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class="vector-menu-content-list"> <li id="pt-sitesupport-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_ko.wikipedia.org&uselang=ko" class=""><span>기부</span></a> </li> <li id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=%ED%8A%B9%EC%88%98:%EA%B3%84%EC%A0%95%EB%A7%8C%EB%93%A4%EA%B8%B0&returnto=%EC%A0%88%EB%8C%93%EA%B0%92" title="계정을 만들고 로그인하는 것이 좋습니다. 하지만 필수는 아닙니다" class=""><span>계정 만들기</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=%ED%8A%B9%EC%88%98:%EB%A1%9C%EA%B7%B8%EC%9D%B8&returnto=%EC%A0%88%EB%8C%93%EA%B0%92" title="위키백과에 로그인하면 여러가지 편리한 기능을 사용할 수 있습니다. [o]" accesskey="o" class=""><span>로그인</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="더 많은 옵션" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="개인 도구" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-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-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">개인 도구</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="사용자 메뉴" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_ko.wikipedia.org&uselang=ko"><span>기부</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=%ED%8A%B9%EC%88%98:%EA%B3%84%EC%A0%95%EB%A7%8C%EB%93%A4%EA%B8%B0&returnto=%EC%A0%88%EB%8C%93%EA%B0%92" title="계정을 만들고 로그인하는 것이 좋습니다. 하지만 필수는 아닙니다"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>계정 만들기</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=%ED%8A%B9%EC%88%98:%EB%A1%9C%EA%B7%B8%EC%9D%B8&returnto=%EC%A0%88%EB%8C%93%EA%B0%92" title="위키백과에 로그인하면 여러가지 편리한 기능을 사용할 수 있습니다. [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>로그인</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> 로그아웃한 편집자를 위한 문서 <a href="/wiki/%EB%8F%84%EC%9B%80%EB%A7%90:%EC%86%8C%EA%B0%9C" aria-label="편집에 관해 더 알아보기"><span>더 알아보기</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/%ED%8A%B9%EC%88%98:%EB%82%B4%EA%B8%B0%EC%97%AC" title="이 IP 주소의 편집 목록 [y]" accesskey="y"><span>기여</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/%ED%8A%B9%EC%88%98:%EB%82%B4%EC%82%AC%EC%9A%A9%EC%9E%90%ED%86%A0%EB%A1%A0" title="현재 사용하는 IP 주소에 대한 토론 문서 [n]" accesskey="n"><span>토론</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="사이트"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="목차" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">목차</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">사이드바로 이동</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">숨기기</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">처음 위치</div> </a> </li> <li id="toc-정의" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#정의"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>정의</span> </div> </a> <button aria-controls="toc-정의-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>정의 하위섹션 토글하기</span> </button> <ul id="toc-정의-sublist" class="vector-toc-list"> <li id="toc-실수의_경우" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#실수의_경우"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>실수의 경우</span> </div> </a> <ul id="toc-실수의_경우-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-복소수의_경우" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#복소수의_경우"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>복소수의 경우</span> </div> </a> <ul id="toc-복소수의_경우-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-성질" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#성질"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>성질</span> </div> </a> <button aria-controls="toc-성질-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>성질 하위섹션 토글하기</span> </button> <ul id="toc-성질-sublist" class="vector-toc-list"> <li id="toc-부등식" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#부등식"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>부등식</span> </div> </a> <ul id="toc-부등식-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-항등식" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#항등식"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>항등식</span> </div> </a> <ul id="toc-항등식-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-미분" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#미분"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>미분</span> </div> </a> <ul id="toc-미분-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-응용" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#응용"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>응용</span> </div> </a> <button aria-controls="toc-응용-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>응용 하위섹션 토글하기</span> </button> <ul id="toc-응용-sublist" class="vector-toc-list"> <li id="toc-복소수의_극형식" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#복소수의_극형식"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>복소수의 극형식</span> </div> </a> <ul id="toc-복소수의_극형식-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-거리_공간_구조" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#거리_공간_구조"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>거리 공간 구조</span> </div> </a> <ul id="toc-거리_공간_구조-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-관련_개념" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#관련_개념"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>관련 개념</span> </div> </a> <button aria-controls="toc-관련_개념-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>관련 개념 하위섹션 토글하기</span> </button> <ul id="toc-관련_개념-sublist" class="vector-toc-list"> <li id="toc-노름" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#노름"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>노름</span> </div> </a> <ul id="toc-노름-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-정역_위의_절댓값" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#정역_위의_절댓값"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>정역 위의 절댓값</span> </div> </a> <ul id="toc-정역_위의_절댓값-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-참고_문헌" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#참고_문헌"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>참고 문헌</span> </div> </a> <ul id="toc-참고_문헌-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="목차" 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="목차 토글" > <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">목차 토글</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">절댓값</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="다른 언어로 문서를 방문합니다. 76개 언어로 읽을 수 있습니다" > <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">76개 언어</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 – 아프리칸스어" lang="af" hreflang="af" data-title="Absolute waarde" data-language-autonym="Afrikaans" data-language-local-name="아프리칸스어" 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 – 암하라어" lang="am" hreflang="am" data-title="ንጥረ እሴት" data-language-autonym="አማርኛ" data-language-local-name="암하라어" 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 – 아랍어" lang="ar" hreflang="ar" data-title="قيمة مطلقة" data-language-autonym="العربية" data-language-local-name="아랍어" 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 – 아제르바이잔어" lang="az" hreflang="az" data-title="Mütləq qiymət" data-language-autonym="Azərbaycanca" data-language-local-name="아제르바이잔어" class="interlanguage-link-target"><span>Azərbaycanca</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 – 벨라루스어" lang="be" hreflang="be" data-title="Абсалютная велічыня" data-language-autonym="Беларуская" data-language-local-name="벨라루스어" 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 – 불가리아어" lang="bg" hreflang="bg" data-title="Абсолютна стойност" data-language-autonym="Български" data-language-local-name="불가리아어" class="interlanguage-link-target"><span>Български</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 – 벵골어" lang="bn" hreflang="bn" data-title="পরম