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class="vector-toc-numb">2</span> <span>Of integers and real numbers</span> </div> </a> <button aria-controls="toc-Of_integers_and_real_numbers-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 Of integers and real numbers subsection</span> </button> <ul id="toc-Of_integers_and_real_numbers-sublist" class="vector-toc-list"> <li id="toc-Integers" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Integers"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Integers</span> </div> </a> <ul id="toc-Integers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Natural_numbers" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Natural_numbers"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Natural numbers</span> </div> </a> <ul id="toc-Natural_numbers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Real_numbers" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Real_numbers"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Real numbers</span> </div> </a> <ul id="toc-Real_numbers-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Properties" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Properties"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Properties</span> </div> </a> <button aria-controls="toc-Properties-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 Properties subsection</span> </button> <ul id="toc-Properties-sublist" class="vector-toc-list"> <li id="toc-Anti-commutativity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Anti-commutativity"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Anti-commutativity</span> </div> </a> <ul id="toc-Anti-commutativity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Non-associativity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Non-associativity"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Non-associativity</span> </div> </a> <ul id="toc-Non-associativity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Predecessor" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Predecessor"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Predecessor</span> </div> </a> <ul id="toc-Predecessor-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Units_of_measurement" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Units_of_measurement"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Units of measurement</span> </div> </a> <button aria-controls="toc-Units_of_measurement-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 Units of measurement subsection</span> </button> <ul id="toc-Units_of_measurement-sublist" class="vector-toc-list"> <li id="toc-Percentages" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Percentages"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Percentages</span> </div> </a> <ul id="toc-Percentages-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-In_computing" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#In_computing"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>In computing</span> </div> </a> <ul id="toc-In_computing-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_teaching_of_subtraction_in_schools" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#The_teaching_of_subtraction_in_schools"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>The teaching of subtraction in schools</span> </div> </a> <button aria-controls="toc-The_teaching_of_subtraction_in_schools-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 The teaching of subtraction in schools subsection</span> </button> <ul id="toc-The_teaching_of_subtraction_in_schools-sublist" class="vector-toc-list"> <li id="toc-In_America" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#In_America"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>In America</span> </div> </a> <ul id="toc-In_America-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-In_Europe" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#In_Europe"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>In Europe</span> </div> </a> <ul id="toc-In_Europe-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Comparing_the_two_main_methods" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Comparing_the_two_main_methods"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.3</span> <span>Comparing the two main methods</span> </div> </a> <ul id="toc-Comparing_the_two_main_methods-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Subtraction_by_hand" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Subtraction_by_hand"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Subtraction by hand</span> </div> </a> <button aria-controls="toc-Subtraction_by_hand-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 Subtraction by hand subsection</span> </button> <ul id="toc-Subtraction_by_hand-sublist" class="vector-toc-list"> <li id="toc-Austrian_method" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Austrian_method"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.1</span> <span>Austrian method</span> </div> </a> <ul id="toc-Austrian_method-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Subtraction_from_left_to_right" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Subtraction_from_left_to_right"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.2</span> <span>Subtraction from left to right</span> </div> </a> <ul id="toc-Subtraction_from_left_to_right-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-American_method" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#American_method"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.3</span> <span>American method</span> </div> </a> <ul id="toc-American_method-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Trade_first" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Trade_first"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.4</span> <span>Trade first</span> </div> </a> <ul id="toc-Trade_first-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Partial_differences" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Partial_differences"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.5</span> <span>Partial differences</span> </div> </a> <ul id="toc-Partial_differences-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Nonvertical_methods" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Nonvertical_methods"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.6</span> <span>Nonvertical methods</span> </div> </a> <ul id="toc-Nonvertical_methods-sublist" class="vector-toc-list"> <li id="toc-Counting_up" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Counting_up"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.6.1</span> <span>Counting up</span> </div> </a> <ul id="toc-Counting_up-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Breaking_up_the_subtraction" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Breaking_up_the_subtraction"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.6.2</span> <span>Breaking up the subtraction</span> </div> </a> <ul id="toc-Breaking_up_the_subtraction-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Same_change" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Same_change"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.6.3</span> <span>Same change</span> </div> </a> <ul id="toc-Same_change-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bibliography" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Bibliography"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>Bibliography</span> </div> </a> <ul id="toc-Bibliography-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</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">Subtraction: Difference between revisions</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 117 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-117" 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">117 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/Aftrek" title="Aftrek – Afrikaans" lang="af" hreflang="af" data-title="Aftrek" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Subtraktion" title="Subtraktion – Alemannic" lang="gsw" hreflang="gsw" data-title="Subtraktion" data-language-autonym="Alemannisch" data-language-local-name="Alemannic" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B7%D8%B1%D8%AD" title="طرح – Arabic" lang="ar" hreflang="ar" data-title="طرح" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Resta" title="Resta – Aragonese" lang="an" hreflang="an" data-title="Resta" data-language-autonym="Aragonés" data-language-local-name="Aragonese" class="interlanguage-link-target"><span>Aragonés</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Resta" title="Resta – Asturian" lang="ast" hreflang="ast" data-title="Resta" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-ay mw-list-item"><a href="https://ay.wikipedia.org/wiki/Jakhuqawi" title="Jakhuqawi – Aymara" lang="ay" hreflang="ay" data-title="Jakhuqawi" data-language-autonym="Aymar aru" data-language-local-name="Aymara" class="interlanguage-link-target"><span>Aymar aru</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/%C3%87%C4%B1xma" title="Çıxma – Azerbaijani" lang="az" hreflang="az" data-title="Çıxma" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-azb mw-list-item"><a href="https://azb.wikipedia.org/wiki/%DA%86%DB%8C%D8%AE%D9%85%D8%A7" title="چیخما – South Azerbaijani" lang="azb" hreflang="azb" data-title="چیخما" data-language-autonym="تۆرکجه" data-language-local-name="South Azerbaijani" class="interlanguage-link-target"><span>تۆرکجه</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AC%E0%A6%BF%E0%A6%AF%E0%A6%BC%E0%A7%8B%E0%A6%97" title="বিয়োগ – Bangla" lang="bn" hreflang="bn" data-title="বিয়োগ" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Ki%C3%A1m-hoat" title="Kiám-hoat – Minnan" lang="nan" hreflang="nan" data-title="Kiám-hoat" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%90%D0%BB%D1%8B%D1%83" title="Алыу – Bashkir" lang="ba" hreflang="ba" data-title="Алыу" data-language-autonym="Башҡортса" data-language-local-name="Bashkir" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%90%D0%B4%D0%BD%D1%96%D0%BC%D0%B0%D0%BD%D0%BD%D0%B5" title="Адніманне – Belarusian" lang="be" hreflang="be" data-title="Адніманне" 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%B4%D1%8B%D0%BC%D0%B0%D0%BD%D1%8C%D0%BD%D0%B5" title="Адыманьне – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Адыманьне" 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-bcl mw-list-item"><a href="https://bcl.wikipedia.org/wiki/Pag-ina" title="Pag-ina – Central Bikol" lang="bcl" hreflang="bcl" data-title="Pag-ina" data-language-autonym="Bikol Central" data-language-local-name="Central Bikol" class="interlanguage-link-target"><span>Bikol Central</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%98%D0%B7%D0%B2%D0%B0%D0%B6%D0%B4%D0%B0%D0%BD%D0%B5" title="Изваждане – Bulgarian" lang="bg" hreflang="bg" data-title="Изваждане" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bo mw-list-item"><a href="https://bo.wikipedia.org/wiki/%E0%BD%A0%E0%BD%95%E0%BE%B2%E0%BD%B2%E0%BC%8B%E0%BD%A2%E0%BE%A9%E0%BD%B2%E0%BD%A6%E0%BC%8D" title="འཕྲི་རྩིས། – Tibetan" lang="bo" hreflang="bo" data-title="འཕྲི་རྩིས།" data-language-autonym="བོད་ཡིག" data-language-local-name="Tibetan" class="interlanguage-link-target"><span>བོད་ཡིག</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Oduzimanje" title="Oduzimanje – Bosnian" lang="bs" hreflang="bs" data-title="Oduzimanje" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-br mw-list-item"><a href="https://br.wikipedia.org/wiki/Lamadur" title="Lamadur – Breton" lang="br" hreflang="br" data-title="Lamadur" data-language-autonym="Brezhoneg" data-language-local-name="Breton" class="interlanguage-link-target"><span>Brezhoneg</span></a></li><li class="interlanguage-link interwiki-bxr mw-list-item"><a href="https://bxr.wikipedia.org/wiki/%D0%A5%D0%B0%D2%BB%D0%B0%D0%BB%D1%82%D0%B0" title="Хаһалта – Russia Buriat" lang="bxr" hreflang="bxr" data-title="Хаһалта" data-language-autonym="Буряад" data-language-local-name="Russia Buriat" class="interlanguage-link-target"><span>Буряад</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Resta" title="Resta – Catalan" lang="ca" hreflang="ca" data-title="Resta" 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%9A%C4%83%D0%BB%D0%B0%D1%80%D0%B0%D1%81%D1%81%D0%B8" title="Кăларасси – Chuvash" lang="cv" hreflang="cv" data-title="Кăларасси" 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/Od%C4%8D%C3%ADt%C3%A1n%C3%AD" title="Odčítání – Czech" lang="cs" hreflang="cs" data-title="Odčítání" 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/Kubvisa" title="Kubvisa – Shona" lang="sn" hreflang="sn" data-title="Kubvisa" 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/Tynnu" title="Tynnu – Welsh" lang="cy" hreflang="cy" data-title="Tynnu" 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/Subtraktion" title="Subtraktion – Danish" lang="da" hreflang="da" data-title="Subtraktion" 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/Subtraktion" title="Subtraktion – German" lang="de" hreflang="de" data-title="Subtraktion" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-dz mw-list-item"><a href="https://dz.wikipedia.org/wiki/%E0%BD%9F%E0%BD%91%E0%BC%8B%E0%BD%A0%E0%BD%82%E0%BE%B2%E0%BD%BC%E0%BC%8B%E0%BD%A6%E0%BE%90%E0%BE%B1%E0%BD%B2%E0%BD%93%E0%BC%8B%E0%BD%A0%E0%BD%82%E0%BE%B2%E0%BD%B4%E0%BD%A3%E0%BC%8B%E0%BD%82%E0%BE%B1%E0%BD%B2%E0%BC%8B%E0%BD%91%E0%BD%BC%E0%BD%93%E0%BC%8B%E0%BD%A3%E0%BD%B4%E0%BC%8B" title="ཟད་འགྲོ་སྐྱིན་འགྲུལ་གྱི་དོན་ལུ་ – Dzongkha" lang="dz" hreflang="dz" data-title="ཟད་འགྲོ་སྐྱིན་འགྲུལ་གྱི་དོན་ལུ་" data-language-autonym="ཇོང་ཁ" data-language-local-name="Dzongkha" class="interlanguage-link-target"><span>ཇོང་ཁ</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Lahutamine" title="Lahutamine – Estonian" lang="et" hreflang="et" data-title="Lahutamine" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%91%CF%86%CE%B1%CE%AF%CF%81%CE%B5%CF%83%CE%B7" title="Αφαίρεση – Greek" lang="el" hreflang="el" data-title="Αφαίρεση" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Resta" title="Resta – Spanish" lang="es" hreflang="es" data-title="Resta" 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/Subtraho" title="Subtraho – Esperanto" lang="eo" hreflang="eo" data-title="Subtraho" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Kenketa" title="Kenketa – Basque" lang="eu" hreflang="eu" data-title="Kenketa" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AA%D9%81%D8%B1%DB%8C%D9%82" title="تفریق – Persian" lang="fa" hreflang="fa" data-title="تفریق" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-hif mw-list-item"><a href="https://hif.wikipedia.org/wiki/Ghataana" title="Ghataana – Fiji Hindi" lang="hif" hreflang="hif" data-title="Ghataana" data-language-autonym="Fiji Hindi" data-language-local-name="Fiji Hindi" class="interlanguage-link-target"><span>Fiji Hindi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Soustraction" title="Soustraction – French" lang="fr" hreflang="fr" data-title="Soustraction" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gd mw-list-item"><a href="https://gd.wikipedia.org/wiki/Toirt_air_falbh" title="Toirt air falbh – Scottish Gaelic" lang="gd" hreflang="gd" data-title="Toirt air falbh" data-language-autonym="Gàidhlig" data-language-local-name="Scottish Gaelic" class="interlanguage-link-target"><span>Gàidhlig</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Subtracci%C3%B3n" title="Subtracción – Galician" lang="gl" hreflang="gl" data-title="Subtracción" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-gan mw-list-item"><a href="https://gan.wikipedia.org/wiki/%E6%B8%9B%E6%B3%95" title="減法 – Gan" lang="gan" hreflang="gan" data-title="減法" data-language-autonym="贛語" data-language-local-name="Gan" class="interlanguage-link-target"><span>贛語</span></a></li><li class="interlanguage-link interwiki-hak mw-list-item"><a href="https://hak.wikipedia.org/wiki/K%C3%A1m-fap" title="Kám-fap – Hakka Chinese" lang="hak" hreflang="hak" data-title="Kám-fap" data-language-autonym="客家語 / Hak-kâ-ngî" data-language-local-name="Hakka Chinese" class="interlanguage-link-target"><span>客家語 / Hak-kâ-ngî</span></a></li><li class="interlanguage-link interwiki-xal mw-list-item"><a