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<span>Binary adders</span> </div> </a> <button aria-controls="toc-Binary_adders-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 Binary adders subsection</span> </button> <ul id="toc-Binary_adders-sublist" class="vector-toc-list"> <li id="toc-Half_adder" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Half_adder"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Half adder</span> </div> </a> <ul id="toc-Half_adder-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Full_adder" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Full_adder"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Full adder</span> </div> </a> <ul id="toc-Full_adder-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Adders_supporting_multiple_bits" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Adders_supporting_multiple_bits"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Adders supporting multiple bits</span> </div> </a> <ul id="toc-Adders_supporting_multiple_bits-sublist" class="vector-toc-list"> <li id="toc-Ripple-carry_adder" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Ripple-carry_adder"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3.1</span> <span>Ripple-carry adder</span> </div> </a> <ul id="toc-Ripple-carry_adder-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Carry-lookahead_adder_(Weinberger_and_Smith,_1958)" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Carry-lookahead_adder_(Weinberger_and_Smith,_1958)"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3.2</span> <span>Carry-lookahead adder (Weinberger and Smith, 1958)</span> </div> </a> <ul id="toc-Carry-lookahead_adder_(Weinberger_and_Smith,_1958)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Carry-save_adders" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Carry-save_adders"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3.3</span> <span>Carry-save adders</span> </div> </a> <ul id="toc-Carry-save_adders-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-3:2_compressors" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#3:2_compressors"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>3:2 compressors</span> </div> </a> <ul id="toc-3:2_compressors-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Quantum_adders" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Quantum_adders"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Quantum adders</span> </div> </a> <ul id="toc-Quantum_adders-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Analog_adders" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Analog_adders"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Analog adders</span> </div> </a> <ul id="toc-Analog_adders-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Adder (electronics)</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 40 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-40" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">40 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AC%D9%88%D8%A7%D9%85%D8%B9_%D9%85%D9%86%D8%B7%D9%82%D9%8A%D8%A9" title="جوامع منطقية – Arabic" lang="ar" hreflang="ar" data-title="جوامع منطقية" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AF%E0%A7%8B%E0%A6%97%E0%A7%87%E0%A6%B0_%E0%A6%AC%E0%A6%B0%E0%A7%8D%E0%A6%A4%E0%A6%A8%E0%A7%80" title="যোগের বর্তনী – Bangla" lang="bn" hreflang="bn" data-title="যোগের বর্তনী" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%A1%D1%83%D0%BC%D0%B0%D1%82%D0%B0%D1%80" title="Суматар – Belarusian" lang="be" hreflang="be" data-title="Суматар" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A1%D1%83%D0%BC%D0%B0%D1%82%D0%BE%D1%80" title="Суматор – Bulgarian" lang="bg" hreflang="bg" data-title="Суматор" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Sumador" title="Sumador – Catalan" lang="ca" hreflang="ca" data-title="Sumador" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Bin%C3%A1rn%C3%AD_s%C4%8D%C3%ADta%C4%8Dka" title="Binární sčítačka – Czech" lang="cs" hreflang="cs" data-title="Binární sčítačka" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Additionskredsl%C3%B8b_(digital_elektronik)" title="Additionskredsløb (digital elektronik) – Danish" lang="da" hreflang="da" data-title="Additionskredsløb (digital elektronik)" 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/Volladdierer" title="Volladdierer – German" lang="de" hreflang="de" data-title="Volladdierer" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Summaator" title="Summaator – Estonian" lang="et" hreflang="et" data-title="Summaator" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%91%CE%B8%CF%81%CE%BF%CE%B9%CF%83%CF%84%CE%AD%CF%82_(%CE%B7%CE%BB%CE%B5%CE%BA%CF%84%CF%81%CE%BF%CE%BD%CE%B9%CE%BA%CE%AC_%CE%BA%CF%85%CE%BA%CE%BB%CF%8E%CE%BC%CE%B1%CF%84%CE%B1)" title="Αθροιστές (ηλεκτρονικά κυκλώματα) – Greek" lang="el" hreflang="el" data-title="Αθροιστές (ηλεκτρονικά κυκλώματα)" 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/Sumador" title="Sumador – Spanish" lang="es" hreflang="es" data-title="Sumador" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Batugailu" title="Batugailu – Basque" lang="eu" hreflang="eu" data-title="Batugailu" 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%AC%D9%85%D8%B9%E2%80%8C%DA%A9%D9%86%D9%86%D8%AF%D9%87" title="جمع‌کننده – Persian" lang="fa" hreflang="fa" data-title="جمع‌کننده" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Additionneur" title="Additionneur – French" lang="fr" hreflang="fr" data-title="Additionneur" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EA%B0%80%EC%82%B0%EA%B8%B0" title="가산기 – Korean" lang="ko" hreflang="ko" data-title="가산기" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%A6%E0%A5%8D%E0%A4%B5%E0%A4%BF-%E0%A4%86%E0%A4%A7%E0%A4%BE%E0%A4%B0%E0%A5%80_%E0%A4%AF%E0%A5%8B%E0%A4%9C%E0%A4%95" title="द्वि-आधारी योजक – Hindi" lang="hi" hreflang="hi" data-title="द्वि-आधारी योजक" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Penambah_(elektronik)" title="Penambah (elektronik) – Indonesian" lang="id" hreflang="id" data-title="Penambah (elektronik)" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Full-adder" title="Full-adder – Italian" lang="it" hreflang="it" data-title="Full-adder" 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%9E%D7%97%D7%91%D7%A8_%D7%9C%D7%95%D7%92%D7%99" title="מחבר לוגי – Hebrew" lang="he" hreflang="he" data-title="מחבר לוגי" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D2%9A%D0%BE%D1%81%D1%8B%D0%BD%D0%B4%D1%8B%D0%BB%D0%B0%D1%83%D1%8B%D1%88" title="Қосындылауыш – Kazakh" lang="kk" hreflang="kk" data-title="Қосындылауыш" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Summators" title="Summators – Latvian" lang="lv" hreflang="lv" data-title="Summators" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Sommator" title="Sommator – Lombard" lang="lmo" hreflang="lmo" data-title="Sommator" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/%C3%96sszead%C3%B3_(elektronika)" title="Összeadó (elektronika) – Hungarian" lang="hu" hreflang="hu" data-title="Összeadó (elektronika)" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Adder_(elektronica)" title="Adder (elektronica) – Dutch" lang="nl" hreflang="nl" data-title="Adder (elektronica)" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%8A%A0%E7%AE%97%E5%99%A8" title="加算器 – Japanese" lang="ja" hreflang="ja" data-title="加算器" 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-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Sumator_(uk%C5%82ad_logiczny)" title="Sumator (układ logiczny) – Polish" lang="pl" hreflang="pl" data-title="Sumator (układ logiczny)" 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/Circuito_aritm%C3%A9tico" title="Circuito aritmético – Portuguese" lang="pt" hreflang="pt" data-title="Circuito aritmético" 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/Sumator_(electronic%C4%83)" title="Sumator (electronică) – Romanian" lang="ro" hreflang="ro" data-title="Sumator (electronică)" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A1%D1%83%D0%BC%D0%BC%D0%B0%D1%82%D0%BE%D1%80" title="Сумматор – Russian" lang="ru" hreflang="ru" data-title="Сумматор" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Binary_adder" title="Binary adder – Simple English" lang="en-simple" hreflang="en-simple" data-title="Binary adder" 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/Bin%C3%A1rna_s%C4%8D%C3%ADta%C4%8Dka" title="Binárna sčítačka – Slovak" lang="sk" hreflang="sk" data-title="Binárna sčítačka" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A1%D0%B0%D0%B1%D0%B8%D1%80%D0%B0%D1%87_(%D0%B5%D0%BB%D0%B5%D0%BA%D1%82%D1%80%D0%BE%D0%BD%D0%B8%D0%BA%D0%B0)" title="Сабирач (електроника) – Serbian" lang="sr" hreflang="sr" data-title="Сабирач (електроника)" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Summain" title="Summain – Finnish" lang="fi" hreflang="fi" data-title="Summain" 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/Adderare" title="Adderare – Swedish" lang="sv" hreflang="sv" data-title="Adderare" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%A7%E0%B8%87%E0%B8%88%E0%B8%A3%E0%B8%9A%E0%B8%A7%E0%B8%81" title="วงจรบวก – Thai" lang="th" hreflang="th" data-title="วงจรบวก" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Toplay%C4%B1c%C4%B1" title="Toplayıcı – Turkish" lang="tr" hreflang="tr" data-title="Toplayıcı" 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%A1%D1%83%D0%BC%D0%B0%D1%82%D0%BE%D1%80" title="Суматор – Ukrainian" lang="uk" hreflang="uk" data-title="Суматор" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link 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.mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><table class="sidebar nomobile nowraplinks" style="width: 17em;"><tbody><tr><td class="sidebar-pretitle" style="_background-color: #CDC6CA; padding: 4px 0;">Part of a series on</td></tr><tr><th class="sidebar-title-with-pretitle" style="_background-color: #CDC6CA; padding: 0 2px;">Arithmetic logic circuits</th></tr><tr><td class="sidebar-above" style="border: none;"> Quick navigation</td></tr><tr><td class="sidebar-content plainlist" style="padding: 0; margin: 0; float:lfet;"> <div class="collapsible-list mw-collapsible mw-collapsed" style="text-align: left;"> <div style="line-height: 1.6em; font-weight: bold; width: 100%; text-align: center; padding: 0; margin: 0;"><div style="margin: 0 4em">Theory</div></div> <ul class="mw-collapsible-content" style="margin-top: 0; margin-bottom: 0; line-height: inherit; list-style: none; margin-left: 0; text-align: center"><li style="line-height: inherit; margin: 0"> <a href="/wiki/Binary_number" title="Binary number">Binary number</a> </li><li style="line-height: inherit; margin: 0"> <a href="/wiki/Boolean_algebra" title="Boolean algebra">Boolean algebra</a> </li><li style="line-height: inherit; margin: 0"> <a href="/wiki/Logic_gate" title="Logic gate">Logic gate</a> </li><li style="line-height: inherit; margin: 0"> <a href="/wiki/Ones%27_complement" title="Ones' complement">Ones' complement number</a> </li><li style="line-height: inherit; margin: 0"> <a href="/wiki/Two%27s_complement" title="Two's complement">Two's complement number</a> </li><li style="line-height: inherit; margin: 0"> <a href="/wiki/Signed_number_representations" title="Signed number representations">Signed number representations</a> </li></ul> </div></td> </tr><tr><td class="sidebar-content-with-subgroup plainlist" style="padding: 0; margin: 0; float:lfet;"> <div class="collapsible-list mw-collapsible" style="text-align: left;"> <div style="line-height: 1.6em; font-weight: bold; width: 100%; text-align: center; padding: 0; margin: 0;"><div style="margin: 0 4em">Components</div></div> <ul class="mw-collapsible-content" style="margin-top: 0; margin-bottom: 0; line-height: inherit; list-style: none; margin-left: 0; text-align: center; padding: 0.4em 0 0.1em;"><li style="line-height: inherit; margin: 0"> <table class="sidebar-subgroup" style=";width: 17em;"><tbody><tr><td class="sidebar-content plainlist"> <div class="collapsible-list mw-collapsible" style="text-align: left; border:none; padding:0"> <div style="line-height: 1.6em; font-weight: bold; margin: 0 !important; padding: 0.1em 0 0.1em;"><div>Adder (+)</div></div> <ul class="mw-collapsible-content" style="margin-top: 0; margin-bottom: 0; line-height: inherit; list-style: none; margin-left: 0; text-align: left; padding-left: 1em;"><li style="line-height: inherit; margin: 0"> <a class="mw-selflink selflink">Adder</a> </li><li style="line-height: inherit; margin: 0"> <a href="/wiki/Half_adder" class="mw-redirect" title="Half adder">Half adder</a> </li><li style="line-height: inherit; margin: 0"> <a href="/wiki/Full_adder" class="mw-redirect" title="Full adder">Full adder</a> </li><li style="line-height: inherit; margin: 0"> <a href="/wiki/Ripple-carry_adder" class="mw-redirect" title="Ripple-carry adder"> Ripple-carry adder</a> </li><li style="line-height: inherit; margin: 0"> <a href="/wiki/Carry-lookahead_adder" title="Carry-lookahead adder">Carry-lookahead adder</a> </li><li style="line-height: inherit; margin: 0"> <a href="/wiki/Brent%E2%80%93Kung_adder" title="Brent–Kung adder">Brent–Kung adder</a> </li><li style="line-height: inherit; margin: 0"> <a href="/wiki/Kogge%E2%80%93Stone_adder" title="Kogge–Stone adder">Kogge–Stone adder</a> </li><li style="line-height: inherit; margin: 0"> <a href="/wiki/Ling_adder" title="Ling adder">Ling adder</a> </li><li style="line-height: inherit; margin: 0"> <a href="/wiki/Carry-save_adder" title="Carry-save adder">Carry-save adder</a> </li><li style="line-height: inherit; margin: 0"> <a href="/wiki/Carry-select_adder" title="Carry-select adder">Carry-select adder</a> </li><li style="line-height: inherit; margin: 0"> <a href="/wiki/Carry-skip_adder" title="Carry-skip adder">Carry-skip adder</a> </li></ul> </div></td> </tr><tr><td class="sidebar-content plainlist"> <div class="collapsible-list mw-collapsible mw-collapsed" style="text-align: left; border:none; padding:0"> <div style="line-height: 1.6em; font-weight: bold; margin: 0 !important; padding: 0.1em 0 0.1em;"><div>Adder–subtractor (±)</div></div> <ul class="mw-collapsible-content" style="margin-top: 0; margin-bottom: 0; line-height: inherit; list-style: none; margin-left: 0; text-align: left; padding-left: 1em;"><li style="line-height: inherit; margin: 0"> <a href="/wiki/Adder%E2%80%93subtractor" title="Adder–subtractor">Adder–subtractor</a> </li></ul> </div></td> </tr><tr><td class="sidebar-content plainlist"> <div class="collapsible-list mw-collapsible mw-collapsed" style="text-align: left; border:none; padding:0"> <div style="line-height: 1.6em; font-weight: bold; margin: 0 !important; padding: 0.1em 0 0.1em;"><div>Subtractor (−)</div></div> <ul class="mw-collapsible-content" style="margin-top: 0; margin-bottom: 0; line-height: inherit; list-style: none; margin-left: 0; text-align: left; padding-left: 1em;"><li style="line-height: inherit; margin: 0"> <a href="/wiki/Subtractor" title="Subtractor">Subtractor</a> </li><li style="line-height: inherit; margin: 0"> <a href="/wiki/Full_subtractor" class="mw-redirect" title="Full subtractor">Full subtractor</a> </li><li style="line-height: inherit; margin: 0"> <a href="/wiki/Half_subtractor" class="mw-redirect" title="Half subtractor">Half subtractor</a> </li></ul> </div></td> </tr><tr><td class="sidebar-content plainlist"> <div class="collapsible-list mw-collapsible mw-collapsed" style="text-align: left; border:none; padding:0"> <div style="line-height: 1.6em; font-weight: bold; margin: 0 !important; padding: 0.1em 0 0.1em;"><div>Multiplier (×)</div></div> <ul class="mw-collapsible-content" style="margin-top: 0; margin-bottom: 0; line-height: inherit; list-style: none; margin-left: 0; text-align: left; padding-left: 1em;"><li style="line-height: inherit; margin: 0"> <a href="/wiki/Binary_multiplier" title="Binary multiplier">Binary multiplier</a> </li><li style="line-height: inherit; margin: 0"> <a href="/wiki/Multiplication_algorithm" title="Multiplication algorithm">Multiplication algorithm</a> </li><li style="line-height: inherit; margin: 0"> <a href="/wiki/Booth%27s_multiplication_algorithm" title="Booth's multiplication algorithm">Booth's multiplication algorithm</a> </li><li style="line-height: inherit; margin: 0"> <a href="/wiki/Wallace_tree" title="Wallace tree">Wallace tree</a> </li><li style="line-height: inherit; margin: 0"> <a href="/wiki/Dadda_multiplier" title="Dadda multiplier">Dadda multiplier</a> </li></ul> </div></td> </tr><tr><td class="sidebar-content plainlist"> <div class="collapsible-list mw-collapsible mw-collapsed" style="text-align: left; border:none; padding:0"> <div style="line-height: 1.6em; font-weight: bold; margin: 0 !important; padding: 0.1em 0 0.1em;"><div>Divider (÷)</div></div> <ul class="mw-collapsible-content" style="margin-top: 0; margin-bottom: 0; line-height: inherit; list-style: none; margin-left: 0; text-align: left; padding-left: 1em;"><li style="line-height: inherit; margin: 0"> <a href="/w/index.php?title=Binary_Divider&action=edit&redlink=1" class="new" title="Binary Divider (page does not exist)">Binary Divider</a> </li><li style="line-height: inherit; margin: 0"> <a href="/wiki/Division_algorithm" title="Division algorithm">Division algorithm</a> </li></ul> </div></td> </tr><tr><td class="sidebar-content plainlist"> <div class="collapsible-list mw-collapsible mw-collapsed" style="text-align: left; border:none; padding:0"> <div style="line-height: 1.6em; font-weight: bold; margin: 0 !important; padding: 0.1em 0 0.1em;"><div>Bitwise ops</div></div> <ul class="mw-collapsible-content" style="margin-top: 0; margin-bottom: 0; line-height: inherit; list-style: none; margin-left: 0; text-align: left; padding-left: 1em;"><li style="line-height: inherit; margin: 0"> <a href="/wiki/Bitwise_operation" title="Bitwise operation">Bitwise operation</a> </li><li style="line-height: inherit; margin: 0"> <a href="/wiki/Bitwise_operation#NOT" title="Bitwise operation">NOT</a> </li><li style="line-height: inherit; margin: 0"> <a href="/wiki/Bitwise_operation#AND" title="Bitwise operation">AND</a> </li><li style="line-height: inherit; margin: 0"> <a href="/wiki/Bitwise_operation#OR" title="Bitwise operation">OR</a> </li><li style="line-height: inherit; margin: 0"> <a href="/wiki/Bitwise_operation#XOR" title="Bitwise operation">XOR</a> </li><li style="line-height: inherit; margin: 0"> <a href="/wiki/Bitwise_operation#Bit_shifts" title="Bitwise operation">Bit shifts</a> </li><li style="line-height: inherit; margin: 0"> <a href="/wiki/Bit_manipulation" title="Bit manipulation">Bit manipulation</a> </li></ul> </div></td> </tr><tr><td class="sidebar-content plainlist"> <div class="collapsible-list mw-collapsible mw-collapsed" style="text-align: left; border:none; padding:0"> <div style="line-height: 1.6em; font-weight: bold; margin: 0 !important; padding: 0.1em 0 0.1em;"><div>See also</div></div> <ul class="mw-collapsible-content" style="margin-top: 0; margin-bottom: 0; line-height: inherit; list-style: none; margin-left: 0; text-align: left; padding-left: 1em;"><li style="line-height: inherit; margin: 0"> <a href="/wiki/Kochanski_multiplication" title="Kochanski multiplication">Kochanski multiplication</a> (exponentiation) </li><li style="line-height: inherit; margin: 0"> <a href="/wiki/Multiply%E2%80%93accumulate_operation" title="Multiply–accumulate operation">Multiply–accumulate operation</a> </li></ul> </div></td> </tr></tbody></table> </li></ul> </div></td> </tr><tr><td class="sidebar-content plainlist" style="padding: 0; margin: 0; float:lfet;"> <div class="collapsible-list mw-collapsible mw-collapsed" style="text-align: left;"> <div style="line-height: 1.6em; font-weight: bold; width: 100%; text-align: center; padding: 0; margin: 0;"><div style="margin: 0 4em">Categories</div></div> <ul class="mw-collapsible-content" style="margin-top: 0; margin-bottom: 0; line-height: inherit; list-style: none; margin-left: 0; text-align: center"><li style="line-height: inherit; margin: 0"> <a href="/wiki/Category:Binary_arithmetic" title="Category:Binary arithmetic">Category:Binary arithmetic</a> </li><li style="line-height: inherit; margin: 0"> <a href="/wiki/Category:Computer_arithmetic" title="Category:Computer arithmetic">Category:Computer arithmetic</a> </li></ul> </div></td> </tr><tr><td class="sidebar-content plainlist" style="padding: 0; margin: 0; float:lfet;"> <div class="collapsible-list mw-collapsible mw-collapsed" style="text-align: left;"> <div style="line-height: 1.6em; font-weight: bold; width: 100%; text-align: center; padding: 0; margin: 0;"><div style="margin: 0 4em">See also</div></div> <ul class="mw-collapsible-content hlist" style="margin-top: 0; margin-bottom: 0; line-height: inherit; text-align: center"><li style="line-height: inherit; margin: 0"> <a href="/wiki/Floating-point_unit" title="Floating-point unit"> FPU</a> </li><li style="line-height: inherit; margin: 0"> <a href="/wiki/Graphics_processing_unit" title="Graphics processing unit"> GPU</a> </li><li style="line-height: inherit; margin: 0"> <a href="/wiki/Address_generation_unit" title="Address generation unit"> AGU</a> </li><li style="line-height: inherit; margin: 0"> <a href="/wiki/Mechanical_calculator" title="Mechanical calculator">Mechanical calculator</a> </li></ul> </div></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .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><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Sidebar_arithmetic_logic_circuits" title="Template:Sidebar arithmetic logic circuits"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Sidebar_arithmetic_logic_circuits" title="Template talk:Sidebar arithmetic logic circuits"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Sidebar_arithmetic_logic_circuits" title="Special:EditPage/Template:Sidebar arithmetic logic circuits"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>An <b>adder</b>, or <b>summer</b>,<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> is a <a href="/wiki/Digital_circuit" class="mw-redirect" title="Digital circuit">digital circuit</a> that performs <a href="/wiki/Addition" title="Addition">addition</a> of numbers. In many <a href="/wiki/Computer" title="Computer">computers</a> and other kinds of <a href="/wiki/Microprocessor" title="Microprocessor">processors</a>, adders are used in the <a href="/wiki/Arithmetic_logic_units" class="mw-redirect" title="Arithmetic logic units">arithmetic logic units</a> (ALUs). They are also used in other parts of the processor, where they are used to calculate <a href="/wiki/Address_space" title="Address space">addresses</a>, <a href="/wiki/Database_index" title="Database index">table indices</a>, <a href="/wiki/Increment_and_decrement_operators" title="Increment and decrement operators">increment and decrement operators</a> and similar operations. </p><p>Although adders can be constructed for many <a href="/wiki/Number_representation" class="mw-redirect" title="Number representation">number representations</a>, such as <a href="/wiki/Binary-coded_decimal" title="Binary-coded decimal">binary-coded decimal</a> or <a href="/wiki/Excess-3" title="Excess-3">excess-3</a>, the most common adders operate on <a href="/wiki/Binary_number" title="Binary number">binary numbers</a>. In cases where <a href="/wiki/Two%27s_complement" title="Two's complement">two's complement</a> or <a href="/wiki/Ones%27_complement" title="Ones' complement">ones' complement</a> is being used to represent <a href="/wiki/Negative_number" title="Negative number">negative numbers</a>, it is trivial to modify an adder into an <a href="/wiki/Adder%E2%80%93subtractor" title="Adder–subtractor">adder–subtractor</a>. Other <a href="/wiki/Signed_number_representations" title="Signed number representations">signed number representations</a> require more logic around the basic adder. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Adder_(electronics)&action=edit&section=1" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/George_Stibitz" title="George Stibitz">George Stibitz</a> invented the 2-bit binary adder (the <a href="/wiki/Model_K_(calculator)" title="Model K (calculator)">Model K</a>) in 1937. </p> <div class="mw-heading mw-heading2"><h2 id="Binary_adders">Binary adders</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Adder_(electronics)&action=edit&section=2" title="Edit section: Binary adders"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Half_adder">Half adder</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Adder_(electronics)&action=edit&section=3" title="Edit section: Half adder"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <b>half adder</b> adds two single binary digits <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span>. It has two outputs, sum (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span>) and carry (<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 C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span>). The carry signal represents an <a href="/wiki/Integer_overflow" title="Integer overflow">overflow</a> into the next digit of a multi-digit addition. The value of the sum is <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 2C+S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>C</mi> <mo>+</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2C+S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/245b019e3a3dc97c009b4a2d4bb7aedd9c111535" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.268ex; height:2.343ex;" alt="{\displaystyle 2C+S}"></span>. The simplest half-adder design, pictured on the right, incorporates an <a href="/wiki/XOR_gate" title="XOR gate">XOR gate</a> for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span> and an <a href="/wiki/AND_gate" title="AND gate">AND gate</a> for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span>. The Boolean logic for the sum (in this case <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span>) will be <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\oplus B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⊕</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\oplus B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0512d6bdd29ff000dea0bf68b853618dcaabc3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.348ex; height:2.343ex;" alt="{\displaystyle A\oplus B}"></span> whereas for the carry (<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 C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span>) will be <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\cdot B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⋅</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\cdot B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a90e903f21f11a0f4ab3caca1e6943ba7a9849" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.186ex; height:2.176ex;" alt="{\displaystyle A\cdot B}"></span>. With the addition of an <a href="/wiki/OR_gate" title="OR gate">OR gate</a> to combine their carry outputs, two half adders can be combined to make a full adder.