মান" data-language-autonym="বাংলা" data-language-local-name="벵골어" class="interlanguage-link-target"><span>বাংলা</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་། – 티베트어" lang="bo" hreflang="bo" data-title="ཚུགས་ཐུབ་རིན་ཐང་།" data-language-autonym="བོད་ཡིག" data-language-local-name="티베트어" 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 – 보스니아어" lang="bs" hreflang="bs" data-title="Apsolutna vrijednost" data-language-autonym="Bosanski" data-language-local-name="보스니아어" 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 – 카탈로니아어" lang="ca" hreflang="ca" data-title="Valor absolut" data-language-autonym="Català" data-language-local-name="카탈로니아어" class="interlanguage-link-target"><span>Català</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 – 소라니 쿠르드어" lang="ckb" hreflang="ckb" data-title="نرخی ڕەھا" data-language-autonym="کوردی" data-language-local-name="소라니 쿠르드어" class="interlanguage-link-target"><span>کوردی</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 – 체코어" lang="cs" hreflang="cs" data-title="Absolutní hodnota" data-language-autonym="Čeština" data-language-local-name="체코어" class="interlanguage-link-target"><span>Čeština</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 – 추바시어" lang="cv" hreflang="cv" data-title="Абсолютлă кап" data-language-autonym="Чӑвашла" data-language-local-name="추바시어" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Gwerth_absoliwt" title="Gwerth absoliwt – 웨일스어" lang="cy" hreflang="cy" data-title="Gwerth absoliwt" data-language-autonym="Cymraeg" data-language-local-name="웨일스어" 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 – 덴마크어" lang="da" hreflang="da" data-title="Numerisk værdi" data-language-autonym="Dansk" data-language-local-name="덴마크어" 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 – 독일어" lang="de" hreflang="de" data-title="Betragsfunktion" data-language-autonym="Deutsch" data-language-local-name="독일어" class="interlanguage-link-target"><span>Deutsch</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 – 그리스어" lang="el" hreflang="el" data-title="Απόλυτη τιμή" data-language-autonym="Ελληνικά" data-language-local-name="그리스어" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Absolute_value" title="Absolute value – 영어" lang="en" hreflang="en" data-title="Absolute value" data-language-autonym="English" data-language-local-name="영어" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Absoluta_valoro" title="Absoluta valoro – 에스페란토어" lang="eo" hreflang="eo" data-title="Absoluta valoro" data-language-autonym="Esperanto" data-language-local-name="에스페란토어" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Valor_absoluto" title="Valor absoluto – 스페인어" lang="es" hreflang="es" data-title="Valor absoluto" data-language-autonym="Español" data-language-local-name="스페인어" class="interlanguage-link-target"><span>Español</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 – 에스토니아어" lang="et" hreflang="et" data-title="Absoluutväärtus" data-language-autonym="Eesti" data-language-local-name="에스토니아어" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Balio_absolutu" title="Balio absolutu – 바스크어" lang="eu" hreflang="eu" data-title="Balio absolutu" data-language-autonym="Euskara" data-language-local-name="바스크어" 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) – 페르시아어" lang="fa" hreflang="fa" data-title="قدر مطلق (ریاضی)" data-language-autonym="فارسی" data-language-local-name="페르시아어" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Itseisarvo" title="Itseisarvo – 핀란드어" lang="fi" hreflang="fi" data-title="Itseisarvo" data-language-autonym="Suomi" data-language-local-name="핀란드어" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Valeur_absolue" title="Valeur absolue – 프랑스어" lang="fr" hreflang="fr" data-title="Valeur absolue" data-language-autonym="Français" data-language-local-name="프랑스어" 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 – 프리울리어" lang="fur" hreflang="fur" data-title="Valôr assolût" data-language-autonym="Furlan" data-language-local-name="프리울리어" 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 – 갈리시아어" lang="gl" hreflang="gl" data-title="Valor absoluto" data-language-autonym="Galego" data-language-local-name="갈리시아어" class="interlanguage-link-target"><span>Galego</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 – 히브리어" lang="he" hreflang="he" data-title="ערך מוחלט" data-language-autonym="עברית" data-language-local-name="히브리어" class="interlanguage-link-target"><span>עברית</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 – 힌디어" lang="hi" hreflang="hi" data-title="निरपेक्ष मान" data-language-autonym="हिन्दी" data-language-local-name="힌디어" 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 – 크로아티아어" lang="hr" hreflang="hr" data-title="Apsolutna vrijednost broja" data-language-autonym="Hrvatski" data-language-local-name="크로아티아어" class="interlanguage-link-target"><span>Hrvatski</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 – 헝가리어" lang="hu" hreflang="hu" data-title="Abszolútérték-függvény" data-language-autonym="Magyar" data-language-local-name="헝가리어" class="interlanguage-link-target"><span>Magyar</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 – 아르메니아어" lang="hy" hreflang="hy" data-title="Բացարձակ արժեք" data-language-autonym="Հայերեն" data-language-local-name="아르메니아어" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Valor_absolute" title="Valor absolute – 인터링구아" lang="ia" hreflang="ia" data-title="Valor absolute" data-language-autonym="Interlingua" data-language-local-name="인터링구아" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Nilai_absolut" title="Nilai absolut – 인도네시아어" lang="id" hreflang="id" data-title="Nilai absolut" data-language-autonym="Bahasa Indonesia" data-language-local-name="인도네시아어" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Algildi" title="Algildi – 아이슬란드어" lang="is" hreflang="is" data-title="Algildi" data-language-autonym="Íslenska" data-language-local-name="아이슬란드어" 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 – 이탈리아어" lang="it" hreflang="it" data-title="Valore assoluto" data-language-autonym="Italiano" data-language-local-name="이탈리아어" class="interlanguage-link-target"><span>Italiano</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 – 일본어" lang="ja" hreflang="ja" data-title="絶対値" data-language-autonym="日本語" data-language-local-name="일본어" class="interlanguage-link-target"><span>日本語</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 – 조지아어" lang="ka" hreflang="ka" data-title="აბსოლუტური სიდიდე" data-language-autonym="ქართული" data-language-local-name="조지아어" 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 – 카자흐어" lang="kk" hreflang="kk" data-title="Абсолютті шама" data-language-autonym="Қазақша" data-language-local-name="카자흐어" class="interlanguage-link-target"><span>Қазақша</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 – 크메르어" lang="km" hreflang="km" data-title="តម្លៃដាច់ខាត" data-language-autonym="ភាសាខ្មែរ" data-language-local-name="크메르어" class="interlanguage-link-target"><span>ភាសាខ្មែរ</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 – 키르기스어" lang="ky" hreflang="ky" data-title="Абсолюттук чоңдук" data-language-autonym="Кыргызча" data-language-local-name="키르기스어" 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 – 라틴어" lang="la" hreflang="la" data-title="Magnitudo absoluta" data-language-autonym="Latina" data-language-local-name="라틴어" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lfn mw-list-item"><a href="https://lfn.wikipedia.org/wiki/Valua_asoluta" title="Valua asoluta – 링구아 프랑카 노바" lang="lfn" hreflang="lfn" data-title="Valua asoluta" data-language-autonym="Lingua Franca Nova" data-language-local-name="링구아 프랑카 노바" class="interlanguage-link-target"><span>Lingua Franca Nova</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Modulis" title="Modulis – 리투아니아어" lang="lt" hreflang="lt" data-title="Modulis" data-language-autonym="Lietuvių" data-language-local-name="리투아니아어" class="interlanguage-link-target"><span>Lietuvių</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 – 라트비아어" lang="lv" hreflang="lv" data-title="Absolūtā vērtība" data-language-autonym="Latviešu" data-language-local-name="라트비아어" class="interlanguage-link-target"><span>Latviešu</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 – 마케도니아어" lang="mk" hreflang="mk" data-title="Апсолутна вредност" data-language-autonym="Македонски" data-language-local-name="마케도니아어" 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 – 말레이어" lang="ms" hreflang="ms" data-title="Nilai mutlak" data-language-autonym="Bahasa Melayu" data-language-local-name="말레이어" 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 – 버마어" lang="my" hreflang="my" data-title="တမ်းပလိတ်:Absolute value" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="버마어" 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 – 네덜란드어" lang="nl" hreflang="nl" data-title="Absolute waarde" data-language-autonym="Nederlands" data-language-local-name="네덜란드어" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Absoluttverdi" title="Absoluttverdi – 노르웨이어(니노르스크)" lang="nn" hreflang="nn" data-title="Absoluttverdi" data-language-autonym="Norsk nynorsk" data-language-local-name="노르웨이어(니노르스크)" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Absoluttverdi" title="Absoluttverdi – 노르웨이어(보크말)" lang="nb" hreflang="nb" data-title="Absoluttverdi" data-language-autonym="Norsk bokmål" data-language-local-name="노르웨이어(보크말)" class="interlanguage-link-target"><span>Norsk bokmål</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 – 폴란드어" lang="pl" hreflang="pl" data-title="Wartość bezwzględna" data-language-autonym="Polski" data-language-local-name="폴란드어" class="interlanguage-link-target"><span>Polski</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-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Fun%C3%A7%C3%A3o_modular" title="Função modular – 포르투갈어" lang="pt" hreflang="pt" data-title="Função modular" data-language-autonym="Português" data-language-local-name="포르투갈어" 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 – 루마니아어" lang="ro" hreflang="ro" data-title="Modul" data-language-autonym="Română" data-language-local-name="루마니아어" 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 – 러시아어" lang="ru" hreflang="ru" data-title="Абсолютная величина" data-language-autonym="Русский" data-language-local-name="러시아어" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Apsolutna_vrijednost" title="Apsolutna vrijednost – 세르비아-크로아티아어" lang="sh" hreflang="sh" data-title="Apsolutna vrijednost" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="세르비아-크로아티아어" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</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) – 슬로바키아어" lang="sk" hreflang="sk" data-title="Absolútna hodnota (reálne a komplexné číslo)" data-language-autonym="Slovenčina" data-language-local-name="슬로바키아어" 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 – 슬로베니아어" lang="sl" hreflang="sl" data-title="Absolutna vrednost" data-language-autonym="Slovenščina" data-language-local-name="슬로베니아어" 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 – 소말리아어" lang="so" hreflang="so" data-title="Qiime sugan" data-language-autonym="Soomaaliga" data-language-local-name="소말리아어" class="interlanguage-link-target"><span>Soomaaliga</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Vlera_absolute" title="Vlera absolute – 알바니아어" lang="sq" hreflang="sq" data-title="Vlera absolute" data-language-autonym="Shqip" data-language-local-name="알바니아어" class="interlanguage-link-target"><span>Shqip</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 – 세르비아어" lang="sr" hreflang="sr" data-title="Апсолутна вредност" data-language-autonym="Српски / srpski" data-language-local-name="세르비아어" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Absolutbelopp" title="Absolutbelopp – 스웨덴어" lang="sv" hreflang="sv" data-title="Absolutbelopp" data-language-autonym="Svenska" data-language-local-name="스웨덴어" class="interlanguage-link-target"><span>Svenska</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 – 타밀어" lang="ta" hreflang="ta" data-title="தனி மதிப்பு" data-language-autonym="தமிழ்" data-language-local-name="타밀어" 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 – 태국어" lang="th" hreflang="th" data-title="ค่าสัมบูรณ์" data-language-autonym="ไทย" data-language-local-name="태국어" class="interlanguage-link-target"><span>ไทย</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 – 투르크멘어" lang="tk" hreflang="tk" data-title="Absolýut ululyk" data-language-autonym="Türkmençe" data-language-local-name="투르크멘어" class="interlanguage-link-target"><span>Türkmençe</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 – 타갈로그어" lang="tl" hreflang="tl" data-title="Ganap na halaga" data-language-autonym="Tagalog" data-language-local-name="타갈로그어" class="interlanguage-link-target"><span>Tagalog</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 – 터키어" lang="tr" hreflang="tr" data-title="Mutlak değer" data-language-autonym="Türkçe" data-language-local-name="터키어" class="interlanguage-link-target"><span>Türkçe</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) – 우크라이나어" lang="uk" hreflang="uk" data-title="Модуль (математика)" data-language-autonym="Українська" data-language-local-name="우크라이나어" 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 – 베트남어" lang="vi" hreflang="vi" data-title="Giá trị tuyệt đối" data-language-autonym="Tiếng Việt" data-language-local-name="베트남어" class="interlanguage-link-target"><span>Tiếng Việt</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 – 우어" lang="wuu" hreflang="wuu" data-title="绝对值" data-language-autonym="吴语" data-language-local-name="우어" class="interlanguage-link-target"><span>吴语</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 – 중국어" lang="zh" hreflang="zh" data-title="绝对值" data-language-autonym="中文" data-language-local-name="중국어" class="interlanguage-link-target"><span>中文</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-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 – 광둥어" lang="yue" hreflang="yue" data-title="絕對值" data-language-autonym="粵語" data-language-local-name="광둥어" 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data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">보이기</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">사이드바로 이동</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">숨기기</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">위키백과, 우리 모두의 백과사전.</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="ko" dir="ltr"><p><span class="nowrap"></span> </p> <div class="dablink hatnote"><span typeof="mw:File"><a href="/wiki/%EC%9C%84%ED%82%A4%EB%B0%B1%EA%B3%BC:%EB%8F%99%EC%9D%8C%EC%9D%B4%EC%9D%98%EC%96%B4_%EB%AC%B8%EC%84%9C" title="위키백과:동음이의어 문서"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Disambig_grey.svg/23px-Disambig_grey.svg.png" decoding="async" width="23" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Disambig_grey.svg/35px-Disambig_grey.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Disambig_grey.svg/46px-Disambig_grey.svg.png 2x" data-file-width="260" data-file-height="200" /></a></span> 이 문서는 실수와 복소수의 표준적인 절댓값에 관한 것입니다. <a href="/wiki/%EB%8C%80%EC%88%98%EC%A0%81_%EC%88%98%EB%A1%A0" title="대수적 수론">대수적 수론</a>의 절댓값에 대해서는 <a href="/wiki/%EC%A0%88%EB%8C%93%EA%B0%92_(%EB%8C%80%EC%88%98%ED%95%99)" title="절댓값 (대수학)">절댓값 (대수학)</a> 문서를 참고하십시오.