>t</mi> <mo>−</mo> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mo>−</mo> <mn>2</mn> <mo movablelimits="true" form="prefix">min</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>s</mi> <mo>+</mo> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |t-s|=-2\min(s,t)+s+t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/351ad0677c04d97678df702ca534fb92aea030de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.78ex; height:2.843ex;" alt="{\displaystyle |t-s|=-2\min(s,t)+s+t}"></span></dd></dl> <p>and </p> <dl><dd><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 |t-s|=2\max(s,t)-s-t.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>t</mi> <mo>−</mo> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mn>2</mn> <mo movablelimits="true" form="prefix">max</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>s</mi> <mo>−</mo> <mi>t</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |t-s|=2\max(s,t)-s-t.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d10f24c2c77c4cbc8425cb3c87923fe9f6daf6de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.069ex; height:2.843ex;" alt="{\displaystyle |t-s|=2\max(s,t)-s-t.}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Derivative">Derivative</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Absolute_value&action=edit&section=8" title="Edit section: Derivative"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The real absolute value function has a <a href="/wiki/Derivative" title="Derivative">derivative</a> for every <span class="texhtml"><i>x</i> ≠ 0</span>, but is not <a href="/wiki/Differentiable" class="mw-redirect" title="Differentiable">differentiable</a> at <span class="texhtml"><i>x</i> = 0</span>. Its derivative for <span class="texhtml"><i>x</i> ≠ 0</span> is given by the <a href="/wiki/Step_function" title="Step function">step function</a>:<sup id="cite_ref-MathWorld_12-0" class="reference"><a href="#cite_note-MathWorld-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-BS163_13-0" class="reference"><a href="#cite_note-BS163-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><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 {d\left|x\right|}{dx}}={\frac {x}{|x|}}={\begin{cases}-1&x<0\\1&x>0.\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mrow> <mo>|</mo> <mi>x</mi> <mo>|</mo> </mrow> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mi>x</mi> <mo><</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <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 {\frac {d\left|x\right|}{dx}}={\frac {x}{|x|}}={\begin{cases}-1&x<0\\1&x>0.\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6c243fff473340f54b4202cfaafcf9d5691d45e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:28.744ex; height:6.509ex;" alt="{\displaystyle {\frac {d\left|x\right|}{dx}}={\frac {x}{|x|}}={\begin{cases}-1&x<0\\1&x>0.\end{cases}}}"></span></dd></dl> <p>The real absolute value function is an example of a continuous function that achieves a <a href="/wiki/Maximum_and_minimum" title="Maximum and minimum">global minimum</a> where the derivative does not exist. </p><p>The <a href="/wiki/Subderivative" title="Subderivative">subdifferential</a> of <span class="texhtml">|<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;"><span class="texhtml mvar" style="font-style:italic;">x</span></span>|</span> at <span class="texhtml"><i>x</i> = 0</span> is the interval <span class="texhtml">[−1, 1]</span>.<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> </p><p>The <a href="/wiki/Complex_number" title="Complex number">complex</a> absolute value function is continuous everywhere but <a href="/wiki/Complex_differentiable" class="mw-redirect" title="Complex differentiable">complex differentiable</a> <i>nowhere</i> because it violates the <a href="/wiki/Cauchy%E2%80%93Riemann_equations" title="Cauchy–Riemann equations">Cauchy–Riemann equations</a>.<sup id="cite_ref-MathWorld_12-1" class="reference"><a href="#cite_note-MathWorld-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p><p>The second derivative of <span class="texhtml">|<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;"><span class="texhtml mvar" style="font-style:italic;">x</span></span>|</span> with respect to <span class="texhtml mvar" style="font-style:italic;">x</span> is zero everywhere except zero, where it does not exist. As a <a href="/wiki/Generalised_function" class="mw-redirect" title="Generalised function">generalised function</a>, the second derivative may be taken as two times the <a href="/wiki/Dirac_delta_function" title="Dirac delta function">Dirac delta function</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Antiderivative">Antiderivative</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Absolute_value&action=edit&section=9" title="Edit section: Antiderivative"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Antiderivative" title="Antiderivative">antiderivative</a> (indefinite <a href="/wiki/Integral" title="Integral">integral</a>) of the real absolute value function is </p> <dl><dd><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 \int \left|x\right|dx={\frac {x\left|x\right|}{2}}+C,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <mrow> <mo>|</mo> <mi>x</mi> <mo>|</mo> </mrow> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mrow> <mo>|</mo> <mi>x</mi> <mo>|</mo> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mi>C</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int \left|x\right|dx={\frac {x\left|x\right|}{2}}+C,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a04925bea7eff6817ad0a6171856e0fa8fc9e0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:21.665ex; height:6.176ex;" alt="{\displaystyle \int \left|x\right|dx={\frac {x\left|x\right|}{2}}+C,}"></span></dd></dl> <p>where <span class="texhtml mvar" style="font-style:italic;">C</span> is an arbitrary <a href="/wiki/Constant_of_integration" title="Constant of integration">constant of integration</a>. This is not a <a href="/wiki/Complex_antiderivative" class="mw-redirect" title="Complex antiderivative">complex antiderivative</a> because complex antiderivatives can only exist for complex-differentiable (<a href="/wiki/Holomorphic" class="mw-redirect" title="Holomorphic">holomorphic</a>) functions, which the complex absolute value function is not. </p> <div class="mw-heading mw-heading3"><h3 