<sup id="cite_ref-Lancaster_2004_2-0" class="reference"><a href="#cite_note-Lancaster_2004-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> The half adder adds two input bits and generates a carry and sum, which are the two outputs of a half adder. The input <a href="/wiki/Variable_(computer_science)" title="Variable (computer science)">variables</a> of a half adder are called the augend and addend bits. The output variables are the sum and carry. </p><p>The <a href="/wiki/Truth_table" title="Truth table">truth table</a> for the half adder is: </p> <dl><dd><table class="wikitable" style="text-align:center"> <tbody><tr> <th colspan="2">Inputs</th> <th colspan="2">Outputs </th></tr> <tr style="background:#def; text-align:center;"> <td><b>A</b></td> <td><b>B</b></td> <td><b>C</b><sub>out</sub></td> <td><b>S</b> </td></tr> <tr> <td style="background: #FFE3E3; color: black; vertical-align: middle; text-align: center;" class="table-no2">0</td> <td style="background: #FFE3E3; color: black; vertical-align: middle; text-align: center;" class="table-no2">0</td> <td style="background: #FFE3E3; color: black; vertical-align: middle; text-align: center;" class="table-no2">0</td> <td style="background: #FFE3E3; color: black; vertical-align: middle; text-align: center;" class="table-no2">0 </td></tr> <tr> <td style="background: #FFE3E3; color: black; vertical-align: middle; text-align: center;" class="table-no2">0</td> <td style="background:#bfd; color:black; vertical-align:middle; text-align:center;" class="table-yes2">1</td> <td style="background: #FFE3E3; color: black; vertical-align: middle; text-align: center;" class="table-no2">0</td> <td style="background:#bfd; color:black; vertical-align:middle; text-align:center;" class="table-yes2">1 </td></tr> <tr> <td style="background:#bfd; color:black; vertical-align:middle; text-align:center;" class="table-yes2">1</td> <td style="background: #FFE3E3; color: black; vertical-align: middle; text-align: center;" class="table-no2">0</td> <td style="background: #FFE3E3; color: black; vertical-align: middle; text-align: center;" class="table-no2">0</td> <td style="background:#bfd; color:black; vertical-align:middle; text-align:center;" class="table-yes2">1 </td></tr> <tr> <td style="background:#bfd; color:black; vertical-align:middle; text-align:center;" class="table-yes2">1</td> <td style="background:#bfd; color:black; vertical-align:middle; text-align:center;" class="table-yes2">1</td> <td style="background:#bfd; color:black; vertical-align:middle; text-align:center;" class="table-yes2">1</td> <td style="background: #FFE3E3; color: black; vertical-align: middle; text-align: center;" class="table-no2">0 </td></tr> </tbody></table></dd></dl> <p>Various half adder digital logic circuits: </p> <ul class="gallery mw-gallery-traditional"> <li class="gallerybox" style="width: 255px"> <div class="thumb" style="width: 250px; height: 195px;"><span typeof="mw:File"><a href="/wiki/File:Halfadder.gif" class="mw-file-description" title="Half adder in action."><img alt="Half adder in action." src="//upload.wikimedia.org/wikipedia/commons/thumb/9/92/Halfadder.gif/218px-Halfadder.gif" decoding="async" width="218" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/92/Halfadder.gif/328px-Halfadder.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/92/Halfadder.gif/437px-Halfadder.gif 2x" data-file-width="480" data-file-height="363" /></a></span></div> <div class="gallerytext">Half adder in action.</div> </li> <li class="gallerybox" style="width: 255px"> <div class="thumb" style="width: 250px; height: 195px;"><span typeof="mw:File"><a href="/wiki/File:Half_Adder.svg" class="mw-file-description" title="Schematic of half adder implemented with one XOR gate and one AND gate."><img alt="Schematic of half adder implemented with one XOR gate and one AND gate." src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Half_Adder.svg/220px-Half_Adder.svg.png" decoding="async" width="220" height="122" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Half_Adder.svg/330px-Half_Adder.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Half_Adder.svg/440px-Half_Adder.svg.png 2x" data-file-width="180" data-file-height="100" /></a></span></div> <div class="gallerytext"><a href="/wiki/Schematic" title="Schematic">Schematic</a> of half adder implemented with one <a href="/wiki/XOR_gate" title="XOR gate">XOR gate</a> and one <a href="/wiki/AND_gate" title="AND gate">AND gate</a>.</div> </li> <li class="gallerybox" style="width: 255px"> <div class="thumb" style="width: 250px; height: 195px;"><span typeof="mw:File"><a href="/wiki/File:Half_adder_using_NAND_gates_only.jpg" class="mw-file-description" title="Schematic of half adder implemented with five NAND gates."><img alt="Schematic of half adder implemented with five NAND gates." src="//upload.wikimedia.org/wikipedia/commons/thumb/5/55/Half_adder_using_NAND_gates_only.jpg/220px-Half_adder_using_NAND_gates_only.jpg" decoding="async" width="220" height="115" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/55/Half_adder_using_NAND_gates_only.jpg/330px-Half_adder_using_NAND_gates_only.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/55/Half_adder_using_NAND_gates_only.jpg/440px-Half_adder_using_NAND_gates_only.jpg 2x" data-file-width="1010" data-file-height="528" /></a></span></div> <div class="gallerytext">Schematic of half adder implemented with five <a href="/wiki/NAND_gate" title="NAND gate">NAND gates</a>.</div> </li> <li class="gallerybox" style="width: 255px"> <div class="thumb" style="width: 250px; height: 195px;"><span typeof="mw:File"><a href="/wiki/File:1-bit_half-adder.svg" class="mw-file-description" title="Schematic symbol for a 1-bit half adder."><img alt="Schematic symbol for a 1-bit half adder." src="//upload.wikimedia.org/wikipedia/commons/thumb/8/80/1-bit_half-adder.svg/192px-1-bit_half-adder.svg.png" decoding="async" width="192" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/80/1-bit_half-adder.svg/288px-1-bit_half-adder.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/80/1-bit_half-adder.svg/384px-1-bit_half-adder.svg.png 2x" data-file-width="215" data-file-height="185" /></a></span></div> <div class="gallerytext"><a href="/wiki/Electronic_symbol" title="Electronic symbol">Schematic symbol</a> for a 1-bit half adder.</div> </li> </ul> <div class="mw-heading mw-heading3"><h3 id="Full_adder">Full adder</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Adder_(electronics)&action=edit&section=4" title="Edit section: Full adder"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <b>full adder</b> adds binary numbers and accounts for values carried in as well as out. A one-bit full-adder adds three one-bit numbers, often written as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{in}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{in}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c89dce295f8c5cacce1b3e1e8be7cf9859fb170a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.448ex; height:2.509ex;" alt="{\displaystyle C_{in}}"></span>; <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> are the operands, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{in}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{in}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c89dce295f8c5cacce1b3e1e8be7cf9859fb170a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.448ex; height:2.509ex;" alt="{\displaystyle C_{in}}"></span> is a bit carried in from the previous less-significant stage.<sup id="cite_ref-Mano_1979_3-0" class="reference"><a href="#cite_note-Mano_1979-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> The circuit produces a two-bit output. Output carry and sum are typically represented by the signals <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 C_{out}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> <mi>u</mi> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{out}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d9603349e295df10584185a5f030673f5f48a55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.225ex; height:2.509ex;" alt="{\displaystyle C_{out}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span>, where the sum equals <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 2C_{out}+S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> <mi>u</mi> <mi>t</mi> </mrow> </msub> <mo>+</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2C_{out}+S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c8f1a09a0d12d47ffd133eb06eb45039cab1df3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.727ex; height:2.509ex;" alt="{\displaystyle 2C_{out}+S}"></span>. The full adder is usually a component in a cascade of adders, which add 8, 16, 32, etc. bit binary numbers. </p><p>A full adder can be implemented in many different ways such as with a custom <a href="/wiki/Transistor" title="Transistor">transistor</a>-level circuit or composed of other gates. The most common implementation is with: </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 S=A\oplus B\oplus C_{in}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mi>A</mi> <mo>⊕</mo> <mi>B</mi> <mo>⊕</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=A\oplus B\oplus C_{in}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/990290db40d42d06d3a1e308db603e962455e729" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.233ex; height:2.509ex;" alt="{\displaystyle S=A\oplus B\oplus C_{in}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{out}=(A\cdot B)+(C_{in}\cdot (A\oplus B))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> <mi>u</mi> <mi>t</mi> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo>⋅</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>n</mi> </mrow> </msub> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo>⊕</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{out}=(A\cdot B)+(C_{in}\cdot (A\oplus B))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e5fc9a394d89a7b7c78a9fe64105abb9fc2a580" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.253ex; height:2.843ex;" alt="{\displaystyle C_{out}=(A\cdot B)+(C_{in}\cdot (A\oplus B))}"></span></dd></dl> <p>The above expressions for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{in}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{in}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c89dce295f8c5cacce1b3e1e8be7cf9859fb170a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.448ex; height:2.509ex;" alt="{\displaystyle C_{in}}"></span> can be derived from using a <a href="/wiki/Karnaugh_map" title="Karnaugh map">Karnaugh map</a> to simplify the truth table. </p><p>In this implementation, the final <a href="/wiki/OR_gate" title="OR gate">OR gate</a> before the carry-out output may be replaced by an <a href="/wiki/XOR_gate" title="XOR gate">XOR gate</a> without altering the resulting logic. This is because when A and B are both 1, the term <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A\oplus B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo>⊕</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A\oplus B)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/966575de8d90ba0483c7eb54bd43ec27f7404e50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.157ex; height:2.843ex;" alt="{\displaystyle (A\oplus B)}"></span> is always 0, and hence <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 (C_{in}\cdot (A\oplus B))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>n</mi> </mrow> </msub> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo>⊕</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (C_{in}\cdot (A\oplus B))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d054eb52746d55ab27530efeb29786d90795dfca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.093ex; height:2.843ex;" alt="{\displaystyle (C_{in}\cdot (A\oplus B))}"></span> can only be 0. Thus, the inputs to the final OR gate can never be both 1's (this is the only combination for which the OR and XOR outputs differ). </p><p>Due to the <a href="/wiki/Functional_completeness" title="Functional completeness">functional completeness</a> property of the NAND and NOR gates, a full adder can also be implemented using nine <a href="/wiki/NAND_gates" class="mw-redirect" title="NAND gates">NAND gates</a>,<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> or nine <a href="/wiki/NOR_gates" class="mw-redirect" title="NOR gates">NOR gates</a>. </p><p>Using only two types of gates is convenient if the circuit is being implemented using simple <a href="/wiki/Integrated_circuit" title="Integrated circuit">integrated circuit</a> chips which contain only one gate type per chip. </p><p>A full adder can also be constructed from two half adders by connecting <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> to the input of one half adder, then taking its sum-output <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span> as one of the inputs to the second half adder and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{in}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{in}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c89dce295f8c5cacce1b3e1e8be7cf9859fb170a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.448ex; height:2.509ex;" alt="{\displaystyle C_{in}}"></span> as its other input, and finally the carry outputs from the two half-adders are connected to an OR gate. The sum-output from the second half adder is the final sum output (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span>) of the full adder and the output from the OR gate is the final carry output (<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 C_{out}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> <mi>u</mi> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{out}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d9603349e295df10584185a5f030673f5f48a55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.225ex; height:2.509ex;" alt="{\displaystyle C_{out}}"></span>). The critical path of a full adder runs through both XOR gates and ends at the sum bit <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span>. Assumed that an XOR gate takes 1 delays to complete, the delay imposed by the critical path of a full adder is equal to: </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 T_{\text{FA}}=2\cdot T_{\text{XOR}}=2D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>FA</mtext> </mrow> </msub> <mo>=</mo> <mn>2</mn> <mo>⋅</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>XOR</mtext> </mrow> </msub> <mo>=</mo> <mn>2</mn> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{\text{FA}}=2\cdot T_{\text{XOR}}=2D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01a19499534da6019dfb4a49ecdce0167ee69e95" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:21.331ex; height:2.509ex;" alt="{\displaystyle T_{\text{FA}}=2\cdot T_{\text{XOR}}=2D}"></span></dd></dl> <p>The critical path of a carry runs through one XOR gate in adder and through 2 gates (AND and OR) in carry-block and therefore, if AND or OR gates take 1 delay to complete, has a delay of: </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 T_{\text{c}}=T_{\text{XOR}}+T_{\text{AND}}+T_{\text{OR}}=D+D+D=3D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> </msub> <mo>=</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>XOR</mtext> </mrow> </msub> <mo>+</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>AND</mtext> </mrow> </msub> <mo>+</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>OR</mtext> </mrow> </msub> <mo>=</mo> <mi>D</mi> <mo>+</mo> <mi>D</mi> <mo>+</mo> <mi>D</mi> <mo>=</mo> <mn>3</mn> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{\text{c}}=T_{\text{XOR}}+T_{\text{AND}}+T_{\text{OR}}=D+D+D=3D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4228d4c1af9ccf5354ba13d56ba2d5c7245c926" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:46.535ex; height:2.509ex;" alt="{\displaystyle T_{\text{c}}=T_{\text{XOR}}+T_{\text{AND}}+T_{\text{OR}}=D+D+D=3D}"></span></dd></dl> <p>The <a href="/wiki/Truth_table" title="Truth table">truth table</a> for the full adder is: </p> <dl><dd><table class="wikitable" style="text-align:center"> <tbody><tr style="background:#def; text-align:center;"> <th colspan="3">Inputs</th> <th colspan="2">Outputs </th></tr> <tr style="background:#def; text-align:center;"> <td><b>A</b></td> <td><b>B</b></td> <td><b>C</b><sub>in</sub></td> <td><b>C</b><sub>out</sub></td> <td><b>S</b> </td></tr> <tr> <td style="background: #FFE3E3; color: black; vertical-align: middle; text-align: center;" class="table-no2">0</td> <td style="background: #FFE3E3; color: black; vertical-align: middle; text-align: center;" class="table-no2">0</td> <td style="background: #FFE3E3; color: black; vertical-align: middle; text-align: center;" class="table-no2">0</td> <td style="background: #FFE3E3; color: black; vertical-align: middle; text-align: center;" class="table-no2">0</td> <td style="background: #FFE3E3; color: black; vertical-align: middle; text-align: center;" class="table-no2">0 </td></tr> <tr> <td style="background: #FFE3E3; color: black; vertical-align: middle; text-align: center;" class="table-no2">0</td> <td style="background: #FFE3E3; color: black; vertical-align: middle; text-align: center;" class="table-no2">0</td> <td style="background:#bfd; color:black; vertical-align:middle; text-align:center;" class="table-yes2">1</td> <td style="background: #FFE3E3; color: black; vertical-align: middle; text-align: center;" class="table-no2">0</td> <td style="background:#bfd; color:black; vertical-align:middle; text-align:center;" class="table-yes2">1 </td></tr> <tr> <td style="background: #FFE3E3; color: black; vertical-align: middle; text-align: center;" class="table-no2">0</td> <td style="background:#bfd; color:black; vertical-align:middle; text-align:center;" class="table-yes2">1</td> <td style="background: #FFE3E3; color: black; vertical-align: middle; text-align: center;" class="table-no2">0</td> <td style="background: #FFE3E3; color: black; vertical-align: middle; text-align: center;" class="table-no2">0</td> <td style="background:#bfd; color:black; vertical-align:middle; text-align:center;" class="table-yes2">1 </td></tr> <tr> <td style="background: #FFE3E3; color: black; vertical-align: middle; text-align: center;" class="table-no2">0</td> <td style="background:#bfd; color:black; vertical-align:middle; text-align:center;" class="table-yes2">1</td> <td style="background:#bfd; color:black; vertical-align:middle; text-align:center;" class="table-yes2">1</td> <td style="background:#bfd; color:black; vertical-align:middle; text-align:center;" class="table-yes2">1</td> <td style="background: #FFE3E3; color: black; vertical-align: middle; text-align: center;" class="table-no2">0 </td></tr> <tr> <td style="background:#bfd; color:black; vertical-align:middle; text-align:center;" class="table-yes2">1</td> <td style="background: #FFE3E3; color: black; vertical-align: middle; text-align: center;" class="table-no2">0</td> <td style="background: #FFE3E3; color: black; vertical-align: middle; text-align: center;" class="table-no2">0</td> <td style="background: #FFE3E3; color: black; vertical-align: middle; text-align: center;" class="table-no2">0</td> <td style="background:#bfd; color:black; vertical-align:middle; text-align:center;" class="table-yes2">1 </td></tr> <tr> <td style="background:#bfd; color:black; vertical-align:middle; text-align:center;" class="table-yes2">1</td> <td style="background: #FFE3E3; color: black; vertical-align: middle; text-align: center;" class="table-no2">0</td> <td style="background:#bfd; color:black; vertical-align:middle; text-align:center;" class="table-yes2">1</td> <td style="background:#bfd; color:black; vertical-align:middle; text-align:center;" class="table-yes2">1</td> <td style="background: #FFE3E3; color: black; vertical-align: middle; text-align: center;" class="table-no2">0 </td></tr> <tr> <td style="background:#bfd; color:black; vertical-align:middle; text-align:center;" class="table-yes2">1</td> <td style="background:#bfd; color:black; vertical-align:middle; text-align:center;" class="table-yes2">1</td> <td style="background: #FFE3E3; color: black; vertical-align: middle; text-align: center;" class="table-no2">0</td> <td style="background:#bfd; color:black; vertical-align:middle; text-align:center;" class="table-yes2">1</td> <td style="background: #FFE3E3; color: black; vertical-align: middle; text-align: center;" class="table-no2">0 </td></tr> <tr> <td style="background:#bfd; color:black; vertical-align:middle; text-align:center;" class="table-yes2">1</td> <td style="background:#bfd; color:black; vertical-align:middle; text-align:center;" class="table-yes2">1</td> <td style="background:#bfd; color:black; vertical-align:middle; text-align:center;" class="table-yes2">1</td> <td style="background:#bfd; color:black; vertical-align:middle; text-align:center;" class="table-yes2">1</td> <td style="background:#bfd; color:black; vertical-align:middle; text-align:center;" class="table-yes2">1 </td></tr> </tbody></table></dd></dl> <p>Inverting all inputs of a full adder also inverts all of its outputs, which can be used in the design of fast ripple-carry adders, because there is no need to invert the carry.<sup id="cite_ref-Fischer_5-0" class="reference"><a href="#cite_note-Fischer-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>Various full adder digital logic circuits: </p> <ul class="gallery mw-gallery-traditional"> <li class="gallerybox" style="width: 255px"> <div class="thumb" style="width: 250px; height: 195px;"><span typeof="mw:File"><a href="/wiki/File:Fulladder.gif" class="mw-file-description" title="Full adder in action."><img alt="Full adder in action." src="//upload.wikimedia.org/wikipedia/commons/thumb/5/57/Fulladder.gif/220px-Fulladder.gif" decoding="async" width="220" height="149" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/57/Fulladder.gif/330px-Fulladder.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/57/Fulladder.gif/440px-Fulladder.gif 2x" data-file-width="480" data-file-height="324" /></a></span></div> <div class="gallerytext">Full adder in action.</div> </li> <li class="gallerybox" style="width: 255px"> <div class="thumb" style="width: 250px; height: 195px;"><span typeof="mw:File"><a href="/wiki/File:Full-adder_logic_diagram.svg" class="mw-file-description" title="Schematic of full adder implemented with two XOR gates, two AND gates, one OR gate."><img alt="Schematic of full adder implemented with two XOR gates, two AND gates, one OR gate." src="//upload.wikimedia.org/wikipedia/commons/thumb/6/69/Full-adder_logic_diagram.svg/220px-Full-adder_logic_diagram.svg.png" decoding="async" width="220" height="99" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/69/Full-adder_logic_diagram.svg/330px-Full-adder_logic_diagram.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/69/Full-adder_logic_diagram.svg/440px-Full-adder_logic_diagram.svg.png 2x" data-file-width="400" data-file-height="180" /></a></span></div> <div class="gallerytext"><a href="/wiki/Schematic" title="Schematic">Schematic</a> of full adder implemented with two <a href="/wiki/XOR_gate" title="XOR gate">XOR gates</a>, two <a href="/wiki/AND_gate" title="AND gate">AND gates</a>, one <a href="/wiki/OR_gate" title="OR gate">OR gate</a>.</div> </li> <li class="gallerybox" style="width: 255px"> <div class="thumb" style="width: 250px; height: 195px;"><span typeof="mw:File"><a href="/wiki/File:Full_Adder_using_NAND_gates.svg" class="mw-file-description" title="Schematic of full adder implemented with nine NAND gates."><img alt="Schematic of full adder implemented with nine NAND gates." src="//upload.wikimedia.org/wikipedia/commons/thumb/6/69/Full_Adder_using_NAND_gates.svg/220px-Full_Adder_using_NAND_gates.svg.png" decoding="async" width="220" height="79" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/69/Full_Adder_using_NAND_gates.svg/330px-Full_Adder_using_NAND_gates.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/69/Full_Adder_using_NAND_gates.svg/440px-Full_Adder_using_NAND_gates.svg.png 2x" data-file-width="568" data-file-height="204" /></a></span></div> <div class="gallerytext">Schematic of full adder implemented with nine <a href="/wiki/NAND_gate" title="NAND gate">NAND gates</a>.</div> </li> <li class="gallerybox" style="width: 255px"> <div class="thumb" style="width: 250px; height: 195px;"><span typeof="mw:File"><a href="/wiki/File:Full_Adder_using_NOR_gates.svg" class="mw-file-description" title="Schematic of full adder implemented with nine NOR gates."