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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%8B%A4%EC%88%98" 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%8B%A4%EC%88%98" 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%8B%A4%EC%88%98" 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%8B%A4%EC%88%98" 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-공리적_정의" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#공리적_정의"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>공리적 정의</span> </div> </a> <ul id="toc-공리적_정의-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-구성적_정의" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#구성적_정의"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>구성적 정의</span> </div> </a> <ul id="toc-구성적_정의-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-연산" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#연산"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>연산</span> </div> </a> <button aria-controls="toc-연산-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>연산 하위섹션 토글하기</span> </button> <ul id="toc-연산-sublist" class="vector-toc-list"> <li id="toc-사칙_연산" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#사칙_연산"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>사칙 연산</span> </div> </a> <ul id="toc-사칙_연산-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-거듭제곱과_거듭제곱근" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#거듭제곱과_거듭제곱근"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>거듭제곱과 거듭제곱근</span> </div> </a> <ul id="toc-거듭제곱과_거듭제곱근-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-순서" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#순서"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>순서</span> </div> </a> <button aria-controls="toc-순서-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>순서 하위섹션 토글하기</span> </button> <ul id="toc-순서-sublist" class="vector-toc-list"> <li id="toc-구간" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#구간"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>구간</span> </div> </a> <ul id="toc-구간-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-상한_공리" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#상한_공리"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>상한 공리</span> </div> </a> <ul id="toc-상한_공리-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-데데킨트_완비성" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#데데킨트_완비성"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>데데킨트 완비성</span> </div> </a> <ul id="toc-데데킨트_완비성-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-기타_성질" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#기타_성질"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>기타 성질</span> </div> </a> <ul id="toc-기타_성질-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-위상" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#위상"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>위상</span> </div> </a> <ul id="toc-위상-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-분류" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#분류"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>분류</span> </div> </a> <ul id="toc-분류-sublist" class="vector-toc-list"> </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">6</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-집합론적_성질" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#집합론적_성질"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>집합론적 성질</span> </div> </a> <ul id="toc-집합론적_성질-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-역사" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#역사"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>역사</span> </div> </a> <ul id="toc-역사-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-같이_보기" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#같이_보기"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>같이 보기</span> </div> </a> <ul id="toc-같이_보기-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-외부_링크" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#외부_링크"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>외부 링크</span> </div> </a> <ul id="toc-외부_링크-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">실수</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. 118개 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-118" 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">118개 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/Re%C3%ABle_getal" title="Reële getal – 아프리칸스어" lang="af" hreflang="af" data-title="Reële getal" data-language-autonym="Afrikaans" data-language-local-name="아프리칸스어" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Reelle_Zahl" title="Reelle Zahl – 독일어(스위스)" lang="gsw" hreflang="gsw" data-title="Reelle Zahl" data-language-autonym="Alemannisch" data-language-local-name="독일어(스위스)" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B9%D8%AF%D8%AF_%D8%AD%D9%82%D9%8A%D9%82%D9%8A" title="عدد حقيقي – 아랍어" lang="ar" hreflang="ar" data-title="عدد حقيقي" data-language-autonym="العربية" data-language-local-name="아랍어" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/N%C3%BAmberu_real" title="Númberu real – 아스투리아어" lang="ast" hreflang="ast" data-title="Númberu real" data-language-autonym="Asturianu" data-language-local-name="아스투리아어" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/H%C9%99qiqi_%C9%99d%C9%99dl%C9%99r" title="Həqiqi ədədlər – 아제르바이잔어" lang="az" hreflang="az" data-title="Həqiqi ədədlər" data-language-autonym="Azərbaycanca" data-language-local-name="아제르바이잔어" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-azb mw-list-item"><a href="https://azb.wikipedia.org/wiki/%D8%AD%D9%82%DB%8C%D9%82%DB%8C_%D8%B3%D8%A7%DB%8C%DB%8C%D9%84%D8%A7%D8%B1" title="حقیقی ساییلار – South Azerbaijani" lang="azb" hreflang="azb" data-title="حقیقی ساییلار" data-language-autonym="تۆرکجه" data-language-local-name="South Azerbaijani" class="interlanguage-link-target"><span>تۆرکجه</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%AB%D1%81%D1%8B%D0%BD_%D2%BB%D0%B0%D0%BD" title="Ысын һан – 바슈키르어" lang="ba" hreflang="ba" data-title="Ысын һан" data-language-autonym="Башҡортса" data-language-local-name="바슈키르어" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-bcl mw-list-item"><a href="https://bcl.wikipedia.org/wiki/Tunay_na_bilang" title="Tunay na bilang – Central Bikol" lang="bcl" hreflang="bcl" data-title="Tunay na bilang" data-language-autonym="Bikol Central" data-language-local-name="Central Bikol" class="interlanguage-link-target"><span>Bikol Central</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%A0%D1%8D%D1%87%D0%B0%D1%96%D1%81%D0%BD%D1%8B_%D0%BB%D1%96%D0%BA" title="Рэчаісны лік – 벨라루스어" lang="be" hreflang="be" data-title="Рэчаісны лік" 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%A0%D1%8D%D1%87%D0%B0%D1%96%D1%81%D0%BD%D1%8B_%D0%BB%D1%96%D0%BA" title="Рэчаісны лік – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Рэчаісны лік" 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%A0%D0%B5%D0%B0%D0%BB%D0%BD%D0%BE_%D1%87%D0%B8%D1%81%D0%BB%D0%BE" title="Реално число – 불가리아어" lang="bg" hreflang="bg" data-title="Реално число" data-language-autonym="Български" data-language-local-name="불가리아어" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bh mw-list-item"><a href="https://bh.wikipedia.org/wiki/%E0%A4%B5%E0%A4%BE%E0%A4%B8%E0%A5%8D%E0%A4%A4%E0%A4%B5%E0%A4%BF%E0%A4%95_%E0%A4%B8%E0%A4%82%E0%A4%96%E0%A5%8D%E0%A4%AF%E0%A4%BE" title="वास्तविक संख्या – Bhojpuri" lang="bh" hreflang="bh" data-title="वास्तविक संख्या" data-language-autonym="भोजपुरी" data-language-local-name="Bhojpuri" class="interlanguage-link-target"><span>भोजपुरी</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AC%E0%A6%BE%E0%A6%B8%E0%A7%8D%E0%A6%A4%E0%A6%AC_%E0%A6%B8%E0%A6%82%E0%A6%96%E0%A7%8D%E0%A6%AF%E0%A6%BE" title="বাস্তব সংখ্যা – 벵골어" lang="bn" hreflang="bn" data-title="বাস্তব সংখ্যা" data-language-autonym="বাংলা" data-language-local-name="벵골어" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Realan_broj" title="Realan broj – 보스니아어" lang="bs" hreflang="bs" data-title="Realan broj" data-language-autonym="Bosanski" data-language-local-name="보스니아어" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-bxr mw-list-item"><a href="https://bxr.wikipedia.org/wiki/%D0%91%D0%BE%D0%B4%D0%BE%D1%82%D0%BE_%D1%82%D0%BE%D0%BE" title="Бодото тоо – Russia Buriat" lang="bxr" hreflang="bxr" data-title="Бодото тоо" data-language-autonym="Буряад" data-language-local-name="Russia Buriat" class="interlanguage-link-target"><span>Буряад</span></a></li><li class="interlanguage-link interwiki-ca badge-Q17437798 badge-goodarticle mw-list-item" title="좋은 글"><a href="https://ca.wikipedia.org/wiki/Nombre_real" title="Nombre real – 카탈로니아어" lang="ca" hreflang="ca" data-title="Nombre real" 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/%DA%98%D9%85%D8%A7%D8%B1%DB%95%DB%8C_%DA%95%D8%A7%D8%B3%D8%AA%DB%95%D9%82%DB%8C%D9%86%DB%95" title="ژمارەی ڕاستەقینە – 소라니 쿠르드어" lang="ckb" hreflang="ckb" data-title="ژمارەی ڕاستەقینە" data-language-autonym="کوردی" data-language-local-name="소라니 쿠르드어" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-crh mw-list-item"><a href="https://crh.wikipedia.org/wiki/Aqiqiy_say%C4%B1" title="Aqiqiy sayı – 크리민 터키어; 크리민 타타르어" lang="crh" hreflang="crh" data-title="Aqiqiy sayı" data-language-autonym="Qırımtatarca" data-language-local-name="크리민 터키어; 크리민 타타르어" class="interlanguage-link-target"><span>Qırımtatarca</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Re%C3%A1ln%C3%A9_%C4%8D%C3%ADslo" title="Reálné číslo – 체코어" lang="cs" hreflang="cs" data-title="Reálné číslo" 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%A7%C4%83%D0%BD_%D1%85%D0%B8%D1%81%D0%B5%D0%BF" title="Чăн хисеп – 추바시어" lang="cv" hreflang="cv" data-title="Чăн хисеп" 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/Rhif_real" title="Rhif real – 웨일스어" lang="cy" hreflang="cy" data-title="Rhif real" 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/Reelle_tal" title="Reelle tal – 덴마크어" lang="da" hreflang="da" data-title="Reelle tal" 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/Reelle_Zahl" title="Reelle Zahl – 독일어" lang="de" hreflang="de" data-title="Reelle Zahl" data-language-autonym="Deutsch" data-language-local-name="독일어" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-diq mw-list-item"><a href="https://diq.wikipedia.org/wiki/Amaro_reel" title="Amaro reel – Zazaki" lang="diq" hreflang="diq" data-title="Amaro reel" data-language-autonym="Zazaki" data-language-local-name="Zazaki" class="interlanguage-link-target"><span>Zazaki</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A0%CF%81%CE%B1%CE%B3%CE%BC%CE%B1%CF%84%CE%B9%CE%BA%CF%8C%CF%82_%CE%B1%CF%81%CE%B9%CE%B8%CE%BC%CF%8C%CF%82" title="Πραγματικός αριθμός – 그리스어" lang="el" hreflang="el" data-title="Πραγματικός αριθμός" data-language-autonym="Ελληνικά" data-language-local-name="그리스어" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-eml mw-list-item"><a href="https://eml.wikipedia.org/wiki/N%C3%B3mmer_re%C3%A8l" title="Nómmer reèl – Emiliano-Romagnolo" lang="egl" hreflang="egl" data-title="Nómmer reèl" data-language-autonym="Emiliàn e rumagnòl" data-language-local-name="Emiliano-Romagnolo" class="interlanguage-link-target"><span>Emiliàn e rumagnòl</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Real_number" title="Real number – 영어" lang="en" hreflang="en" data-title="Real number" 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/Reelo" title="Reelo – 에스페란토어" lang="eo" hreflang="eo" data-title="Reelo" 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/N%C3%BAmero_real" title="Número real – 스페인어" lang="es" hreflang="es" data-title="Número real" 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/Reaalarv" title="Reaalarv – 에스토니아어" lang="et" hreflang="et" data-title="Reaalarv" 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/Zenbaki_erreal" title="Zenbaki erreal – 바스크어" lang="eu" hreflang="eu" data-title="Zenbaki erreal" 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/%D8%B9%D8%AF%D8%AF_%D8%AD%D9%82%DB%8C%D9%82%DB%8C" title="عدد حقیقی – 페르시아어" lang="fa" hreflang="fa" data-title="عدد حقیقی" 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/Reaaliluku" title="Reaaliluku – 핀란드어" lang="fi" hreflang="fi" data-title="Reaaliluku" data-language-autonym="Suomi" data-language-local-name="핀란드어" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fiu-vro mw-list-item"><a href="https://fiu-vro.wikipedia.org/wiki/Reaalarv" title="Reaalarv – Võro" lang="vro" hreflang="vro" data-title="Reaalarv" data-language-autonym="Võro" data-language-local-name="Võro" class="interlanguage-link-target"><span>Võro</span></a></li><li class="interlanguage-link interwiki-fo mw-list-item"><a href="https://fo.wikipedia.org/wiki/Reelt_tal" title="Reelt tal – 페로어" lang="fo" hreflang="fo" data-title="Reelt tal" data-language-autonym="Føroyskt" data-language-local-name="페로어" class="interlanguage-link-target"><span>Føroyskt</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Nombre_r%C3%A9el" title="Nombre réel – 프랑스어" lang="fr" hreflang="fr" data-title="Nombre réel" data-language-autonym="Français" data-language-local-name="프랑스어" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Re%27el_taal" title="Re'el