href="https://xal.wikipedia.org/wiki/%D0%A5%D0%B0%D1%81%D0%BB%D0%B0%D2%BB%D0%B0%D0%BD" title="Хаслаһан – Kalmyk" lang="xal" hreflang="xal" data-title="Хаслаһан" data-language-autonym="Хальмг" data-language-local-name="Kalmyk" class="interlanguage-link-target"><span>Хальмг</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%BA%84%EC%85%88" title="뺄셈 – Korean" lang="ko" hreflang="ko" data-title="뺄셈" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%80%D5%A1%D5%B6%D5%B8%D6%82%D5%B4" title="Հանում – Armenian" lang="hy" hreflang="hy" data-title="Հանում" 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%98%E0%A4%9F%E0%A4%BE%E0%A4%A8%E0%A4%BE" title="घटाना – Hindi" lang="hi" hreflang="hi" data-title="घटाना" 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/Oduzimanje" title="Oduzimanje – Croatian" lang="hr" hreflang="hr" data-title="Oduzimanje" 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/Pengurangan" title="Pengurangan – Indonesian" lang="id" hreflang="id" data-title="Pengurangan" 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-xh mw-list-item"><a href="https://xh.wikipedia.org/wiki/Ukuthabatha" title="Ukuthabatha – Xhosa" lang="xh" hreflang="xh" data-title="Ukuthabatha" data-language-autonym="IsiXhosa" data-language-local-name="Xhosa" class="interlanguage-link-target"><span>IsiXhosa</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Fr%C3%A1dr%C3%A1ttur" title="Frádráttur – Icelandic" lang="is" hreflang="is" data-title="Frádráttur" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Sottrazione" title="Sottrazione – Italian" lang="it" hreflang="it" data-title="Sottrazione" 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%97%D7%99%D7%A1%D7%95%D7%A8" title="חיסור – Hebrew" lang="he" hreflang="he" data-title="חיסור" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-jv mw-list-item"><a href="https://jv.wikipedia.org/wiki/Pangurangan" title="Pangurangan – Javanese" lang="jv" hreflang="jv" data-title="Pangurangan" data-language-autonym="Jawa" data-language-local-name="Javanese" class="interlanguage-link-target"><span>Jawa</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%B5%E0%B3%8D%E0%B2%AF%E0%B2%B5%E0%B2%95%E0%B2%B2%E0%B2%A8" title="ವ್ಯವಕಲನ – Kannada" lang="kn" hreflang="kn" data-title="ವ್ಯವಕಲನ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="Kannada" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%92%E1%83%90%E1%83%9B%E1%83%9D%E1%83%99%E1%83%9A%E1%83%94%E1%83%91%E1%83%90" title="გამოკლება – Georgian" lang="ka" hreflang="ka" data-title="გამოკლება" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%90%D0%B7%D0%B0%D0%B9%D1%82%D1%83_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Азайту (математика) – Kazakh" lang="kk" hreflang="kk" data-title="Азайту (математика)" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Utoaji" title="Utoaji – Swahili" lang="sw" hreflang="sw" data-title="Utoaji" data-language-autonym="Kiswahili" data-language-local-name="Swahili" class="interlanguage-link-target"><span>Kiswahili</span></a></li><li class="interlanguage-link interwiki-gcr mw-list-item"><a href="https://gcr.wikipedia.org/wiki/Soustraksyon" title="Soustraksyon – Guianan Creole" lang="gcr" hreflang="gcr" data-title="Soustraksyon" data-language-autonym="Kriyòl gwiyannen" data-language-local-name="Guianan Creole" class="interlanguage-link-target"><span>Kriyòl gwiyannen</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%9A%D0%B5%D0%BC%D0%B8%D1%82%D2%AF%D2%AF" title="Кемитүү – Kyrgyz" lang="ky" hreflang="ky" data-title="Кемитүү" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-lo mw-list-item"><a href="https://lo.wikipedia.org/wiki/%E0%BA%81%E0%BA%B2%E0%BA%99%E0%BA%AB%E0%BA%B1%E0%BA%81%E0%BA%A5%E0%BA%BB%E0%BA%9A_(%E0%BA%84%E0%BA%B0%E0%BA%99%E0%BA%B4%E0%BA%94%E0%BA%AA%E0%BA%B2%E0%BA%94)" title="ການຫັກລົບ (ຄະນິດສາດ) – Lao" lang="lo" hreflang="lo" data-title="ການຫັກລົບ (ຄະນິດສາດ)" data-language-autonym="ລາວ" data-language-local-name="Lao" class="interlanguage-link-target"><span>ລາວ</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Subtractio" title="Subtractio – Latin" lang="la" hreflang="la" data-title="Subtractio" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/At%C5%86em%C5%A1ana" title="Atņemšana – Latvian" lang="lv" hreflang="lv" data-title="Atņemšana" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Atimtis" title="Atimtis – Lithuanian" lang="lt" hreflang="lt" data-title="Atimtis" 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/Kivon%C3%A1s" title="Kivonás – Hungarian" lang="hu" hreflang="hu" data-title="Kivonás" 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%9E%D0%B4%D0%B7%D0%B5%D0%BC%D0%B0%D1%9A%D0%B5" title="Одземање – Macedonian" lang="mk" hreflang="mk" data-title="Одземање" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%B5%E0%B5%8D%E0%B4%AF%E0%B4%B5%E0%B4%95%E0%B4%B2%E0%B4%A8%E0%B4%82" title="വ്യവകലനം – Malayalam" lang="ml" hreflang="ml" data-title="വ്യവകലനം" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mt mw-list-item"><a href="https://mt.wikipedia.org/wiki/Tnaqqis" title="Tnaqqis – Maltese" lang="mt" hreflang="mt" data-title="Tnaqqis" data-language-autonym="Malti" data-language-local-name="Maltese" class="interlanguage-link-target"><span>Malti</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%B5%E0%A4%9C%E0%A4%BE%E0%A4%AC%E0%A4%BE%E0%A4%95%E0%A5%80" title="वजाबाकी – Marathi" lang="mr" hreflang="mr" data-title="वजाबाकी" data-language-autonym="मराठी" data-language-local-name="Marathi" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-arz mw-list-item"><a href="https://arz.wikipedia.org/wiki/%D8%B7%D8%B1%D8%AD" title="طرح – Egyptian Arabic" lang="arz" hreflang="arz" data-title="طرح" data-language-autonym="مصرى" data-language-local-name="Egyptian Arabic" class="interlanguage-link-target"><span>مصرى</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Penolakan" title="Penolakan – Malay" lang="ms" hreflang="ms" data-title="Penolakan" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-cdo mw-list-item"><a href="https://cdo.wikipedia.org/wiki/G%C4%93ng-hu%C3%A1k" title="Gēng-huák – Mindong" lang="cdo" hreflang="cdo" data-title="Gēng-huák" data-language-autonym="閩東語 / Mìng-dĕ̤ng-ngṳ̄" data-language-local-name="Mindong" class="interlanguage-link-target"><span>閩東語 / Mìng-dĕ̤ng-ngṳ̄</span></a></li><li class="interlanguage-link interwiki-fj mw-list-item"><a href="https://fj.wikipedia.org/wiki/Subtraction" title="Subtraction – Fijian" lang="fj" hreflang="fj" data-title="Subtraction" data-language-autonym="Na Vosa Vakaviti" data-language-local-name="Fijian" class="interlanguage-link-target"><span>Na Vosa Vakaviti</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Aftrekken_(wiskunde)" title="Aftrekken (wiskunde) – Dutch" lang="nl" hreflang="nl" data-title="Aftrekken (wiskunde)" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ne mw-list-item"><a href="https://ne.wikipedia.org/wiki/%E0%A4%98%E0%A4%9F%E0%A4%BE%E0%A4%89" title="घटाउ – Nepali" lang="ne" hreflang="ne" data-title="घटाउ" data-language-autonym="नेपाली" data-language-local-name="Nepali" class="interlanguage-link-target"><span>नेपाली</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E6%B8%9B%E6%B3%95" title="減法 – Japanese" lang="ja" hreflang="ja" data-title="減法" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Subtraksjon" title="Subtraksjon – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Subtraksjon" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Subtraksjon" title="Subtraksjon – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Subtraksjon" 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-nov mw-list-item"><a href="https://nov.wikipedia.org/wiki/Subtraktione" title="Subtraktione – Novial" lang="nov" hreflang="nov" data-title="Subtraktione" data-language-autonym="Novial" data-language-local-name="Novial" class="interlanguage-link-target"><span>Novial</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Sostraccion" title="Sostraccion – Occitan" lang="oc" hreflang="oc" data-title="Sostraccion" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Ayirish" title="Ayirish – Uzbek" lang="uz" hreflang="uz" data-title="Ayirish" 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-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%98%E0%A8%9F%E0%A8%BE%E0%A8%85" title="ਘਟਾਅ – Punjabi" lang="pa" hreflang="pa" data-title="ਘਟਾਅ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-ps mw-list-item"><a href="https://ps.wikipedia.org/wiki/%D8%AA%D9%81%D8%B1%D9%8A%D9%82" title="تفريق – Pashto" lang="ps" hreflang="ps" data-title="تفريق" data-language-autonym="پښتو" data-language-local-name="Pashto" class="interlanguage-link-target"><span>پښتو</span></a></li><li class="interlanguage-link interwiki-jam mw-list-item"><a href="https://jam.wikipedia.org/wiki/Sobchakshan" title="Sobchakshan – Jamaican Creole English" lang="jam" hreflang="jam" data-title="Sobchakshan" data-language-autonym="Patois" data-language-local-name="Jamaican Creole English" class="interlanguage-link-target"><span>Patois</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Sotrassion" title="Sotrassion – Piedmontese" lang="pms" hreflang="pms" data-title="Sotrassion" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Odejmowanie" title="Odejmowanie – Polish" lang="pl" hreflang="pl" data-title="Odejmowanie" 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/Subtra%C3%A7%C3%A3o" title="Subtração – Portuguese" lang="pt" hreflang="pt" data-title="Subtração" 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/Sc%C4%83dere" title="Scădere – Romanian" lang="ro" hreflang="ro" data-title="Scădere" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-qu mw-list-item"><a href="https://qu.wikipedia.org/wiki/Qichuy" title="Qichuy – Quechua" lang="qu" hreflang="qu" data-title="Qichuy" data-language-autonym="Runa Simi" data-language-local-name="Quechua" class="interlanguage-link-target"><span>Runa Simi</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%92%D1%8B%D1%87%D0%B8%D1%82%D0%B0%D0%BD%D0%B8%D0%B5" title="Вычитание – Russian" lang="ru" hreflang="ru" data-title="Вычитание" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sah mw-list-item"><a href="https://sah.wikipedia.org/wiki/%D0%9A%D3%A9%D2%95%D2%AF%D1%80%D1%8D%D1%82%D0%B8%D0%B8" title="Көҕүрэтии – Yakut" lang="sah" hreflang="sah" data-title="Көҕүрэтии" data-language-autonym="Саха тыла" data-language-local-name="Yakut" class="interlanguage-link-target"><span>Саха тыла</span></a></li><li class="interlanguage-link interwiki-sat mw-list-item"><a href="https://sat.wikipedia.org/wiki/%E1%B1%B5%E1%B1%B7%E1%B1%AE%E1%B1%9C%E1%B1%AE%E1%B1%AB" title="ᱵᱷᱮᱜᱮᱫ – Santali" lang="sat" hreflang="sat" data-title="ᱵᱷᱮᱜᱮᱫ" data-language-autonym="ᱥᱟᱱᱛᱟᱲᱤ" data-language-local-name="Santali" class="interlanguage-link-target"><span>ᱥᱟᱱᱛᱟᱲᱤ</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Zbritja_(matematik%C3%AB)" title="Zbritja (matematikë) – Albanian" lang="sq" hreflang="sq" data-title="Zbritja (matematikë)" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Suttrazzioni" title="Suttrazzioni – Sicilian" lang="scn" hreflang="scn" data-title="Suttrazzioni" data-language-autonym="Sicilianu" data-language-local-name="Sicilian" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Subtraction" title="Subtraction – Simple English" lang="en-simple" hreflang="en-simple" data-title="Subtraction" 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/Od%C4%8D%C3%ADtanie" title="Odčítanie – Slovak" lang="sk" hreflang="sk" data-title="Odčítanie" 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/Od%C5%A1tevanje" title="Odštevanje – Slovenian" lang="sl" hreflang="sl" data-title="Odštevanje" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-so mw-list-item"><a href="https://so.wikipedia.org/wiki/Faraq" title="Faraq – Somali" lang="so" hreflang="so" data-title="Faraq" data-language-autonym="Soomaaliga" data-language-local-name="Somali" class="interlanguage-link-target"><span>Soomaaliga</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D9%84%DB%8E%D8%AF%DB%95%D8%B1%DA%A9%D8%B1%D8%AF%D9%86" title="لێدەرکردن – Central Kurdish" lang="ckb" hreflang="ckb" data-title="لێدەرکردن" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9E%D0%B4%D1%83%D0%B7%D0%B8%D0%BC%D0%B0%D1%9A%D0%B5" title="Одузимање – Serbian" lang="sr" hreflang="sr" data-title="Одузимање" 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-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Oduzimanje" title="Oduzimanje – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Oduzimanje" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/V%C3%A4hennyslasku" title="Vähennyslasku – Finnish" lang="fi" hreflang="fi" data-title="Vähennyslasku" 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/Subtraktion" title="Subtraktion – Swedish" lang="sv" hreflang="sv" data-title="Subtraktion" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Pagbabawas" title="Pagbabawas – Tagalog" lang="tl" hreflang="tl" data-title="Pagbabawas" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%95%E0%AE%B4%E0%AE%BF%E0%AE%A4%E0%AF%8D%E0%AE%A4%E0%AE%B2%E0%AF%8D_(%E0%AE%95%E0%AE%A3%E0%AE%BF%E0%AE%A4%E0%AE%AE%E0%AF%8D)" title="கழித்தல் (கணிதம்) – Tamil" lang="ta" hreflang="ta" data-title="கழித்தல் (கணிதம்)" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%A4%E0%B1%80%E0%B0%B8%E0%B0%BF%E0%B0%B5%E0%B1%87%E0%B0%A4" title="తీసివేత – Telugu" lang="te" hreflang="te" data-title="తీసివేత" data-language-autonym="తెలుగు" data-language-local-name="Telugu" class="interlanguage-link-target"><span>తెలుగు</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%81%E0%B8%B2%E0%B8%A3%E0%B8%A5%E0%B8%9A" title="การลบ – Thai" lang="th" hreflang="th" data-title="การลบ" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tg mw-list-item"><a href="https://tg.wikipedia.org/wiki/%D0%A2%D0%B0%D1%80%D2%B3_(%D0%B0%D0%BC%D0%B0%D0%BB%D0%B8_%D1%80%D0%B8%D1%91%D0%B7%D3%A3)" title="Тарҳ (амали риёзӣ) – Tajik" lang="tg" hreflang="tg" data-title="Тарҳ (амали риёзӣ)" data-language-autonym="Тоҷикӣ" data-language-local-name="Tajik" class="interlanguage-link-target"><span>Тоҷикӣ</span></a></li><li class="interlanguage-link interwiki-chr mw-list-item"><a href="https://chr.wikipedia.org/wiki/%E1%8F%97%E1%8E%AA%E1%8F%A3%E1%8E%B4%E1%8F%8D%E1%8F%97" title="ᏗᎪᏣᎴᏍᏗ – Cherokee" lang="chr" hreflang="chr" data-title="ᏗᎪᏣᎴᏍᏗ" data-language-autonym="ᏣᎳᎩ" data-language-local-name="Cherokee" class="interlanguage-link-target"><span>ᏣᎳᎩ</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/%C3%87%C4%B1karma" title="Çıkarma – Turkish" lang="tr" hreflang="tr" data-title="Çıkarma" 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%92%D1%96%D0%B4%D0%BD%D1%96%D0%BC%D0%B0%D0%BD%D0%BD%D1%8F" title="Віднімання – Ukrainian" lang="uk" hreflang="uk" data-title="Віднімання" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D8%AA%D9%81%D8%B1%DB%8C%D9%82_(%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C)" title="تفریق (ریاضی) – Urdu" lang="ur" hreflang="ur" data-title="تفریق (ریاضی)" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-za mw-list-item"><a href="https://za.wikipedia.org/wiki/Gemjfap" title="Gemjfap – Zhuang" lang="za" hreflang="za" data-title="Gemjfap" data-language-autonym="Vahcuengh" data-language-local-name="Zhuang" class="interlanguage-link-target"><span>Vahcuengh</span></a></li><li class="interlanguage-link interwiki-vec mw-list-item"><a href="https://vec.wikipedia.org/wiki/Sotra" title="Sotra – Venetian" lang="vec" hreflang="vec" data-title="Sotra" data-language-autonym="Vèneto" data-language-local-name="Venetian" class="interlanguage-link-target"><span>Vèneto</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Ph%C3%A9p_tr%E1%BB%AB" title="Phép trừ – Vietnamese" lang="vi" hreflang="vi" data-title="Phép trừ" 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-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Pag-iban-iban" title="Pag-iban-iban – Waray" lang="war" hreflang="war" data-title="Pag-iban-iban" data-language-autonym="Winaray" data-language-local-name="Waray" class="interlanguage-link-target"><span>Winaray</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E5%87%8F%E6%B3%95" title="减法 – Wu" lang="wuu" hreflang="wuu" data-title="减法" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%90%D7%A8%D7%90%D7%A4%D7%A0%D7%A2%D7%9D" title="אראפנעם – Yiddish" lang="yi" hreflang="yi" data-title="אראפנעם" data-language-autonym="ייִדיש" data-language-local-name="Yiddish" class="interlanguage-link-target"><span>ייִדיש</span></a></li><li class="interlanguage-link interwiki-yo mw-list-item"><a href="https://yo.wikipedia.org/wiki/%C3%8Cy%E1%BB%8Dk%C3%BAr%C3%B2" title="Ìyọkúrò – Yoruba" lang="yo" hreflang="yo" data-title="Ìyọkúrò" data-language-autonym="Yorùbá" data-language-local-name="Yoruba" class="interlanguage-link-target"><span>Yorùbá</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E6%B8%9B" title="減 – Cantonese" lang="yue" hreflang="yue" data-title="減" 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/%E6%B8%9B%E6%B3%95" title="減法 – Chinese" lang="zh" hreflang="zh" data-title="減法" 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/Q40754#sitelinks-wikipedia" title="Edit interlanguage links" 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title="Wikipedia:Rollback">Rollback</a></span></span></div><div id="mw-diff-otitle4"><a href="/w/index.php?title=Subtraction&diff=prev&oldid=1241976144" title="Subtraction" id="differences-prevlink">← Previous edit</a></div></td> <td colspan="2" class="diff-ntitle diff-side-added"><div id="mw-diff-ntitle1"><strong><a href="/w/index.php?title=Subtraction&oldid=1258597029" title="Subtraction">Revision as of 15:58, 20 November 2024</a> <span class="mw-diff-edit"><a href="/w/index.php?title=Subtraction&action=edit&oldid=1258597029" title="Subtraction">edit</a></span><span class="mw-diff-timestamp" data-timestamp="2024-11-20T15:58:56Z"></span> <span class="mw-diff-undo"><a href="/w/index.php?title=Subtraction&action=edit&undoafter=1241976144&undo=1258597029" title=""Undo" reverts this edit and opens the edit form in preview mode. It allows adding a reason in the summary.">undo</a></span></strong></div><div id="mw-diff-ntitle2"><a href="/wiki/Special:Contributions/84.32.95.247" class="mw-userlink mw-anonuserlink" title="Special:Contributions/84.32.95.247" data-mw-revid="1258597029"><bdi>84.32.95.247</bdi></a> <span class="mw-usertoollinks">(<a href="/wiki/User_talk:84.32.95.247" class="mw-usertoollinks-talk" title="User talk:84.32.95.247">talk</a>)</span><div class="mw-diff-usermetadata"></div></div><div id="mw-diff-ntitle3"> <span class="comment comment--without-parentheses">added content</span></div><div id="mw-diff-ntitle5"><span class="mw-tag-markers"><a href="/wiki/Special:Tags" title="Special:Tags">Tags</a>: <span class="mw-tag-marker mw-tag-marker-mw-reverted">Reverted</span> <span class="mw-tag-marker mw-tag-marker-mobile_edit">Mobile edit</span> <span class="mw-tag-marker mw-tag-marker-mobile_web_edit">Mobile web edit</span> <span class="mw-tag-marker mw-tag-marker-disambiguator-link-added"><a href="/wiki/Wikipedia:Disambiguation#Links_to_disambiguation_pages" title="Wikipedia:Disambiguation">Disambiguation links</a> added</span></span></div><div id="mw-diff-ntitle4"><a href="/w/index.php?title=Subtraction&diff=next&oldid=1258597029" title="Subtraction" id="differences-nextlink">Next edit →</a></div></td> </tr><tr> <td colspan="2" class="diff-lineno">Line 1:</td> <td colspan="2" class="diff-lineno">Line 1:</td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>{{Short description|One of the four basic arithmetic operations}}</div></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div><ins class="diffchange diffchange-inline">{{short description|subtraction means to multiply a number by a number there for if we had the equation 6-4 the answer would be 24</ins>{{Short description|One of the four basic arithmetic operations}}</div></td> </tr> <tr> <td class="diff-marker" data-marker="−"></td> <td class="diff-deletedline diff-side-deleted"><div>{{Redirects|Subtract|other uses|<del class="diffchange diffchange-inline">Subtraction</del> (disambiguation)}}</div></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div>{{Redirects|Subtract|other uses|<ins class="diffchange diffchange-inline">addition</ins> (disambiguation)}}</div></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>{{more citations needed|date=May 2018}}</div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>{{more citations needed|date=May 2018}}</div></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>[[File:Subtraction01.svg|right|thumb|180px|"{{nowrap|5 − 2}} = 3" (verbally, "five minus two equals three")]]</div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>[[File:Subtraction01.svg|right|thumb|180px|"{{nowrap|5 − 2}} = 3" (verbally, "five minus two equals three")]]</div></td> </tr>  </table><hr class='diff-hr' id='mw-oldid' /> <h2 class='diff-currentversion-title'>Revision as of 15:58, 20 November 2024</h2> <div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p>{{short description|subtraction means to multiply a number by a number there for if we had the equation 6-4 the answer would be 24</p><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">One of the four basic arithmetic operations</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">"Subtract" redirects here. For other uses, see <a href="/wiki/Addition_(disambiguation)" class="mw-disambig" title="Addition (disambiguation)">addition (disambiguation)</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/Subtraction" title="Special:EditPage/Subtraction">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=%22Subtraction%22">"Subtraction"</a> – <a rel="nofollow" class="external text" href="https://www.google.com/search?tbm=nws&q=%22Subtraction%22+-wikipedia&tbs=ar:1">news</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?&q=%22Subtraction%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=%22Subtraction%22+-wikipedia">books</a> <b>·</b> <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?q=%22Subtraction%22">scholar</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.jstor.org/action/doBasicSearch?Query=%22Subtraction%22&acc=on&wc=on">JSTOR</a></span></small></span> <span class="date-container"><i>(<span class="date">May 2018</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Subtraction01.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8b/Subtraction01.svg/180px-Subtraction01.svg.png" decoding="async" width="180" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8b/Subtraction01.svg/270px-Subtraction01.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8b/Subtraction01.svg/360px-Subtraction01.svg.png 2x" data-file-width="300" data-file-height="200" /></a><figcaption>"<span class="nowrap">5 − 2</span> = 3" (verbally, "five minus two equals three")</figcaption></figure> <style 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.navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><table class="sidebar nomobile nowraplinks"><tbody><tr><td class="sidebar-above" style="background:#efefef;"> <span style="font-size:130%;"><a href="/wiki/Arithmetic_operations" class="mw-redirect" title="Arithmetic operations">Arithmetic operations</a></span><div class="navbar plainlinks hlist navbar-mini" style="float:right"><ul><li class="nv-view"><a href="/wiki/Template:Arithmetic_operations" title="Template:Arithmetic operations"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Arithmetic_operations" title="Template talk:Arithmetic operations"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Arithmetic_operations" title="Special:EditPage/Template:Arithmetic operations"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr><tr><td class="sidebar-content" style="font-size:130%;"> <table class="infobox-subbox infobox-3cols-child infobox-table"><tbody><tr><th colspan="4" class="infobox-header"><a href="/wiki/Addition" title="Addition">Addition</a> (+)</th></tr><tr><th scope="row" class="infobox-label" style="display:none;"></th><td class="infobox-data infobox-data-a" style="text-align:right; vertical-align:middle;"> <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 \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{term}}\,+\,{\text{term}}\\\scriptstyle {\text{summand}}\,+\,{\text{summand}}\\\scriptstyle {\text{addend}}\,+\,{\text{addend}}\\\scriptstyle {\text{augend}}\,+\,{\text{addend}}\end{matrix}}\right\}\,=\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtext>term</mtext> </mrow> <mspace width="thinmathspace" /> <mo>+</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>term</mtext> </mrow> </mstyle> </mtd> </mtr> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtext>summand</mtext> </mrow> <mspace width="thinmathspace" /> <mo>+</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>summand</mtext> </mrow> </mstyle> </mtd> </mtr> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtext>addend</mtext> </mrow> <mspace width="thinmathspace" /> <mo>+</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>addend</mtext> </mrow> </mstyle> </mtd> </mtr> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtext>augend</mtext> </mrow> <mspace width="thinmathspace" /> <mo>+</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>addend</mtext> </mrow> </mstyle> </mtd> </mtr> </mtable> </mrow> <mo>}</mo> </mrow> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{term}}\,+\,{\text{term}}\\\scriptstyle {\text{summand}}\,+\,{\text{summand}}\\\scriptstyle {\text{addend}}\,+\,{\text{addend}}\\\scriptstyle {\text{augend}}\,+\,{\text{addend}}\end{matrix}}\right\}\,=\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea99a27b5a763ef48889c450ac8157083ea97118" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:20.217ex; height:9.843ex;" alt="{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{term}}\,+\,{\text{term}}\\\scriptstyle {\text{summand}}\,+\,{\text{summand}}\\\scriptstyle {\text{addend}}\,+\,{\text{addend}}\\\scriptstyle {\text{augend}}\,+\,{\text{addend}}\end{matrix}}\right\}\,=\,}"></span></td><td class="infobox-data infobox-data-b" style="text-align:left; vertical-align:middle;"> <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 \scriptstyle {\text{sum}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtext>sum</mtext> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\text{sum}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8609baca9fdbc4c529f5894884a08122d695dad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.931ex; height:1.343ex;" alt="{\displaystyle \scriptstyle {\text{sum}}}"></span></td></tr><tr><th colspan="4" class="infobox-header"><a class="mw-selflink selflink">Subtraction</a> (−)</th></tr><tr><th scope="row" class="infobox-label" style="display:none;"></th><td class="infobox-data infobox-data-a" style="text-align:right; vertical-align:middle;"> <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 \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{term}}\,-\,{\text{term}}\\\scriptstyle {\text{minuend}}\,-\,{\text{subtrahend}}\end{matrix}}\right\}\,=\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtext>term</mtext> </mrow> <mspace width="thinmathspace" /> <mo>−</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>term</mtext> </mrow> </mstyle> </mtd> </mtr> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtext>minuend</mtext> </mrow> <mspace width="thinmathspace" /> <mo>−</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>subtrahend</mtext> </mrow> </mstyle> </mtd> </mtr> </mtable> </mrow> <mo>}</mo> </mrow> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{term}}\,-\,{\text{term}}\\\scriptstyle {\text{minuend}}\,-\,{\text{subtrahend}}\end{matrix}}\right\}\,=\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2780b756445a5f8f95b16c33e3b924f976958ea0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:20.356ex; height:4.843ex;" alt="{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{term}}\,-\,{\text{term}}\\\scriptstyle {\text{minuend}}\,-\,{\text{subtrahend}}\end{matrix}}\right\}\,=\,}"></span></td><td class="infobox-data infobox-data-b" style="text-align:left; vertical-align:middle;"> <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 \scriptstyle {\text{difference}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtext>difference</mtext> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\text{difference}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ac22c4e24eef2036cff5bfea924cc0dbb30c5d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.857ex; height:1.676ex;" alt="{\displaystyle \scriptstyle {\text{difference}}}"></span></td></tr><tr><th colspan="4" class="infobox-header"><a href="/wiki/Multiplication" title="Multiplication">Multiplication</a> (×)</th></tr><tr><th scope="row" class="infobox-label" style="display:none;"></th><td class="infobox-data infobox-data-a" style="text-align:right; vertical-align:middle;"> <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 \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{factor}}\,\times \,{\text{factor}}\\\scriptstyle {\text{multiplier}}\,\times \,{\text{multiplicand}}\end{matrix}}\right\}\,=\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtext>factor</mtext> </mrow> <mspace width="thinmathspace" /> <mo>×</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>factor</mtext> </mrow> </mstyle> </mtd> </mtr> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtext>multiplier</mtext> </mrow> <mspace width="thinmathspace" /> <mo>×</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>multiplicand</mtext> </mrow> </mstyle> </mtd> </mtr> </mtable> </mrow> <mo>}</mo> </mrow> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{factor}}\,\times \,{\text{factor}}\\\scriptstyle {\text{multiplier}}\,\times \,{\text{multiplicand}}\end{matrix}}\right\}\,=\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93f7b476e32221c7b05d356289c8085aef54059b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:22.176ex; height:4.843ex;" alt="{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{factor}}\,\times \,{\text{factor}}\\\scriptstyle {\text{multiplier}}\,\times \,{\text{multiplicand}}\end{matrix}}\right\}\,=\,}"></span></td><td class="infobox-data infobox-data-b" style="text-align:left; vertical-align:middle;"> <a href="/wiki/Product_(mathematics)" title="Product (mathematics)"><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 \scriptstyle {\text{product}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtext>product</mtext> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\text{product}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5c8b7509b8be1043622cb7b1b9a36ca8bfc2616" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.578ex; height:1.843ex;" alt="{\displaystyle \scriptstyle {\text{product}}}"></span></a></td></tr><tr><th colspan="4" class="infobox-header"><a href="/wiki/Division_(mathematics)" title="Division (mathematics)">Division</a> (÷)</th></tr><tr><th scope="row" class="infobox-label" style="display:none;"></th><td class="infobox-data infobox-data-a" style="text-align:right; vertical-align:middle;"> <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 \scriptstyle \left.{\begin{matrix}\scriptstyle {\frac {\scriptstyle {\text{dividend}}}{\scriptstyle {\text{divisor}}}}\\[1ex]\scriptstyle {\frac {\scriptstyle {\text{numerator}}}{\scriptstyle {\text{denominator}}}}\end{matrix}}\right\}\,=\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="0.83em 0.4em" columnspacing="1em"> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtext>dividend</mtext> </mrow> </mstyle> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtext>divisor</mtext> </mrow> </mstyle> </mfrac> </mrow> </mstyle> </mtd> </mtr> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtext>numerator</mtext> </mrow> </mstyle> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtext>denominator</mtext> </mrow> </mstyle> </mfrac> </mrow> </mstyle> </mtd> </mtr> </mtable> </mrow> <mo>}</mo> </mrow> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\frac {\scriptstyle {\text{dividend}}}{\scriptstyle {\text{divisor}}}}\\[1ex]\scriptstyle {\frac {\scriptstyle {\text{numerator}}}{\scriptstyle {\text{denominator}}}}\end{matrix}}\right\}\,=\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d5d22ff59234f0d437be740306e8dd905991e1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:14.15ex; height:8.843ex;" alt="{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\frac {\scriptstyle {\text{dividend}}}{\scriptstyle {\text{divisor}}}}\\[1ex]\scriptstyle {\frac {\scriptstyle {\text{numerator}}}{\scriptstyle {\text{denominator}}}}\end{matrix}}\right\}\,=\,}"></span></td><td class="infobox-data infobox-data-b" style="text-align:left; vertical-align:middle;"> <a href="/wiki/Quotient" title="Quotient"><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 \scriptstyle \left\{{\begin{matrix}\scriptstyle {\text{fraction}}\\\scriptstyle {\text{quotient}}\\\scriptstyle {\text{ratio}}\end{matrix}}\right.