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class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Unicode"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.2</span> <span>Unicode</span> </div> </a> <ul id="toc-Unicode-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Science" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Science"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Science</span> </div> </a> <ul id="toc-Science-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Law" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Law"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Law</span> </div> </a> <ul id="toc-Law-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-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">6</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header 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</div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Approximation</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 44 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-44" 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">44 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AA%D9%82%D8%B1%D9%8A%D8%A8" title="تقريب – Arabic" lang="ar" hreflang="ar" data-title="تقريب" 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/Approksimasiya" title="Approksimasiya – Azerbaijani" lang="az" hreflang="az" data-title="Approksimasiya" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%90%D0%BF%D1%80%D0%B0%D0%BA%D1%81%D1%96%D0%BC%D0%B0%D1%86%D1%8B%D1%8F" title="Апраксімацыя – Belarusian" lang="be" hreflang="be" data-title="Апраксімацыя" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%90%D0%BF%D1%80%D0%B0%D0%BA%D1%81%D1%8B%D0%BC%D0%B0%D1%86%D1%8B%D1%8F" title="Апраксымацыя – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Апраксымацыя" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%90%D0%BF%D1%80%D0%BE%D0%BA%D1%81%D0%B8%D0%BC%D0%B0%D1%86%D0%B8%D1%8F" title="Апроксимация – Bulgarian" lang="bg" hreflang="bg" data-title="Апроксимация" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-br mw-list-item"><a href="https://br.wikipedia.org/wiki/Tostadur" title="Tostadur – Breton" lang="br" hreflang="br" data-title="Tostadur" data-language-autonym="Brezhoneg" data-language-local-name="Breton" class="interlanguage-link-target"><span>Brezhoneg</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%90%D0%BF%D0%BF%D1%80%D0%BE%D0%BA%D1%81%D0%B8%D0%BC%D0%B0%D1%86%D0%B8" title="Аппроксимаци – Chuvash" lang="cv" hreflang="cv" data-title="Аппроксимаци" 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/Aproximace" title="Aproximace – Czech" lang="cs" hreflang="cs" data-title="Aproximace" 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/Chipo_(huwandu)" title="Chipo (huwandu) – Shona" lang="sn" hreflang="sn" data-title="Chipo (huwandu)" data-language-autonym="ChiShona" data-language-local-name="Shona" class="interlanguage-link-target"><span>ChiShona</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Brasamcan" title="Brasamcan – Welsh" lang="cy" hreflang="cy" data-title="Brasamcan" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Approksimation" title="Approksimation – Danish" lang="da" hreflang="da" data-title="Approksimation" 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/Approximation" title="Approximation – German" lang="de" hreflang="de" data-title="Approximation" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Aproximaci%C3%B3n" title="Aproximación – Spanish" lang="es" hreflang="es" data-title="Aproximación" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Proksimuma_kalkulado" title="Proksimuma kalkulado – Esperanto" lang="eo" hreflang="eo" data-title="Proksimuma kalkulado" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AA%D9%82%D8%B1%DB%8C%D8%A8" title="تقریب – Persian" lang="fa" hreflang="fa" data-title="تقریب" 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/Approximation" title="Approximation – French" lang="fr" hreflang="fr" data-title="Approximation" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EA%B7%BC%EC%82%BF%EA%B0%92" title="근삿값 – Korean" lang="ko" hreflang="ko" data-title="근삿값" 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%84%D5%B8%D5%BF%D5%A1%D6%80%D5%AF%D5%B8%D6%82%D5%B4" title="Մոտարկում – Armenian" lang="hy" hreflang="hy" data-title="Մոտարկում" 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%A8%E0%A5%8D%E0%A4%A8%E0%A4%BF%E0%A4%95%E0%A4%9F%E0%A4%A8" title="सन्निकटन – Hindi" lang="hi" hreflang="hi" data-title="सन्निकटन" 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/Aproksimacija" title="Aproksimacija – Croatian" lang="hr" hreflang="hr" data-title="Aproksimacija" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Penghampiran" title="Penghampiran – Indonesian" lang="id" hreflang="id" data-title="Penghampiran" 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-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Approssimazione" title="Approssimazione – Italian" lang="it" hreflang="it" data-title="Approssimazione" 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%A7%D7%99%D7%A8%D7%95%D7%91" title="קירוב – Hebrew" lang="he" hreflang="he" data-title="קירוב" 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%90%D0%BF%D0%BF%D1%80%D0%BE%D0%BA%D1%81%D0%B8%D0%BC%D0%B0%D1%86%D0%B8%D1%8F" title="Аппроксимация – Kazakh" lang="kk" hreflang="kk" data-title="Аппроксимация" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%91%D0%BE%D0%BB%D0%B6%D0%BE%D0%BB%D0%B4%D0%BE%D0%BE" title="Болжолдоо – Kyrgyz" lang="ky" hreflang="ky" data-title="Болжолдоо" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Benadering" title="Benadering – Dutch" lang="nl" hreflang="nl" data-title="Benadering" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-new mw-list-item"><a href="https://new.wikipedia.org/wiki/%E0%A4%85%E0%A4%A8%E0%A5%81%E0%A4%AE%E0%A4%BE%E0%A4%A8%E0%A4%82_(%E0%A4%B8%E0%A4%A8%E0%A5%8D_%E0%A5%A7%E0%A5%AF%E0%A5%AC%E0%A5%A7%E0%A4%AF%E0%A4%BE_%E0%A4%B8%E0%A4%82%E0%A4%95%E0%A4%BF%E0%A4%AA%E0%A4%BE)" title="अनुमानं (सन् १९६१या संकिपा) – Newari" lang="new" hreflang="new" data-title="अनुमानं (सन् १९६१या संकिपा)" data-language-autonym="नेपाल