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class="vector-toc-numb">2</span> <span>Buckingham π theorem</span> </div> </a> <ul id="toc-Buckingham_π_theorem-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Integers" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Integers"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Integers</span> </div> </a> <ul id="toc-Integers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ratios,_proportions,_and_angles" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Ratios,_proportions,_and_angles"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Ratios, proportions, and angles</span> </div> </a> <ul id="toc-Ratios,_proportions,_and_angles-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Dimensionless_physical_constants" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Dimensionless_physical_constants"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Dimensionless physical constants</span> </div> </a> <ul id="toc-Dimensionless_physical_constants-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-List" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#List"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>List</span> </div> </a> <button aria-controls="toc-List-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle List subsection</span> </button> <ul id="toc-List-sublist" class="vector-toc-list"> <li id="toc-Physics_and_engineering" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Physics_and_engineering"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Physics and engineering</span> </div> </a> <ul id="toc-Physics_and_engineering-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Chemistry" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Chemistry"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Chemistry</span> </div> </a> <ul id="toc-Chemistry-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_fields" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Other_fields"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.3</span> <span>Other fields</span> </div> </a> <ul id="toc-Other_fields-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</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">8</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-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">10</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Dimensionless quantity</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 40 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-40" 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">40 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/Dimensieloos" title="Dimensieloos – Afrikaans" lang="af" hreflang="af" data-title="Dimensieloos" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%83%D9%85%D9%8A%D8%A9_%D9%84%D8%A7_%D8%A8%D8%B9%D8%AF%D9%8A%D8%A9" title="كمية لا بعدية – Arabic" lang="ar" hreflang="ar" data-title="كمية لا بعدية" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%91%D0%B5%D0%B7%D1%80%D0%B0%D0%B7%D0%BC%D0%B5%D1%80%D0%BD%D0%B0%D1%8F_%D0%B2%D0%B5%D0%BB%D1%96%D1%87%D1%8B%D0%BD%D1%8F" title="Безразмерная велічыня – Belarusian" lang="be" hreflang="be" data-title="Безразмерная велічыня" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%91%D0%B5%D0%B7%D1%80%D0%B0%D0%B7%D0%BC%D0%B5%D1%80%D0%BD%D0%B0_%D0%B2%D0%B5%D0%BB%D0%B8%D1%87%D0%B8%D0%BD%D0%B0" title="Безразмерна величина – Bulgarian" lang="bg" hreflang="bg" data-title="Безразмерна величина" 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/Bezdimenziona_veli%C4%8Dina" title="Bezdimenziona veličina – Bosnian" lang="bs" hreflang="bs" data-title="Bezdimenziona veličina" 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/Magnitud_adimensional" title="Magnitud adimensional – Catalan" lang="ca" hreflang="ca" data-title="Magnitud adimensional" 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%A5%D0%B0%D0%BF%D0%B0%D1%81%C4%83%D1%80_%D0%BA%D0%B0%D0%BF" title="Хапасăр кап – Chuvash" lang="cv" hreflang="cv" data-title="Хапасăр кап" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Bezrozm%C4%9Brn%C3%A1_veli%C4%8Dina" title="Bezrozměrná veličina – Czech" lang="cs" hreflang="cs" data-title="Bezrozměrná veličina" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Dimensionslose_Gr%C3%B6%C3%9Fe" title="Dimensionslose Größe – German" lang="de" hreflang="de" data-title="Dimensionslose Größe" 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/Dimensioonita_suurus" title="Dimensioonita suurus – Estonian" lang="et" hreflang="et" data-title="Dimensioonita suurus" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Magnitud_adimensional" title="Magnitud adimensional – Spanish" lang="es" hreflang="es" data-title="Magnitud adimensional" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%DA%A9%D9%85%DB%8C%D8%AA_%D8%A8%D8%AF%D9%88%D9%86_%D8%A8%D8%B9%D8%AF" title="کمیت بدون بعد – Persian" lang="fa" hreflang="fa" data-title="کمیت بدون بعد" 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/Grandeur_sans_dimension" title="Grandeur sans dimension – French" lang="fr" hreflang="fr" data-title="Grandeur sans dimension" 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/%EB%AC%B4%EC%B0%A8%EC%9B%90%EB%9F%89" title="무차원량 – Korean" lang="ko" hreflang="ko" data-title="무차원량" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B5%E0%A4%BF%E0%A4%AE%E0%A4%BE%E0%A4%B9%E0%A5%80%E0%A4%A8_%E0%A4%B8%E0%A4%82%E0%A4%96%E0%A5%8D%E0%A4%AF%E0%A4%BE" title="विमाहीन संख्या – Hindi" lang="hi" hreflang="hi" data-title="विमाहीन संख्या" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Besaran_nirdimensi" title="Besaran nirdimensi – Indonesian" lang="id" hreflang="id" data-title="Besaran nirdimensi" 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-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%92%D7%95%D7%93%D7%9C_%D7%97%D7%A1%D7%A8_%D7%9E%D7%9E%D7%93" title="גודל חסר ממד – Hebrew" lang="he" hreflang="he" data-title="גודל חסר ממד" 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/%D3%A8%D0%BB%D1%88%D0%B5%D0%BC%D1%81%D1%96%D0%B7_%D1%88%D0%B0%D0%BC%D0%B0" title="Өлшемсіз шама – Kazakh" lang="kk" hreflang="kk" data-title="Өлшемсіз шама" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Bedimensis_dydis" title="Bedimensis dydis – Lithuanian" lang="lt" hreflang="lt" data-title="Bedimensis dydis" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Dimenzi%C3%B3mentes_mennyis%C3%A9g" title="Dimenziómentes mennyiség – Hungarian" lang="hu" hreflang="hu" data-title="Dimenziómentes mennyiség" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%91%D0%B5%D0%B7%D0%B4%D0%B8%D0%BC%D0%B5%D0%BD%D0%B7%D0%B8%D0%BE%D0%BD%D0%B0%D0%BB%D0%BD%D0%B0_%D0%B2%D0%B5%D0%BB%D0%B8%D1%87%D0%B8%D0%BD%D0%B0" title="Бездимензионална величина – Macedonian" lang="mk" hreflang="mk" data-title="Бездимензионална величина" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ne mw-list-item"><a href="https://ne.wikipedia.org/wiki/%E0%A4%B5%E0%A4%BF%E0%A4%AE%E0%A4%BE%E0%A4%B9%E0%A5%80%E0%A4%A8_%E0%A4%B8%E0%A4%82%E0%A4%96%E0%A5%8D%E0%A4%AF%E0%A4%BE" title="विमाहीन संख्या – Nepali" lang="ne" hreflang="ne" data-title="विमाहीन संख्या" 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/%E7%84%A1%E6%AC%A1%E5%85%83%E9%87%8F" title="無次元量 – Japanese" lang="ja" hreflang="ja" data-title="無次元量" 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/Dimensjonslaus_storleik" title="Dimensjonslaus storleik – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Dimensjonslaus storleik" 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%AA%E0%A8%BE%E0%A8%B8%E0%A8%BE%E0%A8%B0%E0%A8%B0%E0%A8%B9%E0%A8%BF%E0%A8%A4_%E0%A8%AE%E0%A8%BE%E0%A8%A4%E0%A8%B0%E0%A8%BE" title="ਪਾਸਾਰਰਹਿਤ ਮਾਤਰਾ – Punjabi" lang="pa" hreflang="pa" data-title="ਪਾਸਾਰਰਹਿਤ ਮਾਤਰਾ" 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/Wielko%C5%9B%C4%87_bezwymiarowa" title="Wielkość bezwymiarowa – Polish" lang="pl" hreflang="pl" data-title="Wielkość bezwymiarowa" 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/Magnitude_adimensional" title="Magnitude adimensional – Portuguese" lang="pt" hreflang="pt" data-title="Magnitude adimensional" 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/M%C4%83rime_adimensional%C4%83" title="Mărime adimensională – Romanian" lang="ro" hreflang="ro" data-title="Mărime adimensională" 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%91%D0%B5%D0%B7%D1%80%D0%B0%D0%B7%D0%BC%D0%B5%D1%80%D0%BD%D0%B0%D1%8F_%D0%B2%D0%B5%D0%BB%D0%B8%D1%87%D0%B8%D0%BD%D0%B0" title="Безразмерная величина – Russian" lang="ru" hreflang="ru" data-title="Безразмерная величина" 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/Dimensionless_quantity" title="Dimensionless quantity – Simple English" lang="en-simple" hreflang="en-simple" data-title="Dimensionless quantity" 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/Bezrozmern%C3%A1_veli%C4%8Dina" title="Bezrozmerná veličina – Slovak" lang="sk" hreflang="sk" data-title="Bezrozmerná veličina" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Brezrazse%C5%BEna_koli%C4%8Dina" title="Brezrazsežna količina – Slovenian" lang="sl" hreflang="sl" data-title="Brezrazsežna količina" 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/%D8%A8%DA%95%DB%8C_%D8%A8%DB%8E%DA%95%DB%95%DA%BE%DB%95%D9%86%D8%AF" title="بڕی بێڕەھەند – Central Kurdish" lang="ckb" hreflang="ckb" data-title="بڕی بێڕەھەند" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Dimensioton_suure" title="Dimensioton suure – Finnish" lang="fi" hreflang="fi" data-title="Dimensioton suure" 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/Dimensionsl%C3%B6s_storhet" title="Dimensionslös storhet – Swedish" lang="sv" hreflang="sv" data-title="Dimensionslös storhet" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Boyutsuz_nicelik" title="Boyutsuz nicelik – Turkish" lang="tr" hreflang="tr" data-title="Boyutsuz nicelik" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%91%D0%B5%D0%B7%D1%80%D0%BE%D0%B7%D0%BC%D1%96%D1%80%D0%BD%D1%96%D1%81%D0%BD%D0%B0_%D1%84%D1%96%D0%B7%D0%B8%D1%87%D0%BD%D0%B0_%D0%B2%D0%B5%D0%BB%D0%B8%D1%87%D0%B8%D0%BD%D0%B0" title="Безрозмірнісна фізична величина – Ukrainian" lang="uk" hreflang="uk" data-title="Безрозмірнісна фізична величина" data-language-autonym="Українська" data-language-local-name="Ukrainian" 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<div id="contentSub"><div id="mw-content-subtitle"><span class="mw-redirectedfrom">(Redirected from <a href="/w/index.php?title=Dimensionless_number&redirect=no" class="mw-redirect" title="Dimensionless number">Dimensionless number</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">Quantity with no physical dimension</div> <p class="mw-empty-elt"> </p> <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/Dimensionless_quantity" title="Special:EditPage/Dimensionless quantity">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=%22Dimensionless+quantity%22">"Dimensionless quantity"</a> – <a rel="nofollow" class="external text" href="https://www.google.com/search?tbm=nws&q=%22Dimensionless+quantity%22+-wikipedia&tbs=ar:1">news</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?&q=%22Dimensionless+quantity%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=%22Dimensionless+quantity%22+-wikipedia">books</a> <b>·</b> <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?q=%22Dimensionless+quantity%22">scholar</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.jstor.org/action/doBasicSearch?Query=%22Dimensionless+quantity%22&acc=on&wc=on">JSTOR</a></span></small></span> <span class="date-container"><i>(<span class="date">March 2017</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><b>Dimensionless quantities</b>, or quantities of dimension one,<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> are quantities <a href="/wiki/Implicitly_defined" class="mw-redirect" title="Implicitly defined">implicitly defined</a> in a manner that prevents their aggregation into <a href="/wiki/Unit_of_measurement" title="Unit of measurement">units of measurement</a>.