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow> <mo>{</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtext>fraction</mtext> </mrow> </mstyle> </mtd> </mtr> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtext>quotient</mtext> </mrow> </mstyle> </mtd> </mtr> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtext>ratio</mtext> </mrow> </mstyle> </mtd> </mtr> </mtable> </mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \left\{{\begin{matrix}\scriptstyle {\text{fraction}}\\\scriptstyle {\text{quotient}}\\\scriptstyle {\text{ratio}}\end{matrix}}\right.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2359c3ca6e50e7ae8065baa710440b3c79895023" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:8.197ex; height:7.176ex;" alt="{\displaystyle \scriptstyle \left\{{\begin{matrix}\scriptstyle {\text{fraction}}\\\scriptstyle {\text{quotient}}\\\scriptstyle {\text{ratio}}\end{matrix}}\right.}"></span></a></td></tr><tr><th colspan="4" class="infobox-header"><a href="/wiki/Exponentiation" title="Exponentiation">Exponentiation</a> (^)</th></tr><tr><th scope="row" class="infobox-label" style="display:none;"></th><td class="infobox-data infobox-data-a" style="text-align:right; vertical-align:middle;"> <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 \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{base}}^{\text{exponent}}\\\scriptstyle {\text{base}}^{\text{power}}\end{matrix}}\right\}\,=\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mtext>base</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>exponent</mtext> </mrow> </msup> </mstyle> </mtd> </mtr> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mtext>base</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>power</mtext> </mrow> </msup> </mstyle> </mtd> </mtr> </mtable> </mrow> <mo>}</mo> </mrow> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{base}}^{\text{exponent}}\\\scriptstyle {\text{base}}^{\text{power}}\end{matrix}}\right\}\,=\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ecb107371002b62a60fcbd13e742f4d81f872b67" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:12.618ex; height:4.843ex;" alt="{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{base}}^{\text{exponent}}\\\scriptstyle {\text{base}}^{\text{power}}\end{matrix}}\right\}\,=\,}"></span></td><td class="infobox-data infobox-data-b" style="text-align:left; vertical-align:middle;"> <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 \scriptstyle {\text{power}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtext>power</mtext> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\text{power}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0d0a9fbffb659c0055d5ee6fde3f7f28e96f45c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.297ex; height:1.509ex;" alt="{\displaystyle \scriptstyle {\text{power}}}"></span></td></tr><tr><th colspan="4" class="infobox-header"><a href="/wiki/Nth_root" title="Nth root"><i>n</i>th root</a> (√)</th></tr><tr><th scope="row" class="infobox-label" style="display:none;"></th><td class="infobox-data infobox-data-a" style="text-align:right; vertical-align:middle;"> <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 \scriptstyle {\sqrt[{\text{degree}}]{\scriptstyle {\text{radicand}}}}\,=\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtext>radicand</mtext> </mrow> </mstyle> <mrow class="MJX-TeXAtom-ORD"> <mtext>degree</mtext> </mrow> </mroot> </mrow> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\sqrt[{\text{degree}}]{\scriptstyle {\text{radicand}}}}\,=\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5582d567e7e7fbcdb728291770905e09beb0ea18" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.422ex; height:2.676ex;" alt="{\displaystyle \scriptstyle {\sqrt[{\text{degree}}]{\scriptstyle {\text{radicand}}}}\,=\,}"></span></td><td class="infobox-data infobox-data-b" style="text-align:left; vertical-align:middle;"> <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 \scriptstyle {\text{root}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtext>root</mtext> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\text{root}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a015c1122190da3f1f1732d88b8bb03a8d7eb91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.928ex; height:1.676ex;" alt="{\displaystyle \scriptstyle {\text{root}}}"></span></td></tr><tr><th colspan="4" class="infobox-header"><a href="/wiki/Logarithm" title="Logarithm">Logarithm</a> (log)</th></tr><tr><th scope="row" class="infobox-label" style="display:none;"></th><td class="infobox-data infobox-data-a" style="text-align:right; vertical-align:middle;"> <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 \scriptstyle \log _{\text{base}}({\text{anti-logarithm}})\,=\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>base</mtext> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>anti-logarithm</mtext> </mrow> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \log _{\text{base}}({\text{anti-logarithm}})\,=\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2435266fcae4aa91d3d70a74bb91b5b35ef52edd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.454ex; height:2.176ex;" alt="{\displaystyle \scriptstyle \log _{\text{base}}({\text{anti-logarithm}})\,=\,}"></span></td><td class="infobox-data infobox-data-b" style="text-align:left; vertical-align:middle;"> <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 \scriptstyle {\text{logarithm}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtext>logarithm</mtext> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\text{logarithm}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe5d50baa86b950ff6d15760b7a38df1f8d8c868" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.948ex; height:2.009ex;" alt="{\displaystyle \scriptstyle {\text{logarithm}}}"></span></td></tr></tbody></table></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Arithmetic_operations" title="Template:Arithmetic operations"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Arithmetic_operations" title="Template talk:Arithmetic operations"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Arithmetic_operations" title="Special:EditPage/Template:Arithmetic operations"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Shop_placard_showing_20%25_reduction.JPG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2d/Shop_placard_showing_20%25_reduction.JPG/220px-Shop_placard_showing_20%25_reduction.JPG" decoding="async" width="220" height="293" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2d/Shop_placard_showing_20%25_reduction.JPG/330px-Shop_placard_showing_20%25_reduction.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2d/Shop_placard_showing_20%25_reduction.JPG/440px-Shop_placard_showing_20%25_reduction.JPG 2x" data-file-width="2304" data-file-height="3072" /></a><figcaption>Placard outside a shop in <a href="/wiki/Bordeaux" title="Bordeaux">Bordeaux</a> advertising subtraction of 20% from the price of the second perfume purchased.</figcaption></figure> <p><b>Subtraction</b> (which is signified by the <a href="/wiki/Minus_sign" class="mw-redirect" title="Minus sign">minus sign</a> <span class="nounderlines" style="border: 1px solid var(--border-color-muted,#ddd); color: var(--color-base); background-color: var( --background-color-neutral-subtle, #fdfdfd); padding: 1px 1px;">−</span>) is one of the four <a href="/wiki/Arithmetic#Arithmetic_operations" title="Arithmetic">arithmetic operations</a> along with <a href="/wiki/Addition" title="Addition">addition</a>, <a href="/wiki/Multiplication" title="Multiplication">multiplication</a> and <a href="/wiki/Division_(mathematics)" title="Division (mathematics)">division</a>. Subtraction is an operation that represents removal of objects from a collection.<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> For example, in the adjacent picture, there are <span class="nowrap">5 − 2</span> peaches—meaning 5 peaches with 2 taken away, resulting in a total of 3 peaches. Therefore, the <i>difference</i> of 5 and 2 is 3; that is, <span class="nowrap">5 − 2 = 3</span>. While primarily associated with natural numbers in <a href="/wiki/Arithmetic" title="Arithmetic">arithmetic</a>, subtraction can also represent removing or decreasing physical and abstract quantities using different kinds of objects including <a href="/wiki/Negative_number" title="Negative number">negative numbers</a>, <a href="/wiki/Fraction_(mathematics)" class="mw-redirect" title="Fraction (mathematics)">fractions</a>, <a href="/wiki/Irrational_number" title="Irrational number">irrational numbers</a>, <a href="/wiki/Euclidean_vector" title="Euclidean vector">vectors</a>, decimals, functions, and matrices.<sup id="cite_ref-:0_2-0" class="reference"><a href="#cite_note-:0-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>In a sense, subtraction is the inverse of addition. That is, <span class="texhtml"><i>c</i> = <i>a</i> − <i>b</i></span> <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> <span class="texhtml"><i>c</i> + <i>b</i> = <i>a</i></span>. In words: the difference of two numbers is the number that gives the first one when added to the second one. </p><p>Subtraction follows several important patterns. It is <a href="/wiki/Anticommutative" class="mw-redirect" title="Anticommutative">anticommutative</a>, meaning that changing the order changes the sign of the answer. It is also not <a href="/wiki/Associativity" class="mw-redirect" title="Associativity">associative</a>, meaning that when one subtracts more than two numbers, the order in which subtraction is performed matters. Because <a href="/wiki/0_(number)" class="mw-redirect" title="0 (number)">0</a> is the <a href="/wiki/Additive_identity" title="Additive identity">additive identity</a>, subtraction of it does not change a number. Subtraction also obeys predictable rules concerning related operations, such as <a href="/wiki/Addition" title="Addition">addition</a> and <a href="/wiki/Multiplication" title="Multiplication">multiplication</a>. All of these rules can be <a href="/wiki/Mathematical_proof" title="Mathematical proof">proven</a>, starting with the subtraction of <a href="/wiki/Integers" class="mw-redirect" title="Integers">integers</a> and generalizing up through the <a href="/wiki/Real_number" title="Real number">real numbers</a> and beyond. General <a href="/wiki/Binary_operations" class="mw-redirect" title="Binary operations">binary operations</a> that follow these patterns are studied in <a href="/wiki/Abstract_algebra" title="Abstract algebra">abstract algebra</a>. </p><p>In <a href="/wiki/Computability_theory" title="Computability theory">computability theory</a>, considering subtraction is not <a href="/wiki/Well-defined_expression" title="Well-defined expression">well-defined</a> over <a href="/wiki/Natural_number" title="Natural number">natural numbers</a>, operations between numbers are actually defined using "truncated subtraction" or <a href="/wiki/Monus" title="Monus">monus</a>.<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> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Notation_and_terminology">Notation and terminology</h2></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Subtraction_chart.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/29/Subtraction_chart.png/180px-Subtraction_chart.png" decoding="async" width="180" height="201" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/29/Subtraction_chart.png/270px-Subtraction_chart.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/29/Subtraction_chart.png/360px-Subtraction_chart.png 2x" data-file-width="604" data-file-height="676" /></a><figcaption>Subtraction of numbers 0–10. Line labels = minuend. X axis = subtrahend. Y axis = difference.</figcaption></figure> <p>Subtraction is usually written using the <a href="/wiki/Minus_sign" class="mw-redirect" title="Minus sign">minus sign</a> "−" between the terms; that is, in <a href="/wiki/Infix_notation" title="Infix notation">infix notation</a>. The result is expressed with an <a href="/wiki/Equals_sign" title="Equals sign">equals sign</a>. For example, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2-1=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>−</mo> <mn>1</mn> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2-1=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e790a95dd451fdb4c8e2f74f60823807ca50f2b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.426ex; height:2.343ex;" alt="{\displaystyle 2-1=1}"></span> (pronounced as "two minus one equals one")</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4-2=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mo>−</mo> <mn>2</mn> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4-2=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0444de9ba156ac35201702bd456fa0fe052dcca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.426ex; height:2.343ex;" alt="{\displaystyle 4-2=2}"></span> (pronounced as "four minus two equals two")</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 6-3=3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>6</mn> <mo>−</mo> <mn>3</mn> <mo>=</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 6-3=3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b72940ca10f48bd8b0cb353a04f9001edfeb7d15" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.426ex; height:2.343ex;" alt="{\displaystyle 6-3=3}"></span> (pronounced as "six minus three equals three")</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4-6=-2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mo>−</mo> <mn>6</mn> <mo>=</mo> <mo>−</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4-6=-2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b681fa4f73fcb5a0966be6f7f975ec5e36286727" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.234ex; height:2.343ex;" alt="{\displaystyle 4-6=-2}"></span> (pronounced as "four minus six equals negative two")</dd></dl> <p>There are also situations where subtraction is "understood", even though no symbol appears:<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (January 2023)">citation needed</span></a></i>]</sup> </p> <ul><li>A column of two numbers, with the lower number in red, usually indicates that the lower number in the column is to be subtracted, with the difference written below, under a line. This is most common in accounting.</li></ul> <p>Formally, the number being subtracted is known as the <b>subtrahend</b>,<sup id="cite_ref-Schmid_1974_4-0" class="reference"><a href="#cite_note-Schmid_1974-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Schmid_1983_5-0" class="reference"><a href="#cite_note-Schmid_1983-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> while the number it is subtracted from is the <b>minuend</b>.<sup id="cite_ref-Schmid_1974_4-1" class="reference"><a href="#cite_note-Schmid_1974-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Schmid_1983_5-1" class="reference"><a href="#cite_note-Schmid_1983-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> The result is the <b>difference</b>.<sup id="cite_ref-Schmid_1974_4-2" class="reference"><a href="#cite_note-Schmid_1974-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Schmid_1983_5-2" class="reference"><a href="#cite_note-Schmid_1983-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:0_2-1" class="reference"><a href="#cite_note-:0-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><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> That is, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\rm {minuend}}-{\rm {subtrahend}}={\rm {difference}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">b</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">h</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">f</mi> <mi mathvariant="normal">f</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">e</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {minuend}}-{\rm {subtrahend}}={\rm {difference}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a658749b52144559eb4567ca040b89e470b9c686" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:36.117ex; height:2.343ex;" alt="{\displaystyle {\rm {minuend}}-{\rm {subtrahend}}={\rm {difference}}}"></span>.