id="Derivatives_of_compositions">Derivatives of compositions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Absolute_value&action=edit&section=10" title="Edit section: Derivatives of compositions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The following two formulae are special cases of the <a href="/wiki/Chain_rule" title="Chain rule">chain rule</a>: </p><p><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 {d \over dx}f(|x|)={x \over |x|}(f'(|x|))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {d \over dx}f(|x|)={x \over |x|}(f'(|x|))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30b2c6d3fada43ccfba78cedeeb43d4a02e1978e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:23.898ex; height:6.176ex;" alt="{\displaystyle {d \over dx}f(|x|)={x \over |x|}(f'(|x|))}"></span> </p><p>if the absolute value is inside a function, and </p><p><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 {d \over dx}|f(x)|={f(x) \over |f(x)|}f'(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {d \over dx}|f(x)|={f(x) \over |f(x)|}f'(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/444f2f6538d883fc80e6926301335e8577621599" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:23.883ex; height:6.509ex;" alt="{\displaystyle {d \over dx}|f(x)|={f(x) \over |f(x)|}f'(x)}"></span> </p><p>if another function is inside the absolute value. In the first case, the derivative is always discontinuous at <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 x=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle x=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95946e5db8fcf9ae99523bcd4f83433c8b2e441e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\textstyle x=0}"></span> in the first case and 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="{\textstyle f(x)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle f(x)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a69999a3446a1c579f7dcf6aef723ee741c42b8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.678ex; height:2.843ex;" alt="{\textstyle f(x)=0}"></span> in the second case. </p> <div class="mw-heading mw-heading2"><h2 id="Distance">Distance</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Absolute_value&action=edit&section=11" title="Edit section: Distance"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/Metric_space" title="Metric space">Metric space</a></div> <p>The absolute value is closely related to the idea of <a href="/wiki/Distance" title="Distance">distance</a>. As noted above, the absolute value of a real or complex number is the distance from that number to the origin, along the real number line, for real numbers, or in the complex plane, for complex numbers, and more generally, the absolute value of the difference of two real or complex numbers is the distance between them. </p><p>The standard <a href="/wiki/Euclidean_distance" title="Euclidean distance">Euclidean distance</a> between two points </p> <dl><dd><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=(a_{1},a_{2},\dots ,a_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=(a_{1},a_{2},\dots ,a_{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae13f3bc3275e21097960bd2c1692d901d4296df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.366ex; height:2.843ex;" alt="{\displaystyle a=(a_{1},a_{2},\dots ,a_{n})}"></span></dd></dl> <p>and </p> <dl><dd><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 b=(b_{1},b_{2},\dots ,b_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b=(b_{1},b_{2},\dots ,b_{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b09e74ba4c7e782e72c134f8d7ffa0710e7cd016" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.437ex; height:2.843ex;" alt="{\displaystyle b=(b_{1},b_{2},\dots ,b_{n})}"></span></dd></dl> <p>in <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean <span class="texhtml mvar" style="font-style:italic;">n</span>-space</a> is defined as: </p> <dl><dd><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 {\textstyle \sum _{i=1}^{n}(a_{i}-b_{i})^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mstyle displaystyle="false" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {\textstyle \sum _{i=1}^{n}(a_{i}-b_{i})^{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7bf44aa2a097b58d7fcbf063e0d5a8495ef690f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:17.855ex; height:4.843ex;" alt="{\displaystyle {\sqrt {\textstyle \sum _{i=1}^{n}(a_{i}-b_{i})^{2}}}.}"></span></dd></dl> <p>This can be seen as a generalisation, since 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 a_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbf42ecda092975c9c69dae84e16182ba5fe2e07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.284ex; height:2.009ex;" alt="{\displaystyle a_{1}}"></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 b_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9af2720c91be489f57ecde4bb651b95e113d0144" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.052ex; height:2.509ex;" alt="{\displaystyle b_{1}}"></span> real, i.e. in a 1-space, according to the alternative definition of the absolute value, </p> <dl><dd><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_{1}-b_{1}|={\sqrt {(a_{1}-b_{1})^{2}}}={\sqrt {\textstyle \sum _{i=1}^{1}(a_{i}-b_{i})^{2}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mstyle displaystyle="false" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </munderover> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </msqrt> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |a_{1}-b_{1}|={\sqrt {(a_{1}-b_{1})^{2}}}={\sqrt {\textstyle \sum _{i=1}^{1}(a_{i}-b_{i})^{2}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a7bb55a0c265c53089be40d699e309449f58ab9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:44.886ex; height:4.843ex;" alt="{\displaystyle |a_{1}-b_{1}|={\sqrt {(a_{1}-b_{1})^{2}}}={\sqrt {\textstyle \sum _{i=1}^{1}(a_{i}-b_{i})^{2}}},}"></span></dd></dl> <p>and 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 a=a_{1}+ia_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mi>i</mi> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=a_{1}+ia_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0a11615836f3c7cddb7d06b7706ceb575f09250" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.539ex; height:2.509ex;" alt="{\displaystyle a=a_{1}+ia_{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="{\displaystyle