</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/%ED%8C%8C%EC%9D%BC: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>실수 절댓값 함수의 <a href="/wiki/%ED%95%A8%EC%88%98%EC%9D%98_%EA%B7%B8%EB%9E%98%ED%94%84" title="함수의 그래프">그래프</a></figcaption></figure> <p><a href="/wiki/%EC%88%98%ED%95%99" title="수학">수학</a>에서 <b>절댓값</b>(絕對값, <span style="font-size: smaller;"><a href="/wiki/%EC%98%81%EC%96%B4" title="영어">영어</a>: </span><span lang="en">absolute value 또는 modulus</span>)은 <a href="/wiki/%EC%8B%A4%EC%88%98" title="실수">실수</a>나 <a href="/wiki/%EB%B3%B5%EC%86%8C%EC%88%98" title="복소수">복소수</a>가 원점으로부터 떨어진 거리를 나타내는 음이 아닌 실수이다. 실수의 절댓값은 단순히 부호를 무시한 음의 아닌 값이다. 절댓값의 개념의 다양한 일반화가 존재한다. 예를 들어, 실수와 복소수의 절댓값은 1차원 <a href="/wiki/%EB%85%B8%EB%A6%84_%EA%B3%B5%EA%B0%84" title="노름 공간">노름 공간</a> 위의 <a href="/wiki/%EB%85%B8%EB%A6%84" class="mw-redirect" title="노름">노름</a>을 이루며, <a href="/wiki/%EC%8B%A4%EC%88%98%EC%B2%B4" class="mw-redirect" title="실수체">실수체</a>와 <a href="/wiki/%EB%B3%B5%EC%86%8C%EC%88%98%EC%B2%B4" class="mw-redirect" title="복소수체">복소수체</a>의 <a href="/wiki/%EC%A0%88%EB%8C%93%EA%B0%92_(%EB%8C%80%EC%88%98%ED%95%99)" title="절댓값 (대수학)">대수적 절댓값</a>을 이룬다. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="정의"><span id=".EC.A0.95.EC.9D.98"></span>정의</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%A0%88%EB%8C%93%EA%B0%92&action=edit&section=1" title="부분 편집: 정의"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="실수의_경우"><span id=".EC.8B.A4.EC.88.98.EC.9D.98_.EA.B2.BD.EC.9A.B0"></span>실수의 경우</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%A0%88%EB%8C%93%EA%B0%92&action=edit&section=2" title="부분 편집: 실수의 경우"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/%ED%8C%8C%EC%9D%BC: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>실수선 위에서, 실수 -3과 실수 0 사이의 거리는 3이다.</figcaption></figure> <p><a href="/wiki/%EC%8B%A4%EC%88%98" title="실수">실수</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\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9c6d458566aec47a7259762034790c8981aefab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.848ex; height:2.176ex;" alt="{\displaystyle x\in \mathbb {R} }"></span>의 <b>절댓값</b> <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|\in [0,\infty )}"> <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 stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |x|\in [0,\infty )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70c9a298f80d1b65d1dd76c4fcf9b1bb63a4284a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.536ex; height:2.843ex;" alt="{\displaystyle |x|\in [0,\infty )}"></span>은 다음과 같이 정의된다. </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|={\sqrt {x^{2}}}={\begin{cases}x&x>0\\0&x=0\\-x&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"> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </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> </mtd> <mtd> <mi>x</mi> <mo>></mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mi>x</mi> </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 |x|={\sqrt {x^{2}}}={\begin{cases}x&x>0\\0&x=0\\-x&x<0\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2da6c413ce9877b5032908a8de3ee9e65bdf2ae4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:27.396ex; height:8.509ex;" alt="{\displaystyle |x|={\sqrt {x^{2}}}={\begin{cases}x&x>0\\0&x=0\\-x&x<0\end{cases}}}"></span></dd></dl> <p>여기서 </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf0bf28fd28f45d07e1ceb909ce333c18c558c93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.384ex; height:2.676ex;" alt="{\displaystyle x^{2}}"></span>는 <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>의 <a href="/wiki/%EC%A0%9C%EA%B3%B1" title="제곱">제곱</a>이다.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {x^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {x^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bfacde1f08c72c09d7727023c39cd4627a24bd07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.708ex; height:3.343ex;" alt="{\displaystyle {\sqrt {x^{2}}}}"></span>는 <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^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf0bf28fd28f45d07e1ceb909ce333c18c558c93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.384ex; height:2.676ex;" alt="{\displaystyle x^{2}}"></span>의 <a href="/wiki/%EC%A3%BC_%EC%A0%9C%EA%B3%B1%EA%B7%BC" class="mw-redirect" title="주 제곱근">주 제곱근</a>이다.</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 -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>는 <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>의 <a href="/wiki/%EB%B0%98%EC%88%98_(%EC%88%98%ED%95%99)" class="mw-redirect" title="반수 (수학)">반수</a>이다.</li></ul> <p>즉, 실수의 절댓값은 그 실수의 숫자 부분만 남겨두고 부호를 버려 얻는 음이 아닌 실수이다. <a href="/wiki/%EC%8B%A4%EC%88%98%EC%84%A0" class="mw-redirect" title="실수선">실수선</a> 위에서 보면, 이는 실수와 0 사이의 거리와 같다. </p> <div class="mw-heading mw-heading3"><h3 id="복소수의_경우"><span id=".EB.B3.B5.EC.86.8C.EC.88.98.EC.9D.98_.EA.B2.BD.EC.9A.B0"></span>복소수의 경우</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%A0%88%EB%8C%93%EA%B0%92&action=edit&section=3" title="부분 편집: 복소수의 경우"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/%ED%8C%8C%EC%9D%BC: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>복소평면 위에서, 복소수 <i>z</i>의 절댓값은 원점과의 거리 <i>r</i>와 같다. 모든 복소수 <i>z</i>의 절댓값과 그 켤레 복소수 <span class="sfrac nowrap;"><span style="display:none; display:inline-block; text-align:center;"><span style="display:block; line-height:0; font-size:70%; padding:0 0.1em;">—</span><span style="display:block; line-height:1em; padding:0 0.1em;"><i>z</i></span></span></span>의 절댓값은 서로 같다.</figcaption></figure> <p><a href="/wiki/%EB%B3%B5%EC%86%8C%EC%88%98" title="복소수">복소수</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\in \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z\in \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/169fae60c23a2027ece2aa7fd4b5047492887e91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.607ex; height:2.176ex;" alt="{\displaystyle z\in \mathbb {C} }"></span>의 <b>절댓값</b> <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|\in [0,\infty )}"> <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> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |z|\in [0,\infty )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0180089aaf196358fc9dc74325d30033c799e691" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.294ex; height:2.843ex;" alt="{\displaystyle |z|\in [0,\infty )}"></span>은 다음과 같이 정의된다. </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 |z|={\sqrt {z{\bar {z}}}}={\sqrt {(\operatorname {Re} z)^{2}+(\operatorname {Im} 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> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>z</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo stretchy="false">(</mo> <mi>Re</mi> <mo>⁡</mo> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <mi>Im</mi> <mo>⁡</mo> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |z|={\sqrt {z{\bar {z}}}}={\sqrt {(\operatorname {Re} z)^{2}+(\operatorname {Im} z)^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a124187ba9652f5fd17ac536f98e90ca4deba49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:32.259ex; height:4.843ex;" alt="{\displaystyle |z|={\sqrt {z{\bar {z}}}}={\sqrt {(\operatorname {Re} z)^{2}+(\operatorname {Im} z)^{2}}}}"></span></dd></dl> <p>여기서 </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\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>는 <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>의 <a href="/wiki/%EC%BC%A4%EB%A0%88_%EB%B3%B5%EC%86%8C%EC%88%98" title="켤레 복소수">켤레 복소수</a>이다.</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 \operatorname {Re} z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Re</mi> <mo>⁡</mo> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Re} z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed84e2cc84e38be8d0c2823550cdd5de8d87b079" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.218ex; height:2.176ex;" alt="{\displaystyle \operatorname {Re} z}"></span>는 <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>의 <a href="/wiki/%EC%8B%A4%EC%88%98%EB%B6%80" class="mw-redirect" title="실수부">실수부</a>이다.