taal – 북부 프리지아어" lang="frr" hreflang="frr" data-title="Re'el taal" data-language-autonym="Nordfriisk" data-language-local-name="북부 프리지아어" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-fur mw-list-item"><a href="https://fur.wikipedia.org/wiki/Numars_re%C3%A2i" title="Numars reâi – 프리울리어" lang="fur" hreflang="fur" data-title="Numars reâi" data-language-autonym="Furlan" data-language-local-name="프리울리어" class="interlanguage-link-target"><span>Furlan</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/R%C3%A9aduimhir" title="Réaduimhir – 아일랜드어" lang="ga" hreflang="ga" data-title="Réaduimhir" data-language-autonym="Gaeilge" data-language-local-name="아일랜드어" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gan mw-list-item"><a href="https://gan.wikipedia.org/wiki/%E5%AF%A6%E6%95%B8" title="實數 – 간어" lang="gan" hreflang="gan" data-title="實數" data-language-autonym="贛語" data-language-local-name="간어" class="interlanguage-link-target"><span>贛語</span></a></li><li class="interlanguage-link interwiki-gcr mw-list-item"><a href="https://gcr.wikipedia.org/wiki/Nonm_r%C3%A9y%C3%A8l" title="Nonm réyèl – Guianan Creole" lang="gcr" hreflang="gcr" data-title="Nonm réyèl" data-language-autonym="Kriyòl gwiyannen" data-language-local-name="Guianan Creole" class="interlanguage-link-target"><span>Kriyòl gwiyannen</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/N%C3%BAmero_real" title="Número real – 갈리시아어" lang="gl" hreflang="gl" data-title="Número real" data-language-autonym="Galego" data-language-local-name="갈리시아어" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-gv mw-list-item"><a href="https://gv.wikipedia.org/wiki/Feer_earroo" title="Feer earroo – 맹크스어" lang="gv" hreflang="gv" data-title="Feer earroo" data-language-autonym="Gaelg" data-language-local-name="맹크스어" class="interlanguage-link-target"><span>Gaelg</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A1%D7%A4%D7%A8_%D7%9E%D7%9E%D7%A9%D7%99" title="מספר ממשי – 히브리어" lang="he" hreflang="he" data-title="מספר ממשי" 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%B5%E0%A4%BE%E0%A4%B8%E0%A5%8D%E0%A4%A4%E0%A4%B5%E0%A4%BF%E0%A4%95_%E0%A4%B8%E0%A4%82%E0%A4%96%E0%A5%8D%E0%A4%AF%E0%A4%BE" title="वास्तविक संख्या – 힌디어" lang="hi" hreflang="hi" data-title="वास्तविक संख्या" 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/Realni_broj" title="Realni broj – 크로아티아어" lang="hr" hreflang="hr" data-title="Realni broj" 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/Val%C3%B3s_sz%C3%A1mok" title="Valós számok – 헝가리어" lang="hu" hreflang="hu" data-title="Valós számok" 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%BB%D6%80%D5%A1%D5%AF%D5%A1%D5%B6_%D5%A9%D5%AB%D5%BE" title="Իրական թիվ – 아르메니아어" lang="hy" hreflang="hy" data-title="Իրական թիվ" 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/Numero_real" title="Numero real – 인터링구아" lang="ia" hreflang="ia" data-title="Numero real" data-language-autonym="Interlingua" data-language-local-name="인터링구아" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-iba mw-list-item"><a href="https://iba.wikipedia.org/wiki/Lumur_bendar" title="Lumur bendar – 이반어" lang="iba" hreflang="iba" data-title="Lumur bendar" data-language-autonym="Jaku Iban" data-language-local-name="이반어" class="interlanguage-link-target"><span>Jaku Iban</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Bilangan_riil" title="Bilangan riil – 인도네시아어" lang="id" hreflang="id" data-title="Bilangan riil" 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-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Reala_nombro" title="Reala nombro – 이도어" lang="io" hreflang="io" data-title="Reala nombro" data-language-autonym="Ido" data-language-local-name="이도어" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Rauntala" title="Rauntala – 아이슬란드어" lang="is" hreflang="is" data-title="Rauntala" 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/Numero_reale" title="Numero reale – 이탈리아어" lang="it" hreflang="it" data-title="Numero reale" 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/%E5%AE%9F%E6%95%B0" title="実数 – 일본어" lang="ja" hreflang="ja" data-title="実数" 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-jam mw-list-item"><a href="https://jam.wikipedia.org/wiki/Riil_nomba" title="Riil nomba – Jamaican Creole English" lang="jam" hreflang="jam" data-title="Riil nomba" data-language-autonym="Patois" data-language-local-name="Jamaican Creole English" class="interlanguage-link-target"><span>Patois</span></a></li><li class="interlanguage-link interwiki-jbo mw-list-item"><a href="https://jbo.wikipedia.org/wiki/pavycimdyna%27u" title="pavycimdyna'u – 로반어" lang="jbo" hreflang="jbo" data-title="pavycimdyna'u" data-language-autonym="La .lojban." data-language-local-name="로반어" class="interlanguage-link-target"><span>La .lojban.</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%9C%E1%83%90%E1%83%9B%E1%83%93%E1%83%95%E1%83%98%E1%83%9A%E1%83%98_%E1%83%A0%E1%83%98%E1%83%AA%E1%83%AE%E1%83%95%E1%83%98" title="ნამდვილი რიცხვი – 조지아어" lang="ka" hreflang="ka" data-title="ნამდვილი რიცხვი" data-language-autonym="ქართული" data-language-local-name="조지아어" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kbp mw-list-item"><a href="https://kbp.wikipedia.org/wiki/Si%C5%8B%C5%8B_%C3%B1%CA%8A%C5%8B_(t%CA%8A%CA%8Az%CA%8A%CA%8A)" title="Siŋŋ ñʊŋ (tʊʊzʊʊ) – Kabiye" lang="kbp" hreflang="kbp" data-title="Siŋŋ ñʊŋ (tʊʊzʊʊ)" data-language-autonym="Kabɩyɛ" data-language-local-name="Kabiye" class="interlanguage-link-target"><span>Kabɩyɛ</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%9D%D0%B0%D2%9B%D1%82%D1%8B_%D1%81%D0%B0%D0%BD" title="Нақты сан – 카자흐어" lang="kk" hreflang="kk" data-title="Нақты сан" 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%85%E1%9F%86%E1%9E%93%E1%9E%BD%E1%9E%93%E1%9E%96%E1%9E%B7%E1%9E%8F" title="ចំនួនពិត – 크메르어" lang="km" hreflang="km" data-title="ចំនួនពិត" data-language-autonym="ភាសាខ្មែរ" data-language-local-name="크메르어" class="interlanguage-link-target"><span>ភាសាខ្មែរ</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%A8%E0%B3%88%E0%B2%9C_%E0%B2%B8%E0%B2%82%E0%B2%96%E0%B3%8D%E0%B2%AF%E0%B3%86" title="ನೈಜ ಸಂಖ್ಯೆ – 칸나다어" lang="kn" hreflang="kn" data-title="ನೈಜ ಸಂಖ್ಯೆ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="칸나다어" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-ku mw-list-item"><a href="https://ku.wikipedia.org/wiki/Hejmar%C3%AAn_rast%C3%AEn" title="Hejmarên rastîn – 쿠르드어" lang="ku" hreflang="ku" data-title="Hejmarên rastîn" data-language-autonym="Kurdî" data-language-local-name="쿠르드어" class="interlanguage-link-target"><span>Kurdî</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%90%D0%BD%D1%8B%D0%BA_%D1%81%D0%B0%D0%BD" title="Анык сан – 키르기스어" lang="ky" hreflang="ky" data-title="Анык сан" 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/Numerus_realis" title="Numerus realis – 라틴어" lang="la" hreflang="la" data-title="Numerus realis" 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/Numero_real" title="Numero real – 링구아 프랑카 노바" lang="lfn" hreflang="lfn" data-title="Numero real" 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-li mw-list-item"><a href="https://li.wikipedia.org/wiki/Re%C3%ABel_getal" title="Reëel getal – 림버거어" lang="li" hreflang="li" data-title="Reëel getal" data-language-autonym="Limburgs" data-language-local-name="림버거어" class="interlanguage-link-target"><span>Limburgs</span></a></li><li class="interlanguage-link interwiki-lij mw-list-item"><a href="https://lij.wikipedia.org/wiki/Numeri_re%C3%A6" title="Numeri reæ – Ligurian" lang="lij" hreflang="lij" data-title="Numeri reæ" data-language-autonym="Ligure" data-language-local-name="Ligurian" class="interlanguage-link-target"><span>Ligure</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Numer_real" title="Numer real – Lombard" lang="lmo" hreflang="lmo" data-title="Numer real" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-lo mw-list-item"><a href="https://lo.wikipedia.org/wiki/%E0%BA%88%E0%BA%B3%E0%BA%99%E0%BA%A7%E0%BA%99%E0%BA%88%E0%BA%B4%E0%BA%87" title="ຈຳນວນຈິງ – 라오어" lang="lo" hreflang="lo" data-title="ຈຳນວນຈິງ" data-language-autonym="ລາວ" data-language-local-name="라오어" class="interlanguage-link-target"><span>ລາວ</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Realusis_skai%C4%8Dius" title="Realusis skaičius – 리투아니아어" lang="lt" hreflang="lt" data-title="Realusis skaičius" 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/Re%C4%81ls_skaitlis" title="Reāls skaitlis – 라트비아어" lang="lv" hreflang="lv" data-title="Reāls skaitlis" data-language-autonym="Latviešu" data-language-local-name="라트비아어" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mg mw-list-item"><a href="https://mg.wikipedia.org/wiki/Isa_voatsapa" title="Isa voatsapa – 말라가시어" lang="mg" hreflang="mg" data-title="Isa voatsapa" data-language-autonym="Malagasy" data-language-local-name="말라가시어" class="interlanguage-link-target"><span>Malagasy</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%A0%D0%B5%D0%B0%D0%BB%D0%B5%D0%BD_%D0%B1%D1%80%D0%BE%D1%98" title="Реален број – 마케도니아어" lang="mk" hreflang="mk" data-title="Реален број" data-language-autonym="Македонски" data-language-local-name="마케도니아어" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%B5%E0%B4%BE%E0%B4%B8%E0%B5%8D%E0%B4%A4%E0%B4%B5%E0%B4%BF%E0%B4%95%E0%B4%B8%E0%B4%82%E0%B4%96%E0%B5%8D%E0%B4%AF" title="വാസ്തവികസംഖ്യ – 말라얄람어" lang="ml" hreflang="ml" data-title="വാസ്തവികസംഖ്യ" data-language-autonym="മലയാളം" data-language-local-name="말라얄람어" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%B5%E0%A4%BE%E0%A4%B8%E0%A5%8D%E0%A4%A4%E0%A4%B5%E0%A4%BF%E0%A4%95_%E0%A4%B8%E0%A4%82%E0%A4%96%E0%A5%8D%E0%A4%AF%E0%A4%BE" title="वास्तविक संख्या – 마라티어" lang="mr" hreflang="mr" data-title="वास्तविक संख्या" data-language-autonym="मराठी" data-language-local-name="마라티어" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Nombor_nyata" title="Nombor nyata – 말레이어" lang="ms" hreflang="ms" data-title="Nombor nyata" 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%80%E1%80%AD%E1%80%94%E1%80%BA%E1%80%B8%E1%80%85%E1%80%85%E1%80%BA" title="ကိန်းစစ် – 버마어" lang="my" hreflang="my" data-title="ကိန်းစစ်" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="버마어" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-ne mw-list-item"><a href="https://ne.wikipedia.org/wiki/%E0%A4%B5%E0%A4%BE%E0%A4%B8%E0%A5%8D%E0%A4%A4%E0%A4%B5%E0%A4%BF%E0%A4%95_%E0%A4%B8%E0%A4%99%E0%A5%8D%E0%A4%96%E0%A5%8D%E0%A4%AF%E0%A4%BE" title="वास्तविक सङ्ख्या – 네팔어" lang="ne" hreflang="ne" data-title="वास्तविक सङ्ख्या" data-language-autonym="नेपाली" data-language-local-name="네팔어" class="interlanguage-link-target"><span>नेपाली</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Re%C3%ABel_getal" title="Reëel getal – 네덜란드어" lang="nl" hreflang="nl" data-title="Reëel getal" 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/Reelle_tal" title="Reelle tal – 노르웨이어(니노르스크)" lang="nn" hreflang="nn" data-title="Reelle tal" 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/Reelt_tall" title="Reelt tall – 노르웨이어(보크말)" lang="nb" hreflang="nb" data-title="Reelt tall" 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-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Nombre_real" title="Nombre real – 오크어" lang="oc" hreflang="oc" data-title="Nombre real" data-language-autonym="Occitan" data-language-local-name="오크어" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-os mw-list-item"><a href="https://os.wikipedia.org/wiki/%D0%91%C3%A6%D0%BB%D0%B2%D1%8B%D1%80%D0%B4_%D0%BD%D1%8B%D0%BC%C3%A6%D1%86" title="Бæлвырд нымæц – 오세트어" lang="os" hreflang="os" data-title="Бæлвырд нымæц" data-language-autonym="Ирон" data-language-local-name="오세트어" class="interlanguage-link-target"><span>Ирон</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%B5%E0%A8%BE%E0%A8%B8%E0%A8%A4%E0%A8%B5%E0%A8%BF%E0%A8%95_%E0%A8%85%E0%A9%B0%E0%A8%95" title="ਵਾਸਤਵਿਕ ਅੰਕ – 펀잡어" lang="pa" hreflang="pa" data-title="ਵਾਸਤਵਿਕ ਅੰਕ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="펀잡어" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Liczby_rzeczywiste" title="Liczby rzeczywiste – 폴란드어" lang="pl" hreflang="pl" data-title="Liczby rzeczywiste" 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/N%C3%B9mer_real" title="Nùmer real – Piedmontese" lang="pms" hreflang="pms" data-title="Nùmer real" 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/N%C3%BAmero_real" title="Número real – 포르투갈어" lang="pt" hreflang="pt" data-title="Número real" 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/Num%C4%83r_real" title="Număr real – 루마니아어" lang="ro" hreflang="ro" data-title="Număr real" 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%92%D0%B5%D1%89%D0%B5%D1%81%D1%82%D0%B2%D0%B5%D0%BD%D0%BD%D0%BE%D0%B5_%D1%87%D0%B8%D1%81%D0%BB%D0%BE" title="Вещественное число – 러시아어" lang="ru" hreflang="ru" data-title="Вещественное число" data-language-autonym="Русский" data-language-local-name="러시아어" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sah mw-list-item"><a href="https://sah.wikipedia.org/wiki/%D0%94%D1%8C%D0%B8%D2%A5%D0%BD%D1%8D%D1%8D%D1%85_%D1%87%D1%8B%D1%8B%D2%BB%D1%8B%D0%BB%D0%B0%D0%BB%D0%B0%D1%80" title="Дьиҥнээх