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class="vector-toc-numb">3</span> <span>Hyperbolic coordinates</span> </div> </a> <ul id="toc-Hyperbolic_coordinates-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Computer_encoding" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Computer_encoding"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Computer encoding</span> </div> </a> <ul id="toc-Computer_encoding-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Proportionality (mathematics)</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" 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Available in 55 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-55" 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">55 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Eweredigheid_(wiskunde)" title="Eweredigheid (wiskunde) – Afrikaans" lang="af" hreflang="af" data-title="Eweredigheid (wiskunde)" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" 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/Proportionalit%C3%A4t" title="Proportionalität – Alemannic" lang="gsw" hreflang="gsw" data-title="Proportionalität" data-language-autonym="Alemannisch" data-language-local-name="Alemannic" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%89%80%E1%8C%A5%E1%89%B0%E1%8A%9B_%E1%8B%9D%E1%88%9D%E1%8B%B5%E1%8A%93" title="ቀጥተኛ ዝምድና – Amharic" lang="am" hreflang="am" data-title="ቀጥተኛ ዝምድና" data-language-autonym="አማርኛ" data-language-local-name="Amharic" class="interlanguage-link-target"><span>አማርኛ</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AA%D9%86%D8%A7%D8%B3%D8%A8_(%D8%B1%D9%8A%D8%A7%D8%B6%D9%8A%D8%A7%D8%AA)" title="تناسب (رياضيات) – Arabic" lang="ar" hreflang="ar" data-title="تناسب (رياضيات)" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/M%C3%BCt%C9%99nasib" title="Mütənasib – Azerbaijani" lang="az" hreflang="az" data-title="Mütənasib" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9F%D1%80%D0%BE%D0%BF%D0%BE%D1%80%D1%86%D0%B8%D0%BE%D0%BD%D0%B0%D0%BB%D0%BD%D0%BE%D1%81%D1%82" title="Пропорционалност – Bulgarian" lang="bg" hreflang="bg" data-title="Пропорционалност" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Proporcionalnost_(matematika)" title="Proporcionalnost (matematika) – Bosnian" lang="bs" hreflang="bs" data-title="Proporcionalnost (matematika)" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Proporcionalitat" title="Proporcionalitat – Catalan" lang="ca" hreflang="ca" data-title="Proporcionalitat" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9F%D1%80%D0%BE%D0%BF%D0%BE%D1%80%D1%86%D0%B8%D0%BE%D0%BD%D0%B0%D0%BB%D0%BB%C4%83%D1%85" title="Пропорционаллăх – Chuvash" lang="cv" hreflang="cv" data-title="Пропорционаллăх" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/P%C5%99%C3%ADm%C3%A1_a_nep%C5%99%C3%ADm%C3%A1_%C3%BAm%C4%9Brnost" title="Přímá a nepřímá úměrnost – Czech" lang="cs" hreflang="cs" data-title="Přímá a nepřímá úměrnost" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-sn mw-list-item"><a href="https://sn.wikipedia.org/wiki/Muzasedimbu" title="Muzasedimbu – Shona" lang="sn" hreflang="sn" data-title="Muzasedimbu" data-language-autonym="ChiShona" data-language-local-name="Shona" class="interlanguage-link-target"><span>ChiShona</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Proportionalitet" title="Proportionalitet – Danish" lang="da" hreflang="da" data-title="Proportionalitet" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Proportionalit%C3%A4t" title="Proportionalität – German" lang="de" hreflang="de" data-title="Proportionalität" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/V%C3%B5rdelisus" title="Võrdelisus – Estonian" lang="et" hreflang="et" data-title="Võrdelisus" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%91%CE%BD%CE%B1%CE%BB%CE%BF%CE%B3%CE%AF%CE%B1_(%CE%BC%CE%B1%CE%B8%CE%B7%CE%BC%CE%B1%CF%84%CE%B9%CE%BA%CE%AC)" title="Αναλογία (μαθηματικά) – Greek" lang="el" hreflang="el" data-title="Αναλογία (μαθηματικά)" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Proporcionalidad" title="Proporcionalidad – Spanish" lang="es" hreflang="es" data-title="Proporcionalidad" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Proportzionaltasun_(matematika)" title="Proportzionaltasun (matematika) – Basque" lang="eu" hreflang="eu" data-title="Proportzionaltasun (matematika)" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AA%D9%86%D8%A7%D8%B3%D8%A8_(%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C%D8%A7%D8%AA)" title="تناسب (ریاضیات) – Persian" lang="fa" hreflang="fa" data-title="تناسب (ریاضیات)" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Proportionnalit%C3%A9" title="Proportionnalité – French" lang="fr" hreflang="fr" data-title="Proportionnalité" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Proporcionalidade" title="Proporcionalidade – Galician" lang="gl" hreflang="gl" data-title="Proporcionalidade" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%B9%84%EB%A1%80" title="비례 – Korean" lang="ko" hreflang="ko" data-title="비례" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%80%D5%A1%D5%B4%D5%A1%D5%B4%D5%A1%D5%BD%D5%B6%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6" title="Համամասնություն – Armenian" lang="hy" hreflang="hy" data-title="Համամասնություն" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B8%E0%A4%AE%E0%A4%BE%E0%A4%A8%E0%A5%81%E0%A4%AA%E0%A4%BE%E0%A4%A4" title="समानुपात – Hindi" lang="hi" hreflang="hi" data-title="समानुपात" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Proporcionalnost" title="Proporcionalnost – Croatian" lang="hr" hreflang="hr" data-title="Proporcionalnost" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Proporciono" title="Proporciono – Ido" lang="io" hreflang="io" data-title="Proporciono" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Kesebandingan_(matematika)" title="Kesebandingan (matematika) – Indonesian" lang="id" hreflang="id" data-title="Kesebandingan (matematika)" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Hlutfall" title="Hlutfall – Icelandic" lang="is" hreflang="is" data-title="Hlutfall" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Proporzionalit%C3%A0_(matematica)" title="Proporzionalità (matematica) – Italian" lang="it" hreflang="it" data-title="Proporzionalità (matematica)" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%99%D7%97%D7%A1_%D7%99%D7%A9%D7%A8" title="יחס ישר – Hebrew" lang="he" hreflang="he" data-title="יחס ישר" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%A2%D1%83%D1%80%D0%B0_%D1%82%D3%99%D1%83%D0%B5%D0%BB%D0%B4%D1%96%D0%BB%D1%96%D0%BA" title="Тура тәуелділік – Kazakh" lang="kk" hreflang="kk" data-title="Тура тәуелділік" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Egyenes_ar%C3%A1nyoss%C3%A1g" title="Egyenes arányosság – Hungarian" lang="hu" hreflang="hu" data-title="Egyenes arányosság" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Perkadaran" title="Perkadaran – Malay" lang="ms" hreflang="ms" data-title="Perkadaran" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Evenredigheid" title="Evenredigheid – Dutch" lang="nl" hreflang="nl" data-title="Evenredigheid" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ne mw-list-item"><a href="https://ne.wikipedia.org/wiki/%E0%A4%85%E0%A4%A8%E0%A5%81%E0%A4%AA%E0%A4%BE%E0%A4%A4%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="अनुपातिक सङ्ख्या – Nepali" lang="ne" hreflang="ne" data-title="अनुपातिक सङ्ख्या" data-language-autonym="नेपाली" data-language-local-name="Nepali" class="interlanguage-link-target"><span>नेपाली</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E6%AF%94%E4%BE%8B" title="比例 – Japanese" lang="ja" hreflang="ja" data-title="比例" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Proporsjonalitet" title="Proporsjonalitet – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Proporsjonalitet" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Proporsjon_i_matematikk" title="Proporsjon i matematikk – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Proporsjon i matematikk" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%B8%E0%A8%AE%E0%A8%BE%E0%A8%A8_%E0%A8%85%E0%A8%A8%E0%A9%81%E0%A8%AA%E0%A8%BE%E0%A8%A4" title="ਸਮਾਨ ਅਨੁਪਾਤ – Punjabi" lang="pa" hreflang="pa" data-title="ਸਮਾਨ ਅਨੁਪਾਤ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" 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/Proporcjonalno%C5%9B%C4%87_prosta" title="Proporcjonalność prosta – Polish" lang="pl" hreflang="pl" data-title="Proporcjonalność prosta" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Proporcionalidade" title="Proporcionalidade – Portuguese" lang="pt" hreflang="pt" data-title="Proporcionalidade" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Propor%C8%9Bionalitate" title="Proporționalitate – Romanian" lang="ro" hreflang="ro" data-title="Proporționalitate" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9F%D1%80%D0%BE%D0%BF%D0%BE%D1%80%D1%86%D0%B8%D0%BE%D0%BD%D0%B0%D0%BB%D1%8C%D0%BD%D0%BE%D1%81%D1%82%D1%8C" title="Пропорциональность – Russian" lang="ru" hreflang="ru" data-title="Пропорциональность" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Proportionality" title="Proportionality – Simple English" lang="en-simple" hreflang="en-simple" data-title="Proportionality" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Razmerje" title="Razmerje – Slovenian" lang="sl" hreflang="sl" data-title="Razmerje" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%DA%BE%D8%A7%D9%88%DA%95%DB%8E%DA%98%DB%95%DB%8C%DB%8C" title="ھاوڕێژەیی – Central Kurdish" lang="ckb" hreflang="ckb" data-title="ھاوڕێژەیی" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Proporcionalnost_(matematika)" title="Proporcionalnost (matematika) – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Proporcionalnost (matematika)" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Verrannollisuus" title="Verrannollisuus – Finnish" lang="fi" hreflang="fi" data-title="Verrannollisuus" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Proportionalitet_(matematik)" title="Proportionalitet (matematik) – Swedish" lang="sv" hreflang="sv" data-title="Proportionalitet (matematik)" data-language-autonym="Svenska" 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id="mw-content-subtitle"><span class="mw-redirectedfrom">(Redirected from <a href="/w/index.php?title=Inversely_proportional&redirect=no" class="mw-redirect" title="Inversely proportional">Inversely proportional</a>)</span></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Property of two varying quantities with a constant ratio</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">For other uses, see <a href="/wiki/Proportionality_(disambiguation)" class="mw-redirect mw-disambig" title="Proportionality (disambiguation)">Proportionality</a>.</div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-More_footnotes_needed plainlinks metadata ambox ambox-style ambox-More_footnotes_needed" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/40px-Text_document_with_red_question_mark.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/60px-Text_document_with_red_question_mark.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/80px-Text_document_with_red_question_mark.svg.png 2x" data-file-width="48" data-file-height="48" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article includes a list of <a href="/wiki/Wikipedia:Citing_sources#General_references" title="Wikipedia:Citing sources">general references</a>, but <b>it lacks sufficient corresponding <a href="/wiki/Wikipedia:Citing_sources#Inline_citations" title="Wikipedia:Citing sources">inline citations</a></b>.<span class="hide-when-compact"> Please help to <a href="/wiki/Wikipedia:WikiProject_Reliability" title="Wikipedia:WikiProject Reliability">improve</a> this article by <a href="/wiki/Wikipedia:When_to_cite" title="Wikipedia:When to cite">introducing</a> more precise citations.</span> <span class="date-container"><i>(<span class="date">August 2021</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Proportional_variables.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Proportional_variables.svg/300px-Proportional_variables.svg.png" decoding="async" width="300" height="240" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Proportional_variables.svg/450px-Proportional_variables.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Proportional_variables.svg/600px-Proportional_variables.svg.png 2x" data-file-width="375" data-file-height="300" /></a><figcaption>The variable <i>y</i> is directly proportional to the variable <i>x</i> with proportionality constant ~0.6.</figcaption></figure> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Inverse_proportionality_function_plot.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/84/Inverse_proportionality_function_plot.gif/300px-Inverse_proportionality_function_plot.gif" decoding="async" width="300" height="226" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/84/Inverse_proportionality_function_plot.gif/450px-Inverse_proportionality_function_plot.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/84/Inverse_proportionality_function_plot.gif/600px-Inverse_proportionality_function_plot.gif 2x" data-file-width="610" data-file-height="460" /></a><figcaption>The variable <i>y</i> is inversely proportional to the variable <i>x</i> with proportionality constant 1.</figcaption></figure> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, two <a href="/wiki/Sequence" title="Sequence">sequences</a> of numbers, often <a href="/wiki/Experimental_data" title="Experimental data">experimental data</a>, are <b>proportional</b> or <b>directly proportional</b> if their corresponding elements have a <a href="/wiki/Constant_(mathematics)" title="Constant (mathematics)">constant</a> <a href="/wiki/Ratio" title="Ratio">ratio</a>. The ratio is called <i><b>coefficient of proportionality</b></i> (or <i><b>proportionality constant</b></i>) and its reciprocal is known as <i><b>constant of normalization</b></i> (or <i><b>normalizing constant</b></i>). Two sequences are <b>inversely proportional</b> if corresponding elements have a constant product, also called the coefficient of proportionality. </p><p>This definition is commonly extended to related varying quantities, which are often