</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 \operatorname {Im} z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Im</mi> <mo>⁡</mo> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Im} z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b399910e7928f6193bbd01211cf321329b005f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.251ex; height:2.176ex;" alt="{\displaystyle \operatorname {Im} z}"></span>는 <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>의 <a href="/wiki/%ED%97%88%EC%88%98%EB%B6%80" class="mw-redirect" title="허수부">허수부</a>이다.</li></ul> <p>즉, <a href="/wiki/%EB%B3%B5%EC%86%8C%ED%8F%89%EB%A9%B4" title="복소평면">복소평면</a>에 놓인 복소수의 절댓값은 그 복소수와 원점 사이의 거리를 <a href="/wiki/%ED%94%BC%ED%83%80%EA%B3%A0%EB%9D%BC%EC%8A%A4_%EC%A0%95%EB%A6%AC" title="피타고라스 정리">피타고라스 정리</a>를 사용하여 구한 것과 같다. </p><p>이는 실수의 절댓값의 정의와 호환된다. 모든 실수 <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\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9c6d458566aec47a7259762034790c8981aefab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.848ex; height:2.176ex;" alt="{\displaystyle x\in \mathbb {R} }"></span>에 대하여, 이를 복소수로 여겼을 때, 그 켤레 복소수는 자기 자신과 같다. 즉, </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 {\bar {x}}=x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {x}}=x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/993748a7418f68a1cbd3779f1d9bb46ab208babc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.758ex; height:2.009ex;" alt="{\displaystyle {\bar {x}}=x}"></span></dd></dl> <p>이다. 따라서, </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 {x{\bar {x}}}}={\sqrt {xx}}={\sqrt {x^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>x</mi> <mi>x</mi> </msqrt> </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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {x{\bar {x}}}}={\sqrt {xx}}={\sqrt {x^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc2cbb23d50ebd680505112db261300e7b4b41b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.095ex; height:3.676ex;" alt="{\displaystyle {\sqrt {x{\bar {x}}}}={\sqrt {xx}}={\sqrt {x^{2}}}}"></span></dd></dl> <p>이다. 즉, <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>의 실수로서의 절댓값과 복소수로서의 절댓값은 서로 같다. 다른 관점에서, 실수선은 복소평면의 좌표축으로 여길 수 있는데, 이 경우 실수와 원점 사이의 거리는 실수선에 국한되어서 보는지 복소평면에서 보는지와 무관하다. 따라서 실수의 절댓값은 복소수의 절댓값의 특수한 경우이다. </p> <div class="mw-heading mw-heading2"><h2 id="성질"><span id=".EC.84.B1.EC.A7.88"></span>성질</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%A0%88%EB%8C%93%EA%B0%92&action=edit&section=4" title="부분 편집: 성질"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>실수를 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,y,a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y,a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a3f40dc4cbcb75fae15da62f83e1a1f78870579" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.783ex; height:2.009ex;" alt="{\displaystyle x,y,a}"></span>, (실수일 수도 아닐 수도 있는) 복소수를 <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,w}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>,</mo> <mi>w</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z,w}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61184bc010f1c8b20a465ae5c41b013fe1a22abe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.786ex; height:2.009ex;" alt="{\displaystyle z,w}"></span>로 나타내자. 그렇다면, 절댓값의 성질을 다음과 같이 나타낼 수 있다. </p> <div class="mw-heading mw-heading3"><h3 id="부등식"><span id=".EB.B6.80.EB.93.B1.EC.8B.9D"></span>부등식</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%A0%88%EB%8C%93%EA%B0%92&action=edit&section=5" title="부분 편집: 부등식"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>복소수의 절댓값은 0 이상이다. </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 |z|\geq 0}"> <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> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |z|\geq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6506224a1dd2262556a29a789033f192f9d3bfef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.643ex; height:2.843ex;" alt="{\displaystyle |z|\geq 0}"></span></dd></dl> <p>복소수의 절댓값은 <a href="/wiki/%EC%96%91%EC%9D%98_%EC%A0%95%EB%B6%80%ED%98%B8%EC%84%B1" class="mw-redirect" title="양의 정부호성">양의 정부호성</a>을 만족시킨다. </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 |z|=0\iff z=0}"> <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> <mn>0</mn> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> <mi>z</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |z|=0\iff z=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/faf862f272a0a97115ad0fd8060e0015ed41e1fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.889ex; height:2.843ex;" alt="{\displaystyle |z|=0\iff z=0}"></span></dd></dl> <p>복소수의 절댓값은 <a href="/wiki/%EC%82%BC%EA%B0%81_%EB%B6%80%EB%93%B1%EC%8B%9D" title="삼각 부등식">삼각 부등식</a>을 만족시킨다. </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 ||z|-|w||\leq |z+w|\leq |z|+|w|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <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"> <mo stretchy="false">|</mo> </mrow> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mo>+</mo> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ||z|-|w||\leq |z+w|\leq |z|+|w|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94a685df8175d6ca8cbfeb571dfcbaea496ee329" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.737ex; height:2.843ex;" alt="{\displaystyle ||z|-|w||\leq |z+w|\leq |z|+|w|}"></span></dd> <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 |z+w|=|z|+|w|\iff z{\bar {w}}\geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mo>+</mo> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>w</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |z+w|=|z|+|w|\iff z{\bar {w}}\geq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d873a9601d81a4a17db37e6130024b27bf3d4fe9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.075ex; height:2.843ex;" alt="{\displaystyle |z+w|=|z|+|w|\iff z{\bar {w}}\geq 0}"></span></dd></dl> <p>실수의 절댓값을 포함하는 몇 가지 실수 부등식의 해는 다음과 같다. </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|\leq a\iff -a\leq x\leq a}"> <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>a</mi> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> <mo>−</mo> <mi>a</mi> <mo>≤</mo> <mi>x</mi> <mo>≤</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |x|\leq a\iff -a\leq x\leq a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d0393011a3c3a0e92230c7e104194d154729f64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.643ex; height:2.843ex;" alt="{\displaystyle |x|\leq a\iff -a\leq x\leq a}"></span></dd> <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|<a\iff -a<x<a}"> <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>a</mi> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> <mo>−</mo> <mi>a</mi> <mo><</mo> <mi>x</mi> <mo><</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |x|<a\iff -a<x<a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3a01acfe9234f0ff5ae9c1590d433d384b61dcb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.643ex; height:2.843ex;" alt="{\displaystyle |x|<a\iff -a<x<a}"></span></dd> <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|=a\iff {\begin{cases}x=\pm a&a\geq 0\\x\in \varnothing &a<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> <mi>a</mi> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> <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> <mo>±</mo> <mi>a</mi> </mtd> <mtd> <mi>a</mi> <mo>≥</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>x</mi> <mo>∈</mo> <mi