чыыһылалар – 야쿠트어" lang="sah" hreflang="sah" data-title="Дьиҥнээх чыыһылалар" data-language-autonym="Саха тыла" data-language-local-name="야쿠트어" class="interlanguage-link-target"><span>Саха тыла</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/N%C3%B9mmuru_riali" title="Nùmmuru riali – 시칠리아어" lang="scn" hreflang="scn" data-title="Nùmmuru riali" data-language-autonym="Sicilianu" data-language-local-name="시칠리아어" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Realan_broj" title="Realan broj – 세르비아-크로아티아어" lang="sh" hreflang="sh" data-title="Realan broj" 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-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B6%AD%E0%B7%8F%E0%B6%AD%E0%B7%8A%E0%B7%80%E0%B7%92%E0%B6%9A_%E0%B7%83%E0%B6%82%E0%B6%9B%E0%B7%8A%E2%80%8D%E0%B6%BA%E0%B7%8F" title="තාත්වික සංඛ්‍යා – 싱할라어" lang="si" hreflang="si" data-title="තාත්වික සංඛ්‍යා" data-language-autonym="සිංහල" data-language-local-name="싱할라어" class="interlanguage-link-target"><span>සිංහල</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Real_number" title="Real number – Simple English" lang="en-simple" hreflang="en-simple" data-title="Real number" 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/Re%C3%A1lne_%C4%8D%C3%ADslo" title="Reálne číslo – 슬로바키아어" lang="sk" hreflang="sk" data-title="Reálne 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/Realno_%C5%A1tevilo" title="Realno število – 슬로베니아어" lang="sl" hreflang="sl" data-title="Realno število" data-language-autonym="Slovenščina" data-language-local-name="슬로베니아어" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-smn mw-list-item"><a href="https://smn.wikipedia.org/wiki/Reaalloho" title="Reaalloho – 이나리 사미어" lang="smn" hreflang="smn" data-title="Reaalloho" data-language-autonym="Anarâškielâ" data-language-local-name="이나리 사미어" class="interlanguage-link-target"><span>Anarâškielâ</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Numrat_real%C3%AB" title="Numrat realë – 알바니아어" lang="sq" hreflang="sq" data-title="Numrat realë" 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%A0%D0%B5%D0%B0%D0%BB%D0%B0%D0%BD_%D0%B1%D1%80%D0%BE%D1%98" title="Реалан број – 세르비아어" lang="sr" hreflang="sr" data-title="Реалан број" 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/Reella_tal" title="Reella tal – 스웨덴어" lang="sv" hreflang="sv" data-title="Reella tal" data-language-autonym="Svenska" data-language-local-name="스웨덴어" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Namba_halisi" title="Namba halisi – 스와힐리어" lang="sw" hreflang="sw" data-title="Namba halisi" data-language-autonym="Kiswahili" data-language-local-name="스와힐리어" class="interlanguage-link-target"><span>Kiswahili</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%AE%E0%AF%86%E0%AE%AF%E0%AF%8D%E0%AE%AF%E0%AF%86%E0%AE%A3%E0%AF%8D" title="மெய்யெண் – 타밀어" lang="ta" hreflang="ta" data-title="மெய்யெண்" 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%88%E0%B8%B3%E0%B8%99%E0%B8%A7%E0%B8%99%E0%B8%88%E0%B8%A3%E0%B8%B4%E0%B8%87" title="จำนวนจริง – 태국어" lang="th" hreflang="th" data-title="จำนวนจริง" data-language-autonym="ไทย" data-language-local-name="태국어" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Tunay_na_bilang" title="Tunay na bilang – 타갈로그어" lang="tl" hreflang="tl" data-title="Tunay na bilang" 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/Reel_say%C4%B1lar" title="Reel sayılar – 터키어" lang="tr" hreflang="tr" data-title="Reel sayılar" 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%94%D1%96%D0%B9%D1%81%D0%BD%D0%B5_%D1%87%D0%B8%D1%81%D0%BB%D0%BE" title="Дійсне число – 우크라이나어" lang="uk" hreflang="uk" data-title="Дійсне число" data-language-autonym="Українська" data-language-local-name="우크라이나어" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D8%AD%D9%82%DB%8C%D9%82%DB%8C_%D8%B9%D8%AF%D8%AF" title="حقیقی عدد – 우르두어" lang="ur" hreflang="ur" data-title="حقیقی عدد" data-language-autonym="اردو" data-language-local-name="우르두어" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Haqiqiy_sonlar" title="Haqiqiy sonlar – 우즈베크어" lang="uz" hreflang="uz" data-title="Haqiqiy sonlar" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="우즈베크어" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/S%E1%BB%91_th%E1%BB%B1c" title="Số thực – 베트남어" lang="vi" hreflang="vi" data-title="Số thực" 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/%E5%AE%9E%E6%95%B0" title="实数 – 우어" lang="wuu" hreflang="wuu" data-title="实数" 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-xal mw-list-item"><a href="https://xal.wikipedia.org/wiki/%D0%91%D3%99%D3%99%D0%BB%D2%BB%D0%B0%D0%BD_%D1%82%D0%BE%D0%B9%D0%B3" title="Бәәлһан тойг – 칼미크어" lang="xal" hreflang="xal" data-title="Бәәлһан тойг" data-language-autonym="Хальмг" data-language-local-name="칼미크어" class="interlanguage-link-target"><span>Хальмг</span></a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%A8%D7%A2%D7%90%D7%9C%D7%A2_%D7%A6%D7%90%D7%9C" title="רעאלע צאל – 이디시어" lang="yi" hreflang="yi" data-title="רעאלע צאל" data-language-autonym="ייִדיש" data-language-local-name="이디시어" class="interlanguage-link-target"><span>ייִדיש</span></a></li><li class="interlanguage-link interwiki-yo badge-Q17437798 badge-goodarticle mw-list-item" title="좋은 글"><a href="https://yo.wikipedia.org/wiki/N%E1%BB%8D%CC%81mb%C3%A0_gidi" title="Nọ́mbà gidi – 요루바어" lang="yo" hreflang="yo" data-title="Nọ́mbà gidi" data-language-autonym="Yorùbá" data-language-local-name="요루바어" class="interlanguage-link-target"><span>Yorùbá</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%AE%9E%E6%95%B0" title="实数 – 중국어" lang="zh" hreflang="zh" data-title="实数" 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/%E5%AF%A6%E6%95%B8" title="實數 – Literary Chinese" lang="lzh" 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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> 다른 뜻에 대해서는 <a href="/wiki/%EC%8B%A4%EC%88%98_(%EB%8F%99%EC%9D%8C%EC%9D%B4%EC%9D%98)" class="mw-disambig" title="실수 (동음이의)">실수 (동음이의)</a> 문서를 참고하십시오.</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/%ED%8C%8C%EC%9D%BC:Number-line.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/09/Number-line.gif/220px-Number-line.gif" decoding="async" width="220" height="17" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/09/Number-line.gif/330px-Number-line.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/09/Number-line.gif/440px-Number-line.gif 2x" data-file-width="750" data-file-height="57" /></a><figcaption>실수를 수직선으로 나타낸 것</figcaption></figure> <p><a href="/wiki/%EC%88%98%ED%95%99" title="수학">수학</a>에서 <b>실수</b>(實數, <span style="font-size: smaller;"><a href="/wiki/%EC%98%81%EC%96%B4" title="영어">영어</a>: </span><span lang="en">real number</span>)는 주로 <a href="/wiki/%EC%8B%A4%EC%A7%81%EC%84%A0" title="실직선">실직선</a> 위의 점 또는 <a href="/wiki/%EC%8B%AD%EC%A7%84%EB%B2%95" title="십진법">십진법</a> 전개로 표현되는 수 체계이다. 예를 들어, -1, 0, <span class="sfrac nowrap" style="display:inline-block; vertical-align:-0.5em; font-size:80%; text-align:center;"><span style="display:block; line-height:1.15em; padding:0 0.1em;">1</span><span style="position:absolute;left:-10000px;top:auto;width:1px;height:1px;overflow:hidden;">/</span><span style="display:block; line-height:1.15em; padding:0 0.1em; border-top:1px solid;">2</span></span> <a href="/wiki/%EC%A0%9C%EA%B3%B1%EA%B7%BC_2" title="제곱근 2"><style data-mw-deduplicate="TemplateStyles:r25030363">'"`UNIQ--templatestyles-00000002-QINU`"'</style><span class="texhtml"><span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">2</span></span></span></a>, <i><a href="/wiki/%EC%9E%90%EC%97%B0%EB%A1%9C%EA%B7%B8%EC%9D%98_%EB%B0%91" title="자연로그의 밑">e</a></i>, <a href="/wiki/%EC%9B%90%EC%A3%BC%EC%9C%A8" title="원주율">π</a> 등은 모두 실수이다. 즉 좌표축을 꽉 채울 수 있는 수의 집합이라고도 할 수 있다. </p><p>실수에 대하여 <a href="/wiki/%EC%82%AC%EC%B9%99_%EC%97%B0%EC%82%B0" class="mw-redirect" title="사칙 연산">사칙 연산</a>(<a href="/wiki/%EB%8D%A7%EC%85%88" title="덧셈">덧셈</a> · <a href="/wiki/%EB%BA%84%EC%85%88" title="뺄셈">뺄셈</a> · <a href="/wiki/%EA%B3%B1%EC%85%88" title="곱셈">곱셈</a> · <a href="/wiki/%EB%82%98%EB%88%97%EC%85%88" title="나눗셈">나눗셈</a>)을 실행할 수 있다. 실수는 크기비교가 가능하며, 실직선에서 더 왼쪽에 있는 수가 더 오른쪽에 있는 수보다 작다. 특히, 실수는 0보다 큰 <a href="/wiki/%EC%96%91%EC%88%98_(%EC%88%98%ED%95%99)" title="양수 (수학)">양수</a> · 0보다 작은 <a href="/wiki/%EC%9D%8C%EC%88%98" title="음수">음수</a> · 0으로 분류된다. 또한, 실수는 <a href="/wiki/%EC%A0%95%EC%88%98" title="정수">정수</a>의 <a href="/wiki/%EB%B9%84_(%EC%88%98%ED%95%99)" title="비 (수학)">비</a>인 <a href="/wiki/%EC%9C%A0%EB%A6%AC%EC%88%98" title="유리수">유리수</a>와 그렇지 않은 <a href="/wiki/%EB%AC%B4%EB%A6%AC%EC%88%98" title="무리수">무리수</a>로도 분류되며, 정수 계수 <a href="/wiki/%EB%8B%A4%ED%95%AD%EC%8B%9D%EC%9D%98_%EA%B7%BC" class="mw-redirect" title="다항식의 근">다항식의 근</a>인 <a href="/wiki/%EB%8C%80%EC%88%98%EC%A0%81_%EC%88%98" title="대수적 수">대수적 수</a>와 그렇지 않은 <a href="/wiki/%EC%B4%88%EC%9B%94%EC%88%98" title="초월수">초월수</a>로도 분류된다. 실직선은 <a href="/wiki/%EB%B3%B5%EC%86%8C_%ED%8F%89%EB%A9%B4" class="mw-redirect" title="복소 평면">복소 평면</a>의 일부로 볼 수 있으며, 이 경우 실수는 <a href="/wiki/%ED%97%88%EC%88%98" title="허수">허수</a>와 함께 <a href="/wiki/%EB%B3%B5%EC%86%8C%EC%88%98" title="복소수">복소수</a>를 이룬다. </p><p>공리적으로, 실수는 <a href="/wiki/%EC%8B%A4%EC%88%98%EC%9D%98_%EC%99%84%EB%B9%84%EC%84%B1" title="실수의 완비성">완비</a> <a href="/wiki/%EC%88%9C%EC%84%9C%EC%B2%B4" title="순서체">순서체</a>로 정의되고, 이는 <a href="/wiki/%EB%8F%99%ED%98%95" class="mw-redirect" title="동형">동형</a> 의미 아래 유일하다. 구성적으로, 실수는 유리수 <a href="/wiki/%EC%BD%94%EC%8B%9C_%EC%88%98%EC%97%B4" class="mw-redirect" title="코시 수열">코시 수열</a>의 <a href="/wiki/%EB%8F%99%EC%B9%98%EB%A5%98" class="mw-redirect" title="동치류">동치류</a> · <a href="/wiki/%EB%8D%B0%EB%8D%B0%ED%82%A8%ED%8A%B8_%EC%A0%88%EB%8B%A8" title="데데킨트 절단">데데킨트 절단</a> · 십진법 전개의 동치류로서 구성된다. <a href="/wiki/%EC%8B%A4%EC%88%98%EC%9D%98_%EC%99%84%EB%B9%84%EC%84%B1" title="실수의 완비성">실수의 완비성</a>은 공집합이 아닌 실수 <a href="/wiki/%EC%9C%A0%EA%B3%84_%EC%A7%91%ED%95%A9" title="유계 집합">유계 집합</a>이 항상 <a href="/wiki/%EC%83%81%ED%95%9C%EA%B3%BC_%ED%95%98%ED%95%9C" title="상한과 하한">상한과 하한</a>을 갖는다는 성질이다. 이는 <a href="/wiki/%EC%9C%A0%EB%A6%AC%EC%88%98" title="유리수">유리수</a>와 구별되는 중요한 성질이다. </p><p>실수 집합은 <a href="/wiki/%EB%B9%84%EA%B0%80%EC%82%B0_%EC%A7%91%ED%95%A9" class="mw-redirect" title="비가산 집합">비가산 집합</a>이다. 즉, <a href="/wiki/%EC%9E%90%EC%97%B0%EC%88%98" title="자연수">자연수</a> 집합과 실수 집합은 둘다 <a href="/wiki/%EB%AC%B4%ED%95%9C_%EC%A7%91%ED%95%A9" title="무한 집합">무한 집합</a>이나, 그 사이에 <a href="/wiki/%EC%9D%BC%EB%8C%80%EC%9D%BC_%EB%8C%80%EC%9D%91" class="mw-redirect" title="일대일 대응">일대일 대응</a>이 존재하지 않는다. 실수 <a href="/wiki/%EC%A7%91%ED%95%A9%EC%9D%98_%ED%81%AC%EA%B8%B0" title="집합의 크기">집합의 크기</a>는 자연수 집합의 크기보다 크다. <a href="/wiki/%EC%97%B0%EC%86%8D%EC%B2%B4_%EA%B0%80%EC%84%A4" title="연속체 가설">연속체 가설</a>은 자연수 집합보다 크며 실수 집합보다 작은 크기를 갖는 실수 <a href="/wiki/%EB%B6%80%EB%B6%84_%EC%A7%91%ED%95%A9" class="mw-redirect" title="부분 집합">부분 집합</a>이 존재하지 않는다는 명제이다. 연속체 가설은 ZFC(즉, <a href="/wiki/%EC%84%A0%ED%83%9D_%EA%B3%B5%EB%A6%AC" title="선택 공리">선택 공리</a>를 추가한 <a href="/wiki/%EC%B2%B4%EB%A5%B4%EB%A9%9C%EB%A1%9C-%ED%94%84%EB%A0%9D%EC%BC%88_%EC%A7%91%ED%95%A9%EB%A1%A0" title="체르멜로-프렝켈 집합론">체르멜로-프렝켈 집합론</a>)에서 증명할 수도, 반증할 수도 없으며, 연속체 가설을 만족하거나, 그 부정을 만족하는 ZFC의 <a href="/wiki/%EB%AA%A8%ED%98%95_(%EB%85%BC%EB%A6%AC%ED%95%99)" class="mw-redirect" title="모형 (논리학)">모형</a>이 모두 존재한다. </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%8B%A4%EC%88%98&action=edit&section=1" title="부분 편집: 정의"><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/%EC%8B%A4%EC%88%98%EC%9D%98_%EA%B5%AC%EC%84%B1" title="실수의 구성">실수의 구성</a>입니다.</div> <p>실수 체계 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbb {R} ,+,\cdot ,<)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>,</mo> <mo>+</mo> <mo>,</mo> <mo>⋅</mo> <mo>,</mo> <mo><</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbb {R} ,+,\cdot ,<)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce32baaeb9a5a0c8f4e82959e676a3ed0c9691d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.852ex; height:2.843ex;" alt="{\displaystyle (\mathbb {R} ,+,\cdot ,<)}"></span>는 실수의 <a href="/wiki/%EA%B3%B5%EB%A6%AC%EA%B3%84" class="mw-redirect" title="공리계">공리계</a>를 통해 정의하거나, 구체적인 <a href="/wiki/%EB%AA%A8%ED%98%95_(%EB%85%BC%EB%A6%AC%ED%95%99)" class="mw-redirect" title="모형 (논리학)">모형</a>을 구성하여 정의할 수 있다. </p> <div class="mw-heading mw-heading3"><h3 id="공리적_정의"><span id=".EA.B3.B5.EB.A6.AC.EC.A0.81_.EC.A0.95.EC.9D.98"></span>공리적 정의</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%8B%A4%EC%88%98&action=edit&section=2" title="부분 편집: 공리적 정의"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>실수는 다음과 같은 공리를 만족하는 수 체계이다. </p> <ul><li><a href="/wiki/%EC%B2%B4_(%EC%88%98%ED%95%99)" title="체 (수학)">체</a>를 이룬다. 즉, 덧셈과 곱셈이라고 불리는 두 <a href="/wiki/%EC%9D%B4%ED%95%AD_%EC%97%B0%EC%82%B0" class="mw-redirect" title="이항 연산">이항 연산</a>을 갖추며, 이들은 익숙한 규칙대로 작용한다.</li> <li><a href="/wiki/%EC%88%9C%EC%84%9C%EC%B2%B4" title="순서체">순서체</a>를 이룬다. 즉, <a href="/wiki/%EC%A0%84%EC%88%9C%EC%84%9C" class="mw-redirect" title="전순서">전순서</a>를 갖추며, 이는 덧셈 및 곱셈과 호환된다.</li> <li><a href="/wiki/%EC%8B%A4%EC%88%98%EC%9D%98_%EC%99%84%EB%B9%84%EC%84%B1" title="실수의 완비성">완비적</a>이다. 