भाषा" data-language-local-name="Newari" class="interlanguage-link-target"><span>नेपाल भाषा</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E8%BF%91%E4%BC%BC" title="近似 – Japanese" lang="ja" hreflang="ja" data-title="近似" 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-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Approksimasjon" title="Approksimasjon – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Approksimasjon" 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-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Approksimatsiya" title="Approksimatsiya – Uzbek" lang="uz" hreflang="uz" data-title="Approksimatsiya" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Aproksymacja" title="Aproksymacja – Polish" lang="pl" hreflang="pl" data-title="Aproksymacja" 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/Aproxima%C3%A7%C3%A3o" title="Aproximação – Portuguese" lang="pt" hreflang="pt" data-title="Aproximação" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%90%D0%BF%D0%BF%D1%80%D0%BE%D0%BA%D1%81%D0%B8%D0%BC%D0%B0%D1%86%D0%B8%D1%8F" title="Аппроксимация – Russian" lang="ru" hreflang="ru" data-title="Аппроксимация" 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/Approximation" title="Approximation – Simple English" lang="en-simple" hreflang="en-simple" data-title="Approximation" 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/Aproxim%C3%A1cia" title="Aproximácia – Slovak" lang="sk" hreflang="sk" data-title="Aproximácia" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%90%D0%BF%D1%80%D0%BE%D0%BA%D1%81%D0%B8%D0%BC%D0%B0%D1%86%D0%B8%D1%98%D0%B0" title="Апроксимација – Serbian" lang="sr" hreflang="sr" data-title="Апроксимација" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Approksimaatio" title="Approksimaatio – Finnish" lang="fi" hreflang="fi" data-title="Approksimaatio" 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/N%C3%A4rmev%C3%A4rde" title="Närmevärde – Swedish" lang="sv" hreflang="sv" data-title="Närmevärde" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%85%E0%AE%A3%E0%AF%8D%E0%AE%A3%E0%AE%B3%E0%AE%B5%E0%AE%BE%E0%AE%95%E0%AF%8D%E0%AE%95%E0%AE%AE%E0%AF%8D" title="அண்ணளவாக்கம் – Tamil" lang="ta" hreflang="ta" data-title="அண்ணளவாக்கம்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-tg mw-list-item"><a href="https://tg.wikipedia.org/wiki/%D0%90%D0%BF%D0%BF%D1%80%D0%BE%D0%BA%D1%81%D0%B8%D0%BC%D0%B0%D1%82%D1%81%D0%B8%D1%8F" title="Аппроксиматсия – Tajik" lang="tg" hreflang="tg" data-title="Аппроксиматсия" data-language-autonym="Тоҷикӣ" data-language-local-name="Tajik" class="interlanguage-link-target"><span>Тоҷикӣ</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%90%D0%BF%D1%80%D0%BE%D0%BA%D1%81%D0%B8%D0%BC%D0%B0%D1%86%D1%96%D1%8F" title="Апроксимація – Ukrainian" lang="uk" hreflang="uk" data-title="Апроксимація" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Ph%C3%A9p_x%E1%BA%A5p_x%E1%BB%89" title="Phép xấp xỉ – Vietnamese" lang="vi" hreflang="vi" data-title="Phép xấp xỉ" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E8%BF%91%E4%BC%BC%E5%80%BC" title="近似值 – Cantonese" lang="yue" hreflang="yue" data-title="近似值" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E8%BF%91%E4%BC%BC" title="近似 – Chinese" lang="zh" hreflang="zh" data-title="近似" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q27058#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namespaces"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet 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Not to be confused with <a href="/wiki/Approximant" title="Approximant">Approximant</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_citations_needed plainlinks metadata ambox ambox-content ambox-Refimprove" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article <b>needs additional citations for <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">verification</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Approximation" title="Special:EditPage/Approximation">improve this article</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and removed.<br /><small><span class="plainlinks"><i>Find sources:</i> <a rel="nofollow" class="external text" href="https://www.google.com/search?as_eq=wikipedia&q=%22Approximation%22">"Approximation"</a> – <a rel="nofollow" class="external text" href="https://www.google.com/search?tbm=nws&q=%22Approximation%22+-wikipedia&tbs=ar:1">news</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?&q=%22Approximation%22&tbs=bkt:s&tbm=bks">newspapers</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?tbs=bks:1&q=%22Approximation%22+-wikipedia">books</a> <b>·</b> <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?q=%22Approximation%22">scholar</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.jstor.org/action/doBasicSearch?Query=%22Approximation%22&acc=on&wc=on">JSTOR</a></span></small></span> <span class="date-container"><i>(<span class="date">April 2013</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> <p>An <b>approximation</b> is anything that is intentionally similar but not exactly <a href="/wiki/Equality_(mathematics)" title="Equality (mathematics)">equal</a> to something else. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Etymology_and_usage">Etymology and usage</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Approximation&action=edit&section=1" title="Edit section: Etymology and usage"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The word <i>approximation</i> is derived from <a href="/wiki/Latin" title="Latin">Latin</a> <i>approximatus</i>, from <i>proximus</i> meaning <i>very near</i> and the <a href="/wiki/Prefix" title="Prefix">prefix</a> <i>ad-</i> (<i>ad-</i> before <i>p</i> becomes ap- by <a href="/wiki/Assimilation_(phonology)" title="Assimilation (phonology)">assimilation</a>) meaning <i>to</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> Words like <i>approximate</i>, <i>approximately</i> and <i>approximation</i> are used especially in technical or scientific contexts. In everyday English, words such as <i>roughly</i> or <i>around</i> are used with a similar meaning.<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> It is often found abbreviated as <i>approx.