<sup id="cite_ref-SI_Brochure_2-0" class="reference"><a href="#cite_note-SI_Brochure-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><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> Typically expressed as <a href="/wiki/Ratio" title="Ratio">ratios</a> that align with another system, these quantities do not necessitate explicitly defined <a href="/wiki/Unit_of_measurement" title="Unit of measurement">units</a>. For instance, <a href="/wiki/Alcohol_by_volume" title="Alcohol by volume">alcohol by volume</a> (ABV) represents a <a href="/wiki/Volumetric_ratio" class="mw-redirect" title="Volumetric ratio">volumetric ratio</a>; its value remains independent of the specific <a href="/wiki/Unit_of_volume" title="Unit of volume">units of volume</a> used, such as in <a href="/wiki/Milliliter" class="mw-redirect" title="Milliliter">milliliters</a> per milliliter (mL/mL). </p><p>The <a href="/wiki/1" title="1">number one</a> is recognized as a dimensionless <a href="/wiki/Base_unit_of_measurement" title="Base unit of measurement">base quantity</a>.<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> <a href="/wiki/Radian" title="Radian">Radians</a> serve as dimensionless units for <a href="/wiki/Angle" title="Angle">angular measurements</a>, derived from the universal ratio of 2π times the <a href="/wiki/Radius" title="Radius">radius</a> of a circle being equal to its circumference.<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> </p><p>Dimensionless quantities play a crucial role serving as <a href="/wiki/Parameter" title="Parameter">parameters</a> in <a href="/wiki/Differential_equation" title="Differential equation">differential equations</a> in various technical disciplines. In <a href="/wiki/Calculus" title="Calculus">calculus</a>, concepts like the unitless ratios in <a href="/wiki/Limits_of_integration" title="Limits of integration">limits</a> or <a href="/wiki/Derivative" title="Derivative">derivatives</a> often involve dimensionless quantities. In <a href="/wiki/Differential_geometry" title="Differential geometry">differential geometry</a>, the use of dimensionless parameters is evident in geometric relationships and transformations. Physics relies on dimensionless numbers like the <a href="/wiki/Reynolds_number" title="Reynolds number">Reynolds number</a> in <a href="/wiki/Fluid_dynamics" title="Fluid dynamics">fluid dynamics</a>,<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> the <a href="/wiki/Fine-structure_constant" title="Fine-structure constant">fine-structure constant</a> in <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>,<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> and the <a href="/wiki/Lorentz_factor" title="Lorentz factor">Lorentz factor</a> in <a href="/wiki/Special_relativity" title="Special relativity">relativity</a>.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> In <a href="/wiki/Chemistry" title="Chemistry">chemistry</a>, <a href="/wiki/Equation_of_state" title="Equation of state">state properties</a> and ratios such as <a href="/wiki/Mole_fraction" title="Mole fraction">mole fractions</a> <a href="/wiki/Concentration_ratio" title="Concentration ratio">concentration ratios</a> are dimensionless.<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> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dimensionless_quantity&action=edit&section=1" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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">See also: <a href="/wiki/Dimensional_analysis#History" title="Dimensional analysis">Dimensional analysis § History</a></div> <p>Quantities having dimension one, <i>dimensionless quantities</i>, regularly occur in sciences, and are formally treated within the field of <a href="/wiki/Dimensional_analysis" title="Dimensional analysis">dimensional analysis</a>. In the 19th century, French mathematician <a href="/wiki/Joseph_Fourier" title="Joseph Fourier">Joseph Fourier</a> and Scottish physicist <a href="/wiki/James_Clerk_Maxwell" title="James Clerk Maxwell">James Clerk Maxwell</a> led significant developments in the modern concepts of <a href="/wiki/Dimension" title="Dimension">dimension</a> and <a href="/wiki/Unit_(measurement)" class="mw-redirect" title="Unit (measurement)">unit</a>. Later work by British physicists <a href="/wiki/Osborne_Reynolds" title="Osborne Reynolds">Osborne Reynolds</a> and <a href="/wiki/Lord_Rayleigh" class="mw-redirect" title="Lord Rayleigh">Lord Rayleigh</a> contributed to the understanding of dimensionless numbers in physics. Building on Rayleigh's method of dimensional analysis, <a href="/wiki/Edgar_Buckingham" title="Edgar Buckingham">Edgar Buckingham</a> proved the <a href="/wiki/Buckingham_%CF%80_theorem" title="Buckingham π theorem"><span class="texhtml mvar" style="font-style:italic;">π</span> theorem</a> (independently of French mathematician <a href="/wiki/Joseph_Bertrand" title="Joseph Bertrand">Joseph Bertrand</a>'s previous work) to formalize the nature of these quantities.