즉, 공집합이 아닌 실수 부분 집합이 <a href="/wiki/%EC%83%81%EA%B3%84_(%EC%88%98%ED%95%99)" class="mw-redirect" title="상계 (수학)">상계</a>를 갖는다면, 항상 <a href="/wiki/%EC%83%81%ED%95%9C" class="mw-redirect" title="상한">상한</a>을 갖는다.</li></ul> <p>마지막 완비성은 실수를 유리수와 구분짓는 성질이다. 이들 공리를 만족하는 수 체계는 <a href="/wiki/%EB%8F%99%ED%98%95" class="mw-redirect" title="동형">동형</a> 의미 하에 유일하다. </p> <div class="mw-heading mw-heading3"><h3 id="구성적_정의"><span id=".EA.B5.AC.EC.84.B1.EC.A0.81_.EC.A0.95.EC.9D.98"></span>구성적 정의</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%8B%A4%EC%88%98&action=edit&section=3" title="부분 편집: 구성적 정의"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>실수는 다음과 같은 대상으로서 구성할 수 있다. 이렇게 구성한 실수는 실수 공리계의 <a href="/wiki/%EB%AA%A8%ED%98%95_(%EB%85%BC%EB%A6%AC%ED%95%99)" class="mw-redirect" title="모형 (논리학)">모형</a>을 이룬다. 즉, 실수 공리계의 모든 공리들을 만족한다. </p> <ul><li><a href="/wiki/%EC%9C%A0%EB%A6%AC%EC%88%98" title="유리수">유리수</a> <a href="/wiki/%EC%BD%94%EC%8B%9C_%EC%88%98%EC%97%B4" class="mw-redirect" title="코시 수열">코시 수열</a>의 "거리가 0으로 수렴"하는 <a href="/wiki/%EB%8F%99%EC%B9%98_%EA%B4%80%EA%B3%84" title="동치 관계">동치 관계</a>에 대한 <a href="/wiki/%EB%8F%99%EC%B9%98%EB%A5%98" class="mw-redirect" title="동치류">동치류</a>. 즉, 유리수의 표준 <a href="/wiki/%EA%B1%B0%EB%A6%AC_%EA%B3%B5%EA%B0%84" title="거리 공간">거리 공간</a>에 대한 <a href="/wiki/%EC%99%84%EB%B9%84%ED%99%94" class="mw-redirect" title="완비화">완비화</a>이다.</li> <li>유리수에 대한 <a href="/wiki/%EB%8D%B0%EB%8D%B0%ED%82%A8%ED%8A%B8_%EC%A0%88%EB%8B%A8" title="데데킨트 절단">데데킨트 절단</a>.</li> <li>십진법 전개의 <a href="/wiki/%EB%8F%99%EC%B9%98%EB%A5%98" class="mw-redirect" title="동치류">동치류</a>. 예를 들어, 1과 <a href="/wiki/0.999%E2%80%A6" title="0.999…">0.999…</a>는 서로 동치이다.</li></ul> <div class="mw-heading mw-heading2"><h2 id="연산"><span id=".EC.97.B0.EC.82.B0"></span>연산</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%8B%A4%EC%88%98&action=edit&section=4" title="부분 편집: 연산"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="사칙_연산"><span id=".EC.82.AC.EC.B9.99_.EC.97.B0.EC.82.B0"></span>사칙 연산</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%8B%A4%EC%88%98&action=edit&section=5" title="부분 편집: 사칙 연산"><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%82%AC%EC%B9%99_%EC%97%B0%EC%82%B0" class="mw-redirect" title="사칙 연산">사칙 연산</a>입니다.</div> <p>실수 집합 위에는 <a href="/wiki/%EB%8D%A7%EC%85%88" title="덧셈">덧셈</a> +, <a href="/wiki/%EB%BA%84%EC%85%88" title="뺄셈">뺄셈</a> -, <a href="/wiki/%EA%B3%B1%EC%85%88" title="곱셈">곱셈</a> ×, <a href="/wiki/%EB%82%98%EB%88%97%EC%85%88" title="나눗셈">나눗셈</a> ÷이 정의되어 있으며, 이들 중 덧셈과 곱셈은 <a href="/wiki/%EA%B5%90%ED%99%98_%EB%B2%95%EC%B9%99" class="mw-redirect" title="교환 법칙">교환 법칙</a>, <a href="/wiki/%EA%B2%B0%ED%95%A9_%EB%B2%95%EC%B9%99" class="mw-redirect" title="결합 법칙">결합 법칙</a>, <a href="/wiki/%EB%B6%84%EB%B0%B0_%EB%B2%95%EC%B9%99" class="mw-redirect" title="분배 법칙">분배 법칙</a>을 만족한다. 즉, 임의의 실수들에 대하여, 다음 성질들이 성립한다. </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 a+b=b+a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo>=</mo> <mi>b</mi> <mo>+</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+b=b+a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/684f43b5094501674e8314be5e24a80ee64682e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.234ex; height:2.343ex;" alt="{\displaystyle a+b=b+a}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a+b)+c=a+(b+c)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>c</mi> <mo>=</mo> <mi>a</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>+</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a+b)+c=a+(b+c)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46b7b8d31d5845966e6abdbb030c73f343c17d4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.547ex; height:2.843ex;" alt="{\displaystyle (a+b)+c=a+(b+c)}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\times b=b\times a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>×</mo> <mi>b</mi> <mo>=</mo> <mi>b</mi> <mo>×</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\times b=b\times a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28d66fd09c6072e183e3395a174fd66dac99e514" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.234ex; height:2.176ex;" alt="{\displaystyle a\times b=b\times a}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a\times b)\times c=a\times (b\times c)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>×</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>×</mo> <mi>c</mi> <mo>=</mo> <mi>a</mi> <mo>×</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>×</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a\times b)\times c=a\times (b\times c)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b74ad4422b4ae5d85f956edfe3696c5a07311c8c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.547ex; height:2.843ex;" alt="{\displaystyle (a\times b)\times c=a\times (b\times c)}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\times (b+c)=a\times b+a\times c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>×</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>+</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> <mo>×</mo> <mi>b</mi> <mo>+</mo> <mi>a</mi> <mo>×</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\times (b+c)=a\times b+a\times c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e066c17d4beaed810d9b04e2fab1645c02a54af5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.808ex; height:2.843ex;" alt="{\displaystyle a\times (b+c)=a\times b+a\times c}"></span></li></ul> <p>실수 0과 1은 사칙 연산에서 특별한 역할을 맡는다. 즉, 임의의 실수들에 대하여, 다음 성질들이 성립한다. </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 0+a=a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>+</mo> <mi>a</mi> <mo>=</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0+a=a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba565116cf2f2d984f7b8365b054b70eb8f89308" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.561ex; height:2.343ex;" alt="{\displaystyle 0+a=a}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\times a=a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>×</mo> <mi>a</mi> <mo>=</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\times a=a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae8b118db98f67964e772e36bed9f14e3db3437d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.561ex; height:2.176ex;" alt="{\displaystyle 1\times a=a}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\times a=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>×</mo> <mi>a</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\times a=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05eeef4d767aa40a68dcac221dab7564bf7f2e29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.494ex; height:2.176ex;" alt="{\displaystyle 0\times a=0}"></span></li></ul> <p>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>과 그 <a href="/wiki/%EB%B0%98%EC%88%98_(%EC%88%98%ED%95%99)" class="mw-redirect" title="반수 (수학)">반수</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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>를 더하면 0이다. 즉, </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+(-x)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x+(-x)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbddcc47f9ee70348043a6ba7ae80e1c9b5eb6d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.378ex; height:2.843ex;" alt="{\displaystyle x+(-x)=0}"></span></li></ul> <p>0이 아닌 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} \setminus \{0\}}"> <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> <mo class="MJX-variant">∖</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in \mathbb {R} \setminus \{0\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/907ed5d1b1444addc10175220cbbe6d11b09c4e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.53ex; height:2.843ex;" alt="{\displaystyle x\in \mathbb {R} \setminus \{0\}}"></span>과 그 <a href="/wiki/%EC%97%AD%EC%88%98" class="mw-redirect" title="역수">역수</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 {\frac {1}{x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68f89eaf83a3811c69adb4bf1119bafd661a4c08" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:2.166ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{x}}}"></span>를 곱하면 1이다. 즉, </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\times {\frac {1}{x}}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\times {\frac {1}{x}}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa61e95061ed97b5dd736b2299451a0fed0331a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.597ex; height:5.176ex;" alt="{\displaystyle x\times {\frac {1}{x}}=1}"></span></li></ul> <p>뺄셈과 나눗셈은 다음과 같이 덧셈과 곱셈으로 귀결된다. </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 a-b=a+(-b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>−</mo> <mi>b</mi> <mo>=</mo> <mi>a</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a-b=a+(-b)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f03e1ad2ad4d5dbbd763ad4c40a0d7bf250cd208" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.851ex; height:2.843ex;" alt="{\displaystyle a-b=a+(-b)}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {a}{b}}=a\times {\frac {1}{b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mrow> <mo>=</mo> <mi>a</mi> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>b</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {a}{b}}=a\times {\frac {1}{b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08b2afd7bdc7a8015a5316191e90ed6e439fbf84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:11.233ex; height:5.343ex;" alt="{\displaystyle {\frac {a}{b}}=a\times {\frac {1}{b}}}"></span></li></ul> <div class="mw-heading mw-heading3"><h3 id="거듭제곱과_거듭제곱근"><span id=".EA.B1.B0.EB.93.AD.EC.A0.9C.EA.B3.B1.EA.B3.BC_.EA.B1.B0.EB.93.AD.EC.A0.9C.EA.B3.B1.EA.B7.BC"></span>거듭제곱과 거듭제곱근</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%8B%A4%EC%88%98&action=edit&section=6" title="부분 편집: 거듭제곱과 거듭제곱근"><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/%EA%B1%B0%EB%93%AD%EC%A0%9C%EA%B3%B1" title="거듭제곱">거듭제곱</a> 및 <a href="/wiki/%EA%B1%B0%EB%93%AD%EC%A0%9C%EA%B3%B1%EA%B7%BC" title="거듭제곱근">거듭제곱근</a>입니다.</div> <p>양수(=실직선에서 0의 우측의 실수=0보다 큰 수) 밑, 실수 지수의 거듭제곱을 정의할 수 있다. 실수에 대하여 거듭제곱을 정의할 수 있는 건 실수의 완비성이 있기 때문이다. 대략의 정의는 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 a^{n}=\overbrace {aa\cdots a} ^{n}\qquad (a>0,\;n\in \mathbb {Z} ^{+})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mover> <mrow> <mi>a</mi> <mi>a</mi> <mo>⋯</mo> <mi>a</mi> </mrow> <mo>⏞</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mover> <mspace width="2em" /> <mo stretchy="false">(</mo> <mi>a</mi> <mo>></mo> <mn>0</mn> <mo>,</mo> <mspace width="thickmathspace" /> <mi>n</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{n}=\overbrace {aa\cdots a} ^{n}\qquad (a>0,\;n\in \mathbb {Z} ^{+})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05c07c3c6a7968cab1b40b85f1003e174db44eab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.654ex; height:5.009ex;" alt="{\displaystyle a^{n}=\overbrace {aa\cdots a} ^{n}\qquad (a>0,\;n\in \mathbb {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 a^{0}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{0}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/448ca9a3f4ef03c4dfcf69258912d2c90b097842" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.545ex; height:2.676ex;" alt="{\displaystyle a^{0}=1}"></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 a^{-n}={\frac {1}{a^{n}}}={\frac {1}{\underbrace {aa\cdots a} _{n}}}\qquad (a>0,\;n\in \mathbb {Z} ^{+})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <mi>a</mi> <mi>a</mi> <mo>⋯</mo> <mi>a</mi> </mrow> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munder> </mfrac> </mrow> <mspace width="2em" /> <mo stretchy="false">(</mo> <mi>a</mi> <mo>></mo> <mn>0</mn> <mo>,</mo> <mspace width="thickmathspace" /> <mi>n</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{-n}={\frac {1}{a^{n}}}={\frac {1}{\underbrace {aa\cdots a} _{n}}}\qquad (a>0,\;n\in \mathbb {Z} ^{+})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/088fd2a6df7fadb779e78c3d948530d47501be9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.171ex; width:42.152ex; height:8.509ex;" alt="{\displaystyle a^{-n}={\frac {1}{a^{n}}}={\frac {1}{\underbrace {aa\cdots a} _{n}}}\qquad (a>0,\;n\in \mathbb {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 a^{\frac {m}{n}}=\sup\{x\in \mathbb {R} \colon x^{n}<a^{m}\}\qquad (a>0,\;m,n\in \mathbb {Z} ,\;n>0,\;\gcd\{m,n\}=1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>m</mi> <mi>n</mi> </mfrac> </mrow> </msup> <mo>=</mo> <mo movablelimits="true" form="prefix">sup</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>:</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo><</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo fence="false" stretchy="false">}</mo> <mspace width="2em" /> <mo stretchy="false">(</mo> <mi>a</mi> <mo>></mo> <mn>0</mn> <mo>,</mo> <mspace width="thickmathspace" /> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>,</mo> <mspace width="thickmathspace" /> <mi>n</mi> <mo>></mo> <mn>0</mn> <mo>,</mo> <mspace