called <i>variables</i>. This meaning of <i>variable</i> is not the common meaning of the term in mathematics (see <a href="/wiki/Variable_(mathematics)" title="Variable (mathematics)">variable (mathematics)</a>); these two different concepts share the same name for historical reasons. </p><p>Two <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">functions</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 f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6ca91363022bd5e4dcb17e5ef29f78b8ef00b59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.255ex; height:2.843ex;" alt="{\displaystyle g(x)}"></span> are <i>proportional</i> if their ratio <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {f(x)}{g(x)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\frac {f(x)}{g(x)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/102bc2ce010b8b1133e1f4cf7953b2a19e6524d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:3.96ex; height:4.843ex;" alt="{\textstyle {\frac {f(x)}{g(x)}}}"></span> is a <a href="/wiki/Constant_function" title="Constant function">constant function</a>. </p><p>If several pairs of variables share the same direct proportionality constant, the <a href="/wiki/Equation" title="Equation">equation</a> expressing the equality of these ratios is called a <b>proportion</b>, e.g., <span class="texhtml"><style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠<span class="tion"><span class="num"><i>a</i></span><span class="sr-only">/</span><span class="den"><i>b</i></span></span>⁠</span> = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>x</i></span><span class="sr-only">/</span><span class="den"><i>y</i></span></span>⁠</span> = ⋯ = <i>k</i></span> (for details see <a href="/wiki/Ratio" title="Ratio">Ratio</a>). Proportionality is closely related to <i><a href="/wiki/Linearity" title="Linearity">linearity</a></i>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Direct_proportionality">Direct proportionality</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Proportionality_(mathematics)&action=edit&section=1" title="Edit section: Direct proportionality"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Equals_sign" title="Equals sign">Equals sign</a></div> <p>Given an <a href="/wiki/Variable_(mathematics)#Dependent_and_independent_variables" title="Variable (mathematics)">independent variable</a> <i>x</i> and a dependent variable <i>y</i>, <i>y</i> is <b>directly proportional</b> to <i>x</i><sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> if there is a positive constant <i>k</i> such that: </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 y=kx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>k</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=kx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0ce87b0aaacaad25a36d2f52dc1fae2ead92689" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.795ex; height:2.509ex;" alt="{\displaystyle y=kx}"></span></dd></dl> <p>The relation is often denoted using the symbols "∝" (not to be confused with the Greek letter <a href="/wiki/Alpha" title="Alpha">alpha</a>) or "~", with exception of Japanese texts, where "~" is reserved for intervals: </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 y\propto x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>∝</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y\propto x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b6f3478f6e9a02ccaba915d97e73241e540b75c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.584ex; height:2.009ex;" alt="{\displaystyle y\propto x}"></span> (or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\sim x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>∼</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y\sim x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c54fe8e222fd770c44b4df39405b2616532a976" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.584ex; height:2.009ex;" alt="{\displaystyle y\sim x}"></span>)</dd></dl> <p>For <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35a455db7b2aab1b0e72ccbc7385e4424e2372e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.591ex; height:2.676ex;" alt="{\displaystyle x\neq 0}"></span> the <b>proportionality constant</b> can be expressed as the ratio: </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 k={\frac {y}{x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>y</mi> <mi>x</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k={\frac {y}{x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c57718f06c0377edf7d3ffcfa34c4e4c93bc0ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:6.476ex; height:4.843ex;" alt="{\displaystyle k={\frac {y}{x}}}"></span></dd></dl> <p>It is also called the <b>constant of variation</b> or <b>constant