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p><p>Numerous dimensionless numbers, mostly ratios, were coined in the early 1900s, particularly in the areas of <a href="/wiki/Fluid_mechanics" title="Fluid mechanics">fluid mechanics</a> and <a href="/wiki/Heat_transfer" title="Heat transfer">heat transfer</a>. Measuring logarithm of ratios as <a href="/wiki/Level_quantity" class="mw-redirect" title="Level quantity"><i>levels</i></a> in the (derived) unit <a href="/wiki/Decibel" title="Decibel">decibel</a> (dB) finds widespread use nowadays. </p><p>There have been periodic proposals to "patch" the SI system to reduce confusion regarding physical dimensions. For example, a 2017 <a href="/wiki/Op-ed" title="Op-ed">op-ed</a> in <a href="/wiki/Nature_(journal)" title="Nature (journal)">Nature</a><sup id="cite_ref-nature_2017_11-0" class="reference"><a href="#cite_note-nature_2017-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> argued for formalizing the <a href="/wiki/Radian" title="Radian">radian</a> as a physical unit. The idea was rebutted<sup id="cite_ref-wendl_2017_12-0" class="reference"><a href="#cite_note-wendl_2017-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> on the grounds that such a change would raise inconsistencies for both established dimensionless groups, like the <a href="/wiki/Strouhal_number" title="Strouhal number">Strouhal number</a>, and for mathematically distinct entities that happen to have the same units, like <a href="/wiki/Torque" title="Torque">torque</a> (a <a href="/wiki/Cross_product" title="Cross product">vector product</a>) versus energy (a <a href="/wiki/Dot_product" title="Dot product">scalar product</a>). In another instance in the early 2000s, the <a href="/wiki/International_Committee_for_Weights_and_Measures" class="mw-redirect" title="International Committee for Weights and Measures">International Committee for Weights and Measures</a> discussed naming the unit of 1 as the "<a href="/wiki/Uno_(unit)" class="mw-redirect" title="Uno (unit)">uno</a>", but the idea of just introducing a new SI name for 1 was dropped.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p><p><br /> </p> <div class="mw-heading mw-heading2"><h2 id="Buckingham_π_theorem"><span id="Buckingham_.CF.80_theorem"></span>Buckingham π theorem</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dimensionless_quantity&action=edit&section=2" title="Edit section: Buckingham π theorem"><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/Buckingham_%CF%80_theorem" title="Buckingham π theorem">Buckingham π theorem</a></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-Unreferenced_section plainlinks metadata ambox ambox-content ambox-Unreferenced" 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 section <b>does not <a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources">cite</a> any <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">sources</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Dimensionless_quantity" title="Special:EditPage/Dimensionless quantity">improve this section</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 <a href="/wiki/Wikipedia:Verifiability#Burden_of_evidence" title="Wikipedia:Verifiability">removed</a>.</span> <span class="date-container"><i>(<span class="date">April 2022</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>The Buckingham <span class="texhtml mvar" style="font-style:italic;">π</span> theorem <sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> indicates that validity of the laws of physics does not depend on a specific unit system. A statement of this theorem is that any physical law can be expressed as an <a href="/wiki/Identity_(mathematics)" title="Identity (mathematics)">identity</a> involving only dimensionless combinations (ratios or products) of the variables linked by the law (e. g., pressure and volume are linked by <a href="/wiki/Boyle%27s_Law" class="mw-redirect" title="Boyle's Law">Boyle's Law</a> – they are inversely proportional). If the dimensionless combinations' values changed with the systems of units, then the equation would not be an identity, and Buckingham's theorem would not hold. </p><p>Another consequence of the theorem is that the <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">functional</a> dependence between a certain number (say, <i>n</i>) of <a href="/wiki/Variable_(mathematics)" title="Variable (mathematics)">variables</a> can be reduced by the number (say, <i>k</i>) of <a href="/wiki/Independent_variable" class="mw-redirect" title="Independent variable">independent</a> <a href="/wiki/Dimension" title="Dimension">dimensions</a> occurring in those variables to give a set of <i>p</i> = <i>n</i> − <i>k</i> independent, dimensionless <a href="/wiki/Quantity" title="Quantity">quantities</a>. For the purposes of the experimenter, different systems that share the same description by dimensionless <a href="/wiki/Quantity" title="Quantity">quantity</a> are equivalent. </p> <div class="mw-heading mw-heading2"><h2 id="Integers">Integers</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dimensionless_quantity&action=edit&section=3" title="Edit section: Integers"><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"><tbody><tr><th colspan="2" class="infobox-above">Number of entities</th></tr><tr><th scope="row" class="infobox-label"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;">Common symbols</div></th><td class="infobox-data"><i>N</i></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/SI_unit" class="mw-redirect" title="SI unit">SI unit</a></th><td class="infobox-data"><a href="/wiki/Unitless" class="mw-redirect" title="Unitless">Unitless</a></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Dimensional_analysis#Formulation" title="Dimensional analysis">Dimension</a></th><td class="infobox-data"><a href="/wiki/Dimension_one" class="mw-redirect" title="Dimension one">1</a></td></tr></tbody></table> <p><a href="/wiki/Integer_numbers" class="mw-redirect" title="Integer numbers">Integer numbers</a> may represent dimensionless quantities. They can represent discrete quantities, which can also be dimensionless. More specifically, <a href="/wiki/Counting_numbers" class="mw-redirect" title="Counting numbers">counting numbers</a> can be used to express <b>countable quantities</b>.<sup id="cite_ref-Rothstein2017_17-0" class="reference"><a href="#cite_note-Rothstein2017-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> The concept is formalized as quantity <b>number of entities</b> (symbol <i>N</i>) in <a href="/wiki/ISO_80000-1" class="mw-redirect" title="ISO 80000-1">ISO 80000-1</a>.