b=b_{1}+ib_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>=</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mi>i</mi> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b=b_{1}+ib_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2fb52fa6f2b1003ff8c68f25fa3675f6c8aeb887" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.842ex; height:2.509ex;" alt="{\displaystyle b=b_{1}+ib_{2}}"></span> complex numbers, i.e. in a 2-space, </p> <dl><dd><table> <tbody><tr> <td><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-b|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>a</mi> <mo>−</mo> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |a-b|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf0a3ef4404c96ab573022b623f731b4491ed771" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.361ex; height:2.843ex;" alt="{\displaystyle |a-b|}"></span> </td> <td><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_{1}+ia_{2})-(b_{1}+ib_{2})|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mi>i</mi> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mo stretchy="false">(</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mi>i</mi> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =|(a_{1}+ia_{2})-(b_{1}+ib_{2})|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e1298570544f1d3c41016dad10d18667fac5fcf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.163ex; height:2.843ex;" alt="{\displaystyle =|(a_{1}+ia_{2})-(b_{1}+ib_{2})|}"></span> </td></tr> <tr> <td> </td> <td><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_{1}-b_{1})+i(a_{2}-b_{2})|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mi>i</mi> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =|(a_{1}-b_{1})+i(a_{2}-b_{2})|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1638d0dacd0adaaf109f7d6ebbc78f090cc028b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.361ex; height:2.843ex;" alt="{\displaystyle =|(a_{1}-b_{1})+i(a_{2}-b_{2})|}"></span> </td></tr> <tr> <td> </td> <td><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_{1}-b_{1})^{2}+(a_{2}-b_{2})^{2}}}={\sqrt {\textstyle \sum _{i=1}^{2}(a_{i}-b_{i})^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mstyle displaystyle="false" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </munderover> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ={\sqrt {(a_{1}-b_{1})^{2}+(a_{2}-b_{2})^{2}}}={\sqrt {\textstyle \sum _{i=1}^{2}(a_{i}-b_{i})^{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa7c87e758e48db761eb9eccd3f6205484c81cb4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:48.651ex; height:4.843ex;" alt="{\displaystyle ={\sqrt {(a_{1}-b_{1})^{2}+(a_{2}-b_{2})^{2}}}={\sqrt {\textstyle \sum _{i=1}^{2}(a_{i}-b_{i})^{2}}}.}"></span> </td></tr></tbody></table></dd></dl> <p>The above shows that the "absolute value"-distance, for real and complex numbers, agrees with the standard Euclidean distance, which they inherit as a result of considering them as one and two-dimensional Euclidean spaces, respectively. </p><p>The properties of the absolute value of the difference of two real or complex numbers: non-negativity, identity of indiscernibles, symmetry and the triangle inequality given above, can be seen to motivate the more general notion of a <a href="/wiki/Distance_function" class="mw-redirect" title="Distance function">distance function</a> as follows: </p><p>A real valued function <span class="texhtml mvar" style="font-style:italic;">d</span> on a set <span class="texhtml"><i>X</i> × <i>X</i></span> is called a <a href="/wiki/Metric_(mathematics)" class="mw-redirect" title="Metric (mathematics)">metric</a> (or a <i>distance function</i>) on <span class="texhtml mvar" style="font-style:italic;">X</span>, if it satisfies the following four axioms:<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> </p> <dl><dd><table> <tbody><tr> <td style="width:250px"><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 d(a,b)\geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(a,b)\geq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40d228e792a13c2b3e8aaf58b33b111f15292b64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.547ex; height:2.843ex;" alt="{\displaystyle d(a,b)\geq 0}"></span> </td> <td>Non-negativity </td></tr> <tr> <td><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 d(a,b)=0\iff a=b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> <mi>a</mi> <mo>=</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(a,b)=0\iff a=b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1df151412624571a51cc4c54c00798f3cada5a0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.77ex; height:2.843ex;" alt="{\displaystyle d(a,b)=0\iff a=b}"></span> </td> <td>Identity of indiscernibles </td></tr> <tr> <td><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 d(a,b)=d(b,a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo>,</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(a,b)=d(b,a)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23d6e875612d947f656a50a10104eb4cb44c40c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.671ex; height:2.843ex;" alt="{\displaystyle d(a,b)=d(b,a)}"></span> </td> <td>Symmetry </td></tr> <tr> <td><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 d(a,b)\leq d(a,c)+d(c,b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>c</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(a,b)\leq d(a,c)+d(c,b)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6322991d56eb23c6e557481926074b8c9df1dea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.585ex; height:2.843ex;" alt="{\displaystyle d(a,b)\leq d(a,c)+d(c,b)}"></span> </td> <td>Triangle inequality </td></tr></tbody></table></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Generalizations">Generalizations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Absolute_value&action=edit&section=12" title="Edit section: Generalizations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Ordered_rings">Ordered rings</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Absolute_value&action=edit&section=13" title="Edit section: Ordered rings"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The definition of absolute value given for real numbers above can be extended to any <a href="/wiki/Ordered_ring" title="Ordered ring">ordered ring</a>. That is, if <span class="texhtml mvar" style="font-style:italic;">a</span> is an element of an ordered ring <i>R</i>, then the <b>absolute value</b> of <span class="texhtml mvar" style="font-style:italic;">a</span>, denoted by <span class="texhtml">|<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;"><i>a</i></span>|</span>, is defined to be:<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> </p> <dl><dd><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|=\left\{{\begin{array}{rl}a,&{\text{if }}a\geq 0\\-a,&{\text{if }}a<0.