width="thickmathspace" /> <mo movablelimits="true" form="prefix">gcd</mo> <mo fence="false" stretchy="false">{</mo> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{\frac {m}{n}}=\sup\{x\in \mathbb {R} \colon x^{n}<a^{m}\}\qquad (a>0,\;m,n\in \mathbb {Z} ,\;n>0,\;\gcd\{m,n\}=1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b2b37639ce2c502611db03d4cc9e0fe0cad1fb3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:73.868ex; height:3.509ex;" alt="{\displaystyle a^{\frac {m}{n}}=\sup\{x\in \mathbb {R} \colon x^{n}<a^{m}\}\qquad (a>0,\;m,n\in \mathbb {Z} ,\;n>0,\;\gcd\{m,n\}=1)}"></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 a^{r}=\sup\{a^{q}\colon q\in \mathbb {Q} ,\;q<r\}\qquad (a>0,\;r\in \mathbb {R} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msup> <mo>=</mo> <mo movablelimits="true" form="prefix">sup</mo> <mo fence="false" stretchy="false">{</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msup> <mo>:</mo> <mi>q</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo>,</mo> <mspace width="thickmathspace" /> <mi>q</mi> <mo><</mo> <mi>r</mi> <mo fence="false" stretchy="false">}</mo> <mspace width="2em" /> <mo stretchy="false">(</mo> <mi>a</mi> <mo>></mo> <mn>0</mn> <mo>,</mo> <mspace width="thickmathspace" /> <mi>r</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{r}=\sup\{a^{q}\colon q\in \mathbb {Q} ,\;q<r\}\qquad (a>0,\;r\in \mathbb {R} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b288512370e2b864ea060b055fe8a747ccc1bfa6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:46.186ex; height:2.843ex;" alt="{\displaystyle a^{r}=\sup\{a^{q}\colon q\in \mathbb {Q} ,\;q<r\}\qquad (a>0,\;r\in \mathbb {R} )}"></span></dd></dl> <p>음수(=실직선에서 0의 좌측의 실수=0보다 작은 수) 밑의 거듭제곱 역시 정의할 수 있는데, 이는 유리수 지수에 한하며, 또한 이렇게 확장된 거듭제곱은 위의 연산 법칙을 비롯한 좋은 성질들을 만족시키지 못한다. </p> <div class="mw-heading mw-heading2"><h2 id="순서"><span id=".EC.88.9C.EC.84.9C"></span>순서</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%8B%A4%EC%88%98&action=edit&section=7" title="부분 편집: 순서"><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> <a href="/wiki/%EB%B6%80%EB%93%B1%EC%8B%9D" title="부등식">부등식</a> 문서를 참고하십시오.</div> <p>실수들 사이에는 순서(즉, 크기 비교)가 존재한다. 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 a,b\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b\in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe0bf0e2324bac843fab916c46d1b3349017a616" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.78ex; height:2.509ex;" alt="{\displaystyle a,b\in \mathbb {R} }"></span>의 순서 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a<b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo><</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a<b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91a7698e4c7401bb321f97888b872b583a9e4642" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.326ex; height:2.176ex;" alt="{\displaystyle a<b}"></span>의 직관은 <a href="/wiki/%EC%8B%A4%EC%A7%81%EC%84%A0" title="실직선">실직선</a> 위에서 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span>가 더 왼쪽에, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span>가 오른쪽에 있다는 것이다. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\leq b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>≤</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\leq b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41558abc50886fdf38817495b243958d7b3dd39b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.326ex; height:2.343ex;" alt="{\displaystyle a\leq b}"></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 a<b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo><</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a<b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91a7698e4c7401bb321f97888b872b583a9e4642" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.326ex; height:2.176ex;" alt="{\displaystyle a<b}"></span>이거나 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1956b03d1314c7071ac1f45ed7b1e29422dcfcc4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.326ex; height:2.176ex;" alt="{\displaystyle a=b}"></span>라는 뜻이다. 이에 따라, 실수의 순서는 다음 성질들을 만족시킨다. </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 a\nless a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>≮</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\nless a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29ee8eb3dc7c13a0a3b878972f95b46a257ee752" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.558ex; height:2.509ex;" alt="{\displaystyle a\nless a}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a<b\implies b\nless a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo><</mo> <mi>b</mi> <mspace width="thickmathspace" /> <mo stretchy="false">⟹</mo> <mspace width="thickmathspace" /> <mi>b</mi> <mo>≮</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a<b\implies b\nless a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0fe2cc84d71057e3be34187d1e564a9db44831c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.038ex; height:2.509ex;" alt="{\displaystyle a<b\implies b\nless a}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a<b<c\implies a<c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo><</mo> <mi>b</mi> <mo><</mo> <mi>c</mi> <mspace width="thickmathspace" /> <mo stretchy="false">⟹</mo> <mspace width="thickmathspace" /> <mi>a</mi> <mo><</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a<b<c\implies a<c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd22bef42cb2a5a2b1c51ea3d85ddf13e4b231a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:21.152ex; height:2.176ex;" alt="{\displaystyle a<b<c\implies a<c}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a<b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo><</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a<b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91a7698e4c7401bb321f97888b872b583a9e4642" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.326ex; height:2.176ex;" alt="{\displaystyle a<b}"></span>이거나, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1956b03d1314c7071ac1f45ed7b1e29422dcfcc4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.326ex; height:2.176ex;" alt="{\displaystyle a=b}"></span>이거나, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a>b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>></mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a>b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83fc0063781fb9bf4ec7608b2fd11ed6d5b05a13" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.326ex; height:2.176ex;" alt="{\displaystyle a>b}"></span>.</li></ul> <p>또한, 실수의 순서는 실수의 연산과 호환된다. 즉, 임의의 실수들에 대하여, 다음 성질들이 성립한다. </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 a<b\implies a+c<b+c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo><</mo> <mi>b</mi> <mspace width="thickmathspace" /> <mo stretchy="false">⟹</mo> <mspace width="thickmathspace" /> <mi>a</mi> <mo>+</mo> <mi>c</mi> <mo><</mo> <mi>b</mi> <mo>+</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a<b\implies a+c<b+c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e05dfaad5505ba886a53d7381acef1b5deda459e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:24.732ex; height:2.343ex;" alt="{\displaystyle a<b\implies a+c<b+c}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a<b,\;c>0\implies ac<bc}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo><</mo> <mi>b</mi> <mo>,</mo> <mspace width="thickmathspace" /> <mi>c</mi> <mo>></mo> <mn>0</mn> <mspace width="thickmathspace" /> <mo stretchy="false">⟹</mo> <mspace width="thickmathspace" /> <mi>a</mi> <mi>c</mi> <mo><</mo> <mi>b</mi> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a<b,\;c>0\implies ac<bc}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b4130302298697111778844cb2401ddfd742d38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:25.998ex; height:2.509ex;" alt="{\displaystyle a<b,\;c>0\implies ac<bc}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a<b,\;c<0\implies ac>bc}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo><</mo> <mi>b</mi> <mo>,</mo> <mspace width="thickmathspace" /> <mi>c</mi> <mo><</mo> <mn>0</mn> <mspace width="thickmathspace" /> <mo stretchy="false">⟹</mo> <mspace width="thickmathspace" /> <mi>a</mi> <mi>c</mi> <mo>></mo> <mi>b</mi> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a<b,\;c<0\implies ac>bc}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ad00f56ccfb010c283daeba872fd1b08183cb4c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:25.998ex; height:2.509ex;" alt="{\displaystyle a<b,\;c<0\implies ac>bc}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<a<b,\;n>0\implies a^{n}<b^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo><</mo> <mi>a</mi> <mo><</mo> <mi>b</mi> <mo>,</mo> <mspace width="thickmathspace" /> <mi>n</mi> <mo>></mo> <mn>0</mn> <mspace width="thickmathspace" /> <mo stretchy="false">⟹</mo> <mspace width="thickmathspace" /> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo><</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0<a<b,\;n>0\implies a^{n}<b^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59295d24207ecd527540fb983e9e7d4c030ff66c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:31.07ex; height:2.676ex;" alt="{\displaystyle 0<a<b,\;n>0\implies a^{n}<b^{n}}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<a<b,\;n<0\implies a^{n}>b^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo><</mo> <mi>a</mi> <mo><</mo> <mi>b</mi> <mo>,</mo> <mspace width="thickmathspace" /> <mi>n</mi> <mo><</mo> <mn>0</mn> <mspace width="thickmathspace" /> <mo stretchy="false">⟹</mo> <mspace width="thickmathspace" /> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>></mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0<a<b,\;n<0\implies a^{n}>b^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45806ece5fbbb46e738368e81535693f9ff3c3c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:31.07ex; height:2.676ex;" alt="{\displaystyle 0<a<b,\;n<0\implies a^{n}>b^{n}}"></span></li></ul> <p><a href="/wiki/%EC%96%91%EC%88%98_(%EC%88%98%ED%95%99)" title="양수 (수학)">양수</a>(<span style="font-size: smaller;"><a href="/wiki/%EC%98%81%EC%96%B4" title="영어">영어</a>: </span><span lang="en">positive number</span>)는 0보다 큰 실수를 뜻하며, <b><a href="/wiki/%EC%9D%8C%EC%88%98" title="음수">음수</a></b>(<span style="font-size: smaller;"><a href="/wiki/%EC%98%81%EC%96%B4" title="영어">영어</a>: </span><span lang="en">negative number</span>)는 0보다 작은 실수를 뜻한다. 위의 성질들에 따라, 모든 실수는 양수, 음수와 0 가운데 하나에 속한다. 또한, 양수 곱하기 양수는 항상 양수이며, 양수 곱하기 음수는 항상 음수이며, 음수 곱하기 음수는 항상 양수이다. 특히, 임의의 실수의 제곱은 항상 음수가 아닌 실수이다.(<a href="/wiki/%EC%A0%9C%EA%B3%B1" title="제곱">제곱</a>해서 <a href="/wiki/%EC%9D%8C%EC%88%98" title="음수">음수</a>가 되는 수는 <a href="/wiki/%ED%97%88%EC%88%98" title="허수">허수</a>라고 불리고, <a href="/wiki/%EC%88%98%EC%A7%81%EC%84%A0_(%EC%88%98%ED%95%99)" title="수직선 (수학)">수직선</a> 상에 표시할 수 없다.) </p> <div class="mw-heading mw-heading3"><h3 id="구간"><span id=".EA.B5.AC.EA.B0.84"></span>구간</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%8B%A4%EC%88%98&action=edit&section=8" title="부분 편집: 구간"><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/%EA%B5%AC%EA%B0%84" title="구간">구간</a>입니다.</div> <p><a href="/wiki/%EA%B5%AC%EA%B0%84" title="구간">구간</a>은 특별한 실수 <a href="/wiki/%EB%B6%80%EB%B6%84_%EC%A7%91%ED%95%A9" class="mw-redirect" title="부분 집합">부분 집합</a>으로서, 주어진 두 실수 사이의 실수를 원소로 갖거나, 주어진 한 실수를 시작점으로 하는 반직선에 놓인 실수를 원소로 갖는다. 예를 들어, 임의의 <span class="mwe-math-element"><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, 다음과 같다. </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\in (3,5)\iff 3<x<5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mo>,</mo> <mn>5</mn> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> <mn>3</mn> <mo><</mo> <mi>x</mi> <mo><</mo> <mn>5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in (3,5)\iff 3<x<5}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df6c4b4c5119d91de7de1bad90909cada468211d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.087ex; height:2.843ex;" alt="{\displaystyle x\in (3,5)\iff 3<x<5}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in [-2,10]\iff -2\leq x\leq 10}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mo>−</mo> <mn>2</mn> <mo>,</mo> <mn>10</mn> <mo stretchy="false">]</mo> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> <mo>−</mo> <mn>2</mn> <mo>≤</mo> <mi>x</mi> <mo>≤</mo> <mn>10</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in [-2,10]\iff -2\leq x\leq 10}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4b1fcb67b840cabdef1a70080397547fa001d80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.513ex; height:2.843ex;" alt="{\displaystyle x\in [-2,10]\iff -2\leq x\leq 10}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in (6,+\infty )\iff 6<x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>6</mn> <mo>,</mo> <mo>+</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> <mn>6</mn> <mo><</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in (6,+\infty )\iff 6<x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2f959711694022e5287185afdee16c610d7700a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.796ex; height:2.843ex;" alt="{\displaystyle x\in (6,+\infty )\iff 6<x}"></span></li></ul> <p>퇴화 구간은 구간과 비슷한 집합으로서, 두 끝점의 순서가 정상적인 구간의 반대이다. 