<sup id="cite_ref-ISO_80000-1_19-0" class="reference"><a href="#cite_note-ISO_80000-1-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> Examples include <a href="/wiki/Number_of_particles" class="mw-redirect" title="Number of particles">number of particles</a> and <a href="/wiki/Population_size" title="Population size">population size</a>. In mathematics, the "number of elements" in a set is termed <i><a href="/wiki/Cardinality" title="Cardinality">cardinality</a></i>. <i><a href="/wiki/Countable_noun" class="mw-redirect" title="Countable noun">Countable nouns</a></i> is a related linguistics concept. Counting numbers, such as number of <a href="/wiki/Bit" title="Bit">bits</a>, can be compounded with units of frequency (<a href="/wiki/Inverse_second" title="Inverse second">inverse second</a>) to derive units of count rate, such as <a href="/wiki/Bits_per_second" class="mw-redirect" title="Bits per second">bits per second</a>. <a href="/wiki/Count_data" title="Count data">Count data</a> is a related concept in statistics. The concept may be generalized by allowing <a href="/wiki/Non-integer_number" class="mw-redirect" title="Non-integer number">non-integer numbers</a> to account for fractions of a full item, e.g., <a href="/wiki/Number_of_turns" class="mw-redirect" title="Number of turns">number of turns</a> equal to one half. </p> <div class="mw-heading mw-heading2"><h2 id="Ratios,_proportions,_and_angles"><span id="Ratios.2C_proportions.2C_and_angles"></span>Ratios, proportions, and angles</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dimensionless_quantity&action=edit&section=4" title="Edit section: Ratios, proportions, and angles"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dimensionless quantities can be obtained as <a href="/wiki/Ratio" title="Ratio">ratios</a> of quantities that are not dimensionless, but whose dimensions cancel out in the mathematical operation.<sup id="cite_ref-ISO_80000-1_19-1" class="reference"><a href="#cite_note-ISO_80000-1-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> Examples of quotients of dimension one include calculating <a href="/wiki/Slope" title="Slope">slopes</a> or some <a href="/wiki/Conversion_of_units" title="Conversion of units">unit conversion factors</a>. Another set of examples is <a href="/wiki/Mass_fraction_(chemistry)" title="Mass fraction (chemistry)">mass fractions</a> or <a href="/wiki/Mole_fraction" title="Mole fraction">mole fractions</a>, often written using <a href="/wiki/Parts-per_notation" title="Parts-per notation">parts-per notation</a> such as ppm (= 10<sup>−6</sup>), ppb (= 10<sup>−9</sup>), and ppt (= 10<sup>−12</sup>), or perhaps confusingly as ratios of two identical units (<a href="/wiki/Kilogram" title="Kilogram">kg</a>/kg or <a href="/wiki/Mole_(unit)" title="Mole (unit)">mol</a>/mol). For example, <a href="/wiki/Alcohol_by_volume" title="Alcohol by volume">alcohol by volume</a>, which characterizes the concentration of <a href="/wiki/Ethanol" title="Ethanol">ethanol</a> in an <a href="/wiki/Alcoholic_beverage" title="Alcoholic beverage">alcoholic beverage</a>, could be written as <span class="nowrap">mL / 100 mL</span>. </p><p>Other common proportions are percentages <a href="/wiki/%25" class="mw-redirect" title="%">%</a> (= 0.01),  <a href="/wiki/Per_mil" class="mw-redirect" title="Per mil">‰</a> (= 0.001). Some angle units such as <a href="/wiki/Turn_(angle)" title="Turn (angle)">turn</a>, <a href="/wiki/Radian" title="Radian">radian</a>, and <a href="/wiki/Steradian" title="Steradian">steradian</a> are defined as ratios of quantities of the same kind. In <a href="/wiki/Statistics" title="Statistics">statistics</a> the <a href="/wiki/Coefficient_of_variation" title="Coefficient of variation">coefficient of variation</a> is the ratio of the <a href="/wiki/Standard_deviation" title="Standard deviation">standard deviation</a> to the <a href="/wiki/Average" title="Average">mean</a> and is used to measure the <a href="/wiki/Statistical_dispersion" title="Statistical dispersion">dispersion</a> in the <a href="/wiki/Statistical_data" class="mw-redirect" title="Statistical data">data</a>. </p><p>It has been argued that quantities defined as ratios <span class="nowrap"><i>Q</i> = <i>A</i>/<i>B</i></span> having equal dimensions in numerator and denominator are actually only <i>unitless quantities</i> and still have physical dimension defined as <span class="nowrap">dim <i>Q</i> = dim <i>A</i> × dim <i>B</i><span style="padding-left:0.12em;"><sup>−1</sup></span></span>.<sup id="cite_ref-Johansson2010_21-0" class="reference"><a href="#cite_note-Johansson2010-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> For example, <a href="/wiki/Moisture_content" class="mw-redirect" title="Moisture content">moisture content</a> may be defined as a ratio of volumes (volumetric moisture, m<sup>3</sup>⋅m<sup>−3</sup>, dimension L<sup>3</sup>⋅L<sup>−3</sup>) or as a ratio of masses (gravimetric moisture, units kg⋅kg<sup>−1</sup>, dimension M⋅M<sup>−1</sup>); both would be unitless quantities, but of different dimension. </p> <div class="mw-heading mw-heading2"><h2 id="Dimensionless_physical_constants">Dimensionless physical constants</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dimensionless_quantity&action=edit&section=5" title="Edit section: Dimensionless physical constants"><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/Dimensionless_physical_constant" title="Dimensionless physical constant">Dimensionless physical constant</a></div> <p>Certain universal dimensioned physical constants, such as the <a href="/wiki/Speed_of_light" title="Speed of light">speed of light</a> in vacuum, the <a href="/wiki/Universal_gravitational_constant" class="mw-redirect" title="Universal gravitational constant">universal gravitational constant</a>, the <a href="/wiki/Planck_constant" title="Planck constant">Planck constant</a>, the <a href="/wiki/Coulomb_constant" class="mw-redirect" title="Coulomb constant">Coulomb constant</a>, and the <a href="/wiki/Boltzmann_constant" title="Boltzmann constant">Boltzmann constant</a> can be normalized to 1 if appropriate units for <a href="/wiki/Time" title="Time">time</a>, <a href="/wiki/Length" title="Length">length</a>, <a href="/wiki/Mass" title="Mass">mass</a>, <a href="/wiki/Electric_charge" title="Electric charge">charge</a>, and <a href="/wiki/Temperature" title="Temperature">temperature</a> are chosen. The resulting <a href="/wiki/System_of_units" class="mw-redirect" title="System of units">system of units</a> is known as the <a href="/wiki/Natural_units" title="Natural units">natural units</a>, specifically regarding these five constants, <a href="/wiki/Planck_units" title="Planck units">Planck units</a>. However, not all <a href="/wiki/Physical_constant" title="Physical constant">physical constants</a> can be normalized in this fashion. For example, the values of the following constants are independent of the system of units, cannot be defined, and can only be determined experimentally:<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> </p> <ul><li><a href="/wiki/Engineering_strain" class="mw-redirect" title="Engineering strain">engineering strain</a>, a measure of physical deformation defined as a change in length divided by the initial length.</li></ul> <ul><li><a href="/wiki/Fine-structure_constant" title="Fine-structure constant">fine-structure constant</a>, <i>α</i> ≈ 1/137 which characterizes the magnitude of the <a href="/wiki/Electromagnetic_interaction" class="mw-redirect" title="Electromagnetic interaction">electromagnetic interaction</a> between electrons. <sup id="cite_ref-:0_23-0" class="reference"><a href="#cite_note-:0-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup></li> <li><i>β</i> (or <i>μ</i>) ≈ 1836, the <a href="/wiki/Proton-to-electron_mass_ratio" title="Proton-to-electron mass ratio">proton-to-electron mass ratio</a>. This ratio is the <a href="/wiki/Rest_mass" class="mw-redirect" title="Rest mass">rest mass</a> of the <a href="/wiki/Proton" title="Proton">proton</a> divided by that of the <a href="/wiki/Electron" title="Electron">electron</a>. An analogous ratio can be defined for any <a href="/wiki/Elementary_particle" title="Elementary particle">elementary particle</a>.</li> <li><a href="/wiki/Strong_force" class="mw-redirect" title="Strong force">Strong force</a> coupling strength <i>α</i><sub>s</sub> ≈ 1.</li> <li>The tensor-to-scalar 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="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span>, a ratio between the contributions of tensor and scalar modes to the primordial power spectrum observed in the <a href="/wiki/Cosmic_microwave_background" title="Cosmic microwave background">CMB</a>. <sup id="cite_ref-:0_23-1" class="reference"><a href="#cite_note-:0-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup></li> <li>The <a href="/wiki/Immirzi_parameter" title="Immirzi parameter">Immirzi-Barbero</a> parameter <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }"></span>, which characterizes the area gap in <a href="/wiki/Loop_quantum_gravity" title="Loop quantum gravity">loop quantum gravity</a>. <sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup></li></ul> <div class="mw-heading mw-heading2"><h2 id="List">List</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dimensionless_quantity&action=edit&section=6" title="Edit section: List"><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/List_of_dimensionless_quantities" title="List of dimensionless quantities">List of dimensionless quantities</a></div> <div class="mw-heading mw-heading3"><h3 id="Physics_and_engineering">Physics and engineering</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dimensionless_quantity&action=edit&section=7" title="Edit section: Physics and engineering"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Lorentz_factor" title="Lorentz factor">Lorentz Factor</a><sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> – parameter used in the context of special relativity for time dilation, length contraction, and relativistic effects between observers moving at different velocities</li> <li><a href="/wiki/Fresnel_number" title="Fresnel number">Fresnel number</a> – wavenumber (spatial frequency) over distance</li> <li><a href="/wiki/Mach_number" title="Mach number">Mach number</a> – ratio of the speed of an object or flow relative to the speed of sound in the fluid.</li></ul> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Dimensionless_numbers_in_fluid_mechanics" title="Dimensionless numbers in fluid mechanics">Dimensionless numbers in fluid mechanics</a></div> <ul><li><a href="/wiki/Beta_(plasma_physics)" class="mw-redirect" title="Beta (plasma physics)">Beta (plasma physics)</a> – ratio of plasma pressure to magnetic pressure, used in magnetospheric physics as well as fusion plasma physics.</li> <li><a href="/wiki/Damk%C3%B6hler_numbers" title="Damköhler numbers">Damköhler numbers</a> (Da) – used in chemical engineering to relate the chemical reaction timescale (reaction rate) to the transport phenomena rate occurring in a system.</li> <li><a href="/wiki/Thiele_modulus" title="Thiele modulus">Thiele modulus</a> – describes the relationship between diffusion and reaction rate in porous catalyst pellets with no mass transfer limitations.</li> <li><a href="/wiki/Numerical_aperture" title="Numerical aperture">Numerical aperture</a> – characterizes the range of angles over which the system can accept or emit light.</li> <li><a href="/wiki/Sherwood_number" title="Sherwood number">Sherwood number</a> – (also called the mass transfer <a href="/wiki/Nusselt_number" title="Nusselt number">Nusselt number</a>) is a dimensionless number used in mass-transfer operation. It represents the ratio of the convective mass transfer to the rate of diffusive mass transport.