of proportionality</b>. Given such a constant <i>k</i>, the proportionality <a href="/wiki/Binary_relation" title="Binary relation">relation</a> ∝ with proportionality constant <i>k</i> between two sets <i>A</i> and <i>B</i> is the <a href="/wiki/Equivalence_relation" title="Equivalence relation">equivalence relation</a> defined by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{(a,b)\in A\times B:a=kb\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mi>A</mi> <mo>×</mo> <mi>B</mi> <mo>:</mo> <mi>a</mi> <mo>=</mo> <mi>k</mi> <mi>b</mi> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{(a,b)\in A\times B:a=kb\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e493e4f1002b7d1fe8dcd61827f60e65f399496" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.705ex; height:2.843ex;" alt="{\displaystyle \{(a,b)\in A\times B:a=kb\}.}"></span> </p><p>A direct proportionality can also be viewed as a <a href="/wiki/Linear_equation" title="Linear equation">linear equation</a> in two variables with a <a href="/wiki/Y-intercept" title="Y-intercept"><i>y</i>-intercept</a> of <span class="texhtml">0</span> and a <a href="/wiki/Slope" title="Slope">slope</a> of <i>k</i> > 0, which corresponds to <a href="/wiki/Linear_growth" class="mw-redirect" title="Linear growth">linear growth</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Examples">Examples</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Proportionality_(mathematics)&action=edit&section=2" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>If an object travels at a constant <a href="/wiki/Speed" title="Speed">speed</a>, then the <a href="/wiki/Distance" title="Distance">distance</a> traveled is directly proportional to the <a href="/wiki/Time" title="Time">time</a> spent traveling, with the speed being the constant of proportionality.</li> <li>The <a href="/wiki/Circumference" title="Circumference">circumference</a> of a <a href="/wiki/Circle" title="Circle">circle</a> is directly proportional to its <a href="/wiki/Diameter" title="Diameter">diameter</a>, with the constant of proportionality equal to <a href="/wiki/Pi" title="Pi"><span class="texhtml mvar" style="font-style:italic;">π</span></a>.</li> <li>On a <a href="/wiki/Map" title="Map">map</a> of a sufficiently small geographical area, drawn to <a href="/wiki/Scale_(map)" title="Scale (map)">scale</a> distances, the distance between any two points on the map is directly proportional to the beeline distance between the two locations represented by those points; the constant of proportionality is the scale of the map.</li> <li>The <a href="/wiki/Force_(physics)" class="mw-redirect" title="Force (physics)">force</a>, acting on a small object with small <a href="/wiki/Mass" title="Mass">mass</a> by a nearby large extended mass due to <a href="/wiki/Gravity" title="Gravity">gravity</a>, is directly proportional to the object's mass; the constant of proportionality between the force and the mass is known as <a href="/wiki/Gravitational_acceleration" title="Gravitational acceleration">gravitational acceleration</a>.</li> <li>The net force acting on an object is proportional to the acceleration of that object with respect to an inertial frame of reference. The constant of proportionality in this, <a href="/wiki/Newton%27s_second_law" class="mw-redirect" title="Newton's second law">Newton's second law</a>, is the classical mass of the object.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Inverse_proportionality">Inverse proportionality</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Proportionality_(mathematics)&action=edit&section=3" title="Edit section: Inverse proportionality"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Inverse_proportionality_function_plot.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/84/Inverse_proportionality_function_plot.gif/300px-Inverse_proportionality_function_plot.gif" decoding="async" width="300" height="226" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/84/Inverse_proportionality_function_plot.gif/450px-Inverse_proportionality_function_plot.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/84/Inverse_proportionality_function_plot.gif/600px-Inverse_proportionality_function_plot.gif 2x" data-file-width="610" data-file-height="460" /></a><figcaption>Inverse proportionality with product <span class="nowrap"><i>x y</i> = 1 .</span></figcaption></figure> <p>Two variables are <b>inversely proportional</b> (also called <b>varying inversely</b>, in <b>inverse variation</b>, in <b>inverse proportion</b>)<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> if each of the variables is directly proportional to the <a href="/wiki/Multiplicative_inverse" title="Multiplicative inverse">multiplicative inverse</a> (reciprocal) of the other, or equivalently if their <a href="/wiki/Product_(mathematics)" title="Product (mathematics)">product</a> is a constant.