예를 들어, 다음과 같다. </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\in [3,3]\iff 3\leq x\leq 3\iff x=3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>3</mn> <mo>,</mo> <mn>3</mn> <mo stretchy="false">]</mo> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> <mn>3</mn> <mo>≤</mo> <mi>x</mi> <mo>≤</mo> <mn>3</mn> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> <mi>x</mi> <mo>=</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in [3,3]\iff 3\leq x\leq 3\iff x=3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/601ee28e7b59e93411dd28cb6730c5d17a6ea489" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.059ex; height:2.843ex;" alt="{\displaystyle x\in [3,3]\iff 3\leq x\leq 3\iff x=3}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in (9,5)\iff 9<x<5\iff x\in \varnothing }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>9</mn> <mo>,</mo> <mn>5</mn> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> <mn>9</mn> <mo><</mo> <mi>x</mi> <mo><</mo> <mn>5</mn> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> <mi>x</mi> <mo>∈</mo> <mi class="MJX-variant">∅</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in (9,5)\iff 9<x<5\iff x\in \varnothing }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/362eb26f568f2acb07cc6e5c719c9321ef3b817b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.963ex; height:2.843ex;" alt="{\displaystyle x\in (9,5)\iff 9<x<5\iff x\in \varnothing }"></span></li></ul> <div class="mw-heading mw-heading3"><h3 id="상한_공리"><span id=".EC.83.81.ED.95.9C_.EA.B3.B5.EB.A6.AC"></span>상한 공리</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%8B%A4%EC%88%98&action=edit&section=9" title="부분 편집: 상한 공리"><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%83%81%ED%95%9C_%EA%B3%B5%EB%A6%AC" class="mw-redirect" title="상한 공리">상한 공리</a>입니다.</div> <p>수들의 집합(예를 들어, <a href="/wiki/%EC%9C%A0%EB%A6%AC%EC%88%98" title="유리수">유리수</a> 집합이나 실수 집합)의 모든 수들보다 작지 않은 수를 그 집합의 <a href="/wiki/%EC%83%81%EA%B3%84%EC%99%80_%ED%95%98%EA%B3%84" class="mw-redirect" title="상계와 하계">상계</a>라고 한다. 이는 보통 존재하지 않거나, 존재한다면 여럿이 같이 존재한다. 수들의 집합에 상계들이 존재하며, 이들 가운데 가장 작은 하나가 존재한다면, 이를 <a href="/wiki/%EC%83%81%ED%95%9C%EA%B3%BC_%ED%95%98%ED%95%9C" title="상한과 하한">상한</a>이라고 한다. 실수 집합 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span>은 다음 성질을 만족시킨다. </p> <ul><li>공집합이 아닌 실수 부분 집합 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varnothing \neq S\subseteq \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi class="MJX-variant">∅</mi> <mo>≠</mo> <mi>S</mi> <mo>⊆</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varnothing \neq S\subseteq \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7adbcf7090ddf999552829bd343288cf6a6db673" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.182ex; height:2.676ex;" alt="{\displaystyle \varnothing \neq S\subseteq \mathbb {R} }"></span>에 상계가 존재한다면, 상한 역시 존재한다.</li></ul> <p>이를 상한 공리이라고 한다. 상한 공리는 실수의 완비성에 대한 한 가지 표현이다. </p> <div class="mw-heading mw-heading3"><h3 id="데데킨트_완비성"><span id=".EB.8D.B0.EB.8D.B0.ED.82.A8.ED.8A.B8_.EC.99.84.EB.B9.84.EC.84.B1"></span>데데킨트 완비성</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%8B%A4%EC%88%98&action=edit&section=10" title="부분 편집: 데데킨트 완비성"><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> <a href="/wiki/%EB%8D%B0%EB%8D%B0%ED%82%A8%ED%8A%B8_%EC%A0%88%EB%8B%A8" title="데데킨트 절단">데데킨트 절단</a> 및 <a href="/wiki/%EC%8B%A4%EC%88%98%EC%9D%98_%EC%99%84%EB%B9%84%EC%84%B1" title="실수의 완비성">실수의 완비성</a> 문서를 참고하십시오.</div> <p>실수의 완비성은 실수의 가장 중요한 성질의 하나이다. <b>데데킨트 절단</b>(<span style="font-size: smaller;"><a href="/wiki/%EC%98%81%EC%96%B4" title="영어">영어</a>: </span><span lang="en">Dedekind cut</span>)을 통해 서술하는 것이 가장 간단하다. 실수 집합 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span>의 두 부분 집합 <span class="mwe-math-element"><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,E\subseteq \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mo>,</mo> <mi>E</mi> <mo>⊆</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D,E\subseteq \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e2013bc4ff305ae8c4d7f554c1c484ae9b4c742" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.51ex; height:2.509ex;" alt="{\displaystyle D,E\subseteq \mathbb {R} }"></span>의 쌍 <span class="mwe-math-element"><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,E)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>D</mi> <mo>,</mo> <mi>E</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (D,E)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e02f6242e4a29d1664fa7d52813ee3c70523ec8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.543ex; height:2.843ex;" alt="{\displaystyle (D,E)}"></span>이 다음 조건들을 만족시키면, <span class="mwe-math-element"><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,E)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>D</mi> <mo>,</mo> <mi>E</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (D,E)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e02f6242e4a29d1664fa7d52813ee3c70523ec8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.543ex; height:2.843ex;" alt="{\displaystyle (D,E)}"></span>를 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span>의 <b>데데킨트 절단</b>이라고 한다. </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 D,E\neq \varnothing }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mo>,</mo> <mi>E</mi> <mo>≠</mo> <mi class="MJX-variant">∅</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D,E\neq \varnothing }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4afcc809d6bbe49a62bcc981391a85ff646be279" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.64ex; height:2.676ex;" alt="{\displaystyle D,E\neq \varnothing }"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D\cup E=\mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mo>∪</mo> <mi>E</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D\cup E=\mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bece6a3f67560a01484b0f323ea23e59b9a3b58" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.059ex; height:2.176ex;" alt="{\displaystyle D\cup E=\mathbb {R} }"></span></li> <li>임의의 <span class="mwe-math-element"><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\in D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo>∈</mo> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\in D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9091400ce51b9fcc3657b445e5728e1bb80eda4d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.981ex; height:2.176ex;" alt="{\displaystyle d\in D}"></span> 및 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e\in E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mo>∈</mo> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e\in E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34778736d9c6d607a4da3d25594b38dd3e8c82ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.7ex; height:2.176ex;" alt="{\displaystyle e\in E}"></span>에 대하여, <span class="mwe-math-element"><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<e}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo><</mo> <mi>e</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d<e}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d541fcf2066e14a15ea1154b03e1b982ebf720aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.398ex; height:2.176ex;" alt="{\displaystyle d<e}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}"></span>는 <a href="/wiki/%EC%B5%9C%EC%86%8C_%EC%9B%90%EC%86%8C" class="mw-redirect" title="최소 원소">최소 원소</a>를 가지지 않는다.</li></ul> <p>이제, 실수의 <b>데데킨트 완비성 공리</b>를 다음과 같이 서술할 수 있다. </p> <ul><li>실수 집합 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span>의 데데킨트 절단 <span class="mwe-math-element"><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,E)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>D</mi> <mo>,</mo> <mi>E</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (D,E)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e02f6242e4a29d1664fa7d52813ee3c70523ec8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.543ex; height:2.843ex;" alt="{\displaystyle (D,E)}"></span>에 대하여, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"></span>는 항상 최소 원소를 가진다.</li></ul> <p>데데킨트 완비성 공리는 상한 공리와 서로 동치이다. </p> <style data-mw-deduplicate="TemplateStyles:r26858958">.mw-parser-output div.proof{border:1px solid #aaaaaa;background-color:#f9f9f9;padding:5px;font-size:95%;min-width:50%}.mw-parser-output div.proof,.mw-parser-output div.prooftitle,.mw-parser-output div.proofcontent{overflow:auto}.mw-parser-output div.prooftitle span.prooftitletext{font-weight:bold}.mw-parser-output div.proofcontent{margin-top:-0.5em;min-height:0.5em}</style><div class="proof mw-collapsible mw-collapsed"> <div class="prooftitle"> <p><span class="prooftitletext">증명 (상한 공리 ⇒ 데데킨트 완비성 공리):</span> </p> </div> <div class="proofcontent mw-collapsible-content"> </div></div><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r26858958"><div class="proof mw-collapsible mw-collapsed"> <div class="prooftitle"> <p><span class="prooftitletext">증명 (데데킨트 완비성 공리 ⇒ 상한 공리):</span> </p> </div> <div class="proofcontent mw-collapsible-content"> </div></div> <div class="mw-heading mw-heading3"><h3 id="기타_성질"><span id=".EA.B8.B0.ED.83.80_.EC.84.B1.EC.A7.88"></span>기타 성질</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%8B%A4%EC%88%98&action=edit&section=11" title="부분 편집: 기타 성질"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>실수 집합은 <a href="/wiki/%EC%95%84%EB%A5%B4%ED%82%A4%EB%A9%94%EB%8D%B0%EC%8A%A4_%EC%84%B1%EC%A7%88" title="아르키메데스 성질">아르키메데스 성질</a>을 만족한다. 즉, 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>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6338d1f32a43c3d4a8b55b35cdf5a615063036ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.78ex; height:2.509ex;" alt="{\displaystyle x,y>0}"></span>가 있다고 하자. 이 경우 <span class="mwe-math-element"><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>가 아무리 작고 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span>가 아무리 크더라도, <span class="mwe-math-element"><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>를 충분히 많은 횟수 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>만큼 더하면, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span>를 초과한다. 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 \underbrace {x+x+\cdots +x} _{n}>y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <mi>x</mi> <mo>+</mo> <mi>x</mi> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mi>x</mi> </mrow> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munder> <mo>></mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \underbrace {x+x+\cdots +x} _{n}>y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a725957d9ca33d79d04894a108aafbe7319b710" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:19.487ex; height:5.509ex;" alt="{\displaystyle \underbrace {x+x+\cdots +x} _{n}>y}"></span></dd></dl> <p>실수 집합 위의 순서는 <a href="/wiki/%EC%A1%B0%EB%B0%80_%EC%88%9C%EC%84%9C" title="조밀 순서">조밀 순서</a>이다. 즉, 임의의 서로 다른 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/aeb239de6fee56ea8b6a65f7858d95b87632069f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.584ex; height:2.176ex;" alt="{\displaystyle x<y}"></span>에 대하여, 항상 그 사이에 또 다른 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<z<y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo><</mo> <mi>z</mi> <mo><</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x<z<y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0aba76d5ea2f8d2bdf1c518db31f0a80f81b7744" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.77ex; height:2.176ex;" alt="{\displaystyle x<z<y}"></span>가 존재한다. </p> <div class="mw-heading mw-heading2"><h2 id="위상"><span id=".EC.9C.84.EC.83.81"></span>위상</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%8B%A4%EC%88%98&action=edit&section=12" title="부분 편집: 위상"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>실수 집합 위에는 표준적인 <a href="/wiki/%EC%9C%84%EC%83%81_%EA%B3%B5%EA%B0%84_(%EC%88%98%ED%95%99)" title="위상 공간 (수학)">위상 공간</a> · <a href="/wiki/%EA%B1%B0%EB%A6%AC_%EA%B3%B5%EA%B0%84" title="거리 공간">거리 공간</a> · <a href="/wiki/%EB%85%B8%EB%A6%84_%EA%B3%B5%EA%B0%84" title="노름 공간">노름 공간</a> · <a href="/wiki/%EB%82%B4%EC%A0%81_%EA%B3%B5%EA%B0%84" title="내적 공간">내적 공간</a> 구조를 부여할 수 있다. 즉, </p> <ul><li>주어진 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\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y\in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2e1fd3534163cb031d88b529c837e5747ee40fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.038ex; height:2.509ex;" alt="{\displaystyle x,y\in \mathbb {R} }"></span>의 내적은 곱 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle xy}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle xy}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c72eb345e496513fb8b2fa4aa8c4d89b855f9a01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.485ex; height:2.009ex;" alt="{\displaystyle xy}"></span>이다.