\end{array}}\right.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow> <mo>{</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>a</mi> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>a</mi> <mo>≥</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mi>a</mi> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>a</mi> <mo><</mo> <mn>0.</mn> </mtd> </mtr> </mtable> </mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |a|=\left\{{\begin{array}{rl}a,&{\text{if }}a\geq 0\\-a,&{\text{if }}a<0.\end{array}}\right.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f9daf9781f56443e38859079599a0a29a16dd2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:22.201ex; height:6.176ex;" alt="{\displaystyle |a|=\left\{{\begin{array}{rl}a,&{\text{if }}a\geq 0\\-a,&{\text{if }}a<0.\end{array}}\right.}"></span></dd></dl> <p>where <span class="texhtml">−<i>a</i></span> is the <a href="/wiki/Additive_inverse" title="Additive inverse">additive inverse</a> of <span class="texhtml mvar" style="font-style:italic;">a</span>, 0 is the <a href="/wiki/Additive_identity" title="Additive identity">additive identity</a>, and < and ≥ have the usual meaning with respect to the ordering in the ring. </p> <div class="mw-heading mw-heading3"><h3 id="Fields">Fields</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Absolute_value&action=edit&section=14" title="Edit section: Fields"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/Absolute_value_(algebra)" title="Absolute value (algebra)">Absolute value (algebra)</a></div> <p>The four fundamental properties of the absolute value for real numbers can be used to generalise the notion of absolute value to an arbitrary field, as follows. </p><p>A real-valued function <span class="texhtml mvar" style="font-style:italic;">v</span> on a <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a> <span class="texhtml mvar" style="font-style:italic;">F</span> is called an <i>absolute value</i> (also a <i>modulus</i>, <i>magnitude</i>, <i>value</i>, or <i>valuation</i>)<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> if it satisfies the following four axioms: </p> <dl><dd><table cellpadding="10"> <tbody><tr> <td><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 v(a)\geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v(a)\geq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e49ba1e78b523d294104cbe0799364fb692fd4bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.428ex; height:2.843ex;" alt="{\displaystyle v(a)\geq 0}"></span> </td> <td>Non-negativity </td></tr> <tr> <td><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 v(a)=0\iff a=\mathbf {0} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> <mi>a</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v(a)=0\iff a=\mathbf {0} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3f3d7035233c3d43c025cdb191841debfa24665" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.99ex; height:2.843ex;" alt="{\displaystyle v(a)=0\iff a=\mathbf {0} }"></span> </td> <td>Positive-definiteness </td></tr> <tr> <td><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 v(ab)=v(a)v(b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>v</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mi>v</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v(ab)=v(a)v(b)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a50f27be85a8207405a2c42e6dca4d18ea187110" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.364ex; height:2.843ex;" alt="{\displaystyle v(ab)=v(a)v(b)}"></span> </td> <td>Multiplicativity </td></tr> <tr> <td><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 v(a+b)\leq v(a)+v(b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>v</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>v</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v(a+b)\leq v(a)+v(b)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02b218923df50a2bb4e38ebe7c8dd5f1f8a5d32b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.045ex; height:2.843ex;" alt="{\displaystyle v(a+b)\leq v(a)+v(b)}"></span> </td> <td>Subadditivity or the triangle inequality </td></tr></tbody></table></dd></dl> <p>Where <b>0</b> denotes the <a href="/wiki/Additive_identity" title="Additive identity">additive identity</a> of <span class="texhtml mvar" style="font-style:italic;">F</span>. It follows from positive-definiteness and multiplicativity that <span class="texhtml"><i>v</i>(<b>1</b>) = 1</span>, where <b>1</b> denotes the <a href="/wiki/Multiplicative_identity" class="mw-redirect" title="Multiplicative identity">multiplicative identity</a> of <span class="texhtml mvar" style="font-style:italic;">F</span>. The real and complex absolute values defined above are examples of absolute values for an arbitrary field. </p><p>If <span class="texhtml mvar" style="font-style:italic;">v</span> is an absolute value on <span class="texhtml mvar" style="font-style:italic;">F</span>, then the function <span class="texhtml mvar" style="font-style:italic;">d</span> on <span class="texhtml"><i>F</i> × <i>F</i></span>, defined by <span class="texhtml"><i>d</i>(<i>a</i>, <i>b</i>) = <i>v</i>(<i>a</i> − <i>b</i>)</span>, is a metric and the following are equivalent: </p> <ul><li><span class="texhtml mvar" style="font-style:italic;">d</span> satisfies the <a href="/wiki/Ultrametric" class="mw-redirect" title="Ultrametric">ultrametric</a> inequality <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 d(x,y)\leq \max(d(x,z),d(y,z))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mo movablelimits="true" form="prefix">max</mo> <mo stretchy="false">(</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(x,y)\leq \max(d(x,z),d(y,z))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc193540adf4b61dcf16458d59c8be385b388117" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.591ex; height:2.843ex;" alt="{\displaystyle