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> It follows that the variable <i>y</i> is inversely proportional to the variable <i>x</i> if there exists a non-zero constant <i>k</i> such that </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 y={\frac {k}{x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>k</mi> <mi>x</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y={\frac {k}{x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec505faf6069cc9862860354e3208a2223cccf63" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:6.42ex; height:5.343ex;" alt="{\displaystyle y={\frac {k}{x}}}"></span></dd></dl> <p>or equivalently, <span class="mwe-math-element"><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=k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mi>y</mi> <mo>=</mo> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle xy=k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58a68fcbf4e4a86e5465ac0a30d2092b8fab34ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.795ex; height:2.509ex;" alt="{\displaystyle xy=k}"></span>. Hence the constant "<i>k</i>" is the product of <i>x</i> and <i>y</i>. </p><p>The graph of two variables varying inversely on the <a href="/wiki/Cartesian_coordinate" class="mw-redirect" title="Cartesian coordinate">Cartesian coordinate</a> plane is a <a href="/wiki/Rectangular_hyperbola" class="mw-redirect" title="Rectangular hyperbola">rectangular hyperbola</a>. The product of the <i>x</i> and <i>y</i> values of each point on the curve equals the constant of proportionality (<i>k</i>). Since neither <i>x</i> nor <i>y</i> can equal zero (because <i>k</i> is non-zero), the graph never crosses either axis. </p><p>Direct and inverse proportion contrast as follows: in direct proportion the variables increase or decrease together. With inverse proportion, an increase in one variable is associated with a decrease in the other. For instance, in travel, a constant speed dictates a direct proportion between distance and time travelled; in contrast, for a given distance (the constant), the time of travel is inversely proportional to speed: <i>s</i> × <i>t</i> = <i>d</i>. </p> <div class="mw-heading mw-heading2"><h2 id="Hyperbolic_coordinates">Hyperbolic coordinates</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Proportionality_(mathematics)&action=edit&section=4" title="Edit section: Hyperbolic coordinates"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Hyperbolic_coordinates" title="Hyperbolic coordinates">Hyperbolic coordinates</a></div> <p>The concepts of <i>direct</i> and <i>inverse</i> proportion lead to the location of points in the Cartesian plane by <a href="/wiki/Hyperbolic_coordinates" title="Hyperbolic coordinates">hyperbolic coordinates</a>; the two coordinates correspond to the constant of direct proportionality that specifies a point as being on a particular <a href="/wiki/Line_(mathematics)#Ray" class="mw-redirect" title="Line (mathematics)">ray</a> and the <a href="/wiki/Constant_(mathematics)" title="Constant (mathematics)">constant</a> of inverse proportionality that specifies a point as being on a particular <a href="/wiki/Hyperbola" title="Hyperbola">hyperbola</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Computer_encoding">Computer encoding</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Proportionality_(mathematics)&action=edit&section=5" title="Edit section: Computer encoding"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Unicode" title="Unicode">Unicode</a> characters for proportionality are the following: </p> <ul><li><span class="nowrap"><style data-mw-deduplicate="TemplateStyles:r886049734">.mw-parser-output .monospaced{font-family:monospace,monospace}</style><span class="monospaced">U+221D</span> </span><span style="font-size:125%;line-height:1em">∝</span> <span style="font-variant: small-caps; text-transform: lowercase;">PROPORTIONAL TO</span> (<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734"><span class="monospaced">&prop;, &Proportional;, &propto;, &varpropto;, &vprop;</span>)</li> <li><span class="nowrap"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734"><span class="monospaced">U+007E</span> </span><span style="font-size:125%;line-height:1em">~</span> <span style="font-variant: small-caps; text-transform: lowercase;">TILDE</span></li> <li><span class="nowrap"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734"><span class="monospaced">U+2237</span> </span><span style="font-size:125%;line-height:1em">∷</span> <span style="font-variant: small-caps; text-transform: lowercase;">PROPORTION</span></li> <li><span