</dd></dl> <p>All of this terminology derives from <a href="/wiki/Latin" title="Latin">Latin</a>. "<a href="https://en.wiktionary.org/wiki/subtraction" class="extiw" title="wikt:subtraction">Subtraction</a>" is an <a href="/wiki/English_language" title="English language">English</a> word derived from the Latin <a href="/wiki/Verb" title="Verb">verb</a> <i>subtrahere</i>, which in turn is a <a href="/wiki/Compound_(linguistics)" title="Compound (linguistics)">compound</a> of <i>sub</i> "from under" and <i>trahere</i> "to pull". Thus, to subtract is to <i>draw from below</i>, or to <i>take away</i>.<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> Using the <a href="/wiki/Gerundive" title="Gerundive">gerundive</a> <a href="/wiki/Affix" title="Affix">suffix</a> <i>-nd</i> results in "subtrahend", "thing to be subtracted".<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>a<span class="cite-bracket">]</span></a></sup> Likewise, from <i>minuere</i> "to reduce or diminish", one gets "minuend", which means "thing to be diminished". </p> <div class="mw-heading mw-heading2"><h2 id="Of_integers_and_real_numbers">Of integers and real numbers</h2></div> <div class="mw-heading mw-heading3"><h3 id="Integers">Integers</h3></div> <figure class="mw-default-size mw-halign-left" typeof="mw:File"><a href="/wiki/File:Line_Segment_jaredwf.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b2/Line_Segment_jaredwf.svg/142px-Line_Segment_jaredwf.svg.png" decoding="async" width="142" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b2/Line_Segment_jaredwf.svg/213px-Line_Segment_jaredwf.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b2/Line_Segment_jaredwf.svg/284px-Line_Segment_jaredwf.svg.png 2x" data-file-width="142" data-file-height="40" /></a><figcaption> </figcaption></figure> <p>Imagine a <a href="/wiki/Line_segment" title="Line segment">line segment</a> of <a href="/wiki/Length" title="Length">length</a> <i>b</i> with the left end labeled <i>a</i> and the right end labeled <i>c</i>. Starting from <i>a</i>, it takes <i>b</i> steps to the right to reach <i>c</i>. This movement to the right is modeled mathematically by <a href="/wiki/Addition" title="Addition">addition</a>: </p> <dl><dd><i>a</i> + <i>b</i> = <i>c</i>.</dd></dl> <p>From <i>c</i>, it takes <i>b</i> steps to the <i>left</i> to get back to <i>a</i>. This movement to the left is modeled by subtraction: </p> <dl><dd><i>c</i> − <i>b</i> = <i>a</i>.</dd></dl> <figure class="mw-default-size mw-halign-left" typeof="mw:File"><a href="/wiki/File:Subtraction_line_segment.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/ca/Subtraction_line_segment.svg/143px-Subtraction_line_segment.svg.png" decoding="async" width="143" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/ca/Subtraction_line_segment.svg/215px-Subtraction_line_segment.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/ca/Subtraction_line_segment.svg/286px-Subtraction_line_segment.svg.png 2x" data-file-width="143" data-file-height="50" /></a><figcaption> </figcaption></figure> <p>Now, a line segment labeled with the numbers <a href="/wiki/1_(number)" class="mw-redirect" title="1 (number)">1</a>, <a href="/wiki/2_(number)" class="mw-redirect" title="2 (number)">2</a>, and <a href="/wiki/3_(number)" class="mw-redirect" title="3 (number)">3</a>. From position 3, it takes no steps to the left to stay at 3, so <span class="nowrap">3 − 0 = 3</span>. It takes 2 steps to the left to get to position 1, so <span class="nowrap">3 − 2 = 1</span>. This picture is inadequate to describe what would happen after going 3 steps to the left of position 3. To represent such an operation, the line must be extended. </p><p>To subtract arbitrary <a href="/wiki/Natural_number" title="Natural number">natural numbers</a>, one begins with a line containing every natural number (0, 1, 2, 3, 4, 5, 6, ...). From 3, it takes 3 steps to the left to get to 0, so <span class="nowrap">3 − 3 = 0</span>. But <span class="nowrap">3 − 4</span> is still invalid, since it again leaves the line. The natural numbers are not a useful context for subtraction. </p><p>The solution is to consider the <a href="/wiki/Integer" title="Integer">integer</a> <a href="/wiki/Number_line" title="Number line">number line</a> (..., −3, −2, −1, 0, 1, 2, 3, ...). This way, it takes 4 steps to the left from 3 to get to −1: </p> <dl><dd><span class="nowrap">3 − 4 = −1</span>.</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Natural_numbers">Natural numbers</h3></div> <p>Subtraction of <a href="/wiki/Natural_numbers" class="mw-redirect" title="Natural numbers">natural numbers</a> is not <a href="/wiki/Closure_(mathematics)" title="Closure (mathematics)">closed</a>: the difference is not a natural number unless the minuend is greater than or equal to the subtrahend. For example, 26 cannot be subtracted from 11 to give a natural number. Such a case uses one of two approaches: </p> <ol><li>Conclude that 26 cannot be subtracted from 11; subtraction becomes a <a href="/wiki/Partial_function" title="Partial function">partial function</a>.</li> <li>Give the answer as an <a href="/wiki/Integer" title="Integer">integer</a> representing a <a href="/wiki/Negative_number" title="Negative number">negative number</a>, so the result of subtracting 26 from 11 is −15.</li></ol> <div class="mw-heading mw-heading3"><h3 id="Real_numbers">Real numbers</h3></div> <p>The <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a> of real numbers can be defined specifying only two binary operations, addition and multiplication, together with <a href="/wiki/Unary_operations" class="mw-redirect" title="Unary operations">unary operations</a> yielding <a href="/wiki/Additive_inverse" title="Additive inverse">additive</a> and <a href="/wiki/Multiplicative_inverse" title="Multiplicative inverse">multiplicative</a> inverses. The subtraction of a real number (the subtrahend) from another (the minuend) can then be defined as the addition of the minuend and the additive inverse of the subtrahend. For example, <span class="texhtml">3 − <i>π</i> = 3 + (−<i>π</i>)</span>. Alternatively, instead of requiring these unary operations, the binary operations of subtraction and <a href="/wiki/Division_(mathematics)" title="Division (mathematics)">division</a> can be taken as basic. </p> <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2></div> <div class="mw-heading mw-heading3"><h3 id="Anti-commutativity">Anti-commutativity</h3></div> <p>Subtraction is <a href="/wiki/Anti-commutative" class="mw-redirect" title="Anti-commutative">anti-commutative</a>, meaning that if one reverses the terms in a difference left-to-right, the result is the negative of the original result. Symbolically, if <i>a</i> and <i>b</i> are any two numbers, then </p> <dl><dd><i>a</i> − <i>b</i> = −(<i>b</i> − <i>a)</i>.</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Non-associativity">Non-associativity</h3></div> <p>Subtraction is <a href="/wiki/Associativity" class="mw-redirect" title="Associativity">non-associative</a>, which comes up when one tries to define repeated subtraction. In general, the expression </p> <dl><dd>"<i>a</i> − <i>b</i> − <i>c</i>"</dd></dl> <p>can be defined to mean either (<i>a</i> − <i>b</i>) − <i>c</i> or <i>a</i> − (<i>b</i> − <i>c</i>), but these two possibilities lead to different answers. To resolve this issue, one must establish an <a href="/wiki/Order_of_operations" title="Order of operations">order of operations</a>, with different orders yielding different results. </p> <div class="mw-heading mw-heading3"><h3 id="Predecessor">Predecessor</h3></div> <p>In the context of integers, subtraction of <a href="/wiki/1_(number)" class="mw-redirect" title="1 (number)">one</a> also plays a special role: for any integer <i>a</i>, the integer <span class="nowrap">(<i>a</i> − 1)</span> is the largest integer less than <i>a</i>, also known as the predecessor of <i>a</i>. </p> <div class="mw-heading mw-heading2"><h2 id="Units_of_measurement">Units of measurement</h2></div> <p>When subtracting two numbers with units of measurement such as <a href="/wiki/Kilogram" title="Kilogram">kilograms</a> or <a href="/wiki/Pound_(mass)" title="Pound (mass)">pounds</a>, they must have the same unit. In most cases, the difference will have the same unit as the original numbers. </p> <div class="mw-heading mw-heading3"><h3 id="Percentages">Percentages</h3></div> <p>Changes in <a href="/wiki/Percentage" title="Percentage">percentages</a> can be reported in at least two forms, <a href="/wiki/Percentage_change" class="mw-redirect" title="Percentage change">percentage change</a> and <a href="/wiki/Percentage_point" title="Percentage point">percentage point</a> change. Percentage change represents the <a href="/wiki/Relative_change" title="Relative change">relative change</a> between the two quantities as a percentage, while <a href="/wiki/Percentage_point" title="Percentage point">percentage point</a> change is simply the number obtained by subtracting the two percentages.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p><p>As an example, suppose that 30% of widgets made in a factory are defective. Six months later, 20% of widgets are defective. The percentage change is <style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠<span class="tion"><span class="num">20% − 30%</span><span class="sr-only">/</span><span class="den">30%</span></span>⁠</span> = −<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">3</span></span>⁠</span> = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠−33<span class="sr-only">+</span><span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">3</span></span>⁠</span>%, while the percentage point change is −10 percentage points. </p> <div class="mw-heading mw-heading2"><h2 id="In_computing">In computing</h2></div> <p>The <a href="/wiki/Method_of_complements" title="Method of complements">method of complements</a> is a technique used to subtract one number from another using only the addition of positive numbers. This method was commonly used in <a href="/wiki/Mechanical_calculator" title="Mechanical calculator">mechanical calculators</a>, and is still used in modern <a href="/wiki/Computers" class="mw-redirect" title="Computers">computers</a>. </p> <table class="wikitable floatright"> <tbody><tr> <th>Binary<br />digit </th> <th>Ones' <br />complement </th></tr> <tr align="center"> <td>0 </td> <td>1 </td></tr> <tr align="center"> <td>1 </td> <td>0 </td></tr></tbody></table> <p>To subtract a binary number <i>y</i> (the subtrahend) from another number <i>x</i> (the minuend), the ones' complement of <i>y</i> is added to <i>x</i> and one is added to the sum. The leading digit "1" of the result is then discarded. </p><p>The method of complements is especially useful in binary (radix 2) since the ones' complement is very easily obtained by inverting each bit (changing "0" to "1" and vice versa). And adding 1 to get the two's complement can be done by simulating a carry into the least significant bit. For example: </p> <pre> 01100100 (x, equals decimal 100) - 00010110 (y, equals decimal 22) </pre> <p>becomes the sum: </p> <pre> 01100100 (x) + 11101001 (ones' complement of y) + 1 (to get the two's complement) —————————— 101001110 </pre> <p>Dropping the initial "1" gives the answer: 01001110 (equals decimal 78) </p> <div class="mw-heading mw-heading2"><h2 id="The_teaching_of_subtraction_in_schools">The teaching of subtraction in schools</h2></div> <p>Methods used to teach subtraction to <a href="/wiki/Elementary_school" class="mw-redirect" title="Elementary school">elementary school</a> vary from country to country, and within a country, different methods are adopted at different times. In what is known in the United States as <a href="/wiki/Traditional_mathematics" title="Traditional mathematics">traditional mathematics</a>, a specific process is taught to students at the end of the 1st year (or during the 2nd year) for use with multi-digit whole numbers, and is extended in either the fourth or fifth grade to include <a href="/wiki/Decimal_representation" title="Decimal representation">decimal representations</a> of fractional numbers. </p> <div class="mw-heading mw-heading3"><h3 id="In_America">In America</h3></div> <p>Almost all American schools currently teach a method of subtraction using borrowing or regrouping (the decomposition algorithm) and a system of markings called crutches.<sup id="cite_ref-Klapper1916_12-0" class="reference"><a href="#cite_note-Klapper1916-12"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> Although a method of borrowing had been known and published in textbooks previously, the use of crutches in American schools spread after <a href="/wiki/William_A._Brownell" title="William A. Brownell">William A. Brownell</a> published a study—claiming that crutches were beneficial to students using this method.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> This system caught on rapidly, displacing the other methods of subtraction in use in America at that time. </p> <div class="mw-heading mw-heading3"><h3 id="In_Europe">In Europe</h3></div> <p>Some European schools employ a method of subtraction called the Austrian method, also known as the additions method. There is no borrowing in this method. There are also crutches (markings to aid memory), which vary by country.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Smith1913_16-0" class="reference"><a href="#cite_note-Smith1913-16"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Comparing_the_two_main_methods">Comparing the two main methods</h3></div> <p>Both these methods break up the subtraction as a process of one digit subtractions by place value. Starting with a least significant digit, a subtraction of the subtrahend: </p> <dl><dd><i>s</i><sub><i>j</i></sub> <i>s</i><sub><i>j</i>−1</sub> ... <i>s</i><sub>1</sub></dd></dl> <p>from the minuend </p> <dl><dd><i>m</i><sub><i>k</i></sub> <i>m</i><sub><i>k</i>−1</sub> ... <i>m</i><sub>1</sub>,</dd></dl> <p>where each <i>s</i><sub><i>i</i></sub> and <i>m</i><sub><i>i</i></sub> is a digit, proceeds by writing down <span class="nowrap"><i>m</i><sub>1</sub> − <i>s</i><sub>1</sub></span>, <span class="nowrap"><i>m</i><sub>2</sub> − <i>s</i><sub>2</sub></span>, and so forth, as long as <i>s</i><sub><i>i</i></sub> does not exceed <i>m</i><sub><i>i</i></sub>. Otherwise, <i>m</i><sub><i>i</i></sub> is increased by 10 and some other digit is modified to correct for this increase. The American method corrects by attempting to decrease the minuend digit <i>m</i><sub><i>i</i>+1</sub> by one (or continuing the borrow leftwards until there is a non-zero digit from which to borrow). The European method corrects by increasing the subtrahend digit <i>s</i><sub><i>i</i>+1</sub> by one. </p><p><b>Example:</b> 704 − 512. </p> <p style="position:relative; text-align:center;"> <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 {\begin{array}{rrrr}&\color {Red}-1\\&C&D&U\\&7&0&4\\&5&1&2\\\hline &1&9&2\\\end{array}}{\begin{array}{l}{\color {Red}\longleftarrow {\rm {carry}}}\\\\\longleftarrow \;{\rm {Minuend}}\\\longleftarrow \;{\rm {Subtrahend}}\\\longleftarrow {\rm {Rest\;or\;Difference}}\\\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right right right right" rowspacing="4pt" columnspacing="1em" rowlines="none none none solid"> <mtr> <mtd /> <mtd> <mstyle mathcolor="#ED1B23"> <mo>−</mo> <mn>1</mn> </mstyle> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi>C</mi> </mtd> <mtd> <mi>D</mi> </mtd> <mtd> <mi>U</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mn>7</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>4</mn> </mtd> </mtr> <mtr> <mtd /> <mtd> <mn>5</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> </mtr> <mtr> <mtd /> <mtd> <mn>1</mn> </mtd> <mtd> <mn>9</mn> </mtd> <mtd> <mn>2</mn> </mtd> </mtr> </mtable> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#ED1B23"> <mo stretchy="false">⟵</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">y</mi> </mrow> </mrow> </mstyle> </mrow> </mtd> </mtr> <mtr> <mtd /> </mtr> <mtr> <mtd> <mo stretchy="false">⟵</mo> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">M</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">d</mi> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mo stretchy="false">⟵</mo> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">b</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">h</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">d</mi> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mo stretchy="false">⟵</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">t</mi> <mspace width="thickmathspace" /> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">r</mi> <mspace width="thickmathspace" /> <mi mathvariant="normal">D</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">f</mi> <mi mathvariant="normal">f</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">e</mi> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{rrrr}&\color {Red}-1\\&C&D&U\\&7&0&4\\&5&1&2\\\hline &1&9&2\\\end{array}}{\begin{array}{l}{\color {Red}\longleftarrow {\rm {carry}}}\\\\\longleftarrow \;{\rm {Minuend}}\\\longleftarrow \;{\rm {Subtrahend}}\\\longleftarrow {\rm {Rest\;or\;Difference}}\\\end{array}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c29ffaf2684f6bb68e6053fd22c29b9ef5273615" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.838ex; width:39.804ex; height:16.843ex;" alt="{\displaystyle {\begin{array}{rrrr}&\color {Red}-1\\&C&D&U\\&7&0&4\\&5&1&2\\\hline &1&9&2\\\end{array}}{\begin{array}{l}{\color {Red}\longleftarrow {\rm {carry}}}\\\\\longleftarrow \;{\rm {Minuend}}\\\longleftarrow \;{\rm {Subtrahend}}\\\longleftarrow {\rm {Rest\;or\;Difference}}\\\end{array}}}"></span> </p> <p>The minuend is 704, the subtrahend is 512. The minuend digits are <span class="nowrap"><i>m</i><sub>3</sub> = 7</span>, <span class="nowrap"><i>m</i><sub>2</sub> = 0</span> and <span class="nowrap"><i>m</i><sub>1</sub> = 4</span>. The subtrahend digits are <span class="nowrap"><i>s</i><sub>3</sub> = 5</span>, <span class="nowrap"><i>s</i><sub>2</sub> = 1</span> and <span class="nowrap"><i>s</i><sub>1</sub> = 2</span>. Beginning at the one's place, 4 is not less than 2 so the difference 2 is written down in the result's one's place. In the ten's place, 0 is less than 1, so the 0 is increased by 10, and the difference with 1, which is 9, is written down in the ten's place. The American method corrects for the increase of ten by reducing the digit in the minuend's hundreds place by one. That is, the 7 is struck through and replaced by a 6. The subtraction then proceeds in the hundreds place, where 6 is not less than 5, so the difference is written down in the result's hundred's place. We are now done, the result is 192. </p><p>The Austrian method does not reduce the 7 to 6. Rather it increases the subtrahend hundreds digit by one. A small mark is made near or below this digit (depending on the school). Then the subtraction proceeds by asking what number when increased by 1, and 5 is added to it, makes 7. The answer is 1, and is written down in the result's hundreds place. </p><p>There is an additional subtlety in that the student always employs a mental subtraction table in the American method. The Austrian method often encourages the student to mentally use the addition table in reverse. In the example above, rather than adding 1 to 5, getting 6, and subtracting that from 7, the student is asked to consider what number, when increased by 1, and 5 is added to it, makes 7. </p> <div class="mw-heading mw-heading2"><h2 id="Subtraction_by_hand">Subtraction by hand</h2></div> <div class="mw-heading mw-heading3"><h3 id="Austrian_method">Austrian method</h3></div> <p>Example:<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (February 2023)">citation needed</span></a></i>]</sup> </p> <ul class="gallery mw-gallery-traditional"> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Vertical_Subtraction_Method_B_Step_1.JPG" class="mw-file-description" title="1 + ... = 3"><img alt="1 + ... = 3" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Vertical_Subtraction_Method_B_Step_1.JPG/76px-Vertical_Subtraction_Method_B_Step_1.JPG" decoding="async" width="76" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Vertical_Subtraction_Method_B_Step_1.JPG/114px-Vertical_Subtraction_Method_B_Step_1.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Vertical_Subtraction_Method_B_Step_1.JPG/152px-Vertical_Subtraction_Method_B_Step_1.JPG 2x" data-file-width="271" data-file-height="427" /></a></span></div> <div class="gallerytext">1 + ... = 3</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Vertical_Subtraction_Method_B_Step_2.JPG" class="mw-file-description" title="The difference is written under the line."><img alt="The difference is written under the line." src="//upload.wikimedia.org/wikipedia/commons/thumb/8/89/Vertical_Subtraction_Method_B_Step_2.JPG/76px-Vertical_Subtraction_Method_B_Step_2.JPG" decoding="async" width="76" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/89/Vertical_Subtraction_Method_B_Step_2.JPG/114px-Vertical_Subtraction_Method_B_Step_2.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/89/Vertical_Subtraction_Method_B_Step_2.JPG/152px-Vertical_Subtraction_Method_B_Step_2.JPG 2x" data-file-width="272" data-file-height="428" /></a></span></div> <div class="gallerytext">The difference is written under the line.</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Vertical_Subtraction_Method_B_Step_3.JPG" class="mw-file-description" title="9 + ... = 5 The required sum (5) is too small."><img alt="9 + ... = 5 The required sum (5) is too small." src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ae/Vertical_Subtraction_Method_B_Step_3.JPG/76px-Vertical_Subtraction_Method_B_Step_3.JPG" decoding="async" width="76" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ae/Vertical_Subtraction_Method_B_Step_3.JPG/114px-Vertical_Subtraction_Method_B_Step_3.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ae/Vertical_Subtraction_Method_B_Step_3.JPG/152px-Vertical_Subtraction_Method_B_Step_3.JPG 2x" data-file-width="272" data-file-height="428" /></a></span></div> <div class="gallerytext">9 + ... = 5<br />The required sum (5) is too small.</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Vertical_Subtraction_Method_B_Step_4.JPG" class="mw-file-description" title="So, we add 10 to it and put a 1 under the next higher place in the subtrahend."><img alt="So, we add 10 to it and put a 1 under the next higher place in the subtrahend." src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Vertical_Subtraction_Method_B_Step_4.JPG/76px-Vertical_Subtraction_Method_B_Step_4.JPG" decoding="async" width="76" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Vertical_Subtraction_Method_B_Step_4.JPG/114px-Vertical_Subtraction_Method_B_Step_4.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Vertical_Subtraction_Method_B_Step_4.JPG/152px-Vertical_Subtraction_Method_B_Step_4.JPG 2x" data-file-width="272" data-file-height="428" /></a></span></div> <div class="gallerytext">So, we add 10 to it and put a 1 under the next higher place in the subtrahend.</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Vertical_Subtraction_Method_B_Step_5.JPG" class="mw-file-description" title="9 + ... = 15 Now we can find the difference as before."><img alt="9 + ... = 15 Now we can find the difference as before." src="//upload.wikimedia.org/wikipedia/commons/thumb/7/75/Vertical_Subtraction_Method_B_Step_5.JPG/76px-Vertical_Subtraction_Method_B_Step_5.JPG" decoding="async" width="76" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/75/Vertical_Subtraction_Method_B_Step_5.JPG/114px-Vertical_Subtraction_Method_B_Step_5.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/75/Vertical_Subtraction_Method_B_Step_5.JPG/152px-Vertical_Subtraction_Method_B_Step_5.JPG 2x" data-file-width="272" data-file-height="428" /></a></span></div> <div class="gallerytext">9 + ... = 15<br />Now we can find the difference as before.</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Vertical_Subtraction_Method_B_Step_6.JPG" class="mw-file-description" title="(4 + 1) + ... = 7"><img alt="(4 + 1) + ... = 7" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Vertical_Subtraction_Method_B_Step_6.JPG/76px-Vertical_Subtraction_Method_B_Step_6.JPG" decoding="async" width="76" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Vertical_Subtraction_Method_B_Step_6.JPG/114px-Vertical_Subtraction_Method_B_Step_6.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Vertical_Subtraction_Method_B_Step_6.JPG/152px-Vertical_Subtraction_Method_B_Step_6.JPG 2x" data-file-width="271" data-file-height="428" /></a></span></div> <div class="gallerytext">(4 + 1) + ... = 7</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Vertical_Subtraction_Method_B_Step_7.JPG" class="mw-file-description" title="The difference is written under the line."><img alt="The difference is written under the line." src="//upload.wikimedia.org/wikipedia/commons/thumb/2/25/Vertical_Subtraction_Method_B_Step_7.JPG/76px-Vertical_Subtraction_Method_B_Step_7.JPG" decoding="async" width="76" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/25/Vertical_Subtraction_Method_B_Step_7.JPG/114px-Vertical_Subtraction_Method_B_Step_7.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/25/Vertical_Subtraction_Method_B_Step_7.JPG/151px-Vertical_Subtraction_Method_B_Step_7.JPG 2x" data-file-width="271" data-file-height="429" /></a></span></div> <div class="gallerytext">The difference is written under the line.</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Vertical_Subtraction_Method_B_Step_8.JPG" class="mw-file-description" title="The total difference."><img alt="The total difference." src="//upload.wikimedia.org/wikipedia/commons/thumb/5/59/Vertical_Subtraction_Method_B_Step_8.JPG/76px-Vertical_Subtraction_Method_B_Step_8.JPG" decoding="async" width="76" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/59/Vertical_Subtraction_Method_B_Step_8.JPG/114px-Vertical_Subtraction_Method_B_Step_8.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/59/Vertical_Subtraction_Method_B_Step_8.JPG/152px-Vertical_Subtraction_Method_B_Step_8.JPG 2x" data-file-width="271" data-file-height="427" /></a></span></div> <div class="gallerytext">The total difference.</div> </li> </ul> <div class="mw-heading mw-heading3"><h3 id="Subtraction_from_left_to_right">Subtraction from left to right</h3></div> <p>Example:<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (February 2023)">citation needed</span></a></i>]</sup> </p> <ul class="gallery mw-gallery-traditional"> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:LeftToRight_Subtraction_Step_1.JPG" class="mw-file-description" title="7 − 4 = 3 This result is only penciled in."><img alt="7 − 4 = 3 This result is only penciled in." src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1d/LeftToRight_Subtraction_Step_1.JPG/88px-LeftToRight_Subtraction_Step_1.JPG" decoding="async" width="88" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1d/LeftToRight_Subtraction_Step_1.JPG/133px-LeftToRight_Subtraction_Step_1.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1d/LeftToRight_Subtraction_Step_1.JPG/177px-LeftToRight_Subtraction_Step_1.JPG 2x" data-file-width="273" data-file-height="370" /></a></span></div> <div class="gallerytext">7 − 4 = 3<br />This result is only penciled in.</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:LeftToRight_Subtraction_Step_2.JPG" class="mw-file-description" title="Because the next digit of the minuend is smaller than the subtrahend, we subtract one from our penciled-in-number and mentally add ten to the next."><img alt="Because the next digit of the minuend is smaller than the subtrahend, we subtract one from our penciled-in-number and mentally add ten to the next." src="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/LeftToRight_Subtraction_Step_2.JPG/88px-LeftToRight_Subtraction_Step_2.JPG" decoding="async" width="88" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/LeftToRight_Subtraction_Step_2.JPG/131px-LeftToRight_Subtraction_Step_2.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/65/LeftToRight_Subtraction_Step_2.JPG/175px-LeftToRight_Subtraction_Step_2.JPG 2x" data-file-width="272" data-file-height="372" /></a></span></div> <div class="gallerytext">Because the next digit of the minuend is smaller than the subtrahend, we subtract one from our penciled-in-number and mentally add ten to the next.</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:LeftToRight_Subtraction_Step_3.JPG" class="mw-file-description" title="15 − 9 = 6"><img alt="15 − 9 = 6" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8e/LeftToRight_Subtraction_Step_3.JPG/88px-LeftToRight_Subtraction_Step_3.JPG" decoding="async" width="88" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8e/LeftToRight_Subtraction_Step_3.JPG/132px-LeftToRight_Subtraction_Step_3.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8e/LeftToRight_Subtraction_Step_3.JPG/176px-LeftToRight_Subtraction_Step_3.JPG 2x" data-file-width="272" data-file-height="371" /></a></span></div> <div class="gallerytext">15 − 9 = 6</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:LeftToRight_Subtraction_Step_4.JPG" class="mw-file-description" title="Because the next digit in the minuend is not smaller than the subtrahend, we keep this number."><img alt="Because the next digit in the minuend is not smaller than the subtrahend, we keep this number." src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3f/LeftToRight_Subtraction_Step_4.JPG/88px-LeftToRight_Subtraction_Step_4.JPG" decoding="async" width="88" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3f/LeftToRight_Subtraction_Step_4.JPG/131px-LeftToRight_Subtraction_Step_4.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3f/LeftToRight_Subtraction_Step_4.JPG/175px-LeftToRight_Subtraction_Step_4.JPG 2x" data-file-width="271" data-file-height="371" /></a></span></div> <div class="gallerytext">Because the next digit in the minuend is not smaller than the subtrahend, we keep this number.</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:LeftToRight_Subtraction_Step_5.JPG" class="mw-file-description" title="3 − 1 = 2"><img alt="3 − 1 = 2" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a7/LeftToRight_Subtraction_Step_5.JPG/88px-LeftToRight_Subtraction_Step_5.JPG" decoding="async" width="88" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a7/LeftToRight_Subtraction_Step_5.JPG/131px-LeftToRight_Subtraction_Step_5.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a7/LeftToRight_Subtraction_Step_5.JPG/175px-LeftToRight_Subtraction_Step_5.JPG 2x" data-file-width="271" data-file-height="371" /></a></span></div> <div class="gallerytext">3 − 1 = 2</div> </li> </ul> <div class="mw-heading mw-heading3"><h3 id="American_method">American method</h3></div> <p>In this method, each digit of the subtrahend is subtracted from the digit above it starting from right to left. If the top number is too small to subtract the bottom number from it, we add 10 to it; this 10 is "borrowed" from the top digit to the left, which we subtract 1 from. Then we move on to subtracting the next digit and borrowing as needed, until every digit has been subtracted. Example:<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (February 2023)">citation needed</span></a></i>]</sup> </p> <ul class="gallery mw-gallery-traditional"> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Vertical_Subtraction_Method_A_Step_1.JPG" class="mw-file-description" title="3 − 1 = ..."><img alt="3 − 1 = ..." src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8f/Vertical_Subtraction_Method_A_Step_1.JPG/76px-Vertical_Subtraction_Method_A_Step_1.JPG" decoding="async" width="76" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8f/Vertical_Subtraction_Method_A_Step_1.JPG/114px-Vertical_Subtraction_Method_A_Step_1.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8f/Vertical_Subtraction_Method_A_Step_1.JPG/153px-Vertical_Subtraction_Method_A_Step_1.JPG 2x" data-file-width="273" data-file-height="429" /></a></span></div> <div class="gallerytext">3 − 1 = ...</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Vertical_Subtraction_Method_A_Step_2.JPG" class="mw-file-description" title="We write the difference under the line."