><img alt="Schematic of full adder implemented with nine NOR gates." src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ad/Full_Adder_using_NOR_gates.svg/220px-Full_Adder_using_NOR_gates.svg.png" decoding="async" width="220" height="79" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ad/Full_Adder_using_NOR_gates.svg/330px-Full_Adder_using_NOR_gates.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ad/Full_Adder_using_NOR_gates.svg/440px-Full_Adder_using_NOR_gates.svg.png 2x" data-file-width="631" data-file-height="228" /></a></span></div> <div class="gallerytext">Schematic of full adder implemented with nine <a href="/wiki/NOR_gate" title="NOR gate">NOR gates</a>.</div> </li> <li class="gallerybox" style="width: 255px"> <div class="thumb" style="width: 250px; height: 195px;"><span typeof="mw:File"><a href="/wiki/File:Inverting_full_adder_CMOS_24T.svg" class="mw-file-description" title="Full adder with inverted outputs with single-transistor carry propagation delay in CMOS[5]"><img alt="Full adder with inverted outputs with single-transistor carry propagation delay in CMOS[5]" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Inverting_full_adder_CMOS_24T.svg/220px-Inverting_full_adder_CMOS_24T.svg.png" decoding="async" width="220" height="156" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Inverting_full_adder_CMOS_24T.svg/330px-Inverting_full_adder_CMOS_24T.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Inverting_full_adder_CMOS_24T.svg/440px-Inverting_full_adder_CMOS_24T.svg.png 2x" data-file-width="550" data-file-height="390" /></a></span></div> <div class="gallerytext">Full adder with inverted outputs with single-transistor carry propagation delay in <a href="/wiki/CMOS" title="CMOS">CMOS</a><sup id="cite_ref-Fischer_5-1" class="reference"><a href="#cite_note-Fischer-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup></div> </li> <li class="gallerybox" style="width: 255px"> <div class="thumb" style="width: 250px; height: 195px;"><span typeof="mw:File"><a href="/wiki/File:1-bit_full-adder.svg" class="mw-file-description" title="Schematic symbol for a 1-bit full adder with Cin and Cout drawn on sides of block to emphasize their use in a multi-bit adder"><img alt="Schematic symbol for a 1-bit full adder with Cin and Cout drawn on sides of block to emphasize their use in a multi-bit adder" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/1-bit_full-adder.svg/192px-1-bit_full-adder.svg.png" decoding="async" width="192" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/1-bit_full-adder.svg/288px-1-bit_full-adder.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/1-bit_full-adder.svg/384px-1-bit_full-adder.svg.png 2x" data-file-width="215" data-file-height="185" /></a></span></div> <div class="gallerytext"><a href="/wiki/Electronic_symbol" title="Electronic symbol">Schematic symbol</a> for a 1-bit full adder with <i>C</i><sub>in</sub> and <i>C</i><sub>out</sub> drawn on sides of block to emphasize their use in a multi-bit adder</div> </li> </ul> <div class="mw-heading mw-heading3"><h3 id="Adders_supporting_multiple_bits">Adders supporting multiple bits</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Adder_(electronics)&action=edit&section=5" title="Edit section: Adders supporting multiple bits"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading4"><h4 id="Ripple-carry_adder">Ripple-carry adder</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Adder_(electronics)&action=edit&section=6" title="Edit section: Ripple-carry adder"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:4-bit_ripple_carry_adder.svg" class="mw-file-description"><img alt="4-bit adder with logical block diagram shown" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5d/4-bit_ripple_carry_adder.svg/220px-4-bit_ripple_carry_adder.svg.png" decoding="async" width="220" height="88" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5d/4-bit_ripple_carry_adder.svg/330px-4-bit_ripple_carry_adder.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5d/4-bit_ripple_carry_adder.svg/440px-4-bit_ripple_carry_adder.svg.png 2x" data-file-width="500" data-file-height="200" /></a><figcaption>4-bit adder with logical block diagram shown</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:RippleCarry2.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/85/RippleCarry2.gif/220px-RippleCarry2.gif" decoding="async" width="220" height="60" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/85/RippleCarry2.gif/330px-RippleCarry2.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/85/RippleCarry2.gif/440px-RippleCarry2.gif 2x" data-file-width="480" data-file-height="130" /></a><figcaption>Decimal 4-digit ripple carry adder. FA = full adder, HA = half adder.</figcaption></figure> <p>It is possible to create a logical circuit using multiple full adders to add <i>N</i>-bit numbers. Each full adder inputs a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{in}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{in}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c89dce295f8c5cacce1b3e1e8be7cf9859fb170a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.448ex; height:2.509ex;" alt="{\displaystyle C_{in}}"></span>, which is the <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 C_{out}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> <mi>u</mi> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{out}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d9603349e295df10584185a5f030673f5f48a55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.225ex; height:2.509ex;" alt="{\displaystyle C_{out}}"></span> of the previous adder. This kind of adder is called a <b>ripple-carry adder</b> (RCA), since each carry bit "ripples" to the next full adder. The first (and only the first) full adder may be replaced by a half adder (under the assumption that <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 C_{in}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{in}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19705bad4890d4935d18ce610a6c5d60736221ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.709ex; height:2.509ex;" alt="{\displaystyle C_{in}=0}"></span>). </p><p>The layout of a ripple-carry adder is simple, which allows fast design time; however, the ripple-carry adder is relatively slow, since each full adder must wait for the carry bit to be calculated from the previous full adder. The <a href="/wiki/Gate_delay" class="mw-redirect" title="Gate delay">gate delay</a> can easily be calculated by inspection of the full adder circuit. Each full adder requires three levels of logic. In a 32-bit ripple-carry adder, there are 32 full adders, so the critical path (worst case) delay is 3 (from input to carry in first adder) + 31 × 2 (for carry propagation in latter adders) = 65 gate delays.<sup id="cite_ref-Adder_6-0" class="reference"><a href="#cite_note-Adder-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> The general equation for the worst-case delay for a <i>n</i>-bit carry-ripple adder, accounting for both the sum and carry bits, 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 T_{\text{CRA}}(n)=T_{\text{HA}}+(n-1)\cdot T_{\text{c}}+T_{\text{s}}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>CRA</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>HA</mtext> </mrow> </msub> <mo>+</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>⋅</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> </msub> <mo>+</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>s</mtext> </mrow> </msub> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{\text{CRA}}(n)=T_{\text{HA}}+(n-1)\cdot T_{\text{c}}+T_{\text{s}}=}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26da5a90eba71eacba0baffc6bf1c9a1c0a6bbb5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.154ex; height:2.843ex;" alt="{\displaystyle T_{\text{CRA}}(n)=T_{\text{HA}}+(n-1)\cdot T_{\text{c}}+T_{\text{s}}=}"></span><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 T_{\text{FA}}+(n-1)\cdot T_{c}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>FA</mtext> </mrow> </msub> <mo>+</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>⋅</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{\text{FA}}+(n-1)\cdot T_{c}=}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11c7d99f1f283d1f5e65be8207b4b384ffd17a5a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.377ex; height:2.843ex;" alt="{\displaystyle T_{\text{FA}}+(n-1)\cdot T_{c}=}"></span><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 3D+(n-1)\cdot 2D=(2n+1)\cdot D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mi>D</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>⋅</mo> <mn>2</mn> <mi>D</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3D+(n-1)\cdot 2D=(2n+1)\cdot D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/942f75f5cbe38cfa328a7e3db3a036c35cc2d7d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.971ex; height:2.843ex;" alt="{\displaystyle 3D+(n-1)\cdot 2D=(2n+1)\cdot D}"></span></dd></dl> <p>A design with alternating carry polarities and optimized <a href="/wiki/AND-OR-Invert" class="mw-redirect" title="AND-OR-Invert">AND-OR-Invert</a> gates can be about twice as fast.<sup id="cite_ref-Burgess_2011_7-0" class="reference"><a href="#cite_note-Burgess_2011-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Fischer_5-2" class="reference"><a href="#cite_note-Fischer-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="Carry-lookahead_adder_(Weinberger_and_Smith,_1958)"><span id="Carry-lookahead_adder_.28Weinberger_and_Smith.2C_1958.29"></span>Carry-lookahead adder (Weinberger and Smith, 1958)</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Adder_(electronics)&action=edit&section=7" title="Edit section: Carry-lookahead adder (Weinberger and Smith, 1958)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Carry-lookahead_adder" title="Carry-lookahead adder">Carry-lookahead adder</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:4-bit_carry_lookahead_adder.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/04/4-bit_carry_lookahead_adder.svg/220px-4-bit_carry_lookahead_adder.svg.png" decoding="async" width="220" height="132" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/04/4-bit_carry_lookahead_adder.svg/330px-4-bit_carry_lookahead_adder.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/04/4-bit_carry_lookahead_adder.svg/440px-4-bit_carry_lookahead_adder.svg.png 2x" data-file-width="500" data-file-height="300" /></a><figcaption>4-bit adder with carry lookahead</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:64-bit_lookahead_carry_unit.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/58/64-bit_lookahead_carry_unit.svg/220px-64-bit_lookahead_carry_unit.svg.png" decoding="async" width="220" height="132" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/58/64-bit_lookahead_carry_unit.svg/330px-64-bit_lookahead_carry_unit.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/58/64-bit_lookahead_carry_unit.svg/440px-64-bit_lookahead_carry_unit.svg.png 2x" data-file-width="500" data-file-height="300" /></a><figcaption>64-bit adder with carry lookahead</figcaption></figure> <p>To reduce the computation time, Weinberger and Smith invented a faster way to add two binary numbers by using <a href="/wiki/Carry-lookahead_adder" title="Carry-lookahead adder">carry-lookahead adders</a> (CLA).<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> They introduced two signals (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span>) for each bit position, based on whether a carry is propagated through from a less significant bit position (at least one input is a 1), generated in that bit position (both inputs are 1), or killed in that bit position (both inputs are 0). In most cases, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span> is simply the sum output of a half adder and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> is the carry output of the same adder. After <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> are generated, the carries for every bit position are created. </p><p>Mere derivation of Weinberger-Smith CLA recurrence, are: <a href="/wiki/Brent%E2%80%93Kung_adder" title="Brent–Kung adder">Brent–Kung adder</a> (BKA),<sup id="cite_ref-Brent-Kung_1982_9-0" class="reference"><a href="#cite_note-Brent-Kung_1982-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> and the <a href="/wiki/Kogge%E2%80%93Stone_adder" title="Kogge–Stone adder">Kogge–Stone adder</a> (KSA).<sup id="cite_ref-Kogge-Stone_1973_10-0" class="reference"><a href="#cite_note-Kogge-Stone_1973-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-ULVD_2015_11-0" class="reference"><a href="#cite_note-ULVD_2015-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> This was shown in Oklobdzija and Zeydel paper in IEEE Journal of Solid-State Circutis.