</i> </p><p>The term can be applied to various properties (e.g., value, quantity, image, description) that are nearly, but not exactly correct; similar, but not exactly the same (e.g., the approximate time was 10 o'clock). </p><p>Although approximation is most often applied to <a href="/wiki/Number" title="Number">numbers</a>, it is also frequently applied to such things as <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">mathematical functions</a>, <a href="/wiki/Shape" title="Shape">shapes</a>, and <a href="/wiki/Physical_law" class="mw-redirect" title="Physical law">physical laws</a>. </p><p>In science, approximation can refer to using a simpler process or model when the correct model is difficult to use. An approximate model is used to make calculations easier. Approximations might also be used if incomplete <a href="/wiki/Information" title="Information">information</a> prevents use of exact representations. </p><p>The type of approximation used depends on the available <a href="/wiki/Information" title="Information">information</a>, <a href="/wiki/Order_of_approximation" title="Order of approximation">the degree of accuracy required</a>, the sensitivity of the problem to this data, and the savings (usually in time and effort) that can be achieved by approximation. </p> <div class="mw-heading mw-heading2"><h2 id="Mathematics">Mathematics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Approximation&action=edit&section=2" title="Edit section: Mathematics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Approximation_theory" title="Approximation theory">Approximation theory</a> is a branch of mathematics, and a quantitative part of <a href="/wiki/Functional_analysis" title="Functional analysis">functional analysis</a>. <a href="/wiki/Diophantine_approximation" title="Diophantine approximation">Diophantine approximation</a> deals with approximations of <a href="/wiki/Real_number" title="Real number">real numbers</a> by <a href="/wiki/Rational_number" title="Rational number">rational numbers</a>. </p><p>Approximation usually occurs when an exact form or an exact numerical number is unknown or difficult to obtain. However some known form may exist and may be able to represent the real form so that no significant deviation can be found. For example, 1.5 × 10<sup>6</sup> means that the true value of something being measured is 1,500,000 to the nearest hundred thousand (so the actual value is somewhere between 1,450,000 and 1,550,000); this is in contrast to the notation 1.500 × 10<sup>6</sup>, which means that the true value is 1,500,000 to the nearest thousand (implying that the true value is somewhere between 1,499,500 and 1,500,500). </p><p><a href="/wiki/Numerical_approximation" class="mw-redirect" title="Numerical approximation">Numerical approximations</a> sometimes result from using a small number of <a href="/wiki/Significant_figures" title="Significant figures">significant digits</a>. Calculations are likely to involve <a href="/wiki/Round-off_error" title="Round-off error">rounding errors</a> and other <a href="/wiki/Approximation_error" title="Approximation error">approximation errors</a>. <a href="/wiki/Logarithm" title="Logarithm">Log tables</a>, slide rules and calculators produce approximate answers to all but the simplest calculations. The results of computer calculations are normally an approximation expressed in a limited number of significant digits, although they can be programmed to produce more precise results.<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> Approximation can occur when a decimal number cannot be expressed in a finite number of binary digits. </p><p>Related to approximation of functions is the <a href="/wiki/Asymptotic_analysis" title="Asymptotic analysis">asymptotic</a> value of a function, i.e. the value as one or more of a function's parameters becomes arbitrarily large. For example, the sum <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k/2+k/4+k/8+\cdots +k/2^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo>+</mo> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> <mo>+</mo> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>8</mn> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k/2+k/4+k/8+\cdots +k/2^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da2d92e6fa50e63ba5148643c20faa73e9811744" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.448ex; height:2.843ex;" alt="{\displaystyle k/2+k/4+k/8+\cdots +k/2^{n}}"></span>⁠</span> is asymptotically equal to <i>k</i>. No consistent notation is used throughout mathematics and some texts use ≈ to mean approximately equal and ~ to mean asymptotically equal whereas other texts use the symbols the other way around. </p> <div class="mw-heading mw-heading3"><h3 id="Typography">Typography <span class="anchor" id="approximately_equals_sign"></span><span class="anchor" id="approximately_equal_to"></span><span class="anchor" id="almost_equal_to"></span></h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Approximation&action=edit&section=3" title="Edit section: Typography"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1257001546">.mw-parser-output .infobox-subbox{padding:0;border:none;margin:-3px;width:auto;min-width:100%;font-size:100%;clear:none;float:none;background-color:transparent}.mw-parser-output .infobox-3cols-child{margin:auto}.mw-parser-output .infobox .navbar{font-size:100%}@media screen{html.skin-theme-clientpref-night .mw-parser-output .infobox-full-data:not(.notheme)>div:not(.notheme)[style]{background:#1f1f23!important;color:#f8f9fa}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .infobox-full-data:not(.notheme) div:not(.notheme){background:#1f1f23!important;color:#f8f9fa}}@media(min-width:640px){body.skin--responsive .mw-parser-output .infobox-table{display:table!important}body.skin--responsive .mw-parser-output .infobox-table>caption{display:table-caption!important}body.skin--responsive .mw-parser-output .infobox-table>tbody{display:table-row-group}body.skin--responsive .mw-parser-output .infobox-table tr{display:table-row!important}body.skin--responsive .mw-parser-output .infobox-table th,body.skin--responsive .mw-parser-output .infobox-table