class="MJX-variant">∅</mi> </mtd> <mtd> <mi>a</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|=a\iff {\begin{cases}x=\pm a&a\geq 0\\x\in \varnothing &a<0\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0295c32d6944df26403ef122763a4623ce0d8639" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:31.623ex; height:6.176ex;" alt="{\displaystyle |x|=a\iff {\begin{cases}x=\pm a&a\geq 0\\x\in \varnothing &a<0\end{cases}}}"></span></dd> <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|>a\iff x>a\lor x<-a}"> <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>a</mi> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> <mi>x</mi> <mo>></mo> <mi>a</mi> <mo>∨</mo> <mi>x</mi> <mo><</mo> <mo>−</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |x|>a\iff x>a\lor x<-a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1673933a9c8a63b10d96f0eab3dba84e0e8eee31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.555ex; height:2.843ex;" alt="{\displaystyle |x|>a\iff x>a\lor x<-a}"></span></dd> <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|\geq a\iff x\geq a\lor x\leq -a}"> <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>a</mi> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> <mi>x</mi> <mo>≥</mo> <mi>a</mi> <mo>∨</mo> <mi>x</mi> <mo>≤</mo> <mo>−</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |x|\geq a\iff x\geq a\lor x\leq -a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72f0f819d9d5e4aa99e129854036ba978d5d3fc3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.555ex; height:2.843ex;" alt="{\displaystyle |x|\geq a\iff x\geq a\lor x\leq -a}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="항등식"><span id=".ED.95.AD.EB.93.B1.EC.8B.9D"></span>항등식</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%A0%88%EB%8C%93%EA%B0%92&action=edit&section=6" title="부분 편집: 항등식"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>복소수의 절댓값은 곱셈·나눗셈을 보존한다. </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 |zw|=|z||w|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |zw|=|z||w|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5df220deb42cd816f43478c2c5a17f7ceecb6650" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.484ex; height:2.843ex;" alt="{\displaystyle |zw|=|z||w|}"></span></dd> <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 |z/w|=|z|/|w|\qquad (w\neq 0)}"> <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>/</mo> </mrow> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mspace width="2em" /> <mo stretchy="false">(</mo> <mi>w</mi> <mo>≠</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |z/w|=|z|/|w|\qquad (w\neq 0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2bb8704380b36d21d4b4680e08f86853ff45891" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.188ex; height:2.843ex;" alt="{\displaystyle |z/w|=|z|/|w|\qquad (w\neq 0)}"></span></dd></dl> <p>복소수 절댓값 함수는 <a href="/wiki/%EB%A9%B1%EB%93%B1_%ED%95%A8%EC%88%98" class="mw-redirect" title="멱등 함수">멱등 함수</a>이며, <a href="/wiki/%ED%9A%8C%EC%A0%84_%EB%8C%80%EC%B9%AD" class="mw-redirect" title="회전 대칭">회전 대칭</a>과 <a href="/w/index.php?title=%EB%B0%98%EC%82%AC_%EB%8C%80%EC%B9%AD&action=edit&redlink=1" class="new" title="반사 대칭 (없는 문서)">반사 대칭</a>을 만족시킨다. 특히, 실수 절댓값 함수는 <a href="/wiki/%EC%A7%9D%ED%95%A8%EC%88%98" class="mw-redirect" title="짝함수">짝함수</a>이다. </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 ||z||=|z|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ||z||=|z|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4836e3e44a755f87334e9d323bf377e4e7b637d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.156ex; height:2.843ex;" alt="{\displaystyle ||z||=|z|}"></span></dd> <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 |-z|=|z|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>−</mo> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |-z|=|z|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5df3f1aa768bb6ef61307814ffe9ea6ddc89421e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.702ex; height:2.843ex;" alt="{\displaystyle |-z|=|z|}"></span></dd> <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 |{\bar {z}}|=|z|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>z</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |{\bar {z}}|=|z|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60f099ef66bdf1c47ea6879cc940edfcbf7c833e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.07ex; height:2.843ex;" alt="{\displaystyle |{\bar {z}}|=|z|}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="미분"><span id=".EB.AF.B8.EB.B6.84"></span>미분</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%A0%88%EB%8C%93%EA%B0%92&action=edit&section=7" title="부분 편집: 미분"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>실수 절댓값 함수는 0이 아닌 모든 실수점에서 <a href="/wiki/%ED%95%B4%EC%84%9D_%ED%95%A8%EC%88%98" title="해석 함수">해석 함수</a>이다. 그 <a href="/wiki/%EB%8F%84%ED%95%A8%EC%88%98" class="mw-redirect" title="도함수">도함수</a>는 다음과 같다. </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}{dx}}|x|=\operatorname {sgn} x\qquad (x\neq 0)}"> <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>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mi>sgn</mi> <mo>⁡</mo> <mi>x</mi> <mspace width="2em" /> <mo stretchy="false">(</mo> <mi>x</mi> <mo>≠</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dx}}|x|=\operatorname {sgn} x\qquad (x\neq 0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fda31a5e97b4fb282aa0d778850ea9bcb0b44b60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:26.237ex; height:5.509ex;" alt="{\displaystyle {\frac {d}{dx}}|x|=\operatorname {sgn} x\qquad (x\neq 0)}"></span></dd></dl> <p>복소수 절댓값 함수는 모든 복소수에서 <a href="/wiki/%EC%97%B0%EC%86%8D_%ED%95%A8%EC%88%98" title="연속 함수">연속 함수</a>이지만, 모든 복소수점에서 비(非) <a href="/wiki/%EB%B3%B5%EC%86%8C_%EB%AF%B8%EB%B6%84_%EA%B0%80%EB%8A%A5_%ED%95%A8%EC%88%98" class="mw-redirect" title="복소 미분 가능 함수">복소 미분 가능 함수</a>이다. 이는 </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 {|z|-|z_{0}|}{z-z_{0}}}={\frac {1}{|z|+|z_{0}|}}\left(z{\frac {\overline {z-z_{0}}}{z-z_{0}}}+{\bar {z}}_{0}\right)\qquad (z_{0}\in \mathbb {C} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <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"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mrow> <mi>z</mi> <mo>−</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <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"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mover> <mrow> <mi>z</mi> <mo>−</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mo accent="false">¯</mo> </mover> <mrow> <mi>z</mi> <mo>−</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>z</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mspace width="2em" /> <mo stretchy="false">(</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {|z|-|z_{0}|}{z-z_{0}}}={\frac {1}{|z|+|z_{0}|}}\left(z{\frac {\overline {z-z_{0}}}{z-z_{0}}}+{\bar {z}}_{0}\right)\qquad (z_{0}\in \mathbb {C} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/def0bfe5112d7cf4e41b533b11540197804c367f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:52.931ex; height:7.509ex;" alt="{\displaystyle {\frac {|z|-|z_{0}|}{z-z_{0}}}={\frac {1}{|z|+|z_{0}|}}\left(z{\frac {\overline {z-z_{0}}}{z-z_{0}}}+{\bar {z}}_{0}\right)\qquad (z_{0}\in \mathbb {C} )}"></span></dd></dl> <p>가 <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\to z_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo stretchy="false">→</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z\to z_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56d44b10d6439812cb218e4af3c17347557e0bde" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.838ex; height:2.176ex;" alt="{\displaystyle z\to z_{0}}"></span>에서 항상 발산하기 때문이다. </p> <div class="mw-heading mw-heading2"><h2 id="응용"><span id=".EC.9D.91.EC.9A.A9"></span>응용</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%A0%88%EB%8C%93%EA%B0%92&action=edit&section=8" title="부분 편집: 응용"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="복소수의_극형식"><span id=".EB.B3.B5.EC.86.8C.EC.88.98.EC.9D.98_.EA.B7.B9.ED.98.95.EC.8B.9D"></span>복소수의 극형식</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%A0%88%EB%8C%93%EA%B0%92&action=edit&section=9" title="부분 편집: 복소수의 극형식"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r34311305">.mw-parser-output .hatnote{}.