</li> <li><a href="/wiki/Schmidt_number" title="Schmidt number">Schmidt number</a> – defined as the ratio of momentum diffusivity (kinematic viscosity) and mass diffusivity, and is used to characterize fluid flows in which there are simultaneous momentum and mass diffusion convection processes.</li> <li><a href="/wiki/Reynolds_number" title="Reynolds number">Reynolds number</a> is commonly used in fluid mechanics to characterize flow, incorporating both properties of the fluid and the flow. It is interpreted as the ratio of inertial forces to viscous forces and can indicate flow regime as well as correlate to frictional heating in application to flow in pipes.<sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup></li> <li>Zukoski number, usually noted <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1785c0a77ab5a06684e8a7ac4f5e59d59ec0319" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.893ex; height:2.676ex;" alt="{\displaystyle Q^{*}}"></span>, is the ratio of the heat release rate of a fire to the enthalpy of the gas flow rate circulating through the fire. Accidental and natural fires usually have 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 Q^{*}\approx 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>≈</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q^{*}\approx 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2b2d70a74fd5f2e20e9cd662387f2d284b02f25" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.153ex; height:2.676ex;" alt="{\displaystyle Q^{*}\approx 1}"></span>. Flat spread fires such as forest fires have <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q^{*}<1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo><</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q^{*}<1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7eb7c2fa9eca270533fdc14fa814611445fe089f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.153ex; height:2.676ex;" alt="{\displaystyle Q^{*}<1}"></span>. Fires originating from pressured vessels or pipes, with additional momentum caused by pressure, have <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q^{*}\gg 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>≫</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q^{*}\gg 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77feeaa07e3aade4e992bd540d2437ca9f62db20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.669ex; height:2.676ex;" alt="{\displaystyle Q^{*}\gg 1}"></span>.<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup></li> <li><a href="/wiki/Eckert_number" title="Eckert number">Eckert number</a></li> <li><a href="/wiki/Biot_number" title="Biot number">Biot number</a></li> <li><a href="/wiki/Grashof_number" title="Grashof number">Grashof number</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Chemistry">Chemistry</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dimensionless_quantity&action=edit&section=8" title="Edit section: Chemistry"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Relative_density" title="Relative density">Relative density</a> – density relative to <a href="/wiki/Water" title="Water">water</a></li> <li><a href="/wiki/Relative_atomic_mass" title="Relative atomic mass">Relative atomic mass</a>, <a href="/wiki/Standard_atomic_weight" title="Standard atomic weight">Standard atomic weight</a></li> <li><a href="/wiki/Equilibrium_constant" title="Equilibrium constant">Equilibrium constant</a> (which is sometimes dimensionless)</li></ul> <div class="mw-heading mw-heading3"><h3 id="Other_fields">Other fields</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dimensionless_quantity&action=edit&section=9" title="Edit section: Other fields"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Cost_of_transport" title="Cost of transport">Cost of transport</a> is the <a href="/wiki/Efficiency" title="Efficiency">efficiency</a> in moving from one place to another</li> <li><a href="/wiki/Elasticity_(economics)" title="Elasticity (economics)">Elasticity</a> is the measurement of the proportional change of an economic variable in response to a change in another</li> <li><a href="/wiki/Basic_reproduction_number" title="Basic reproduction number">Basic reproduction number</a> is a dimensionless ratio used in epidemiology to quantify the transmissibility of an infection.</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=Dimensionless_quantity&action=edit&section=10" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/List_of_dimensionless_quantities" title="List of dimensionless quantities">List of dimensionless quantities</a></li> <li><a href="/wiki/Arbitrary_unit" title="Arbitrary unit">Arbitrary unit</a></li> <li><a href="/wiki/Dimensional_analysis" title="Dimensional analysis">Dimensional analysis</a></li> <li><a href="/wiki/Normalization_(statistics)" title="Normalization (statistics)">Normalization (statistics)</a> and <a href="/wiki/Standardized_moment" title="Standardized moment">standardized moment</a>, the analogous concepts in <a href="/wiki/Statistics" title="Statistics">statistics</a></li> <li><a href="/wiki/Orders_of_magnitude_(numbers)" title="Orders of magnitude (numbers)">Orders of magnitude (numbers)</a></li> <li><a href="/wiki/Similitude_(model)" class="mw-redirect" title="Similitude (model)">Similitude (model)</a></li></ul> <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=Dimensionless_quantity&action=edit&section=11" 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 mw-references-columns"><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"><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 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Retrieved <span class="nowrap">2015-10-07</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=NRL+Plasma+Formulary%3A+Dimensionless+Numbers+of+Fluid+Mechanics&rft.pages=23-25&rft.pub=Naval+Research+Laboratory&rft.date=2007&rft.aulast=Huba&rft.aufirst=Joseph+D.