td{padding-left:inherit;padding-right:inherit}}</style><table class="infobox" style="max-width:20em;"><tbody><tr><th colspan="2" class="infobox-above" style="background: #ccc; font-size: 500%; line-height:1.3; font-weight:normal; font-family:'times new roman', times, georgia, serif, sans-serif;"> ≅ ≈  </th></tr><tr><th colspan="2" class="infobox-header" style="background: #ccc;"><div style="background: #ccc; font-size: 125%; font-weight: bold; padding: 0.1em 0.25em 0.2em; line-height: 1.2em;">Approximately equal to<br />Almost equal to</div></th></tr><tr><th scope="row" class="infobox-label">In <a href="/wiki/Unicode" title="Unicode">Unicode</a></th><td class="infobox-data"><span class="nowrap"><style data-mw-deduplicate="TemplateStyles:r886049734">.mw-parser-output .monospaced{font-family:monospace,monospace}</style><span class="monospaced">U+2245</span> </span><span style="font-size:125%;line-height:1em">≅</span> <span style="font-variant: small-caps; text-transform: lowercase;">APPROXIMATELY EQUAL TO</span> (<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734"><span class="monospaced">&cong;, &TildeFullEqual;</span>)<br /><span class="nowrap"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734"><span class="monospaced">U+2248</span> </span><span style="font-size:125%;line-height:1em">≈</span> <span style="font-variant: small-caps; text-transform: lowercase;">ALMOST EQUAL TO</span> (<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734"><span class="monospaced">&ap;, &approx;, &asymp;, &thickapprox;, &thkap;, &TildeTilde;</span>)</td></tr><tr><th colspan="2" class="infobox-header" style="background: #ccc;">Different from</th></tr><tr><th scope="row" class="infobox-label">Different from</th><td class="infobox-data"><span class="nowrap"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734"><span class="monospaced">U+2242</span> </span><span style="font-size:125%;line-height:1em">≂</span> <span style="font-variant: small-caps; text-transform: lowercase;">MINUS TILDE</span></td></tr><tr><th colspan="2" class="infobox-header" style="background: #ccc;">Related</th></tr><tr><th scope="row" class="infobox-label">See also</th><td class="infobox-data"><span class="nowrap"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734"><span class="monospaced">U+2249</span> </span><span style="font-size:125%;line-height:1em">≉</span> <span style="font-variant: small-caps; text-transform: lowercase;"><a class="mw-selflink-fragment" href="#Unicode">NOT ALMOST EQUAL TO</a></span><br /><span class="nowrap"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734"><span class="monospaced">U+003D</span> </span><span style="font-size:125%;line-height:1em">=</span> <span style="font-variant: small-caps; text-transform: lowercase;"><a href="/wiki/Equals_sign" title="Equals sign">EQUALS SIGN</a></span><br /><span class="nowrap"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734"><span class="monospaced">U+2243</span> </span><span style="font-size:125%;line-height:1em">≃</span> <span style="font-variant: small-caps; text-transform: lowercase;">ASYMPTOTICALLY EQUAL TO</span></td></tr></tbody></table> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Glossary_of_mathematical_symbols#Equality,_equivalence_and_similarity" title="Glossary of mathematical symbols">Glossary of mathematical symbols § Equality, equivalence and similarity</a></div> <p>The <b>approximately equals sign</b>, <b>≈</b>, was introduced by British mathematician <a href="/wiki/Alfred_Greenhill" class="mw-redirect" title="Alfred Greenhill">Alfred Greenhill</a>.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="The citation that was provided did not mention Alfred Greenhill (October 2024)">citation needed</span></a></i>]</sup> </p> <div class="mw-heading mw-heading4"><h4 id="LaTeX_symbols">LaTeX symbols</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Approximation&action=edit&section=4" title="Edit section: LaTeX symbols"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Symbols used in <a href="/wiki/LaTeX" title="LaTeX">LaTeX</a> markup. </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 \approx }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≈</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \approx }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f58f4c2b73283ce8a5ad28fb3746f2a8c998789" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.125ex; margin-bottom: -0.297ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \approx }"></span> (<code>\approx</code>), usually to indicate approximation between numbers, like <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi \approx 3.14}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo>≈</mo> <mn>3.14</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi \approx 3.14}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1ebf415189f77817a10ef0e344fa92de8be11ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.565ex; height:2.176ex;" alt="{\displaystyle \pi \approx 3.14}"></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 \not \approx }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≉</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \not \approx }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29325ae4f84a9793819e275b3b978733f7877317" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.809ex; height:2.676ex;" alt="{\displaystyle \not \approx }"></span> (<code>\not\approx</code>), usually to indicate that numbers are not approximately equal (<span class="mwe-math-element"><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\not \approx 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>≉</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\not \approx 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ceb3e46934b27886bc575634d6acf5c47bffd1c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.425ex; height:2.676ex;" alt="{\displaystyle 1\not \approx 2}"></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 \simeq }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≃</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \simeq }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65b9738551241417d16d9843525ed52410af4dc9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.081ex; margin-bottom: -0.253ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \simeq }"></span> (<code>\simeq</code>), usually to indicate asymptotic equivalence between functions, like <span class="mwe-math-element"><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(n)\simeq 3n^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>≃</mo> <mn>3</mn> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(n)\simeq 3n^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac1ebd869d14900651ca6be5a56bffef1b18f680" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.192ex; height:3.176ex;" alt="{\displaystyle f(n)\simeq 3n^{2}}"></span>. <ul><li>So writing <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi \simeq 3.14}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo>≃</mo> <mn>3.14</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi \simeq 3.14}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/178740bd526c0da54494dfec910012bc4d646d8f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.565ex; height:2.176ex;" alt="{\displaystyle \pi \simeq 3.14}"></span> would be wrong under this definition, despite wide use.</li></ul></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 \sim }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∼</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sim }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afcc42adfcfdc24d5c4c474869e5d8eaa78d1173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle \sim }"></span> (<code>\sim</code>), usually to indicate proportionality between functions, the same <span class="mwe-math-element"><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(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1c49fad1eccc4e9af1e4f23f32efdc3ac4da973" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.483ex; height:2.843ex;" alt="{\displaystyle f(n)}"></span> of the line above will be <span class="mwe-math-element"><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(n)\sim n^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>∼</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(n)\sim n^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/acf586a9dfd9d7f43343f3ca49b65a19707baeda" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.03ex; height:3.176ex;" alt="{\displaystyle f(n)\sim n^{2}}"></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 \cong }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≅</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cong }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a725ebc5ab8de11d7b71a8aa5a3706c2ea467885" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.049ex; margin-bottom: -0.22ex; width:1.808ex; height:1.843ex;" alt="{\displaystyle \cong }"></span> (<code>\cong</code>), usually to indicate congruence between figures, like <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta ABC\cong \Delta A'B'C'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>A</mi> <mi>B</mi> <mi>C</mi> <mo>≅</mo> <mi mathvariant="normal">Δ</mi> <msup> <mi>A</mi> <mo>′</mo> </msup> <msup> <mi>B</mi> <mo>′</mo> </msup> <msup> <mi>C</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta ABC\cong \Delta A'B'C'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/777cb65f88e0797811ae9e3126078d9c8e679084" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:19.603ex; height:2.509ex;" alt="{\displaystyle \Delta ABC\cong \Delta 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 \eqsim }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≂</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eqsim }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7575ecea581fb80faf30c9706065726a90cbf1bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.077ex; margin-bottom: -0.248ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \eqsim }"></span> (<code>\eqsim</code>), usually to indicate that two quantities are equal up to constants.</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 \lessapprox }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⪅</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lessapprox }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/991226da2309bcf8e3618996b0ff9689077c27d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:1.808ex; height:3.009ex;" alt="{\displaystyle \lessapprox }"></span> (<code>\lessapprox</code>) 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 \gtrapprox }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⪆</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gtrapprox }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/239fc92495834d2c3302bcf1eaae01f5888320f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:1.808ex; height:3.009ex;" alt="{\displaystyle \gtrapprox }"></span> (<code>\gtrapprox</code>), usually to indicate that either the inequality holds or the two values are approximately equal.</li></ul> <div class="mw-heading mw-heading4"><h4 id="Unicode">Unicode</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Approximation&action=edit&section=5" title="Edit section: Unicode"><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/Unicode_mathematical_operators" class="mw-redirect" title="Unicode mathematical operators">Unicode mathematical operators</a></div> <p>Symbols used to denote items that are approximately equal are wavy or dotted equals signs.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <table class="wikitable"><tbody><tr style="vertical-align:top"><td> <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></td><td> Which is also sometimes used to indicate <a href="/wiki/Proportionality_(mathematics)" title="Proportionality (mathematics)">proportionality.</a></td></tr><tr style="vertical-align:top"><td> <span class="nowrap"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734"><span class="monospaced">U+223D</span> </span><span style="font-size:125%;line-height:1em">∽</span> <span style="font-variant: small-caps; text-transform: lowercase;">REVERSED TILDE</span></td><td> Which is also sometimes used to indicate proportionality.