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p><p>Some other multi-bit adder architectures break the adder into blocks. It is possible to vary the length of these blocks based on the <a href="/wiki/Propagation_delay" title="Propagation delay">propagation delay</a> of the circuits to optimize computation time. These block based adders include the <a href="/wiki/Carry-skip_adder" title="Carry-skip adder">carry-skip (or carry-bypass) adder</a> which will determine <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> values for each block rather than each bit, and the <a href="/wiki/Carry-select_adder" title="Carry-select adder">carry-select adder</a> which pre-generates the sum and carry values for either possible carry input (0 or 1) to the block, using multiplexers to select the appropriate result <i>when</i> the carry bit is known. </p><p>By combining multiple carry-lookahead adders, even larger adders can be created. This can be used at multiple levels to make even larger adders. For example, the following adder is a 64-bit adder that uses four 16-bit CLAs with two levels of <a href="/wiki/Lookahead_carry_unit" title="Lookahead carry unit">lookahead carry units</a>. </p><p>Other adder designs include the <a href="/wiki/Carry-select_adder" title="Carry-select adder">carry-select adder</a>, <a href="/wiki/Conditional_sum_adder" class="mw-redirect" title="Conditional sum adder">conditional sum adder</a>, <a href="/wiki/Carry-skip_adder" title="Carry-skip adder">carry-skip adder</a>, and carry-complete adder. </p> <div class="mw-heading mw-heading4"><h4 id="Carry-save_adders">Carry-save adders</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Adder_(electronics)&action=edit&section=8" title="Edit section: Carry-save adders"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Carry-save_adder" title="Carry-save adder">Carry-save adder</a></div> <p>If an adding circuit is to compute the sum of three or more numbers, it can be advantageous to not propagate the carry result. Instead, three-input adders are used, generating two results: a sum and a carry. The sum and the carry may be fed into two inputs of the subsequent 3-number adder without having to wait for propagation of a carry signal. After all stages of addition, however, a conventional adder (such as the ripple-carry or the lookahead) must be used to combine the final sum and carry results. </p> <div class="mw-heading mw-heading3"><h3 id="3:2_compressors">3:2 compressors</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Adder_(electronics)&action=edit&section=9" title="Edit section: 3:2 compressors"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A full adder can be viewed as a <i>3:2 lossy compressor</i>: it sums three one-bit inputs and returns the result as a single two-bit number; that is, it maps 8 input values to 4 output values. (the term "compressor" instead of "counter" was introduced in<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup>)Thus, for example, a binary input of 101 results in an output of <span class="nowrap">1 + 0 + 1 = 10</span> (decimal number 2). The carry-out represents bit one of the result, while the sum represents bit zero. Likewise, a half adder can be used as a <i>2:2 lossy compressor</i>, compressing four possible inputs into three possible outputs. </p><p>Such compressors can be used to speed up the summation of three or more addends. If the number of addends is exactly three, the layout is known as the <a href="/wiki/Carry-save_adder" title="Carry-save adder">carry-save adder</a>. If the number of addends is four or more, more than one layer of compressors is necessary, and there are various possible designs for the circuit: the most common are <a href="/wiki/Dadda_tree" class="mw-redirect" title="Dadda tree">Dadda</a> and <a href="/wiki/Wallace_tree" title="Wallace tree">Wallace trees</a>. This kind of circuit is most notably used in <a href="/wiki/Binary_multiplier" title="Binary multiplier">multiplier circuits</a>, which is why these circuits are also known as Dadda and Wallace multipliers. </p> <div class="mw-heading mw-heading2"><h2 id="Quantum_adders">Quantum adders</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Adder_(electronics)&action=edit&section=10" title="Edit section: Quantum adders"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Quantum_Full_Adder.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0a/Quantum_Full_Adder.png/220px-Quantum_Full_Adder.png" decoding="async" width="220" height="81" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0a/Quantum_Full_Adder.png/330px-Quantum_Full_Adder.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0a/Quantum_Full_Adder.png/440px-Quantum_Full_Adder.png 2x" data-file-width="891" data-file-height="328" /></a><figcaption>Quantum full adder, using <a href="/wiki/Toffoli_gate" title="Toffoli gate">Toffoli</a> and <a href="/wiki/CNOT" class="mw-redirect" title="CNOT">CNOT</a> gates. The CNOT-gate that is surrounded by a dotted square in this picture can be omitted if <a href="/wiki/Uncomputation" title="Uncomputation">uncomputation</a> to restore the <i>B output</i> is not required.</figcaption></figure><p> Using only the <a href="/wiki/Toffoli_gate" title="Toffoli gate">Toffoli</a> and <a href="/wiki/CNOT" class="mw-redirect" title="CNOT">CNOT</a> <a href="/wiki/Quantum_logic_gate" title="Quantum logic gate">quantum logic gates</a>, it is possible to produce quantum full- and half-adders.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> The same circuits can also be implemented in classical <a href="/wiki/Reversible_computation" class="mw-redirect" title="Reversible computation">reversible computation</a>, as both CNOT and Toffoli are also classical <a href="/wiki/Logic_gate" title="Logic gate">logic gates</a>. </p><p>Since the <a href="/wiki/Quantum_Fourier_transform" title="Quantum Fourier transform">quantum Fourier transform</a> has a low <a href="/wiki/Circuit_complexity" title="Circuit complexity">circuit complexity</a>, it can efficiently be used for adding numbers as well.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Analog_adders">Analog adders</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Adder_(electronics)&action=edit&section=11" title="Edit section: Analog adders"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Just as in Binary adders, combining two input currents effectively adds those currents together. Within the constraints of the hardware, non-binary signals (i.e. with a base higher than 2) can be added together to calculate a sum. Also known as a "summing amplifier",<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> this technique can be used to reduce the number of transistors in an addition circuit. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Adder_(electronics)&action=edit&section=12" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Binary_multiplier" title="Binary multiplier">Binary multiplier</a></li> <li><a href="/wiki/Subtractor" title="Subtractor">Subtractor</a></li> <li><a href="/wiki/Electronic_mixer" title="Electronic mixer">Electronic mixer</a> — for adding analog signals</li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Adder_(electronics)&action=edit&section=13" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFSingh2010" class="citation book cs1">Singh, Ajay Kumar (2010). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Gr023E8Kd5QC&dq=adder+summer+circuit&pg=PA321"><i>Digital VLSI Design</i></a>. Prentice Hall India. p. 321. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9788120341876" title="Special:BookSources/9788120341876"><bdi>9788120341876</bdi></a> – via Google Books.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Digital+VLSI+Design&rft.pages=321&rft.pub=Prentice+Hall+India&rft.date=2010&rft.isbn=9788120341876&rft.aulast=Singh&rft.aufirst=Ajay+Kumar&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DGr023E8Kd5QC%26dq%3Dadder%2Bsummer%2Bcircuit%26pg%3DPA321&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAdder+%28electronics%29" class="Z3988"></span></span> </li> <li id="cite_note-Lancaster_2004-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-Lancaster_2004_2-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLancaster2004" class="citation book cs1">Lancaster, Geoffrey A. 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Pascal Press. p. 180. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-74125175-3" title="Special:BookSources/978-1-74125175-3"><bdi>978-1-74125175-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Excel+HSC+Software+Design+and+Development&rft.pages=180&rft.pub=Pascal+Press&rft.date=2004&rft.isbn=978-1-74125175-3&rft.aulast=Lancaster&rft.aufirst=Geoffrey+A.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DPZkDpS4m0fMC%26pg%3DPA180&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAdder+%28electronics%29" class="Z3988"></span></span> </li> <li id="cite_note-Mano_1979-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-Mano_1979_3-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMano1979" class="citation book cs1">Mano, M. Morris (1979). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/digitallogiccomp00mano/page/119"><i>Digital Logic and Computer Design</i></a></span>. <a href="/wiki/Prentice-Hall" class="mw-redirect" title="Prentice-Hall">Prentice-Hall</a>. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/digitallogiccomp00mano/page/119">119–123</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-13-214510-7" title="Special:BookSources/978-0-13-214510-7"><bdi>978-0-13-214510-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Digital+Logic+and+Computer+Design&rft.pages=119-123&rft.pub=Prentice-Hall&rft.date=1979&rft.isbn=978-0-13-214510-7&rft.aulast=Mano&rft.aufirst=M.+Morris&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fdigitallogiccomp00mano%2Fpage%2F119&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAdder+%28electronics%29" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTeja2021" class="citation cs2">Teja, Ravi (2021-04-15), <a rel="nofollow" class="external text" href="https://www.electronicshub.org/half-adder-and-full-adder-circuits/#Full_Adder_using_NAND_Gates"><i>Half Adder and Full Adder Circuits</i></a><span class="reference-accessdate">, retrieved <span class="nowrap">2021-07-27</span></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Half+Adder+and+Full+Adder+Circuits&rft.date=2021-04-15&rft.aulast=Teja&rft.aufirst=Ravi&rft_id=https%3A%2F%2Fwww.electronicshub.org%2Fhalf-adder-and-full-adder-circuits%2F%23Full_Adder_using_NAND_Gates&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAdder+%28electronics%29" class="Z3988"></span></span> </li> <li id="cite_note-Fischer-5"><span class="mw-cite-backlink">^ <a href="#cite_ref-Fischer_5-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Fischer_5-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Fischer_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="CITEREFFischer" class="citation web cs1">Fischer, P. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20210905175605/https://sus.ziti.uni-heidelberg.de/Lehre/WS1617_DST/DST_Fischer_06_Einfache_Bloecke_FF.pptx.pdf">"Einfache Schaltungsblöcke"</a> <span class="cs1-format">(PDF)</span>. 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Anchor Academic Publishing. p. 22. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-96067058-2" title="Special:BookSources/978-3-96067058-2"><bdi>978-3-96067058-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Design+and+Implementation+of+Carry+Select+Adder+Using+T-Spice&rft.pages=22&rft.pub=Anchor+Academic+Publishing&rft.date=2016&rft.isbn=978-3-96067058-2&rft.aulast=Satpathy&rft.aufirst=Pinaki&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DF_AWDQAAQBAJ%26pg%3DPA22&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAdder+%28electronics%29" class="Z3988"></span></span> </li> <li id="cite_note-Burgess_2011-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-Burgess_2011_7-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBurgess2011" class="citation conference cs1">Burgess, Neil (2011). <a rel="nofollow" class="external text" href="https://ieeexplore.ieee.org/Xplore/login.jsp?url=http%3A%2F%2Fieeexplore.ieee.org%2Fiel5%2F5991607%2F5992089%2F05992115.pdf%3Farnumber%3D5992115&authDecision=-203"><i>Fast Ripple-Carry Adders in Standard-Cell CMOS VLSI</i></a>. <a href="/wiki/20th_IEEE_Symposium_on_Computer_Arithmetic" class="mw-redirect" title="20th IEEE Symposium on Computer Arithmetic">20th IEEE Symposium on Computer Arithmetic</a>. pp. 103–111.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.btitle=Fast+Ripple-Carry+Adders+in+Standard-Cell+CMOS+VLSI&rft.pages=103-111&rft.date=2011&rft.aulast=Burgess&rft.aufirst=Neil&rft_id=https%3A%2F%2Fieeexplore.ieee.org%2FXplore%2Flogin.jsp%3Furl%3Dhttp%253A%252F%252Fieeexplore.ieee.org%252Fiel5%252F5991607%252F5992089%252F05992115.pdf%253Farnumber%253D5992115%26authDecision%3D-203&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAdder+%28electronics%29" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://nvlpubs.nist.gov/nistpubs/Legacy/circ/nbscircular591.pdf">Weinberger and J. 