d(x,y)\leq \max(d(x,z),d(y,z))}"></span> for all <span class="texhtml mvar" style="font-style:italic;">x</span>, <span class="texhtml mvar" style="font-style:italic;">y</span>, <span class="texhtml mvar" style="font-style:italic;">z</span> in <span class="texhtml mvar" style="font-style:italic;">F</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="{\textstyle \left\{v\left(\sum _{k=1}^{n}\mathbf {1} \right):n\in \mathbb {N} \right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow> <mo>{</mo> <mrow> <mi>v</mi> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> </mrow> <mo>)</mo> </mrow> <mo>:</mo> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \left\{v\left(\sum _{k=1}^{n}\mathbf {1} \right):n\in \mathbb {N} \right\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f00ae47556e7044c3dcf39b508cb0c02a12d032b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.866ex; height:3.176ex;" alt="{\textstyle \left\{v\left(\sum _{k=1}^{n}\mathbf {1} \right):n\in \mathbb {N} \right\}}"></span> is <a href="/wiki/Bounded_set" title="Bounded set">bounded</a> in <b>R</b>.</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 v\left({\textstyle \sum _{k=1}^{n}}\mathbf {1} \right)\leq 1\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> </mrow> <mo>)</mo> </mrow> <mo>≤</mo> <mn>1</mn> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v\left({\textstyle \sum _{k=1}^{n}}\mathbf {1} \right)\leq 1\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d42a27a1e72317ba5d1c96d03272212814a45c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.145ex; height:3.176ex;" alt="{\displaystyle v\left({\textstyle \sum _{k=1}^{n}}\mathbf {1} \right)\leq 1\ }"></span> for every <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 n\in \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\in \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d059936e77a2d707e9ee0a1d9575a1d693ce5d0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.913ex; height:2.176ex;" alt="{\displaystyle n\in \mathbb {N} }"></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 v(a)\leq 1\Rightarrow v(1+a)\leq 1\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mn>1</mn> <mo stretchy="false">⇒</mo> <mi>v</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mn>1</mn> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v(a)\leq 1\Rightarrow v(1+a)\leq 1\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1137fcd5536a02877120c254dab723dc7bc5924a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.053ex; height:2.843ex;" alt="{\displaystyle v(a)\leq 1\Rightarrow v(1+a)\leq 1\ }"></span> for all <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\in F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∈</mo> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\in F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/130d30b75a5437ad01787d25462043ac3a9aee3c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.811ex; height:2.176ex;" alt="{\displaystyle a\in F}"></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 v(a+b)\leq \max\{v(a),v(b)\}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mo movablelimits="true" form="prefix">max</mo> <mo fence="false" stretchy="false">{</mo> <mi>v</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>v</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v(a+b)\leq \max\{v(a),v(b)\}\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d60498c0e38d748c97abdfa20328115a6176d36c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.47ex; height:2.843ex;" alt="{\displaystyle v(a+b)\leq \max\{v(a),v(b)\}\ }"></span> for all <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,b\in F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>∈</mo> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b\in F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40b0ed72d499490ba23dc89160fb8dac6664949b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.843ex; height:2.509ex;" alt="{\displaystyle a,b\in F}"></span>.</li></ul> <p>An absolute value which satisfies any (hence all) of the above conditions is said to be <b>non-Archimedean</b>, otherwise it is said to be <a href="/wiki/Archimedean_field" class="mw-redirect" title="Archimedean field">Archimedean</a>.<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> <div class="mw-heading mw-heading3"><h3 id="Vector_spaces">Vector spaces</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Absolute_value&action=edit&section=15" title="Edit section: Vector spaces"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/Norm_(mathematics)" title="Norm (mathematics)">Norm (mathematics)</a></div> <p>Again the fundamental properties of the absolute value for real numbers can be used, with a slight modification, to generalise the notion to an arbitrary vector space. </p><p>A real-valued function on a <a href="/wiki/Vector_space" title="Vector space">vector space</a> <span class="texhtml mvar" style="font-style:italic;">V</span> over a field <span class="texhtml mvar" style="font-style:italic;">F</span>, represented as <span class="texhtml">‖ · ‖</span>, is called an <b>absolute value</b>, but more usually a <a href="/wiki/Norm_(mathematics)" title="Norm (mathematics)"><b>norm</b></a>, if it satisfies the following axioms: </p><p>For all <span class="texhtml mvar" style="font-style:italic;">a</span> in <span class="texhtml mvar" style="font-style:italic;">F</span>, and <span class="texhtml"><b>v</b></span>, <span class="texhtml"><b>u</b></span> in <span class="texhtml mvar" style="font-style:italic;">V</span>, </p> <dl><dd><table cellpadding="10"> <tbody><tr> <td><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 \|\mathbf {v} \|\geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo fence="false" stretchy="false">‖</mo> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|\mathbf {v} \|\geq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c08f44e9bc20e9053f60d3a820e1dd14080abae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.997ex; height:2.843ex;" alt="{\displaystyle \|\mathbf {v} \|\geq 0}"></span> </td> <td>Non-negativity </td></tr> <tr> <td><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 \|\mathbf {v} \|=0\iff \mathbf {v} =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo fence="false" stretchy="false">‖</mo> <mo>=</mo> <mn>0</mn> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|\mathbf {v} \|=0\iff \mathbf {v} =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d82e9fbdf61d65342fa3eac47c5808d4d4641535" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.566ex; height:2.843ex;" alt="{\displaystyle \|\mathbf {v} \|=0\iff \mathbf {v} =0}"></span> </td> <td>Positive-definiteness </td></tr> <tr> <td><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\mathbf {v} \|=\left|a\right|\left\|\mathbf {v} \right\|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo fence="false" stretchy="false">‖</mo> <mo>=</mo> <mrow> <mo>|</mo> <mi>a</mi> <mo>|</mo> </mrow> <mrow> <mo symmetric="true">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo symmetric="true">‖</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|a\mathbf {v} \|=\left|a\right|\left\|\mathbf {v} \right\|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd027544d07d1dd17d5fd74a1f2596831c0745a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.711ex; height:2.843ex;" alt="{\displaystyle \|a\mathbf {v} \|=\left|a\right|\left\|\mathbf {v} \right\|}"></span> </td> <td>Positive homogeneity or positive scalability </td></tr> <tr> <td><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 \|\mathbf {v} +\mathbf {u} \|\leq \|\mathbf {v} \|+\|\mathbf {u} \|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo fence="false" stretchy="false">‖</mo> <mo>≤</mo> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo fence="false" stretchy="false">‖</mo> <mo>+</mo> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo fence="false" stretchy="false">‖</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|\mathbf {v} +\mathbf {u} \|\leq \|\mathbf {v} \|+\|\mathbf {u} \|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9afab35e2383e4c975840eb8d563888f62240518" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.547ex; height:2.843ex;" alt="{\displaystyle \|\mathbf {v} +\mathbf {u} \|\leq \|\mathbf {v} \|+\|\mathbf {u} \|}"></span> </td> <td>Subadditivity or the triangle inequality </td></tr></tbody></table></dd></dl> <p>The norm of a vector is also called its <i>length</i> or <i>magnitude</i>. </p><p>In the case of <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</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 {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"></span>, the function defined by </p> <dl><dd><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},x_{2},\dots ,x_{n})\|={\sqrt {\textstyle \sum _{i=1}^{n}x_{i}^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">‖</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mstyle displaystyle="false" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|(x_{1},x_{2},\dots ,x_{n})\|={\sqrt {\textstyle \sum _{i=1}^{n}x_{i}^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c88821f2938eea6793c1dd96fece846e445cd718" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:31.21ex; height:4.843ex;" alt="{\displaystyle \|(x_{1},x_{2},\dots ,x_{n})\|={\sqrt {\textstyle \sum _{i=1}^{n}x_{i}^{2}}}}"></span></dd></dl> <p>is a norm called the Euclidean norm. When the real 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 \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span> are considered as the one-dimensional vector space <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 {R} ^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec2e282911e406fc800fb1095093667d66f18c7f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{1}}"></span>, the absolute value is a <a href="/wiki/Norm_(mathematics)" title="Norm (mathematics)">norm</a>, and is the <span class="texhtml mvar" style="font-style:italic;">p</span>-norm (see <a href="/wiki/L%5Ep_space#Definition" class="mw-redirect" title="L^p space">L<sup>p</sup> space</a>) for any <span class="texhtml mvar" style="font-style:italic;">p</span>. In fact the absolute value is the "only" norm on <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 {R} ^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec2e282911e406fc800fb1095093667d66f18c7f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{1}}"></span>, in the sense that, for every norm <span class="texhtml">‖ · ‖</span> on <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 {R} ^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec2e282911e406fc800fb1095093667d66f18c7f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{1}}"></span>, <span class="texhtml">‖<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;"><i>x</i></span>‖ = ‖<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;">1</span>‖ ⋅ |<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;"><i>x</i></span>|</span>. </p><p>The complex absolute value is a special case of the norm in an <a href="/wiki/Inner_product_space" title="Inner product space">inner product space</a>, which is identical to the Euclidean norm when the complex plane is identified as the <a href="/wiki/Euclidean_plane" title="Euclidean plane">Euclidean plane</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 {R} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e150115ab9f63023215109595b76686a1ff890fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2}}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Composition_algebras">Composition algebras</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Absolute_value&action=edit&section=16" title="Edit section: Composition algebras"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/Composition_algebra" title="Composition algebra">Composition algebra</a></div> <p>Every composition algebra <i>A</i> has an <a href="/wiki/Involution_(mathematics)" title="Involution (mathematics)">involution</a> <i>x</i> → <i>x</i>* called its <b>conjugation</b>. The product in <i>A</i> of an element <i>x</i> and its conjugate <i>x</i>* is written <i>N</i>(<i>x</i>) = <i>x x</i>* and called the <b>norm of x</b>. </p><p>The real 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 \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span>, 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 \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }"></span>, and quaternions <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> </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/e050965453c42bcc6bd544546703c836bdafeac9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \mathbb {H} }"></span> are all composition