</td></tr><tr style="vertical-align:top"><td> <span class="nowrap"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734"><span class="monospaced">U+2243</span> </span><span style="font-size:125%;line-height:1em">≃</span> <span style="font-variant: small-caps; text-transform: lowercase;">ASYMPTOTICALLY EQUAL TO</span></td><td> A combination of "≈" and "=", which is used to indicate <a href="/wiki/Asymptotic_analysis" title="Asymptotic analysis">asymptotic equality</a>.</td></tr><tr style="vertical-align:top"><td> <span class="nowrap"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734"><span class="monospaced">U+2245</span> </span><span style="font-size:125%;line-height:1em">≅</span> <span style="font-variant: small-caps; text-transform: lowercase;">APPROXIMATELY EQUAL TO</span></td><td> Another combination of "≈" and "=", which is used to indicate <a href="/wiki/Isomorphism" title="Isomorphism">isomorphism</a> or <a href="/wiki/Congruence_relation" title="Congruence relation">congruence</a>.</td></tr><tr style="vertical-align:top"><td> <span class="nowrap"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734"><span class="monospaced">U+2246</span> </span><span style="font-size:125%;line-height:1em">≆</span> <span style="font-variant: small-caps; text-transform: lowercase;">APPROXIMATELY BUT NOT ACTUALLY EQUAL TO</span></td><td></td></tr><tr style="vertical-align:top"><td> <span class="nowrap"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734"><span class="monospaced">U+2247</span> </span><span style="font-size:125%;line-height:1em">≇</span> <span style="font-variant: small-caps; text-transform: lowercase;">NEITHER APPROXIMATELY NOR ACTUALLY EQUAL TO</span></td><td></td></tr><tr style="vertical-align:top"><td> <span class="nowrap"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734"><span class="monospaced">U+2248</span> </span><span style="font-size:125%;line-height:1em">≈</span> <span style="font-variant: small-caps; text-transform: lowercase;">ALMOST EQUAL TO</span></td><td></td></tr><tr style="vertical-align:top"><td> <span class="nowrap"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734"><span class="monospaced">U+2249</span> </span><span style="font-size:125%;line-height:1em">≉</span> <span style="font-variant: small-caps; text-transform: lowercase;">NOT ALMOST EQUAL TO</span></td><td></td></tr><tr style="vertical-align:top"><td> <span class="nowrap"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734"><span class="monospaced">U+224A</span> </span><span style="font-size:125%;line-height:1em">≊</span> <span style="font-variant: small-caps; text-transform: lowercase;">ALMOST EQUAL OR EQUAL TO</span></td><td> Another combination of "≈" and "=", used to indicate equivalence or approximate equivalence.</td></tr><tr style="vertical-align:top"><td> <span class="nowrap"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734"><span class="monospaced">U+2250</span> </span><span style="font-size:125%;line-height:1em">≐</span> <span style="font-variant: small-caps; text-transform: lowercase;">APPROACHES THE LIMIT</span></td><td> Which can be used to represent the approach of a variable, <span class="texhtml mvar" style="font-style:italic;">y</span>, to a <a href="/wiki/Limit_(mathematics)" title="Limit (mathematics)">limit</a>; like the common syntax, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{x\to \infty }y(x)\doteq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mi>y</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≐</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{x\to \infty }y(x)\doteq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/508dd1ee25782a1f95e7cc811e0fef153b7e8baa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.169ex; height:3.843ex;" alt="{\displaystyle \lim _{x\to \infty }y(x)\doteq 0}"></span>.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup></td></tr><tr style="vertical-align:top"><td> <span class="nowrap"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734"><span class="monospaced">U+2252</span> </span><span style="font-size:125%;line-height:1em">≒</span> <span style="font-variant: small-caps; text-transform: lowercase;">APPROXIMATELY EQUAL TO OR THE IMAGE OF</span></td><td> Which is used like "<big>≈</big>" or "<big>≃</big>" in <a href="/wiki/Japanese_language" title="Japanese language">Japan</a>, <a href="/wiki/Taiwanese_Mandarin" title="Taiwanese Mandarin">Taiwan</a>, and <a href="/wiki/Korean_language" title="Korean language">Korea</a>.</td></tr><tr style="vertical-align:top"><td> <span class="nowrap"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734"><span class="monospaced">U+2253</span> </span><span style="font-size:125%;line-height:1em">≓</span> <span style="font-variant: small-caps; text-transform: lowercase;">IMAGE OF OR APPROXIMATELY EQUAL TO</span></td><td> A reversed variation of <span class="nowrap"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734"><span class="monospaced">U+2252</span> </span><span style="font-size:125%;line-height:1em">≒</span> <span style="font-variant: small-caps; text-transform: lowercase;">APPROXIMATELY EQUAL TO OR THE IMAGE OF</span>.</td></tr><tr style="vertical-align:top"><td> <span class="nowrap"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734"><span class="monospaced">U+225F</span> </span><span style="font-size:125%;line-height:1em">≟</span> <span style="font-variant: small-caps; text-transform: lowercase;"><a href="/wiki/%E2%89%9F" class="mw-redirect" title="≟">QUESTIONED EQUAL TO</a></span></td><td></td></tr><tr style="vertical-align:top"><td> <span class="nowrap"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734"><span class="monospaced">U+2A85</span> </span><span style="font-size:125%;line-height:1em">⪅</span> <span style="font-variant: small-caps; text-transform: lowercase;">LESS-THAN OR APPROXIMATE</span></td><td></td></tr><tr style="vertical-align:top"><td> <span class="nowrap"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734"><span class="monospaced">U+2A86</span> </span><span style="font-size:125%;line-height:1em">⪆</span> <span style="font-variant: small-caps; text-transform: lowercase;">GREATER-THAN OR APPROXIMATE</span></td><td></td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Science">Science</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Approximation&action=edit&section=6" title="Edit