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R. Zeydel, D. Baran, V. G. Oklobdzija, "Energy Efficient Design of High-Performance VLSI Adders ", IEEE Journal of Solid-State Circuits , Vol 45, Issue 6. June 2010.</a></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"><a rel="nofollow" class="external text" href="https://www.acsel-lab.com/Publications/Papers/47-multipl-ieee-vlsi-95.pdf">V. G. Oklobdzija and D. Villeger, "Improving Multiplier Design By Using Improved Column Compression Tree And Optimized Final Adder In CMOS Technology", IEEE Transactions on VLSI Systems, Vol. 3, No. 2, June, 1995, 10 pages.</a></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="CITEREFFeynman1986" class="citation journal cs1">Feynman, Richard P. (1986). "Quantum mechanical computers". <i>Foundations of Physics</i>. <b>16</b> (6). Springer Science and Business Media LLC: 507–531. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1986FoPh...16..507F">1986FoPh...16..507F</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fbf01886518">10.1007/bf01886518</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0015-9018">0015-9018</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:122076550">122076550</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Foundations+of+Physics&rft.atitle=Quantum+mechanical+computers&rft.volume=16&rft.issue=6&rft.pages=507-531&rft.date=1986&rft_id=info%3Adoi%2F10.1007%2Fbf01886518&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A122076550%23id-name%3DS2CID&rft.issn=0015-9018&rft_id=info%3Abibcode%2F1986FoPh...16..507F&rft.aulast=Feynman&rft.aufirst=Richard+P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAdder+%28electronics%29" 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"><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.quantum-inspire.com/kbase/full-adder/">"Code example: Quantum full adder"</a>. 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"Addition on a Quantum Computer". <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/quant-ph/0008033">quant-ph/0008033</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=Addition+on+a+Quantum+Computer&rft.date=2000-08-07&rft_id=info%3Aarxiv%2Fquant-ph%2F0008033&rft.aulast=Draper&rft.aufirst=Thomas+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAdder+%28electronics%29" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRuiz-PerezJuan_Carlos2017" class="citation journal cs1">Ruiz-Perez, Lidia; Juan Carlos, Garcia-Escartin (2 May 2017). "Quantum arithmetic with the quantum Fourier transform". <i>Quantum Information Processing</i>. <b>16</b> (6): 152. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1411.5949v2">1411.5949v2</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2017QuIP...16..152R">2017QuIP...16..152R</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs11128-017-1603-1">10.1007/s11128-017-1603-1</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:10948948">10948948</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Quantum+Information+Processing&rft.atitle=Quantum+arithmetic+with+the+quantum+Fourier+transform&rft.volume=16&rft.issue=6&rft.pages=152&rft.date=2017-05-02&rft_id=info%3Aarxiv%2F1411.5949v2&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A10948948%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2Fs11128-017-1603-1&rft_id=info%3Abibcode%2F2017QuIP...16..152R&rft.aulast=Ruiz-Perez&rft.aufirst=Lidia&rft.au=Juan+Carlos%2C+Garcia-Escartin&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAdder+%28electronics%29" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFŞahin2020" class="citation journal cs1">Şahin, Engin (2020). "Quantum arithmetic operations based on quantum Fourier transform on signed integers". <i>International Journal of Quantum Information</i>. <b>18</b> (6): 2050035. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/2005.00443v3">2005.00443v3</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2020IJQI...1850035S">2020IJQI...1850035S</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1142%2Fs0219749920500355">10.1142/s0219749920500355</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1793-6918">1793-6918</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=International+Journal+of+Quantum+Information&rft.atitle=Quantum+arithmetic+operations+based+on+quantum+Fourier+transform+on+signed+integers&rft.volume=18&rft.issue=6&rft.pages=2050035&rft.date=2020&rft_id=info%3Aarxiv%2F2005.00443v3&rft.issn=1793-6918&rft_id=info%3Adoi%2F10.1142%2Fs0219749920500355&rft_id=info%3Abibcode%2F2020IJQI...1850035S&rft.aulast=%C5%9Eahin&rft.aufirst=Engin&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAdder+%28electronics%29" class="Z3988"></span></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"><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.electronics-tutorials.ws/opamp/opamp_4.html">"Summing Amplifier is an Op-amp Voltage Adder"</a>. 22 August 2013.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Summing+Amplifier+is+an+Op-amp+Voltage+Adder&rft.date=2013-08-22&rft_id=https%3A%2F%2Fwww.electronics-tutorials.ws%2Fopamp%2Fopamp_4.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAdder+%28electronics%29" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Adder_(electronics)&action=edit&section=14" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLiuHohulinShiauMuroga1974" class="citation journal cs1">Liu, Tso-Kai; Hohulin, Keith R.; Shiau, Lih-Er; Muroga, Saburo (January 1974). "Optimal One-Bit Full-Adders with Different Types of Gates". <i><a href="/wiki/IEEE_Transactions_on_Computers" title="IEEE Transactions on Computers">IEEE Transactions on Computers</a></i>. <b>C-23</b> (1). Bell Laboratories: <a href="/wiki/IEEE" class="mw-redirect" title="IEEE">IEEE</a>: 63–70. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FT-C.1974.223778">10.1109/T-C.1974.223778</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0018-9340">0018-9340</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:7746693">7746693</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=IEEE+Transactions+on+Computers&rft.atitle=Optimal+One-Bit+Full-Adders+with+Different+Types+of+Gates&rft.volume=C-23&rft.issue=1&rft.pages=63-70&rft.date=1974-01&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A7746693%23id-name%3DS2CID&rft.issn=0018-9340&rft_id=info%3Adoi%2F10.1109%2FT-C.1974.223778&rft.aulast=Liu&rft.aufirst=Tso-Kai&rft.au=Hohulin%2C+Keith+R.&rft.au=Shiau%2C+Lih-Er&rft.au=Muroga%2C+Saburo&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAdder+%28electronics%29" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLaiMuroga1979" class="citation journal cs1">Lai, Hung Chi; Muroga, Saburo (September 1979). "Minimum Binary Parallel Adders with NOR (NAND) Gates". <i><a href="/wiki/IEEE_Transactions_on_Computers" title="IEEE Transactions on Computers">IEEE Transactions on Computers</a></i>. <b>C-28</b> (9). <a href="/wiki/IEEE" class="mw-redirect" title="IEEE">IEEE</a>: 648–659. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FTC.1979.1675433">10.1109/TC.1979.1675433</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:23026844">23026844</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=IEEE+Transactions+on+Computers&rft.atitle=Minimum+Binary+Parallel+Adders+with+NOR+%28NAND%29+Gates&rft.volume=C-28&rft.issue=9&rft.pages=648-659&rft.date=1979-09&rft_id=info%3Adoi%2F10.1109%2FTC.1979.1675433&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A23026844%23id-name%3DS2CID&rft.aulast=Lai&rft.aufirst=Hung+Chi&rft.au=Muroga%2C+Saburo&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAdder+%28electronics%29" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMeadConway1980" class="citation book cs1">Mead, Carver; Conway, Lynn (1980) [December 1979]. <a rel="nofollow" class="external text" href="https://archive.org/details/introductiontovl00mead"><i>Introduction to VLSI Systems</i></a> (1 ed.). Reading, MA, USA: <a href="/wiki/Addison-Wesley" title="Addison-Wesley">Addison-Wesley</a>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1980aw...book.....M">1980aw...book.....M</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-20104358-7" title="Special:BookSources/978-0-20104358-7"><bdi>978-0-20104358-7</bdi></a><span class="reference-accessdate">. Retrieved <span class="nowrap">2018-05-12</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+VLSI+Systems&rft.place=Reading%2C+MA%2C+USA&rft.edition=1&rft.pub=Addison-Wesley&rft.date=1980&rft_id=info%3Abibcode%2F1980aw...book.....M&rft.isbn=978-0-20104358-7&rft.aulast=Mead&rft.aufirst=Carver&rft.au=Conway%2C+Lynn&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fintroductiontovl00mead&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAdder+%28electronics%29" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDavioDechampsThayse1983" class="citation book cs1">Davio, Marc; Dechamps, Jean-Pierre; Thayse, André (1983). <i>Digital Systems, with algorithm implementation</i> (1 ed.). <a href="/wiki/Philips_Research_Laboratory" class="mw-redirect" title="Philips Research Laboratory">Philips Research Laboratory</a>, Brussels, Belgium: <a href="/wiki/John_Wiley_%26_Sons" class="mw-redirect" title="John Wiley & Sons">John Wiley & Sons</a>, a Wiley-Interscience Publication. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-10413-1" title="Special:BookSources/978-0-471-10413-1"><bdi>978-0-471-10413-1</bdi></a>. <a href="/wiki/LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://lccn.loc.gov/82-2710">82-2710</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Digital+Systems%2C+with+algorithm+implementation&rft.place=Philips+Research+Laboratory%2C+Brussels%2C+Belgium&rft.edition=1&rft.pub=John+Wiley+%26+Sons%2C+a+Wiley-Interscience+Publication&rft.date=1983&rft_id=info%3Alccn%2F82-2710&rft.isbn=978-0-471-10413-1&rft.aulast=Davio&rft.aufirst=Marc&rft.au=Dechamps%2C+Jean-Pierre&rft.au=Thayse%2C+Andr%C3%A9&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAdder+%28electronics%29" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGosling1971" class="citation journal cs1">Gosling, John (January 1971). "Review of High-Speed Addition Techniques". <i><a href="/wiki/Proceedings_of_the_Institution_of_Electrical_Engineers" title="Proceedings of the Institution of Electrical Engineers">Proc. IEE</a></i>. <b>188</b> (1): 29–35. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1049%2Fpiee.1971.0004">10.1049/piee.1971.0004</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proc.+IEE&rft.atitle=Review+of+High-Speed+Addition+Techniques&rft.volume=188&rft.issue=1&rft.pages=29-35&rft.date=1971-01&rft_id=info%3Adoi%2F10.1049%2Fpiee.1971.0004&rft.aulast=Gosling&rft.aufirst=John&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAdder+%28electronics%29" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Adder_(electronics)&action=edit&section=15" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Commons-logo.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/12px-Commons-logo.svg.png" decoding="async" width="12" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/18px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/24px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></a></span> Media related to <a href="https://commons.wikimedia.org/wiki/Category:Adders_(digital_circuits)" class="extiw" title="commons:Category:Adders (digital circuits)">Adders (digital circuits)</a> at Wikimedia Commons</li> <li><a rel="nofollow" class="external text" href="http://dev.code.ultimater.net/electronics/8-bit-full-adder-and-subtractor/">8-bit Full Adder and Subtractor</a>, a demonstration of an interactive Full Adder built in JavaScript solely for learning purposes.</li> <li><a rel="nofollow" class="external text" href="https://sean.brunnock.com/React/Circuits/Gates/adders.html">Interactive demonstrations of half and full adders in HTML5</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFShirriff2020" class="citation web cs1">Shirriff, Ken (November 2020). <a rel="nofollow" class="external text" href="http://www.righto.com/2020/11/reverse-engineering-carry-lookahead.html">"Reverse-engineering the carry-lookahead circuit in the Intel 8008 processor"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Reverse-engineering+the+carry-lookahead+circuit+in+the+Intel+8008+processor&rft.date=2020-11&rft.aulast=Shirriff&rft.aufirst=Ken&rft_id=http%3A%2F%2Fwww.righto.com%2F2020%2F11%2Freverse-engineering-carry-lookahead.