algebras with norms given by <a href="/wiki/Definite_quadratic_form" title="Definite quadratic form">definite quadratic forms</a>. The absolute value in these <a href="/wiki/Division_algebra" title="Division algebra">division algebras</a> is given by the square root of the composition algebra norm. </p><p>In general the norm of a composition algebra may be a <a href="/wiki/Quadratic_form" title="Quadratic form">quadratic form</a> that is not definite and has <a href="/wiki/Null_vector" title="Null vector">null vectors</a>. However, as in the case of division algebras, when an element <i>x</i> has a non-zero norm, then <i>x</i> has a <a href="/wiki/Multiplicative_inverse" title="Multiplicative inverse">multiplicative inverse</a> given by <i>x</i>*/<i>N</i>(<i>x</i>). </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Absolute_value&action=edit&section=17" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Least_absolute_values" class="mw-redirect" title="Least absolute values">Least absolute values</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Absolute_value&action=edit&section=18" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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 reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-oed-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-oed_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-oed_1-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-oed_1-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-oed_1-3"><sup><i><b>d</b></i></sup></a></span> <span class="reference-text"><a href="/wiki/Oxford_English_Dictionary" title="Oxford English Dictionary">Oxford English Dictionary</a>, Draft Revision, June 2008</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">Nahin, <a rel="nofollow" class="external text" href="http://www-history.mcs.st-andrews.ac.uk/Mathematicians/Argand.html">O'Connor and Robertson</a>, and <a rel="nofollow" class="external text" href="http://functions.wolfram.com/ComplexComponents/Abs/35/">functions.Wolfram.com.</a>; for the French sense, see <a href="/wiki/Dictionnaire_de_la_langue_fran%C3%A7aise_(Littr%C3%A9)" class="mw-redirect" title="Dictionnaire de la langue française (Littré)">Littré</a>, 1877</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"><a href="/wiki/Lazare_Nicolas_Marguerite_Carnot" class="mw-redirect" title="Lazare Nicolas Marguerite Carnot">Lazare Nicolas M. Carnot</a>, <i>Mémoire sur la relation qui existe entre les distances respectives de cinq point quelconques pris dans l'espace</i>, p. 105 <a rel="nofollow" class="external text" href="https://books.google.com/books?id=YyIOAAAAQAAJ&pg=PA105">at Google Books</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">James Mill Peirce, <i>A Text-book of Analytic Geometry</i> <a rel="nofollow" class="external text" href="https://archive.org/details/atextbookanalyt00peirgoog/page/n60">at Internet Archive</a>. The oldest citation in the 2nd edition of the Oxford English Dictionary is from 1907. The term <i>absolute value</i> is also used in contrast to <i>relative value</i>.</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text">Nicholas J. Higham, <i>Handbook of writing for the mathematical sciences</i>, SIAM. <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><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-89871-420-6" title="Special:BookSources/0-89871-420-6">0-89871-420-6</a>, p. 25</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSpivak1965" class="citation book cs1">Spivak, Michael (1965). <i>Calculus on Manifolds</i>. Boulder, CO: Westview. p. 1. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0805390219" title="Special:BookSources/0805390219"><bdi>0805390219</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+on+Manifolds&rft.place=Boulder%2C+CO&rft.pages=1&rft.pub=Westview&rft.date=1965&rft.isbn=0805390219&rft.aulast=Spivak&rft.aufirst=Michael&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAbsolute+value" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMunkres1991" class="citation book cs1">Munkres, James (1991). <i>Analysis on Manifolds</i>. Boulder, CO: Westview. p. 4. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0201510359" title="Special:BookSources/0201510359"><bdi>0201510359</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Analysis+on+Manifolds&rft.place=Boulder%2C+CO&rft.pages=4&rft.pub=Westview&rft.date=1991&rft.isbn=0201510359&rft.aulast=Munkres&rft.aufirst=James&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAbsolute+value" 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">Mendelson, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=A8hAm38zsCMC&pg=PA2">p. 2</a>.</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="CITEREFSmith2013" class="citation book cs1">Smith, Karl (2013). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ZUJbVQN37bIC&pg=PA8"><i>Precalculus: A Functional Approach to Graphing and Problem Solving</i></a>. Jones & Bartlett Publishers. p. 8. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7637-5177-7" title="Special:BookSources/978-0-7637-5177-7"><bdi>978-0-7637-5177-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Precalculus%3A+A+Functional+Approach+to+Graphing+and+Problem+Solving&rft.pages=8&rft.pub=Jones+%26+Bartlett+Publishers&rft.date=2013&rft.isbn=978-0-7637-5177-7&rft.aulast=Smith&rft.aufirst=Karl&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DZUJbVQN37bIC%26pg%3DPA8&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAbsolute+value" 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="CITEREFStewart,_James_B.2001" class="citation book cs1">Stewart, James B. (2001). <i>Calculus: concepts and contexts</i>. Australia: Brooks/Cole. p. A5. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-534-37718-1" title="Special:BookSources/0-534-37718-1"><bdi>0-534-37718-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Calculus%3A+concepts+and+contexts&rft.place=Australia&rft.pages=A5&rft.pub=Brooks%2FCole&rft.date=2001&rft.isbn=0-534-37718-1&rft.au=Stewart%2C+James+B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAbsolute+value" 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="CITEREFGonzález,_Mario_O.1992" class="citation book cs1">González, Mario O. (1992). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ncxL7EFr7GsC&pg=PA19"><i>Classical Complex Analysis</i></a>. CRC Press. p. 19. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780824784157" title="Special:BookSources/9780824784157"><bdi>9780824784157</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Classical+Complex+Analysis&rft.pages=19&rft.pub=CRC+Press&rft.date=1992&rft.isbn=9780824784157&rft.au=Gonz%C3%A1lez%2C+Mario+O.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DncxL7EFr7GsC%26pg%3DPA19&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAbsolute+value" class="Z3988"></span></span> </li> <li id="cite_note-MathWorld-12"><span class="mw-cite-backlink">^ <a href="#cite_ref-MathWorld_12-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-MathWorld_12-1"><sup><i><b>b</b></i></sup></a></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://mathworld.wolfram.com/AbsoluteValue.html">"Weisstein, Eric W. <i>Absolute Value.</i> From MathWorld – A Wolfram Web Resource"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Weisstein%2C+Eric+W.+Absolute+Value.+From+MathWorld+%E2%80%93+A+Wolfram+Web+Resource.&rft_id=http%3A%2F%2Fmathworld.wolfram.com%2FAbsoluteValue.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAbsolute+value" class="Z3988"></span></span> </li> <li id="cite_note-BS163-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-BS163_13-0">^</a></b></span> <span class="reference-text">Bartle and Sherbert, p. 163</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">Peter Wriggers, Panagiotis Panatiotopoulos, eds., <i>New Developments in Contact Problems</i>, 1999, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3-211-83154-1" title="Special:BookSources/3-211-83154-1">3-211-83154-1</a>, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=tiBtC4GmuKcC&pg=PA31">p. 31–32</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">These axioms are not minimal; for instance, non-negativity can be derived from the other three: <span class="texhtml">0 = <i>d</i>(<i>a</i>, <i>a</i>) ≤ <i>d</i>(<i>a</i>, <i>b</i>) + <i>d</i>(<i>b</i>, <i>a</i>) = 2<i>d</i>(<i>a</i>, <i>b</i>)</span>.</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">Mac Lane, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=L6FENd8GHIUC&pg=PA264">p. 264</a>.</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">Shechter, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=eqUv3Bcd56EC&pg=PA260">p. 260</a>. This meaning of <i>valuation</i> is rare. Usually, a <a href="/wiki/Valuation_(algebra)" title="Valuation (algebra)">valuation</a> is the logarithm of the inverse of an absolute value</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">Shechter, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=eqUv3Bcd56EC&pg=PA260">pp. 260–261</a>.</span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Absolute_value&action=edit&section=19" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Bartle; Sherbert; <i>Introduction to real analysis</i> (4th ed.), John Wiley & Sons, 2011 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-43331-6" title="Special:BookSources/978-0-471-43331-6">978-0-471-43331-6</a>.</li> <li>Nahin, Paul J.; <i>An Imaginary Tale</i>; Princeton University Press; (hardcover, 1998). <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-691-02795-1" title="Special:BookSources/0-691-02795-1">0-691-02795-1</a>.</li> <li>Mac Lane, Saunders, Garrett Birkhoff, <i>Algebra</i>, American Mathematical Soc., 1999. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-1646-2" title="Special:BookSources/978-0-8218-1646-2">978-0-8218-1646-2</a>.</li> <li>Mendelson, Elliott, <i>Schaum's Outline of Beginning Calculus</i>, McGraw-Hill Professional, 2008. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-07-148754-2" title="Special:BookSources/978-0-07-148754-2">978-0-07-148754-2</a>.</li> <li>O'Connor, J.J. and Robertson, E.F.; <a rel="nofollow" class="external text" href="http://www-history.mcs.st-andrews.ac.uk/Mathematicians/Argand.html">"Jean Robert Argand"</a>.</li> <li>Schechter, Eric; <i>Handbook of Analysis and Its Foundations</i>, pp. 259–263, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=eqUv3Bcd56EC&pg=PA259">"Absolute Values"</a>, Academic Press (1997) <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-12-622760-8" title="Special:BookSources/0-12-622760-8">0-12-622760-8</a>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Absolute_value&action=edit&section=20" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs1"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Absolute_value">"Absolute value"</a>. <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>. <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>. 2001 [1994].</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Absolute+value&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DAbsolute_value&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAbsolute+value" class="Z3988"></span></li> <li><a rel="nofollow" class="external text" href="https://planetmath.org/AbsoluteValue">absolute value</a> at <a href="/wiki/PlanetMath" title="PlanetMath">PlanetMath</a>.</li> <li><span class="citation mathworld" id="Reference-Mathworld-Absolute_Value"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/AbsoluteValue.html">"Absolute Value"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Absolute+Value&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FAbsoluteValue.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAbsolute+value" class="Z3988"></span></span></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Absolute_value&oldid=1255369755">https://en.wikipedia.org/w/index.php?title=Absolute_value&oldid=1255369755</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Special_functions" title="Category:Special functions">Special functions</a></li><li><a href="/wiki/Category:Real_numbers" title="Category:Real numbers">Real numbers</a></li><li><a href="/wiki/Category:Norms_(mathematics)" title="Category:Norms (mathematics)">Norms (mathematics)</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li><li><a href="/wiki/Category:Use_dmy_dates_from_December_2020" title="Category:Use dmy dates from December 2020">Use dmy dates from December 2020</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 4 November 2024, at 16:10<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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Absolute value - Wikipedia