><img alt="We write the difference under the line." src="//upload.wikimedia.org/wikipedia/commons/thumb/0/05/Vertical_Subtraction_Method_A_Step_2.JPG/76px-Vertical_Subtraction_Method_A_Step_2.JPG" decoding="async" width="76" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/05/Vertical_Subtraction_Method_A_Step_2.JPG/114px-Vertical_Subtraction_Method_A_Step_2.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/05/Vertical_Subtraction_Method_A_Step_2.JPG/152px-Vertical_Subtraction_Method_A_Step_2.JPG 2x" data-file-width="272" data-file-height="428" /></a></span></div> <div class="gallerytext">We write the difference under the line.</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Vertical_Subtraction_Method_A_Step_3.JPG" class="mw-file-description" title="5 − 9 = ... The minuend (5) is too small!"><img alt="5 − 9 = ... The minuend (5) is too small!" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Vertical_Subtraction_Method_A_Step_3.JPG/76px-Vertical_Subtraction_Method_A_Step_3.JPG" decoding="async" width="76" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Vertical_Subtraction_Method_A_Step_3.JPG/114px-Vertical_Subtraction_Method_A_Step_3.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Vertical_Subtraction_Method_A_Step_3.JPG/152px-Vertical_Subtraction_Method_A_Step_3.JPG 2x" data-file-width="271" data-file-height="428" /></a></span></div> <div class="gallerytext">5 − 9 = ...<br /> The minuend (5) is too small!</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Vertical_Subtraction_Method_A_Step_4.JPG" class="mw-file-description" title="So, we add 10 to it. The 10 is "borrowed" from the digit on the left, which goes down by 1."><img alt="So, we add 10 to it. The 10 is "borrowed" from the digit on the left, which goes down by 1." src="//upload.wikimedia.org/wikipedia/commons/thumb/4/45/Vertical_Subtraction_Method_A_Step_4.JPG/76px-Vertical_Subtraction_Method_A_Step_4.JPG" decoding="async" width="76" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/45/Vertical_Subtraction_Method_A_Step_4.JPG/114px-Vertical_Subtraction_Method_A_Step_4.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/45/Vertical_Subtraction_Method_A_Step_4.JPG/152px-Vertical_Subtraction_Method_A_Step_4.JPG 2x" data-file-width="271" data-file-height="428" /></a></span></div> <div class="gallerytext">So, we add 10 to it. The 10 is "borrowed" from the digit on the left, which goes down by 1.</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Vertical_Subtraction_Method_A_Step_5.JPG" class="mw-file-description" title="15 − 9 = ... Now the subtraction works, and we write the difference under the line."><img alt="15 − 9 = ... Now the subtraction works, and we write the difference under the line." src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f4/Vertical_Subtraction_Method_A_Step_5.JPG/76px-Vertical_Subtraction_Method_A_Step_5.JPG" decoding="async" width="76" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f4/Vertical_Subtraction_Method_A_Step_5.JPG/114px-Vertical_Subtraction_Method_A_Step_5.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f4/Vertical_Subtraction_Method_A_Step_5.JPG/152px-Vertical_Subtraction_Method_A_Step_5.JPG 2x" data-file-width="272" data-file-height="429" /></a></span></div> <div class="gallerytext">15 − 9 = ...<br /> Now the subtraction works, and we write the difference under the line.</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Vertical_Subtraction_Method_A_Step_6.JPG" class="mw-file-description" title="6 − 4 = ..."><img alt="6 − 4 = ..." src="//upload.wikimedia.org/wikipedia/commons/thumb/5/54/Vertical_Subtraction_Method_A_Step_6.JPG/76px-Vertical_Subtraction_Method_A_Step_6.JPG" decoding="async" width="76" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/54/Vertical_Subtraction_Method_A_Step_6.JPG/114px-Vertical_Subtraction_Method_A_Step_6.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/54/Vertical_Subtraction_Method_A_Step_6.JPG/153px-Vertical_Subtraction_Method_A_Step_6.JPG 2x" data-file-width="273" data-file-height="429" /></a></span></div> <div class="gallerytext">6 − 4 = ...</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Vertical_Subtraction_Method_A_Step_7.JPG" class="mw-file-description" title="We write the difference under the line."><img alt="We write the difference under the line." src="//upload.wikimedia.org/wikipedia/commons/thumb/7/72/Vertical_Subtraction_Method_A_Step_7.JPG/76px-Vertical_Subtraction_Method_A_Step_7.JPG" decoding="async" width="76" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/72/Vertical_Subtraction_Method_A_Step_7.JPG/114px-Vertical_Subtraction_Method_A_Step_7.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/72/Vertical_Subtraction_Method_A_Step_7.JPG/153px-Vertical_Subtraction_Method_A_Step_7.JPG 2x" data-file-width="273" data-file-height="429" /></a></span></div> <div class="gallerytext">We write the difference under the line.</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Vertical_Subtraction_Method_A_Step_8.JPG" class="mw-file-description" title="The total difference."><img alt="The total difference." src="//upload.wikimedia.org/wikipedia/commons/thumb/5/53/Vertical_Subtraction_Method_A_Step_8.JPG/76px-Vertical_Subtraction_Method_A_Step_8.JPG" decoding="async" width="76" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/53/Vertical_Subtraction_Method_A_Step_8.JPG/114px-Vertical_Subtraction_Method_A_Step_8.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/53/Vertical_Subtraction_Method_A_Step_8.JPG/152px-Vertical_Subtraction_Method_A_Step_8.JPG 2x" data-file-width="272" data-file-height="428" /></a></span></div> <div class="gallerytext">The total difference.</div> </li> </ul> <div class="mw-heading mw-heading3"><h3 id="Trade_first">Trade first</h3></div> <p>A variant of the American method where all borrowing is done before all subtraction.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p><p>Example: </p> <ul class="gallery mw-gallery-traditional"> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Trade_First_Subtraction_Step_1.JPG" class="mw-file-description" title="1 − 3 = not possible. We add a 10 to the 1. Because the 10 is "borrowed" from the nearby 5, the 5 is lowered by 1."><img alt="1 − 3 = not possible. We add a 10 to the 1. Because the 10 is "borrowed" from the nearby 5, the 5 is lowered by 1." src="//upload.wikimedia.org/wikipedia/commons/thumb/7/78/Trade_First_Subtraction_Step_1.JPG/69px-Trade_First_Subtraction_Step_1.JPG" decoding="async" width="69" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/78/Trade_First_Subtraction_Step_1.JPG/104px-Trade_First_Subtraction_Step_1.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/78/Trade_First_Subtraction_Step_1.JPG/139px-Trade_First_Subtraction_Step_1.JPG 2x" data-file-width="235" data-file-height="406" /></a></span></div> <div class="gallerytext">1 − 3 = not possible.<br />We add a 10 to the 1. Because the 10 is "borrowed" from the nearby 5, the 5 is lowered by 1.</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Trade_First_Subtraction_Step_2.JPG" class="mw-file-description" title="4 − 9 = not possible. So we proceed as in step 1."><img alt="4 − 9 = not possible. So we proceed as in step 1." src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b0/Trade_First_Subtraction_Step_2.JPG/69px-Trade_First_Subtraction_Step_2.JPG" decoding="async" width="69" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b0/Trade_First_Subtraction_Step_2.JPG/104px-Trade_First_Subtraction_Step_2.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b0/Trade_First_Subtraction_Step_2.JPG/139px-Trade_First_Subtraction_Step_2.JPG 2x" data-file-width="234" data-file-height="404" /></a></span></div> <div class="gallerytext">4 − 9 = not possible.<br />So we proceed as in step 1.</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Trade_First_Subtraction_Step_3.JPG" class="mw-file-description" title="Working from right to left: 11 − 3 = 8"><img alt="Working from right to left: 11 − 3 = 8" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5d/Trade_First_Subtraction_Step_3.JPG/70px-Trade_First_Subtraction_Step_3.JPG" decoding="async" width="70" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5d/Trade_First_Subtraction_Step_3.JPG/104px-Trade_First_Subtraction_Step_3.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5d/Trade_First_Subtraction_Step_3.JPG/139px-Trade_First_Subtraction_Step_3.JPG 2x" data-file-width="235" data-file-height="404" /></a></span></div> <div class="gallerytext">Working from right to left:<br />11 − 3 = 8</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Trade_First_Subtraction_Step_4.JPG" class="mw-file-description" title="14 − 9 = 5"><img alt="14 − 9 = 5" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d5/Trade_First_Subtraction_Step_4.JPG/69px-Trade_First_Subtraction_Step_4.JPG" decoding="async" width="69" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d5/Trade_First_Subtraction_Step_4.JPG/104px-Trade_First_Subtraction_Step_4.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d5/Trade_First_Subtraction_Step_4.JPG/138px-Trade_First_Subtraction_Step_4.JPG 2x" data-file-width="234" data-file-height="406" /></a></span></div> <div class="gallerytext">14 − 9 = 5</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Trade_First_Subtraction_Step_5.JPG" class="mw-file-description" title="6 − 4 = 2"><img alt="6 − 4 = 2" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/72/Trade_First_Subtraction_Step_5.JPG/69px-Trade_First_Subtraction_Step_5.JPG" decoding="async" width="69" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/72/Trade_First_Subtraction_Step_5.JPG/103px-Trade_First_Subtraction_Step_5.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/72/Trade_First_Subtraction_Step_5.JPG/138px-Trade_First_Subtraction_Step_5.JPG 2x" data-file-width="233" data-file-height="406" /></a></span></div> <div class="gallerytext">6 − 4 = 2</div> </li> </ul> <div class="mw-heading mw-heading3"><h3 id="Partial_differences">Partial differences</h3></div> <p>The partial differences method is different from other vertical subtraction methods because no borrowing or carrying takes place. In their place, one places plus or minus signs depending on whether the minuend is greater or smaller than the subtrahend. The sum of the partial differences is the total difference.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p><p>Example: </p> <ul class="gallery mw-gallery-traditional"> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Partial-Differences_Subtraction_Step_1.JPG" class="mw-file-description" title="The smaller number is subtracted from the greater: 700 − 400 = 300 Because the minuend is greater than the subtrahend, this difference has a plus sign."><img alt="The smaller number is subtracted from the greater: 700 − 400 = 300 Because the minuend is greater than the subtrahend, this difference has a plus sign." src="//upload.wikimedia.org/wikipedia/commons/thumb/2/29/Partial-Differences_Subtraction_Step_1.JPG/59px-Partial-Differences_Subtraction_Step_1.JPG" decoding="async" width="59" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/29/Partial-Differences_Subtraction_Step_1.JPG/88px-Partial-Differences_Subtraction_Step_1.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/29/Partial-Differences_Subtraction_Step_1.JPG/117px-Partial-Differences_Subtraction_Step_1.JPG 2x" data-file-width="216" data-file-height="441" /></a></span></div> <div class="gallerytext">The smaller number is subtracted from the greater:<br />700 − 400 = 300<br />Because the minuend is greater than the subtrahend, this difference has a plus sign.</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Partial-Differences_Subtraction_Step_2.JPG" class="mw-file-description" title="The smaller number is subtracted from the greater: 90 − 50 = 40 Because the minuend is smaller than the subtrahend, this difference has a minus sign."><img alt="The smaller number is subtracted from the greater: 90 − 50 = 40 Because the minuend is smaller than the subtrahend, this difference has a minus sign." src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0c/Partial-Differences_Subtraction_Step_2.JPG/59px-Partial-Differences_Subtraction_Step_2.JPG" decoding="async" width="59" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0c/Partial-Differences_Subtraction_Step_2.JPG/88px-Partial-Differences_Subtraction_Step_2.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0c/Partial-Differences_Subtraction_Step_2.JPG/117px-Partial-Differences_Subtraction_Step_2.JPG 2x" data-file-width="216" data-file-height="441" /></a></span></div> <div class="gallerytext">The smaller number is subtracted from the greater:<br />90 − 50 = 40<br />Because the minuend is smaller than the subtrahend, this difference has a minus sign.</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Partial-Differences_Subtraction_Step_3.JPG" class="mw-file-description" title="The smaller number is subtracted from the greater: 3 − 1 = 2 Because the minuend is greater than the subtrahend, this difference has a plus sign."><img alt="The smaller number is subtracted from the greater: 3 − 1 = 2 Because the minuend is greater than the subtrahend, this difference has a plus sign." src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Partial-Differences_Subtraction_Step_3.JPG/59px-Partial-Differences_Subtraction_Step_3.JPG" decoding="async" width="59" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Partial-Differences_Subtraction_Step_3.JPG/88px-Partial-Differences_Subtraction_Step_3.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Partial-Differences_Subtraction_Step_3.JPG/117px-Partial-Differences_Subtraction_Step_3.JPG 2x" data-file-width="216" data-file-height="441" /></a></span></div> <div class="gallerytext">The smaller number is subtracted from the greater:<br />3 − 1 = 2<br />Because the minuend is greater than the subtrahend, this difference has a plus sign.</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Partial-Differences_Subtraction_Step_4.JPG" class="mw-file-description" title="+300 − 40 + 2 = 262"><img alt="+300 − 40 + 2 = 262" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/08/Partial-Differences_Subtraction_Step_4.JPG/58px-Partial-Differences_Subtraction_Step_4.JPG" decoding="async" width="58" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/08/Partial-Differences_Subtraction_Step_4.JPG/88px-Partial-Differences_Subtraction_Step_4.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/08/Partial-Differences_Subtraction_Step_4.JPG/117px-Partial-Differences_Subtraction_Step_4.JPG 2x" data-file-width="215" data-file-height="440" /></a></span></div> <div class="gallerytext">+300 − 40 + 2 = 262</div> </li> </ul> <div class="mw-heading mw-heading3"><h3 id="Nonvertical_methods">Nonvertical methods</h3></div> <div class="mw-heading mw-heading4"><h4 id="Counting_up">Counting up</h4></div> <p>Instead of finding the difference digit by digit, one can count up the numbers between the subtrahend and the minuend.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </p><p>Example: 1234 − 567 = can be found by the following steps: </p> <ul><li><span class="nowrap">567 + <b>3</b> = 570</span></li> <li><span class="nowrap">570 + <b>30</b> = 600</span></li> <li><span class="nowrap">600 + <b>400</b> = 1000</span></li> <li><span class="nowrap">1000 + <b>234</b> = 1234</span></li></ul> <p>Add up the value from each step to get the total difference: <span class="nowrap">3 + 30 + 400 + 234 = 667</span>. </p> <div class="mw-heading mw-heading4"><h4 id="Breaking_up_the_subtraction">Breaking up the subtraction</h4></div> <p>Another method that is useful for <a href="/wiki/Mental_calculation" title="Mental calculation">mental arithmetic</a> is to split up the subtraction into small steps.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> </p><p>Example: 1234 − 567 = can be solved in the following way: </p> <ul><li>1234 − <b>500</b> = 734</li> <li>734 − <b>60</b> = 674</li> <li>674 − <b>7</b> = 667</li></ul> <div class="mw-heading mw-heading4"><h4 id="Same_change">Same change</h4></div> <p>The same change method uses the fact that adding or subtracting the same number from the minuend and subtrahend does not change the answer. One simply adds the amount needed to get zeros in the subtrahend.