</li> <li>주어진 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>의 노름은 <a href="/wiki/%EC%A0%88%EB%8C%93%EA%B0%92" title="절댓값">절댓값</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|={\sqrt {x^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |x|={\sqrt {x^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7cf4f434beec0527bf368c9691e30c76acbef05a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.429ex; height:3.509ex;" alt="{\displaystyle |x|={\sqrt {x^{2}}}}"></span>이다.</li> <li>주어진 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\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y\in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2e1fd3534163cb031d88b529c837e5747ee40fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.038ex; height:2.509ex;" alt="{\displaystyle x,y\in \mathbb {R} }"></span>의 거리는 <span class="mwe-math-element"><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"> <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|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9eca622011e2b10520e69a7fa83f3bd75159ab66" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.619ex; height:2.843ex;" alt="{\displaystyle |x-y|}"></span>이다.</li> <li>실수 집합 위의 표준적인 위상은 <a href="/wiki/%EA%B1%B0%EB%A6%AC_%EC%9C%84%EC%83%81" class="mw-redirect" title="거리 위상">거리 위상</a>이자 <a href="/wiki/%EC%88%9C%EC%84%9C_%EC%9C%84%EC%83%81" title="순서 위상">순서 위상</a>이다.</li></ul> <p>실수 부분 집합 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\subseteq \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>⊆</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S\subseteq \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a8d8e251de9443239f3393459a3f0558a37507a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.276ex; height:2.343ex;" alt="{\displaystyle S\subseteq \mathbb {R} }"></span>에 대하여, 다음 조건들이 서로 <a href="/wiki/%EB%8F%99%EC%B9%98" title="동치">동치</a>이다. </p> <ul><li><a href="/wiki/%EC%BD%A4%ED%8C%A9%ED%8A%B8_%EC%A7%91%ED%95%A9" class="mw-redirect" title="콤팩트 집합">콤팩트 집합</a>이다. 즉, 모든 <a href="/wiki/%EC%97%B4%EB%A6%B0%EC%A7%91%ED%95%A9" title="열린집합">열린</a> <a href="/wiki/%EB%8D%AE%EA%B0%9C_(%EC%9C%84%EC%83%81%EC%88%98%ED%95%99)" title="덮개 (위상수학)">덮개</a>가 유한 부분 덮개를 갖는다.</li> <li><a href="/w/index.php?title=%EC%A0%90%EB%A0%AC_%EC%BD%A4%ED%8C%A9%ED%8A%B8_%EC%A7%91%ED%95%A9&action=edit&redlink=1" class="new" title="점렬 콤팩트 집합 (없는 문서)">점렬 콤팩트 집합</a>이다. 즉, 그 속의 모든 수열은 수렴 부분 수열을 갖는다.</li> <li><a href="/w/index.php?title=%EA%B7%B9%ED%95%9C%EC%A0%90_%EC%BD%A4%ED%8C%A9%ED%8A%B8_%EC%A7%91%ED%95%A9&action=edit&redlink=1" class="new" title="극한점 콤팩트 집합 (없는 문서)">극한점 콤팩트 집합</a>이다. 즉, 모든 무한 부분 집합이 <a href="/wiki/%EA%B7%B9%ED%95%9C%EC%A0%90" class="mw-redirect" title="극한점">극한점</a>을 갖는다.</li> <li><a href="/wiki/%EC%9C%A0%EA%B3%84_%EC%A7%91%ED%95%A9" title="유계 집합">유계</a> <a href="/wiki/%EB%8B%AB%ED%9E%8C%EC%A7%91%ED%95%A9" class="mw-redirect" title="닫힌집합">닫힌집합</a>이다.</li></ul> <p>사실, 모든 <a href="/wiki/%EC%9C%A0%ED%81%B4%EB%A6%AC%EB%93%9C_%EA%B3%B5%EA%B0%84" title="유클리드 공간">유클리드 공간</a>에 대하여, 위 네 조건은 서로 동치이며, 모든 <a href="/wiki/%EA%B1%B0%EB%A6%AC_%EA%B3%B5%EA%B0%84" title="거리 공간">거리 공간</a>에 대하여, 앞에 세 조건은 서로 동치이다. </p><p>또한, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\subseteq \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>⊆</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S\subseteq \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a8d8e251de9443239f3393459a3f0558a37507a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.276ex; height:2.343ex;" alt="{\displaystyle S\subseteq \mathbb {R} }"></span>에 대하여, 다음 조건들이 서로 <a href="/wiki/%EB%8F%99%EC%B9%98" title="동치">동치</a>이다. </p> <ul><li><a href="/wiki/%EC%97%B0%EA%B2%B0_%EA%B3%B5%EA%B0%84" title="연결 공간">연결 공간</a>이다.</li> <li><a href="/wiki/%EA%B2%BD%EB%A1%9C_%EC%97%B0%EA%B2%B0_%EA%B3%B5%EA%B0%84" class="mw-redirect" title="경로 연결 공간">경로 연결 공간</a>이다.</li> <li><a href="/wiki/%ED%98%B8_%EC%97%B0%EA%B2%B0_%EA%B3%B5%EA%B0%84" class="mw-redirect" title="호 연결 공간">호 연결 공간</a>이다.</li> <li>중간값 성질을 만족한다. 즉, 임의의 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b\in S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>∈</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b\in S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e9e26744e55d92ec56132c49a80435df0ef4cb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.601ex; height:2.509ex;" alt="{\displaystyle a,b\in S}"></span>에 대하여, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a,b)\subseteq S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>⊆</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a,b)\subseteq S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6191d8614d1bb601431ecfbd744b53251182ec3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.668ex; height:2.843ex;" alt="{\displaystyle (a,b)\subseteq S}"></span>이다.</li> <li>(퇴화 또는 비퇴화) <a href="/wiki/%EA%B5%AC%EA%B0%84" title="구간">구간</a>이다.</li></ul> <div class="mw-heading mw-heading2"><h2 id="분류"><span id=".EB.B6.84.EB.A5.98"></span>분류</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%8B%A4%EC%88%98&action=edit&section=13" title="부분 편집: 분류"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>실수는 <a href="/wiki/%EC%9C%A0%EB%A6%AC%EC%88%98" title="유리수">유리수</a>와 <a href="/wiki/%EB%AC%B4%EB%A6%AC%EC%88%98" title="무리수">무리수</a>로 분류된다. 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 q\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q\in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f39e23ac6313003bf972f808184882d89cc48d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.588ex; height:2.509ex;" alt="{\displaystyle q\in \mathbb {R} }"></span>에 대하여, 다음 조건들이 서로 동치이다. </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}"> <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>는 유리수이다. 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={\frac {m}{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>m</mi> <mi>n</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x={\frac {m}{n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46a3b6cd57a1cc6dd26b851bb8e8383f1650b2f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.305ex; height:4.676ex;" alt="{\displaystyle x={\frac {m}{n}}}"></span>인 정수 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\in \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\in \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12e01e20e2bc24ab21cca6a82edddec5c1a20ec5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.431ex; height:2.176ex;" alt="{\displaystyle m\in \mathbb {Z} }"></span> 및 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\in \mathbb {Z} \setminus \{0\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo class="MJX-variant">∖</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\in \mathbb {Z} \setminus \{0\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/519982201af39ce48f597891c44a98167ecd5013" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.468ex; height:2.843ex;" alt="{\displaystyle n\in \mathbb {Z} \setminus \{0\}}"></span>이 존재한다.</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"> <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>는 <a href="/wiki/%EC%9C%A0%ED%95%9C_%EC%86%8C%EC%88%98" class="mw-redirect" title="유한 소수">유한 소수</a>이거나, <a href="/wiki/%EB%AC%B4%ED%95%9C_%EC%88%9C%ED%99%98_%EC%86%8C%EC%88%98" class="mw-redirect" title="무한 순환 소수">무한 순환 소수</a>이다. 즉, 다음을 만족시키는 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p,q,r\in \{0,1,\dots \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo>,</mo> <mi>r</mi> <mo>∈</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p,q,r\in \{0,1,\dots \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54473f61d32681ecf18da56b0ce38a6b65e63791" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:17.727ex; height:2.843ex;" alt="{\displaystyle p,q,r\in \{0,1,\dots \}}"></span> 및 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{i},b_{i},c_{i}\in \{0,1,\dotsc ,9\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∈</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mn>9</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{i},b_{i},c_{i}\in \{0,1,\dotsc ,9\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3419f1c8e34216e9e691edf628956eb750b1e279" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.566ex; height:2.843ex;" alt="{\displaystyle a_{i},b_{i},c_{i}\in \{0,1,\dotsc ,9\}}"></span>가 존재한다. <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=a_{p}a_{p-1}\cdots a_{0}.b_{1}b_{2}\cdots b_{q}{\dot {c}}_{1}c_{2}\cdots {\dot {c}}_{r}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>⋯</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>.</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⋯</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>c</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⋯</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>c</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=a_{p}a_{p-1}\cdots a_{0}.b_{1}b_{2}\cdots b_{q}{\dot {c}}_{1}c_{2}\cdots {\dot {c}}_{r}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89a5ddd8875c1a8166e4c307a48232236ee92339" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:37.679ex; height:2.843ex;" alt="{\displaystyle x=a_{p}a_{p-1}\cdots a_{0}.b_{1}b_{2}\cdots b_{q}{\dot {c}}_{1}c_{2}\cdots {\dot {c}}_{r}}"></span></dd></dl></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"> <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>는 <a href="/wiki/%EC%9C%A0%ED%95%9C_%EC%97%B0%EB%B6%84%EC%88%98" class="mw-redirect" title="유한 연분수">유한 연분수</a>이다. 즉, 다음을 만족시키는 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{i}\in \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{i}\in \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/726e84f251531b61460cd1cc8cc77cdc1ee4adae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.42ex; height:2.509ex;" alt="{\displaystyle a_{i}\in \mathbb {Z} }"></span> 및 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\in \{0,1,\dots \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>∈</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\in \{0,1,\dots \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e18378e75b69b481dbab79eee68d1366f81e549" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.676ex; height:2.843ex;" alt="{\displaystyle n\in \{0,1,\dots \}}"></span>가 존재한다. <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=[a_{0};a_{1},\dotsc ,a_{n}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mo stretchy="false">[</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>;</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=[a_{0};a_{1},\dotsc ,a_{n}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f39f2dbd632ea2fb369c56e695bb3ef6240d0b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.95ex; height:2.843ex;" alt="{\displaystyle x=[a_{0};a_{1},\dotsc ,a_{n}]}"></span></dd></dl></li></ul> <p>예를 들어, 1/3 = 0.333...은 유리수이며, <a href="/wiki/%EC%9E%90%EC%97%B0%EB%A1%9C%EA%B7%B8%EC%9D%98_%EB%B0%91" title="자연로그의 밑"><i>e</i> = 2.7182...</a>와 <a href="/wiki/%EC%9B%90%EC%A3%BC%EC%9C%A8" title="원주율">π = 3.1415...</a>는 무리수이다. </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%8B%A4%EC%88%98&action=edit&section=14" title="부분 편집: 성질"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="집합론적_성질"><span id=".EC.A7.91.ED.95.A9.EB.A1.A0.EC.A0.81_.EC.84.B1.EC.A7.88"></span>집합론적 성질</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%8B%A4%EC%88%98&action=edit&section=15" title="부분 편집: 집합론적 성질"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>실수 <a href="/wiki/%EC%A7%91%ED%95%A9%EC%9D%98_%ED%81%AC%EA%B8%B0" title="집합의 크기">집합의 크기</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 |\mathbb {R} |=2^{\aleph _{0}}}"> <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"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="normal">ℵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\mathbb {R} |=2^{\aleph _{0}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d03e48338145227cf783e0c5bf75a99ded7d856" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.301ex; height:3.176ex;" alt="{\displaystyle |\mathbb {R} |=2^{\aleph _{0}}}"></span></dd></dl> <p>여기서 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \aleph _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">ℵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \aleph _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/721cd7f8c15a2e72ad162bdfa5baea8eef98aab1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.475ex; height:2.509ex;" alt="{\displaystyle \aleph _{0}}"></span>은 <a href="/wiki/%EC%95%8C%EB%A0%88%ED%94%84_0" class="mw-redirect" title="알레프 0">알레프 0</a>이다. 달리 말해, 실수는 자연수 부분 집합과 일대일 대응한다. 이 둘 사이의 일대일 대응은 여러 가지 만들 수 있다. </p> <div class="mw-heading mw-heading2"><h2 id="역사"><span id=".EC.97.AD.EC.82.AC"></span>역사</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%8B%A4%EC%88%98&action=edit&section=16" title="부분 편집: 역사"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>실수에 대한 엄밀한 정의는 <a href="/wiki/%EA%B2%8C%EC%98%A4%EB%A5%B4%ED%81%AC_%EC%B9%B8%ED%86%A0%EC%96%B4" title="게오르크 칸토어">게오르크 칸토어</a>에 의해 이루어졌다. 