section: Science"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Approximation arises naturally in <a href="/wiki/Scientific_experiment" class="mw-redirect" title="Scientific experiment">scientific experiments</a>. The predictions of a scientific theory can differ from actual measurements. This can be because there are factors in the real situation that are not included in the theory. For example, simple calculations may not include the effect of air resistance. Under these circumstances, the theory is an approximation to reality. Differences may also arise because of limitations in the measuring technique. In this case, the measurement is an approximation to the actual value. </p><p>The <a href="/wiki/History_of_science" title="History of science">history of science</a> shows that earlier theories and laws can be <i>approximations</i> to some deeper set of laws. Under the <a href="/wiki/Correspondence_principle" title="Correspondence principle">correspondence principle</a>, a new scientific theory should reproduce the results of older, well-established, theories in those domains where the old theories work.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> The old theory becomes an approximation to the new theory. </p><p>Some problems in physics are too complex to solve by direct analysis, or progress could be limited by available analytical tools. Thus, even when the exact representation is known, an approximation may yield a sufficiently accurate solution while reducing the complexity of the problem significantly. <a href="/wiki/Physicists" class="mw-redirect" title="Physicists">Physicists</a> often approximate the <a href="/wiki/Shape_of_the_Earth" class="mw-redirect" title="Shape of the Earth">shape of the Earth</a> as a <a href="/wiki/Sphere" title="Sphere">sphere</a> even though more accurate representations are possible, because many physical characteristics (e.g., <a href="/wiki/Gravity" title="Gravity">gravity</a>) are much easier to calculate for a sphere than for other shapes. </p><p>Approximation is also used to analyze the motion of several planets orbiting a star. This is extremely difficult due to the complex interactions of the planets' gravitational effects on each other.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> An approximate solution is effected by performing <a href="/wiki/Iteration" title="Iteration">iterations</a>. In the first iteration, the planets' gravitational interactions are ignored, and the star is assumed to be fixed. If a more precise solution is desired, another iteration is then performed, using the positions and motions of the planets as identified in the first iteration, but adding a first-order gravity interaction from each planet on the others. This process may be repeated until a satisfactorily precise solution is obtained. </p><p>The use of <a href="/wiki/Perturbation_theory" title="Perturbation theory">perturbations</a> to correct for the errors can yield more accurate solutions. Simulations of the motions of the planets and the star also yields more accurate solutions. </p><p>The most common versions of <a href="/wiki/Philosophy_of_science" title="Philosophy of science">philosophy of science</a> accept that empirical <a href="/wiki/Measurement" title="Measurement">measurements</a> are always <i>approximations</i> — they do not perfectly represent what is being measured. </p> <div class="mw-heading mw-heading2"><h2 id="Law">Law</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Approximation&action=edit&section=7" title="Edit section: Law"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Within the <a href="/wiki/European_Union" title="European Union">European Union</a> (EU), "approximation" refers to a process through which EU legislation is implemented and incorporated within <a href="/wiki/Member_state_of_the_European_Union" title="Member state of the European Union">Member States</a>' national laws, despite variations in the existing legal framework in each country. Approximation is required as part of the <a href="/wiki/EU_accession" class="mw-redirect" title="EU accession">pre-accession process</a> for new member states,<sup id="cite_ref-env_8-0" class="reference"><a href="#cite_note-env-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> and as a continuing process when required by an <a href="/wiki/Directive_(European_Union)" title="Directive (European Union)">EU Directive</a>. <i>Approximation</i> is a key word generally employed within the title of a directive, for example the Trade Marks Directive of 16 December 2015 serves "to approximate the laws of the Member States relating to trade marks".<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> The <a href="/wiki/European_Commission" title="European Commission">European Commission</a> describes approximation of law as "a unique obligation of membership in the European Union".<sup id="cite_ref-env_8-1" class="reference"><a href="#cite_note-env-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Approximation&action=edit&section=8" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col"> <ul><li><a href="/wiki/Approximation_algorithm" title="Approximation algorithm">Approximation algorithm</a> – Class of algorithms that find approximate solutions to optimization problems</li> <li><a href="/wiki/Approximate_computing" title="Approximate computing">Approximate computing</a> – Computation of nearly accurate results</li> <li><a href="/wiki/Approximations_of_%CF%80" title="Approximations of π">Approximations of π</a> – Varying methods used to calculate pi</li> <li><a href="/wiki/Binomial_approximation" title="Binomial approximation">Binomial approximation</a> – Approximation of powers of some binomials</li> <li><a href="/wiki/Congruence_relation" title="Congruence relation">Congruence relation</a> – Equivalence relation in algebra</li> <li><a href="/wiki/Double_tilde_(disambiguation)" class="mw-redirect mw-disambig" title="Double tilde (disambiguation)">Double tilde (disambiguation)</a> – Various meanings of ~~ or ≈</li> <li><a href="/wiki/Estimation" title="Estimation">Estimation</a> – Process of finding an approximation</li> <li><a href="/wiki/Fermi_problem" title="Fermi