&rft_id=http%3A%2F%2Fwww.ipp.mpg.de%2F~dpc%2Fnrl%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADimensionless+quantity" class="Z3988"></span></span> </li> <li id="cite_note-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-27">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFZukoski1986" class="citation web cs1">Zukoski, Edward E. (1986). <a rel="nofollow" class="external text" href="https://authors.library.caltech.edu/21188/1/287_Zukoski_EE_1985.pdf">"Fluid Dynamic Aspects of Room Fires"</a> <span class="cs1-format">(PDF)</span>. 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Retrieved <span class="nowrap">2022-06-13</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Fluid+Dynamic+Aspects+of+Room+Fires&rft.pub=Fire+Safety+Science&rft.date=1986&rft.aulast=Zukoski&rft.aufirst=Edward+E.&rft_id=https%3A%2F%2Fauthors.library.caltech.edu%2F21188%2F1%2F287_Zukoski_EE_1985.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADimensionless+quantity" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dimensionless_quantity&action=edit&section=12" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFlater2017" class="citation journal cs1 cs1-prop-location-test">Flater, David (October 2017) [2017-05-20, 2017-03-23, 2016-11-22]. Written at <a href="/wiki/National_Institute_of_Standards_and_Technology" title="National Institute of Standards and Technology">National Institute of Standards and Technology</a>, Gaithersburg, Maryland, USA. <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7727271">"Redressing grievances with the treatment of dimensionless quantities in SI"</a>. <i><a href="/wiki/Measurement_(journal)" title="Measurement (journal)">Measurement</a></i>. <b>109</b>. London, UK: <a href="/wiki/Elsevier_Ltd." class="mw-redirect" title="Elsevier Ltd.">Elsevier Ltd.</a>: 105–110. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2017Meas..109..105F">2017Meas..109..105F</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.measurement.2017.05.043">10.1016/j.measurement.2017.05.043</a>. <a href="/wiki/EISSN_(identifier)" class="mw-redirect" title="EISSN (identifier)">eISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1873-412X">1873-412X</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0263-2241">0263-2241</a>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7727271">7727271</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/33311828">33311828</a>. NIHMS1633436.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Measurement&rft.atitle=Redressing+grievances+with+the+treatment+of+dimensionless+quantities+in+SI&rft.volume=109&rft.pages=105-110&rft.date=2017-10&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC7727271%23id-name%3DPMC&rft.eissn=1873-412X&rft_id=info%3Abibcode%2F2017Meas..109..105F&rft.issn=0263-2241&rft_id=info%3Adoi%2F10.1016%2Fj.measurement.2017.05.043&rft_id=info%3Apmid%2F33311828&rft.aulast=Flater&rft.aufirst=David&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC7727271&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADimensionless+quantity" class="Z3988"></span> <a rel="nofollow" class="external autonumber" href="https://web.archive.org/web/20221220195918/https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7727271/pdf/nihms-1633436.pdf">[1]</a> (15 pages)</li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dimensionless_quantity&action=edit&section=13" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Commons-logo.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/12px-Commons-logo.svg.png" decoding="async" width="12" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/18px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/24px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></a></span> Media related to <a href="https://commons.wikimedia.org/wiki/Category:Dimensionless_numbers" class="extiw" title="commons:Category:Dimensionless numbers">Dimensionless numbers</a> at Wikimedia Commons</li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Dimensionless_quantity&oldid=1260705291">https://en.wikipedia.org/w/index.php?title=Dimensionless_quantity&oldid=1260705291</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Category</a>: <ul><li><a href="/wiki/Category:Dimensionless_quantities" title="Category:Dimensionless quantities">Dimensionless quantities</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_matches_Wikidata" title="Category:Short description matches Wikidata">Short description matches Wikidata</a></li><li><a href="/wiki/Category:Use_dmy_dates_from_December_2022" title="Category:Use dmy dates from December 2022">Use dmy dates from December 2022</a></li><li><a href="/wiki/Category:Articles_needing_additional_references_from_March_2017" title="Category:Articles needing additional references from March 2017">Articles needing additional references from March 2017</a></li><li><a href="/wiki/Category:All_articles_needing_additional_references" title="Category:All articles needing additional references">All articles needing additional references</a></li><li><a href="/wiki/Category:Articles_needing_additional_references_from_April_2022" title="Category:Articles needing additional references from April 2022">Articles needing additional references from April 2022</a></li><li><a href="/wiki/Category:CS1_location_test" title="Category:CS1 location test">CS1 location test</a></li><li><a href="/wiki/Category:Commons_category_link_is_locally_defined" title="Category:Commons category link is locally defined">Commons category link is locally defined</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 2 December 2024, at 04:34<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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