유리수로부터 실수를 이론적으로 확장하여 그 성질을 규정짓게 된 것은 <a href="/wiki/%EC%B9%B4%EB%A5%BC_%EB%B0%94%EC%9D%B4%EC%96%B4%EC%8A%88%ED%8A%B8%EB%9D%BC%EC%8A%A4" title="카를 바이어슈트라스">카를 바이어슈트라스</a>, <a href="/wiki/%EA%B2%8C%EC%98%A4%EB%A5%B4%ED%81%AC_%EC%B9%B8%ED%86%A0%EC%96%B4" title="게오르크 칸토어">게오르크 칸토어</a>, <a href="/wiki/%EB%A6%AC%ED%95%98%EB%A5%B4%ED%8A%B8_%EB%8D%B0%EB%8D%B0%ED%82%A8%ED%8A%B8" title="리하르트 데데킨트">리하르트 데데킨트</a>와 같은 수학자들의 공이 지대하였다. </p> <div class="mw-heading mw-heading2"><h2 id="같이_보기"><span id=".EA.B0.99.EC.9D.B4_.EB.B3.B4.EA.B8.B0"></span>같이 보기</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%8B%A4%EC%88%98&action=edit&section=17" title="부분 편집: 같이 보기"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r36480479">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:#f9f9f9;display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><div class="side-box metadata side-box-right plainlinks"><style data-mw-deduplicate="TemplateStyles:r36480595">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span></div> <div class="side-box-text plainlist"><b><a href="/wiki/%EC%9C%84%ED%82%A4%EB%AF%B8%EB%94%94%EC%96%B4_%EA%B3%B5%EC%9A%A9" title="위키미디어 공용">위키미디어 공용</a></b>에 관련된<br />미디어 분류가 있습니다.<div style="padding-left:1em;"><b><a class="external text" href="https://commons.wikimedia.org/wiki/Category:Real_numbers?uselang=ko">실수</a></b></div></div></div> </div> <ul><li><a href="/wiki/%EC%97%B0%EB%B6%84%EC%88%98" title="연분수">연분수</a></li> <li><a href="/wiki/%EC%8B%A4%ED%95%B4%EC%84%9D%ED%95%99" title="실해석학">실해석학</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="외부_링크"><span id=".EC.99.B8.EB.B6.80_.EB.A7.81.ED.81.AC"></span>외부 링크</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%8B%A4%EC%88%98&action=edit&section=18" title="부분 편집: 외부 링크"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><cite class="citation web"><a rel="nofollow" class="external text" href="http://navercast.naver.com/contents.nhn?contents_id=2083">“자연수 VS 실수”</a>. 《<a href="/wiki/%EB%84%A4%EC%9D%B4%EB%B2%84%EC%BA%90%EC%8A%A4%ED%8A%B8" title="네이버캐스트">네이버캐스트</a>》.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=%EB%84%A4%EC%9D%B4%EB%B2%84%EC%BA%90%EC%8A%A4%ED%8A%B8&rft.atitle=%EC%9E%90%EC%97%B0%EC%88%98+VS+%EC%8B%A4%EC%88%98&rft_id=http%3A%2F%2Fnavercast.naver.com%2Fcontents.nhn%3Fcontents_id%3D2083&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%EC%8B%A4%EC%88%98" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite class="citation web"><a rel="nofollow" class="external text" href="https://encyclopediaofmath.org/wiki/Real_number">“Real number”</a>. 《Encyclopedia of Mathematics》 (영어). Springer-Verlag. 2001. <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%B0/978-1-55608-010-4" title="특수:책찾기/978-1-55608-010-4"><bdi>978-1-55608-010-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Encyclopedia+of+Mathematics&rft.atitle=Real+number&rft.date=2001&rft.isbn=978-1-55608-010-4&rft_id=https%3A%2F%2Fencyclopediaofmath.org%2Fwiki%2FReal_number&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%EC%8B%A4%EC%88%98" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite class="citation web">Weisstein, Eric Wolfgang. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/RealNumber.html">“Real number”</a>. 《<a href="/wiki/%EB%A7%A4%EC%8A%A4%EC%9B%94%EB%93%9C" title="매스월드">Wolfram MathWorld</a>》 (영어). Wolfram Research.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Wolfram+MathWorld&rft.atitle=Real+number&rft.aulast=Weisstein&rft.aufirst=Eric+Wolfgang&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FRealNumber.html&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%EC%8B%A4%EC%88%98" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite class="citation web"><a rel="nofollow" class="external text" href="https://ncatlab.org/nlab/show/real+number">“Real number”</a>. 《nLab》 (영어).</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=nLab&rft.atitle=Real+number&rft_id=https%3A%2F%2Fncatlab.org%2Fnlab%2Fshow%2Freal%2Bnumber&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%EC%8B%A4%EC%88%98" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite class="citation web"><a rel="nofollow" class="external text" href="https://planetmath.org/RealNumber">“Real number”</a>. 《PlanetMath》 (영어).</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=PlanetMath&rft.atitle=Real+number&rft_id=http%3A%2F%2Fplanetmath.org%2FRealNumber&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%EC%8B%A4%EC%88%98" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite class="citation web"><a rel="nofollow" class="external text" href="https://planetmath.org/11RealNumbers">“11. Real numbers”</a>. 《PlanetMath》 (영어).</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=PlanetMath&rft.atitle=11.+Real+numbers&rft_id=http%3A%2F%2Fplanetmath.org%2F11RealNumbers&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%EC%8B%A4%EC%88%98" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite class="citation web"><a rel="nofollow" class="external text" href="https://planetmath.org/representationofrealnumbers">“Representation of real numbers”</a>. 《PlanetMath》 (영어).</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=PlanetMath&rft.atitle=Representation+of+real+numbers&rft_id=http%3A%2F%2Fplanetmath.org%2Frepresentationofrealnumbers&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%EC%8B%A4%EC%88%98" class="Z3988"><span style="display:none;"> </span></span></li></ul> <div class="navbox-styles"><style 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tr+tr>.navbox-list{border-top:2px solid #fdfdfd}.mw-parser-output .navbox-title{background-color:#ccf}.mw-parser-output .navbox-abovebelow,.mw-parser-output .navbox-group,.mw-parser-output .navbox-subgroup .navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}</style></div><div role="navigation" class="navbox" aria-labelledby="수_체계" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r36480591"><style data-mw-deduplicate="TemplateStyles:r34311309">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini 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style="font-size:114%;margin:0 4em"><a href="/wiki/%EC%88%98_(%EC%88%98%ED%95%99)" title="수 (수학)">수</a> 체계</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">복소수</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/%EC%9E%90%EC%97%B0%EC%88%98" title="자연수">자연수</a> (ℕ)</li> <li><a href="/wiki/%EC%A0%95%EC%88%98" title="정수">정수</a> (ℤ)</li> <li><a href="/wiki/%EC%9C%A0%EB%A6%AC%EC%88%98" title="유리수">유리수</a> (ℚ)</li> <li><a href="/wiki/%EB%AC%B4%EB%A6%AC%EC%88%98" title="무리수">무리수</a> (ℝ∖ℚ)</li> <li><a class="mw-selflink selflink">실수</a> (ℝ)</li> <li><a href="/wiki/%ED%97%88%EC%88%98" title="허수">허수</a> (ℂ∖ℝ)</li> <li><a href="/wiki/%EB%B3%B5%EC%86%8C%EC%88%98" title="복소수">복소수</a> (ℂ)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">자연수의 분류</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/%EC%86%8C%EC%88%98_(%EC%88%98%EB%A1%A0)" title="소수 (수론)">소수</a> (ℙ)</li> <li><a href="/wiki/%ED%95%A9%EC%84%B1%EC%88%98" title="합성수">합성수</a> (ℕ∖ℙ∖{0,1})</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">유리수의 분류</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/%EB%B0%98%EC%A0%95%EC%88%98" title="반정수">반정수</a> (1/2+ℤ)</li> <li><a href="/wiki/%EC%9D%B4%EC%A7%84_%EC%9C%A0%EB%A6%AC%EC%88%98" title="이진 유리수">이진 유리수</a> ((2<sup>ℕ</sup>)<sup>-1</sup>ℤ)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">실수의 분류</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/%EC%96%91%EC%88%98_(%EC%88%98%ED%95%99)" title="양수 (수학)">양수</a> (ℝ<sup>+</sup>)</li> <li><a href="/wiki/%EC%9D%8C%EC%88%98" title="음수">음수</a> (ℝ<sup>-</sup>)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">복소수의 분류</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/%EB%8C%80%EC%88%98%EC%A0%81_%EC%88%98" title="대수적 수">대수적 수</a> (<span class="sfrac nowrap;"><span style="display:none; display:inline-block; text-align:center;"><span style="display:block; line-height:0.4em; font-size:70%; padding:0 0.1em;">—</span><span style="display:block; line-height:1em; padding:0 0.1em;">ℚ</span></span></span>)</li> <li><a href="/wiki/%EC%B4%88%EC%9B%94%EC%88%98" title="초월수">초월수</a> (ℂ∖<span class="sfrac nowrap;"><span style="display:none; display:inline-block; text-align:center;"><span style="display:block; line-height:0.4em; font-size:70%; padding:0 0.1em;">—</span><span style="display:block; line-height:1em; padding:0 0.1em;">ℚ</span></span></span>)</li> <li><a href="/wiki/%EB%8C%80%EC%88%98%EC%A0%81_%EC%A0%95%EC%88%98" class="mw-redirect" title="대수적 정수">대수적 정수</a> (<span class="mathcal" style="font-family: 'Lucida Calligraphy', 'Monotype Corsiva', 'URW Chancery L', 'Apple Chancery', serif;">O</span><sub><span class="sfrac nowrap;"><span style="display:none; display:inline-block; text-align:center;"><span style="display:block; line-height:0.4em; font-size:70%; padding:0 0.1em;">—</span><span style="display:block; line-height:1em; padding:0 0.1em;">ℚ</span></span></span></sub>)</li> <li><a href="/wiki/%EA%B0%80%EC%9A%B0%EC%8A%A4_%EC%A0%95%EC%88%98" title="가우스 정수">가우스 정수</a> (ℤ[i])</li> <li><a href="/wiki/%EC%95%84%EC%9D%B4%EC%A0%A0%EC%8A%88%ED%83%80%EC%9D%B8_%EC%A0%95%EC%88%98" title="아이젠슈타인 정수">아이젠슈타인 정수</a> (<span class="mathcal" style="font-family: 'Lucida Calligraphy', 'Monotype Corsiva', 'URW Chancery L', 'Apple Chancery', serif;">O</span><sub>ℚ(√-3)</sub>)</li> <li><a href="/wiki/%EC%9E%91%EB%8F%84_%EA%B0%80%EB%8A%A5%ED%95%9C_%EC%88%98" title="작도 가능한 수">작도 가능한 수</a></li> <li><a href="/wiki/%EA%B3%84%EC%82%B0_%EA%B0%80%EB%8A%A5%ED%95%9C_%EC%88%98" title="계산 가능한 수">계산 가능한 수</a></li> <li><a href="/wiki/%EC%A0%95%EC%9D%98_%EA%B0%80%EB%8A%A5%ED%95%9C_%EC%88%98" title="정의 가능한 수">정의 가능한 수</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">기타</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/%EC%9D%B4%EC%9B%90%EC%88%98_(%EC%88%98%ED%95%99)" title="이원수 (수학)">이원수</a> (ℝ[x]/(x<sup>2</sup>))</li> <li><a href="/w/index.php?title=%EB%8B%A4%EC%9B%90%EC%88%98&action=edit&redlink=1" class="new" title="다원수 (없는 문서)">다원수</a></li> <li><a href="/wiki/%EC%82%AC%EC%9B%90%EC%88%98" title="사원수">사원수</a> (ℍ)</li> <li><a href="/wiki/%ED%8C%94%EC%9B%90%EC%88%98" title="팔원수">팔원수</a> (𝕆)</li> <li><a href="/wiki/%EC%8B%AD%EC%9C%A1%EC%9B%90%EC%88%98" title="십육원수">십육원수</a> (𝕊)</li> <li><a href="/wiki/P%EC%A7%84%EC%88%98" title="P진수">p진수</a> (ℚ<sub>p</sub>)</li> <li><a href="/wiki/%EC%B4%88%EC%8B%A4%EC%88%98" title="초실수">초실수</a></li> <li><a href="/w/index.php?title=%EC%83%81%EC%B4%88%EC%8B%A4%EC%88%98&action=edit&redlink=1" class="new" title="상초실수 (없는 문서)">상초실수</a></li> <li><a href="/wiki/%EC%B4%88%ED%98%84%EC%8B%A4%EC%88%98" title="초현실수">초현실수</a></li> <li><a href="/wiki/%EC%88%9C%EC%84%9C%EC%88%98" title="순서수">순서수</a> (Ord)</li> <li><a href="/wiki/%EA%B8%B0%EC%88%98_(%EC%88%98%ED%95%99)" title="기수 (수학)">기수</a> (Card)</li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r36480591"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r36429174"></div><div role="navigation" class="navbox authority-control" aria-label="Navbox" style="padding:3px"><table class="nowraplinks hlist navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="row" 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style="padding:0 0.25em"> <ul><li><span class="uid"><abbr title="Nombres réels"><a rel="nofollow" class="external text" href="https://catalogue.bnf.fr/ark:/12148/cb11977586x">프랑스</a></abbr></span></li> <li><span class="uid"><abbr title="Nombres réels"><a rel="nofollow" class="external text" href="https://data.bnf.fr/ark:/12148/cb11977586x">BnF 데이터</a></abbr></span></li> <li><span class="uid"><a rel="nofollow" class="external text" href="https://d-nb.info/gnd/4202628-3">독일</a></span></li> <li><span class="uid"><a rel="nofollow" class="external text" href="http://uli.nli.org.il/F/?func=find-b&local_base=NLX10&find_code=UID&request=987007538747105171">이스라엘</a></span></li> <li><span class="uid"><a rel="nofollow" class="external text" href="https://id.loc.gov/authorities/sh85093221">미국</a></span></li> <li><span class="uid"><a rel="nofollow" class="external text" href="https://id.ndl.go.jp/auth/ndlna/00574870">일본</a></span></li> <li><span class="uid"><abbr title="reálná čísla"><a 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href="/wiki/%EB%B6%84%EB%A5%98:%EC%88%98_%EC%B2%B4%EA%B3%84" title="분류:수 체계">수 체계</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">숨은 분류: <ul><li><a href="/wiki/%EB%B6%84%EB%A5%98:%ED%95%B4%EA%B2%B0%EB%90%98%EC%A7%80_%EC%95%8A%EC%9D%80_%EC%86%8D%EC%84%B1%EC%9D%B4_%EC%9E%88%EB%8A%94_%EB%AC%B8%EC%84%9C" title="분류:해결되지 않은 속성이 있는 문서">해결되지 않은 속성이 있는 문서</a></li><li><a href="/wiki/%EB%B6%84%EB%A5%98:%EC%9C%84%ED%82%A4%EB%8D%B0%EC%9D%B4%ED%84%B0_%EC%86%8D%EC%84%B1_P18%EC%9D%84_%EC%82%AC%EC%9A%A9%ED%95%98%EB%8A%94_%EB%AC%B8%EC%84%9C" title="분류:위키데이터 속성 P18을 사용하는 문서">위키데이터 속성 P18을 사용하는 문서</a></li><li><a href="/wiki/%EB%B6%84%EB%A5%98:%EC%9C%84%ED%82%A4%EB%8D%B0%EC%9D%B4%ED%84%B0_%EC%86%8D%EC%84%B1_P373%EC%9D%84_%EC%82%AC%EC%9A%A9%ED%95%98%EB%8A%94_%EB%AC%B8%EC%84%9C" title="분류:위키데이터 속성 P373을 사용하는 문서">위키데이터 속성 P373을 사용하는 문서</a></li><li><a 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