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}</style><div role="note" class="hatnote navigation-not-searchable"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/18px-Icons8_flat_search.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/27px-Icons8_flat_search.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/36px-Icons8_flat_search.svg.png 2x" data-file-width="512" data-file-height="512" /></span></span> 이 부분의 본문은 <a href="/wiki/%EA%B7%B9%ED%98%95%EC%8B%9D" class="mw-redirect" title="극형식">극형식</a>입니다.</div> <p>0이 아닌 복소수에 대하여, 절댓값은 복소수가 원점으로부터 떨어진 거리, 편각은 복소수가 가로축으로부터 회전한 각도를 뜻하므로, 0이 아닌 복소수는 절댓값과 편각으로부터 유일하게 결정된다. 구체적으로, 복소수 <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\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z\neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b8b7eb2d2a30057811a7835502717d3d6ece962" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.349ex; height:2.676ex;" alt="{\displaystyle z\neq 0}"></span>는 절댓값 <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>과 <a href="/wiki/%ED%8E%B8%EA%B0%81_(%EC%88%98%ED%95%99)" title="편각 (수학)">편각</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 \operatorname {arg} z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arg</mi> <mo>⁡</mo> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {arg} z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f08b2c439c1afe3614ac0029eab9720ab9fcd23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.712ex; height:2.009ex;" alt="{\displaystyle \operatorname {arg} z}"></span>을 사용하여 다음과 같은 꼴로 나타낼 수 있다. 이를 복소수의 극형식이라고 한다. </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 z=|z|(\cos \operatorname {arg} z+i\sin \operatorname {arg} z)=|z|e^{i\operatorname {arg} z}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mi>arg</mi> <mo>⁡</mo> <mi>z</mi> <mo>+</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mi>arg</mi> <mo>⁡</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>arg</mi> <mo>⁡</mo> <mi>z</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=|z|(\cos \operatorname {arg} z+i\sin \operatorname {arg} z)=|z|e^{i\operatorname {arg} z}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf6a0cc92ab1b1714d6dfe06bc0a81b7ad3212f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.768ex; height:3.176ex;" alt="{\displaystyle z=|z|(\cos \operatorname {arg} z+i\sin \operatorname {arg} z)=|z|e^{i\operatorname {arg} z}}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="거리_공간_구조"><span id=".EA.B1.B0.EB.A6.AC_.EA.B3.B5.EA.B0.84_.EA.B5.AC.EC.A1.B0"></span>거리 공간 구조</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%A0%88%EB%8C%93%EA%B0%92&action=edit&section=10" title="부분 편집: 거리 공간 구조"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>실수의 절댓값이 0과의 거리를 뜻하듯이, 실수선 위의 두 실수 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ea0abffd33a692ded22accc104515a032851dff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.519ex; height:2.009ex;" alt="{\displaystyle x,y}"></span> 사이의 거리 <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)}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(x,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3772957879a8bbf7946bddf5743c508a1d5072c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.544ex; height:2.843ex;" alt="{\displaystyle d(x,y)}"></span>는 절댓값을 통해 다음과 같이 나타낼 수 있다. </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 d(x,y)=|x-y|={\begin{cases}x-y&x>y\\0&x=y\\y-x&x<y\end{cases}}}"> <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> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mo>−</mo> <mi>y</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> <mi>y</mi> </mtd> <mtd> <mi>x</mi> <mo>></mo> <mi>y</mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>x</mi> <mo>=</mo> <mi>y</mi> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> <mo>−</mo> <mi>x</mi> </mtd> <mtd> <mi>x</mi> <mo><</mo> <mi>y</mi> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(x,y)=|x-y|={\begin{cases}x-y&x>y\\0&x=y\\y-x&x<y\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4adef706f899ffe553d1e453960b1022baa3b43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:35.41ex; height:8.509ex;" alt="{\displaystyle d(x,y)=|x-y|={\begin{cases}x-y&x>y\\0&x=y\\y-x&x<y\end{cases}}}"></span></dd></dl> <p>보다 일반적으로, 복소수의 절댓값이 원점과의 거리를 뜻하듯이, 복소평면 위의 두 복소수 <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,w}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>,</mo> <mi>w</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z,w}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61184bc010f1c8b20a465ae5c41b013fe1a22abe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.786ex; height:2.009ex;" alt="{\displaystyle z,w}"></span> 사이의 거리 <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(z,w)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo>,</mo> <mi>w</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(z,w)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d50afa30e8892ec2e66f6f682b17565a105c9945" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.811ex; height:2.843ex;" alt="{\displaystyle d(z,w)}"></span>는 절댓값을 통해 다음과 같이 나타낼 수 있다. </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 d(z,w)=|z-w|={\sqrt {(\operatorname {Re} z-\operatorname {Re} w)^{2}+(\operatorname {Im} z-\operatorname {Im} w)^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo>,</mo> <mi>w</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mo>−</mo> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo stretchy="false">(</mo> <mi>Re</mi> <mo>⁡</mo> <mi>z</mi> <mo>−</mo> <mi>Re</mi> <mo>⁡</mo> <mi>w</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <mi>Im</mi> <mo>⁡</mo> <mi>z</mi> <mo>−</mo> <mi>Im</mi> <mo>⁡</mo> <mi>w</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(z,w)=|z-w|={\sqrt {(\operatorname {Re} z-\operatorname {Re} w)^{2}+(\operatorname {Im} z-\operatorname {Im} w)^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01bce023d5c9513b68e85982ea07901df7f39bbd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:54.557ex; height:4.843ex;" alt="{\displaystyle d(z,w)=|z-w|={\sqrt {(\operatorname {Re} z-\operatorname {Re} w)^{2}+(\operatorname {Im} z-\operatorname {Im} w)^{2}}}}"></span></dd></dl> <p>이는 복소평면 위의 두 점의 연결선을 빗변으로 하고, 두 빗변이 각각 두 좌표축과 평행하는 직각 삼각형에 <a href="/wiki/%ED%94%BC%ED%83%80%EA%B3%A0%EB%9D%BC%EC%8A%A4_%EC%A0%95%EB%A6%AC" title="피타고라스 정리">피타고라스 정리</a>를 적용한 결과와 같다. <a href="/wiki/%EC%B6%94%EC%83%81%EB%8C%80%EC%88%98%ED%95%99" title="추상대수학">추상대수학</a>의 관점에서, 실수와 복소수의 절댓값은 모두 <a href="/wiki/%EA%B1%B0%EB%A6%AC_%EA%B3%B5%EA%B0%84" title="거리 공간">거리 공간</a> 구조를 부여한다. 사실, 절댓값은 <a href="/wiki/%EB%85%B8%EB%A6%84_%EA%B3%B5%EA%B0%84" title="노름 공간">노름 공간</a> 구조를 부여하며, 모든 노름 공간은 표준적인 거리 공간 구조를 갖춘다. </p> <div class="mw-heading mw-heading2"><h2 id="관련_개념"><span id=".EA.B4.80.EB.A0.A8_.EA.B0.9C.EB.85.90"></span>관련 개념</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%A0%88%EB%8C%93%EA%B0%92&action=edit&section=11" title="부분 편집: 관련 개념"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="노름"><span id=".EB.85.B8.EB.A6.84"></span>노름</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%A0%88%EB%8C%93%EA%B0%92&action=edit&section=12" title="부분 편집: 노름"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r34311305"><div role="note" class="hatnote navigation-not-searchable"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/18px-Icons8_flat_search.