<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </p><p>Example: </p><p>"1234 − 567 =" can be solved as follows: </p> <ul><li><span class="nowrap">1234 − 567 = 1237 − 570 =</span> <span class="nowrap">1267 − 600 = 667</span></li></ul> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div> <ul><li><a href="/wiki/Absolute_difference" title="Absolute difference">Absolute difference</a></li> <li><a href="https://en.wiktionary.org/wiki/decrement" class="extiw" title="wikt:decrement">Decrement</a></li> <li><a href="/wiki/Elementary_arithmetic" title="Elementary arithmetic">Elementary arithmetic</a></li> <li><a href="/wiki/Method_of_complements" title="Method of complements">Method of complements</a></li> <li><a href="/wiki/Negative_number" title="Negative number">Negative number</a></li> <li><a href="/wiki/Plus_and_minus_signs" title="Plus and minus signs">Plus and minus signs</a></li> <li><a href="/wiki/Monus" title="Monus">Monus</a> (truncated subtraction)</li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></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 reflist-lower-alpha"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text">"Subtrahend" is shortened by the inflectional Latin suffix -us, e.g. remaining un-declined as in <i>numerus subtrahendus</i> "the number to be subtracted".</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><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 rel="nofollow" class="external text" href="https://www.splashlearn.com/math-vocabulary/subtraction/subtract">"What is to Subtract?"</a>. <i>SplashLearn</i>. 28 April 2022<span class="reference-accessdate">. Retrieved <span class="nowrap">2022-12-13</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=SplashLearn&rft.atitle=What+is+to+Subtract%3F&rft.date=2022-04-28&rft_id=https%3A%2F%2Fwww.splashlearn.com%2Fmath-vocabulary%2Fsubtraction%2Fsubtract&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASubtraction" class="Z3988"></span></span> </li> <li id="cite_note-:0-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-:0_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:0_2-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1">Weisstein, Eric W. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Subtraction.html">"Subtraction"</a>. <i>mathworld.wolfram.com</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2020-08-26</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=mathworld.wolfram.com&rft.atitle=Subtraction&rft.aulast=Weisstein&rft.aufirst=Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FSubtraction.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASubtraction" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCutland" class="citation book cs1">Cutland, Nigel. <i>Computability: an introduction to recursive function theory</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Computability%3A+an+introduction+to+recursive+function+theory&rft.aulast=Cutland&rft.aufirst=Nigel&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASubtraction" class="Z3988"></span></span> </li> <li id="cite_note-Schmid_1974-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-Schmid_1974_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Schmid_1974_4-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Schmid_1974_4-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchmid1974" class="citation book cs1"><a href="/wiki/Hermann_Schmid_(computer_scientist)" class="mw-redirect" title="Hermann Schmid (computer scientist)">Schmid, Hermann</a> (1974). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/decimalcomputati0000schm"><i>Decimal Computation</i></a></span> (1 ed.). Binghamton, NY: <a href="/wiki/John_Wiley_%26_Sons" class="mw-redirect" title="John Wiley & Sons">John Wiley & Sons</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-76180-8" title="Special:BookSources/978-0-471-76180-8"><bdi>978-0-471-76180-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Decimal+Computation&rft.place=Binghamton%2C+NY&rft.edition=1&rft.pub=John+Wiley+%26+Sons&rft.date=1974&rft.isbn=978-0-471-76180-8&rft.aulast=Schmid&rft.aufirst=Hermann&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fdecimalcomputati0000schm&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASubtraction" class="Z3988"></span></span> </li> <li id="cite_note-Schmid_1983-5"><span class="mw-cite-backlink">^ <a href="#cite_ref-Schmid_1983_5-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Schmid_1983_5-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Schmid_1983_5-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchmid1983" class="citation book cs1"><a href="/wiki/Hermann_Schmid_(computer_scientist)" class="mw-redirect" title="Hermann Schmid (computer scientist)">Schmid, Hermann</a> (1983) [1974]. <i>Decimal Computation</i> (1 (reprint) ed.). Malabar, FL: Robert E. Krieger Publishing Company. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-89874-318-0" title="Special:BookSources/978-0-89874-318-0"><bdi>978-0-89874-318-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Decimal+Computation&rft.place=Malabar%2C+FL&rft.edition=1+%28reprint%29&rft.pub=Robert+E.+Krieger+Publishing+Company&rft.date=1983&rft.isbn=978-0-89874-318-0&rft.aulast=Schmid&rft.aufirst=Hermann&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASubtraction" 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"><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.mathsisfun.com/numbers/subtraction.html">"Subtraction"</a>. <i>www.mathsisfun.com</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2020-08-26</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.mathsisfun.com&rft.atitle=Subtraction&rft_id=https%3A%2F%2Fwww.mathsisfun.com%2Fnumbers%2Fsubtraction.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASubtraction" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFReference-OED-Subtraction" class="citation encyclopaedia cs1"><span class="id-lock-subscription" title="Paid subscription required"><a rel="nofollow" class="external text" href="https://www.oed.com/search/dictionary/?q=Subtraction">"Subtraction"</a></span>. <i><a href="/wiki/Oxford_English_Dictionary" title="Oxford English Dictionary">Oxford English Dictionary</a></i> (Online ed.). <a href="/wiki/Oxford_University_Press" title="Oxford University Press">Oxford University Press</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Subtraction&rft.btitle=Oxford+English+Dictionary&rft.edition=Online&rft.pub=Oxford+University+Press&rft_id=https%3A%2F%2Fwww.oed.com%2Fsearch%2Fdictionary%2F%3Fq%3DSubtraction&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASubtraction" class="Z3988"></span> <span style="font-size:0.95em; font-size:95%; color: var( --color-subtle, #555 )">(Subscription or <a rel="nofollow" class="external text" href="https://www.oed.com/public/login/loggingin#withyourlibrary">participating institution membership</a> required.)</span></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">Paul E. Peterson, Michael Henderson, Martin R. West (2014) <i>Teachers Versus the Public: What Americans Think about Schools and How to Fix Them</i> Brookings Institution Press, p. 163</span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text">Janet Kolodzy (2006) <i>Convergence Journalism: Writing and Reporting across the News Media</i> Rowman & Littlefield Publishers, p. 180</span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text">David Gillborn (2008) <i>Racism and Education: Coincidence Or Conspiracy?</i> Routledge p. 46</span> </li> <li id="cite_note-Klapper1916-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-Klapper1916_12-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKlapper1916" class="citation book cs1"><a href="/wiki/Paul_Klapper" title="Paul Klapper">Klapper, Paul</a> (1916). <a rel="nofollow" class="external text" href="https://archive.org/details/teachingarithme00klapgoog"><i>The Teaching of Arithmetic: A Manual for Teachers</i></a>. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/teachingarithme00klapgoog/page/n94">80</a>–<span class="reference-accessdate">. Retrieved <span class="nowrap">2016-03-11</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Teaching+of+Arithmetic%3A+A+Manual+for+Teachers&rft.pages=80-&rft.date=1916&rft.aulast=Klapper&rft.aufirst=Paul&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fteachingarithme00klapgoog&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASubtraction" class="Z3988"></span></span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text">Susan Ross and Mary Pratt-Cotter. 2000. "Subtraction in the United States: An Historical Perspective," <i>The Mathematics Educator</i> 8(1):4–11. p. 8: "This new version of the decomposition algorithm [i.e., using Brownell's crutch] has so completely dominated the field that it is rare to see any other algorithm used to teach subtraction today [in America]."</span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRossPratt-Cotter1999" class="citation journal cs1">Ross, Susan C.; Pratt-Cotter, Mary (1999). "Subtraction From a Historical Perspective". <i>School Science and Mathematics</i>. <b>99</b> (7): 389–93. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1111%2Fj.1949-8594.1999.tb17499.x">10.1111/j.1949-8594.1999.tb17499.x</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=School+Science+and+Mathematics&rft.atitle=Subtraction+From+a+Historical+Perspective&rft.volume=99&rft.issue=7&rft.pages=389-93&rft.date=1999&rft_id=info%3Adoi%2F10.1111%2Fj.1949-8594.1999.tb17499.x&rft.aulast=Ross&rft.aufirst=Susan+C.&rft.au=Pratt-Cotter%2C+Mary&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASubtraction" class="Z3988"></span></span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text">Klapper 1916, pp. 177–.</span> </li> <li id="cite_note-Smith1913-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-Smith1913_16-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDavid_Eugene_Smith1913" class="citation book cs1">David Eugene Smith (1913). <a rel="nofollow" class="external text" href="https://archive.org/details/bub_gb_A7NJAAAAIAAJ"><i>The Teaching of Arithmetic</i></a>. Ginn. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/bub_gb_A7NJAAAAIAAJ/page/n85">77</a>–<span class="reference-accessdate">. Retrieved <span class="nowrap">2016-03-11</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Teaching+of+Arithmetic&rft.pages=77-&rft.pub=Ginn&rft.date=1913&rft.au=David+Eugene+Smith&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fbub_gb_A7NJAAAAIAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASubtraction" class="Z3988"></span></span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://sites.google.com/a/oswego308.org/msimester/home/math/algorithms/subtraction">The Many Ways of Arithmetic in UCSMP Everyday Mathematics</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20140225135251/https://sites.google.com/a/oswego308.org/msimester/home/math/algorithms/subtraction">Archived</a> 2014-02-25 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> Subtraction: Trade First</span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://ouronlineschools.org/Schools/NC/Demoschool/4thGrade/Math/PartialDifferences.htm">Partial-Differences Subtraction</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20140623021239/http://ouronlineschools.org/Schools/NC/Demoschool/4thGrade/Math/PartialDifferences.htm">Archived</a> 2014-06-23 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>; <a rel="nofollow" class="external text" href="https://sites.google.com/a/oswego308.org/msimester/home/math/algorithms/subtraction">The Many Ways of Arithmetic in UCSMP Everyday Mathematics</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20140225135251/https://sites.google.com/a/oswego308.org/msimester/home/math/algorithms/subtraction">Archived</a> 2014-02-25 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> Subtraction: Partial Differences</span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://sites.google.com/a/oswego308.org/msimester/home/math/algorithms/subtraction">The Many Ways of Arithmetic in UCSMP Everyday Mathematics</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20140225135251/https://sites.google.com/a/oswego308.org/msimester/home/math/algorithms/subtraction">Archived</a> 2014-02-25 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> Subtraction: Counting Up</span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://sites.google.com/a/oswego308.org/msimester/home/math/algorithms/subtraction">The Many Ways of Arithmetic in UCSMP Everyday Mathematics</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20140225135251/https://sites.google.com/a/oswego308.org/msimester/home/math/algorithms/subtraction">Archived</a> 2014-02-25 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> Subtraction: Left to Right Subtraction</span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://sites.google.com/a/oswego308.org/msimester/home/math/algorithms/subtraction">The Many Ways of Arithmetic in UCSMP Everyday Mathematics</a> Subtraction: Same Change Rule</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Bibliography">Bibliography</h2></div> <ul><li>Brownell, W.A. (1939). Learning as reorganization: An experimental study in third-grade arithmetic, Duke University Press.</li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20170811133911/http://math.coe.uga.edu/tme/issues/v10n2/5ross.pdf">Subtraction in the United States: An Historical Perspective, Susan Ross, Mary Pratt-Cotter, <i>The Mathematics Educator</i>, Vol. 8, No. 1 (original publication) and Vol. 10, No. 1 (reprint.)</a> <a href="/wiki/PDF" title="PDF">PDF</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></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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class="extiw" title="wiktionary:Special:Search/subtraction">subtraction</a></b></i> in Wiktionary, the free dictionary.</div></div> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1235681985"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237033735"><div class="side-box side-box-right plainlinks sistersitebox"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span></div> <div class="side-box-text plainlist">Wikimedia Commons has media related to <span style="font-weight: bold; font-style: italic;"><a href="https://commons.wikimedia.org/wiki/Category:Subtraction" class="extiw" title="commons:Category:Subtraction">Subtraction</a></span>.</div></div> </div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Subtraction">"Subtraction"</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Subtraction&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DSubtraction&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASubtraction" class="Z3988"></span></li> <li>Printable Worksheets: <a rel="nofollow" class="external text" href="https://web.archive.org/web/20121119024904/http://www.math-drills.com/subtraction.shtml">Subtraction Worksheets</a>, <a rel="nofollow" class="external text" href="http://www.kwiznet.com/p/takeQuiz.php?ChapterID=1214&CurriculumID=2&Method=Worksheet&NQ=24&NQ4P=3">One Digit Subtraction</a>, <a rel="nofollow" class="external text" href="http://www.kwiznet.com/p/takeQuiz.php?ChapterID=1202&CurriculumID=2&Method=Worksheet&NQ=24&NQ4P=3">Two Digit Subtraction</a>, <a rel="nofollow" class="external text" href="http://www.kwiznet.com/p/takeQuiz.php?ChapterID=1273&CurriculumID=3&Method=Worksheet&NQ=24&NQ4P=3">Four Digit Subtraction</a>, and <a rel="nofollow" class="external text" href="http://www.dadsworksheets.com/worksheets/subtraction.html">More Subtraction Worksheets</a></li> <li><a rel="nofollow" class="external text" href="http://www.cut-the-knot.org/Curriculum/Arithmetic/SubtractionGame.shtml">Subtraction Game</a> at <a href="/wiki/Cut-the-Knot" class="mw-redirect" title="Cut-the-Knot">cut-the-knot</a></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20090416010325/http://webhome.idirect.com/~totton/abacus/pages.htm#Subtraction1">Subtraction on a Japanese abacus</a> selected from <a rel="nofollow" class="external text" href="http://webhome.idirect.com/~totton/abacus/">Abacus: Mystery of the Bead</a></li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style 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