class="nowrap"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734"><span class="monospaced">U+223C</span> </span><span style="font-size:125%;line-height:1em">∼</span> <span style="font-variant: small-caps; text-transform: lowercase;">TILDE OPERATOR</span> (<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734"><span class="monospaced">&sim;, &thicksim;, &thksim;, &Tilde;</span>)</li> <li><span class="nowrap"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734"><span class="monospaced">U+223A</span> </span><span style="font-size:125%;line-height:1em">∺</span> <span style="font-variant: small-caps; text-transform: lowercase;">GEOMETRIC PROPORTION</span> (<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734"><span class="monospaced">&mDDot;</span>)</li></ul> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Proportionality_(mathematics)&action=edit&section=6" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Linear_map" title="Linear map">Linear map</a></li> <li><a href="/wiki/Correlation" title="Correlation">Correlation</a></li> <li><a href="/wiki/Eudoxus_of_Cnidus" title="Eudoxus of Cnidus">Eudoxus of Cnidus</a></li> <li><a href="/wiki/Golden_ratio" title="Golden ratio">Golden ratio</a></li> <li><a href="/wiki/Inverse-square_law" title="Inverse-square law">Inverse-square law</a></li> <li><a href="/wiki/Proportional_font" class="mw-redirect" title="Proportional font">Proportional font</a></li> <li><a href="/wiki/Ratio" title="Ratio">Ratio</a></li> <li><a href="/wiki/Rule_of_three_(mathematics)" class="mw-redirect" title="Rule of three (mathematics)">Rule of three (mathematics)</a></li> <li><a href="/wiki/Sample_size" class="mw-redirect" title="Sample size">Sample size</a></li> <li><a href="/wiki/Similarity_(geometry)" title="Similarity (geometry)">Similarity</a></li> <li><a href="/wiki/Trair%C4%81%C5%9Bika" title="Trairāśika">Trairāśika</a></li> <li><a href="/wiki/Basic_proportionality_theorem" class="mw-redirect" title="Basic proportionality theorem">Basic proportionality theorem</a></li></ul> <dl><dt>Growth</dt></dl> <ul><li><a href="/wiki/Linear_growth" class="mw-redirect" title="Linear growth">Linear growth</a></li> <li><a href="/wiki/Hyperbolic_growth" title="Hyperbolic growth">Hyperbolic growth</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Proportionality_(mathematics)&action=edit&section=7" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">Weisstein, Eric W. <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/DirectlyProportional.html">"Directly Proportional"</a>. <i>MathWorld</i> – A Wolfram Web Resource.</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.math.net/inverse-variation">"Inverse variation"</a>. <i>math.net</i><span class="reference-accessdate">. Retrieved <span class="nowrap">October 31,</span> 2021</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=math.net&rft.atitle=Inverse+variation&rft_id=https%3A%2F%2Fwww.math.net%2Finverse-variation&rfr_id=info%3Asid%2Fen.wikipedia.org%3AProportionality+%28mathematics%29" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">Weisstein, Eric W. <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/InverselyProportional.html">"Inversely Proportional"</a>. <i>MathWorld</i> – A Wolfram Web Resource.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Proportionality_(mathematics)&action=edit&section=8" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Ya. B. Zeldovich, <a href="/wiki/Isaak_Yaglom" title="Isaak Yaglom">I. M. Yaglom</a>: <i>Higher math for beginners</i>, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=dUB8BwAAQBAJ&pg=PA35">p. 34–35</a>.</li> <li>Brian Burrell: <i>Merriam-Webster's Guide to Everyday Math: A Home and Business Reference</i>. Merriam-Webster, 1998, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780877796213" title="Special:BookSources/9780877796213">9780877796213</a>, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=XeaorGgYAXsC&pg=PA85">p. 85–101</a>.</li> <li>Lanius, Cynthia S.; Williams Susan E.: <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/41181344"><i>PROPORTIONALITY: A Unifying Theme for the Middle Grades</i></a>. Mathematics Teaching in the Middle School 8.8 (2003), p. 392–396.</li> <li>Seeley, Cathy; Schielack Jane F.: <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/41182513"><i>A Look at the Development of Ratios, Rates, and Proportionality</i></a>. Mathematics Teaching in the Middle School, 13.3, 2007, p. 140–142.</li> <li>Van Dooren, Wim; De Bock Dirk; Evers Marleen; Verschaffel Lieven : <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/40539331"><i>Students' Overuse of Proportionality on Missing-Value Problems: How Numbers May Change Solutions</i></a>. 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