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/27px-Icons8_flat_search.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/36px-Icons8_flat_search.svg.png 2x" data-file-width="512" data-file-height="512" /></span></span> 이 부분의 본문은 <a href="/wiki/%EB%85%B8%EB%A6%84" class="mw-redirect" title="노름">노름</a>입니다.</div> <p><a href="/wiki/%EB%85%B8%EB%A6%84" class="mw-redirect" title="노름">노름</a>은 <a href="/wiki/%EB%B2%A1%ED%84%B0_%EA%B3%B5%EA%B0%84" title="벡터 공간">벡터 공간</a>에 정의되며, 음이 아닌 실숫값을 취하며, <a href="/wiki/%EC%96%91%EC%9D%98_%EC%A0%95%EB%B6%80%ED%98%B8%EC%84%B1" class="mw-redirect" title="양의 정부호성">양의 정부호성</a>을 만족시키며, <a href="/w/index.php?title=%EC%96%91%EC%9D%98_%EB%8F%99%EC%B0%A8%EC%84%B1&action=edit&redlink=1" class="new" title="양의 동차성 (없는 문서)">양의 동차성</a>을 만족시키며, <a href="/wiki/%EC%82%BC%EA%B0%81_%EB%B6%80%EB%93%B1%EC%8B%9D" title="삼각 부등식">삼각 부등식</a>을 만족시키는 <a href="/wiki/%ED%95%A8%EC%88%98" title="함수">함수</a>이다. 실수체와 복소수체는 벡터 공간의 특수한 경우이므로, 실수 또는 복소수의 절댓값은 노름의 특수한 경우이다. 모든 노름 <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\mapsto \Vert x\Vert }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">↦</mo> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo fence="false" stretchy="false">‖</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\mapsto \Vert x\Vert }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/222f28353aae2a563648c2e07fd1a5c61087565f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.598ex; height:2.843ex;" alt="{\displaystyle x\mapsto \Vert x\Vert }"></span>는 표준적인 <a href="/wiki/%EA%B1%B0%EB%A6%AC_%ED%95%A8%EC%88%98" title="거리 함수">거리 함수</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,y)\mapsto \Vert x-y\Vert }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">↦</mo> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo fence="false" stretchy="false">‖</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y)\mapsto \Vert x-y\Vert }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/658427d64ca68575d2df5306a5c3deb42dc752e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.593ex; height:2.843ex;" alt="{\displaystyle (x,y)\mapsto \Vert x-y\Vert }"></span>를 유도한다. </p> <div class="mw-heading mw-heading3"><h3 id="정역_위의_절댓값"><span id=".EC.A0.95.EC.97.AD_.EC.9C.84.EC.9D.98_.EC.A0.88.EB.8C.93.EA.B0.92"></span>정역 위의 절댓값</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%A0%88%EB%8C%93%EA%B0%92&action=edit&section=13" title="부분 편집: 정역 위의 절댓값"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r34311305"><div role="note" class="hatnote navigation-not-searchable"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/18px-Icons8_flat_search.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/27px-Icons8_flat_search.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/36px-Icons8_flat_search.svg.png 2x" data-file-width="512" data-file-height="512" /></span></span> 이 부분의 본문은 <a href="/wiki/%EC%A0%88%EB%8C%93%EA%B0%92_(%EB%8C%80%EC%88%98%ED%95%99)" title="절댓값 (대수학)">절댓값 (대수학)</a>입니다.</div> <p><a href="/wiki/%EC%A0%95%EC%97%AD" title="정역">정역</a> 위의 절댓값은, <a href="/wiki/%EC%A0%95%EC%97%AD" title="정역">정역</a>에 정의되며, 음이 아닌 실숫값을 취하며, <a href="/wiki/%EC%96%91%EC%9D%98_%EC%A0%95%EB%B6%80%ED%98%B8%EC%84%B1" class="mw-redirect" title="양의 정부호성">양의 정부호성</a>을 만족시키며, 곱셈을 보존하며, <a href="/wiki/%EC%82%BC%EA%B0%81_%EB%B6%80%EB%93%B1%EC%8B%9D" title="삼각 부등식">삼각 부등식</a>을 만족시키는 함수이다. 모든 <a href="/wiki/%EC%B2%B4_(%EC%88%98%ED%95%99)" title="체 (수학)">체</a>는 <a href="/wiki/%EC%A0%95%EC%97%AD" title="정역">정역</a>이므로, 실수 또는 복소수의 절댓값은 정역 위의 절댓값의 특수한 경우이다. 모든 정역 위의 절댓값 <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\mapsto |x|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\mapsto |x|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04f7cb7bcc9c126b2d18614065cfeeb20f8e5514" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.567ex; height:2.843ex;" alt="{\displaystyle x\mapsto |x|}"></span>는 표준적인 <a href="/wiki/%EA%B1%B0%EB%A6%AC_%ED%95%A8%EC%88%98" title="거리 함수">거리 함수</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,y)\mapsto |x-y|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">↦</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y)\mapsto |x-y|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/def7ca84dd617644bff81c8e3f058e4529116842" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.562ex; height:2.843ex;" alt="{\displaystyle (x,y)\mapsto |x-y|}"></span>를 유도한다. </p> <div class="mw-heading mw-heading2"><h2 id="참고_문헌"><span id=".EC.B0.B8.EA.B3.A0_.EB.AC.B8.ED.97.8C"></span>참고 문헌</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%A0%88%EB%8C%93%EA%B0%92&action=edit&section=14" title="부분 편집: 참고 문헌"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Nahin, Paul J.; <a rel="nofollow" class="external text" href="https://www.amazon.com/gp/reader/0691027951"><i>An Imaginary Tale</i></a>; Princeton University Press; (hardcover, 1998). <style data-mw-deduplicate="TemplateStyles:r38117996">.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/%EA%B5%AD%EC%A0%9C_%ED%91%9C%EC%A4%80_%EB%8F%84%EC%84%9C_%EB%B2%88%ED%98%B8" class="mw-redirect" title="국제 표준 도서 번호">ISBN</a> <a href="/wiki/%ED%8A%B9%EC%88%98:%EC%B1%85%EC%B0%BE%EA%B8%B0/0-691-02795-1" title="특수:책찾기/0-691-02795-1">0-691-02795-1</a></li> <li><cite class="citation web">O’Connor, John J.; Robertson, Edmund F. <a rel="nofollow" class="external text" href="http://www-history.mcs.st-andrews.ac.uk/Biographies/Argand.html">“Jean Robert Argand”</a>. 《MacTutor History of Mathematics Archive》 (영어). <a href="/wiki/%EC%84%B8%EC%9D%B8%ED%8A%B8%EC%95%A4%EB%93%9C%EB%A3%A8%EC%8A%A4_%EB%8C%80%ED%95%99%EA%B5%90" title="세인트앤드루스 대학교">세인트앤드루스 대학교</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MacTutor+History+of+Mathematics+Archive&rft.atitle=Jean+Robert+Argand&rft.aulast=O%E2%80%99Connor&rft.aufirst=John+J.&rft.au=Robertson%2C+Edmund+F.&rft_id=http%3A%2F%2Fwww-history.mcs.st-andrews.ac.uk%2FBiographies%2FArgand.html&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%EC%A0%88%EB%8C%93%EA%B0%92" class="Z3988"><span style="display:none;"> </span></span></li> <li>Schechter, Eric; <i>Handbook of Analysis and Its Foundations</i>, pp 259–263, <a rel="nofollow" class="external text" href="https://www.amazon.com/gp/reader/0126227608/?keywords=absolute%20value&v=search-inside">"Absolute Values"</a>, Academic Press (1997) <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r38117996"><a href="/wiki/%EA%B5%AD%EC%A0%9C_%ED%91%9C%EC%A4%80_%EB%8F%84%EC%84%9C_%EB%B2%88%ED%98%B8" class="mw-redirect" title="국제 표준 도서 번호">ISBN</a> <a href="/wiki/%ED%8A%B9%EC%88%98:%EC%B1%85%EC%B0%BE%EA%B8%B0/0-12-622760-8" title="특수:책찾기/0-12-622760-8">0-12-622760-8</a></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="">원본 주소 "<a dir="ltr" href="https://ko.wikipedia.org/w/index.php?title=절댓값&oldid=37542928">https://ko.wikipedia.org/w/index.php?title=절댓값&oldid=37542928</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/%ED%8A%B9%EC%88%98:%EB%B6%84%EB%A5%98" title="특수:분류">분류</a>: <ul><li><a href="/wiki/%EB%B6%84%EB%A5%98:%EB%85%B8%EB%A6%84" title="분류:노름">노름</a></li><li><a href="/wiki/%EB%B6%84%EB%A5%98:%ED%8A%B9%EC%88%98_%ED%95%A8%EC%88%98" title="분류:특수 함수">특수 함수</a></li><li><a href="/wiki/%EB%B6%84%EB%A5%98:%EC%8B%A4%EC%88%98" title="분류:실수">실수</a></li><li><a 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