problem">Fermi problem</a> – Estimation problem in physics or engineering education</li> <li><a href="/wiki/Idealization_(philosophy_of_science)" title="Idealization (philosophy of science)">Idealization (philosophy of science)</a> – Process by which a scientific model is simplified by assuming strictly false facts to be true</li> <li><a href="/wiki/Least_squares" title="Least squares">Least squares</a> – Approximation method in statistics</li> <li><a href="/wiki/Linear_approximation" title="Linear approximation">Linear approximation</a> – Approximation of a function by its tangent line at a point</li> <li><a href="/wiki/Newton%27s_method" title="Newton's method">Newton's method</a> – Algorithm for finding zeros of functions</li> <li><a href="/wiki/Order_of_approximation" title="Order of approximation">Order of approximation</a> – Expressions for approximation accuracy</li> <li><a href="/wiki/Rough_set" title="Rough set">Rough set</a> – Approximation of a mathematical set</li> <li><a href="/wiki/Runge%E2%80%93Kutta_methods" title="Runge–Kutta methods">Runge–Kutta methods</a> – Family of implicit and explicit iterative methods</li> <li><a href="/wiki/Significant_figures" title="Significant figures">Significant figures</a> – Any digit of a number within its measurement resolution, as opposed to spurious digits</li> <li><a href="/wiki/Small-angle_approximation" title="Small-angle approximation">Small-angle approximation</a> – Simplification of the basic trigonometric functions</li> <li><a href="/wiki/Successive-approximation_ADC" title="Successive-approximation ADC">Successive-approximation ADC</a> – Type of analog-to-digital converter</li> <li><a href="/wiki/Taylor_series" title="Taylor series">Taylor series</a> – Mathematical approximation of a function</li> <li><a href="/wiki/Tolerance_relation" title="Tolerance relation">Tolerance relation</a> – Math relation that is reflexive and symmetric</li> <li><a href="/wiki/Intuition" title="Intuition">Intuition</a> – Ability to acquire knowledge without conscious reasoning</li></ul> </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=Approximation&action=edit&section=9" title="Edit section: References"><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">The Concise Oxford Dictionary, <i>Eighth edition 1990, <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-19-861243-5" title="Special:BookSources/0-19-861243-5">0-19-861243-5</a></i></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">Longman Dictionary of Contemporary English, <i>Pearson Education Ltd 2009, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978%2B1%2B4082%2B1532%2B6" title="Special:BookSources/978+1+4082+1532+6">978 1 4082 1532 6</a></i></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20160406101256/http://docs.oracle.com/cd/E19957-01/806-3568/ncg_goldberg.html">"Numerical Computation Guide"</a>. Archived from <a rel="nofollow" class="external text" href="http://docs.oracle.com/cd/E19957-01/806-3568/ncg_goldberg.html">the original</a> on 2016-04-06<span class="reference-accessdate">. Retrieved <span class="nowrap">2013-06-16</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Numerical+Computation+Guide&rft_id=http%3A%2F%2Fdocs.oracle.com%2Fcd%2FE19957-01%2F806-3568%2Fncg_goldberg.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AApproximation" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.unicode.org/charts/PDF/U2200.pdf">"Mathematical Operators – Unicode"</a> <span class="cs1-format">(PDF)</span><span class="reference-accessdate">. Retrieved <span class="nowrap">2013-04-20</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Mathematical+Operators+%E2%80%93+Unicode&rft_id=https%3A%2F%2Fwww.unicode.org%2Fcharts%2FPDF%2FU2200.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AApproximation" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1"><a rel="nofollow" class="external text" href="https://books.google.com/books?id=7FPtZp8abSAC&dq=%22%E2%89%90%22+approach+limit&pg=PA366"><i>D & D Standard Oil & Gas Abbreviator</i></a>. PennWell. 2006. p. 366. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781593701086" title="Special:BookSources/9781593701086"><bdi>9781593701086</bdi></a><span class="reference-accessdate">. Retrieved <span class="nowrap">May 21,</span> 2020</span>. <q>≐ approaches a limit</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=D+%26+D+Standard+Oil+%26+Gas+Abbreviator&rft.pages=366&rft.pub=PennWell&rft.date=2006&rft.isbn=9781593701086&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D7FPtZp8abSAC%26dq%3D%2522%25E2%2589%2590%2522%2Bapproach%2Blimit%26pg%3DPA366&rfr_id=info%3Asid%2Fen.wikipedia.org%3AApproximation" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://www.britannica.com/EBchecked/topic/138678/correspondence-principle">Correspondence principle</a> – <i><a href="/wiki/Encyclop%C3%A6dia_Britannica" title="Encyclopædia Britannica">Encyclopædia Britannica</a></i></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://plus.maths.org/content/mathematical-mysteries-three-body-problem">The three body problem</a></span> </li> <li id="cite_note-env-8"><span class="mw-cite-backlink">^ <a href="#cite_ref-env_8-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-env_8-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">European Commission, <a rel="nofollow" class="external text" href="https://ec.europa.eu/environment/archives/guide/part1.htm">Guide to the Approximation of European Union Environmental Legislation</a>, last updated 2 August 2019, accessed 15 November 2022</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text">EUR-Lex, <a rel="nofollow" class="external text" href="https://eur-lex.europa.eu/legal-content/EN/TXT/?uri=CELEX%3A32015L2436&qid=1668514262360">Directive (EU) 2015/2436 of the European Parliament and of the Council of 16 December 2015 to approximate the laws of the Member States relating to trade marks (recast) (Text with EEA relevance)</a>, published 23 December 2015, accessed 15 November 2022</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Approximation&action=edit&section=10" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 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