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAdder+%28electronics%29" class="Z3988"></span></li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output .navbox-list{line-height:1.5em;border-color:#fdfdfd}.mw-parser-output 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.navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Processor_technologies" style="padding:3px"><table class="nowraplinks mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles: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:Processor_technologies" title="Template:Processor technologies"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Processor_technologies" title="Template talk:Processor technologies"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Processor_technologies" title="Special:EditPage/Template:Processor technologies"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Processor_technologies" style="font-size:114%;margin:0 4em"><a href="/wiki/Processor_(computing)" title="Processor (computing)">Processor technologies</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Model_of_computation" title="Model of computation">Models</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Abstract_machine" title="Abstract machine">Abstract machine</a></li> <li><a href="/wiki/Stored-program_computer" title="Stored-program computer">Stored-program computer</a></li> <li><a href="/wiki/Finite-state_machine" title="Finite-state machine">Finite-state machine</a> <ul><li><a href="/wiki/Finite-state_machine_with_datapath" class="mw-redirect" title="Finite-state machine with datapath">with datapath</a></li> <li><a href="/wiki/Hierarchical_state_machine" class="mw-redirect" title="Hierarchical state machine">Hierarchical</a></li> <li><a href="/wiki/Deterministic_finite_automaton" title="Deterministic finite automaton">Deterministic finite automaton</a></li> <li><a href="/wiki/Queue_automaton" title="Queue automaton">Queue automaton</a></li> <li><a href="/wiki/Cellular_automaton" title="Cellular automaton">Cellular automaton</a></li> <li><a href="/wiki/Quantum_cellular_automaton" title="Quantum cellular automaton">Quantum cellular automaton</a></li></ul></li> <li><a href="/wiki/Turing_machine" title="Turing machine">Turing machine</a> <ul><li><a href="/wiki/Alternating_Turing_machine" title="Alternating Turing machine">Alternating Turing machine</a></li> <li><a href="/wiki/Universal_Turing_machine" title="Universal Turing machine">Universal</a></li> <li><a href="/wiki/Post%E2%80%93Turing_machine" title="Post–Turing machine">Post–Turing</a></li> <li><a href="/wiki/Quantum_Turing_machine" title="Quantum Turing machine">Quantum</a></li> <li><a href="/wiki/Nondeterministic_Turing_machine" title="Nondeterministic Turing machine">Nondeterministic Turing machine</a></li> <li><a href="/wiki/Probabilistic_Turing_machine" title="Probabilistic Turing machine">Probabilistic Turing machine</a></li> <li><a href="/wiki/Hypercomputation" title="Hypercomputation">Hypercomputation</a></li> <li><a href="/wiki/Zeno_machine" title="Zeno machine">Zeno machine</a></li></ul></li> <li><a href="/wiki/History_of_general-purpose_CPUs#Belt_machine_architecture" title="History of general-purpose CPUs">Belt machine</a></li> <li><a href="/wiki/Stack_machine" title="Stack machine">Stack machine</a></li> <li><a href="/wiki/Register_machine" title="Register machine">Register machines</a> <ul><li><a href="/wiki/Counter_machine" title="Counter machine">Counter</a></li> <li><a href="/wiki/Pointer_machine" title="Pointer machine">Pointer</a></li> <li><a href="/wiki/Random-access_machine" title="Random-access machine">Random-access</a></li> <li><a href="/wiki/Random-access_stored-program_machine" title="Random-access stored-program machine">Random-access stored program</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Computer_architecture" title="Computer architecture">Architecture</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Microarchitecture" title="Microarchitecture">Microarchitecture</a></li> <li><a href="/wiki/Von_Neumann_architecture" title="Von Neumann architecture">Von Neumann</a></li> <li><a href="/wiki/Harvard_architecture" title="Harvard architecture">Harvard</a> <ul><li><a href="/wiki/Modified_Harvard_architecture" title="Modified Harvard architecture">modified</a></li></ul></li> <li><a href="/wiki/Dataflow_architecture" title="Dataflow architecture">Dataflow</a></li> <li><a href="/wiki/Transport_triggered_architecture" title="Transport triggered architecture">Transport-triggered</a></li> <li><a href="/wiki/Cellular_architecture" title="Cellular architecture">Cellular</a></li> <li><a href="/wiki/Endianness" title="Endianness">Endianness</a></li> <li><a href="/wiki/Computer_data_storage" title="Computer data storage">Memory access</a> <ul><li><a href="/wiki/Non-uniform_memory_access" title="Non-uniform memory access">NUMA</a></li> <li><a href="/wiki/Uniform_memory_access" title="Uniform memory access">HUMA</a></li> <li><a href="/wiki/Load%E2%80%93store_architecture" title="Load–store architecture">Load–store</a></li> <li><a href="/wiki/Register%E2%80%93memory_architecture" title="Register–memory architecture">Register/memory</a></li></ul></li> <li><a href="/wiki/Cache_hierarchy" title="Cache hierarchy">Cache hierarchy</a></li> <li><a href="/wiki/Memory_hierarchy" title="Memory hierarchy">Memory hierarchy</a> <ul><li><a href="/wiki/Virtual_memory" title="Virtual memory">Virtual memory</a></li> <li><a href="/wiki/Secondary_storage" class="mw-redirect" title="Secondary storage">Secondary storage</a></li></ul></li> <li><a href="/wiki/Heterogeneous_System_Architecture" title="Heterogeneous System Architecture">Heterogeneous</a></li> <li><a href="/wiki/Fabric_computing" title="Fabric computing">Fabric</a></li> <li><a href="/wiki/Multiprocessing" title="Multiprocessing">Multiprocessing</a></li> <li><a href="/wiki/Cognitive_computing" title="Cognitive computing">Cognitive</a></li> <li><a href="/wiki/Neuromorphic_engineering" class="mw-redirect" title="Neuromorphic engineering">Neuromorphic</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Instruction_set_architecture" title="Instruction set architecture">Instruction set<br />architectures</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Types</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Orthogonal_instruction_set" title="Orthogonal instruction set">Orthogonal instruction set</a></li> <li><a href="/wiki/Complex_instruction_set_computer" title="Complex instruction set computer">CISC</a></li> <li><a href="/wiki/Reduced_instruction_set_computer" title="Reduced instruction set computer">RISC</a></li> <li><a href="/wiki/Application-specific_instruction_set_processor" title="Application-specific instruction set processor">Application-specific</a></li> <li><a href="/wiki/Explicit_data_graph_execution" title="Explicit data graph execution">EDGE</a> <ul><li><a href="/wiki/TRIPS_architecture" title="TRIPS architecture">TRIPS</a></li></ul></li> <li><a href="/wiki/Very_long_instruction_word" title="Very long instruction word">VLIW</a> <ul><li><a href="/wiki/Explicitly_parallel_instruction_computing" title="Explicitly parallel instruction computing">EPIC</a></li></ul></li> <li><a href="/wiki/Minimal_instruction_set_computer" title="Minimal instruction set computer">MISC</a></li> <li><a href="/wiki/One-instruction_set_computer" title="One-instruction set computer">OISC</a></li> <li><a href="/wiki/No_instruction_set_computing" title="No instruction set computing">NISC</a></li> <li><a href="/wiki/Zero_instruction_set_computer" class="mw-redirect" title="Zero instruction set computer">ZISC</a></li> <li><a href="/wiki/VISC_architecture" title="VISC architecture">VISC architecture</a></li> <li><a href="/wiki/Quantum_computing" title="Quantum computing">Quantum computing</a></li> <li><a href="/wiki/Comparison_of_instruction_set_architectures" title="Comparison of instruction set architectures">Comparison</a> <ul><li><a href="/wiki/Addressing_mode" title="Addressing mode">Addressing modes</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Instruction<br />sets</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Motorola_68000_series" title="Motorola 68000 series">Motorola 68000 series</a></li> <li><a href="/wiki/VAX" title="VAX">VAX</a></li> <li><a href="/wiki/PDP-11_architecture" title="PDP-11 architecture">PDP-11</a></li> <li><a href="/wiki/X86" title="X86">x86</a></li> <li><a href="/wiki/ARM_architecture_family" title="ARM architecture family">ARM</a></li> <li><a href="/wiki/Stanford_MIPS" title="Stanford MIPS">Stanford MIPS</a></li> <li><a href="/wiki/MIPS_architecture" title="MIPS architecture">MIPS</a></li> <li><a href="/wiki/MIPS-X" title="MIPS-X">MIPS-X</a></li> <li>Power <ul><li><a href="/wiki/IBM_POWER_architecture" title="IBM POWER architecture">POWER</a></li> <li><a href="/wiki/PowerPC" title="PowerPC">PowerPC</a></li> <li><a href="/wiki/Power_ISA" title="Power ISA">Power ISA</a></li></ul></li> <li><a href="/wiki/Clipper_architecture" title="Clipper architecture">Clipper architecture</a></li> <li><a href="/wiki/SPARC" title="SPARC">SPARC</a></li> <li><a href="/wiki/SuperH" title="SuperH">SuperH</a></li> <li><a href="/wiki/DEC_Alpha" title="DEC Alpha">DEC Alpha</a></li> <li><a href="/wiki/ETRAX_CRIS" title="ETRAX CRIS">ETRAX CRIS</a></li> <li><a href="/wiki/M32R" title="M32R">M32R</a></li> <li><a href="/wiki/Unicore" title="Unicore">Unicore</a></li> <li><a href="/wiki/IA-64" title="IA-64">Itanium</a></li> <li><a href="/wiki/OpenRISC" title="OpenRISC">OpenRISC</a></li> <li><a href="/wiki/RISC-V" title="RISC-V">RISC-V</a></li> <li><a href="/wiki/MicroBlaze" title="MicroBlaze">MicroBlaze</a></li> <li><a href="/wiki/Little_man_computer" title="Little man computer">LMC</a></li> <li>System/3x0 <ul><li><a href="/wiki/IBM_System/360_architecture" title="IBM System/360 architecture">S/360</a></li> <li><a href="/wiki/IBM_System/370" title="IBM System/370">S/370</a></li> <li><a href="/wiki/IBM_System/390" title="IBM System/390">S/390</a></li> <li><a href="/wiki/Z/Architecture" title="Z/Architecture">z/Architecture</a></li></ul></li> <li>Tilera ISA</li> <li><a href="/wiki/VISC_architecture" title="VISC architecture">VISC architecture</a></li> <li><a href="/wiki/Adapteva#Products" class="mw-redirect" title="Adapteva">Epiphany architecture</a></li> <li><a href="/wiki/Comparison_of_instruction_set_architectures" title="Comparison of instruction set architectures">Others</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Instruction_cycle" title="Instruction cycle">Execution</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Instruction_pipelining" title="Instruction pipelining">Instruction pipelining</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Pipeline_stall" title="Pipeline stall">Pipeline stall</a></li> <li><a href="/wiki/Operand_forwarding" title="Operand forwarding">Operand forwarding</a></li> <li><a href="/wiki/Classic_RISC_pipeline" title="Classic RISC pipeline">Classic RISC pipeline</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Hazard_(computer_architecture)" title="Hazard (computer architecture)">Hazards</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Data_dependency" title="Data dependency">Data dependency</a></li> <li><a href="/wiki/Structural_hazard" class="mw-redirect" title="Structural hazard">Structural</a></li> <li><a href="/wiki/Control_hazard" class="mw-redirect" title="Control hazard">Control</a></li> <li><a href="/wiki/False_sharing" title="False sharing">False sharing</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Out-of-order_execution" title="Out-of-order execution">Out-of-order</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Scoreboarding" title="Scoreboarding">Scoreboarding</a></li> <li><a href="/wiki/Tomasulo%27s_algorithm" title="Tomasulo's algorithm">Tomasulo's algorithm</a> <ul><li><a href="/wiki/Reservation_station" title="Reservation station">Reservation station</a></li> <li><a href="/wiki/Re-order_buffer" title="Re-order buffer">Re-order buffer</a></li></ul></li> <li><a href="/wiki/Register_renaming" title="Register renaming">Register renaming</a></li> <li><a href="/wiki/Wide-issue" title="Wide-issue">Wide-issue</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Speculative_execution" title="Speculative execution">Speculative</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Branch_predictor" title="Branch predictor">Branch prediction</a></li> <li><a href="/wiki/Memory_dependence_prediction" title="Memory dependence prediction">Memory dependence prediction</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Parallel_computing" title="Parallel computing">Parallelism</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Level</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bit-level_parallelism" title="Bit-level parallelism">Bit</a> <ul><li><a href="/wiki/Bit-serial_architecture" title="Bit-serial architecture">Bit-serial</a></li> <li><a href="/wiki/Word_(computer_architecture)" title="Word (computer architecture)">Word</a></li></ul></li> <li><a href="/wiki/Instruction-level_parallelism" title="Instruction-level parallelism">Instruction</a></li> <li><a href="/wiki/Instruction_pipelining" title="Instruction pipelining">Pipelining</a> <ul><li><a href="/wiki/Scalar_processor" title="Scalar processor">Scalar</a></li> <li><a href="/wiki/Superscalar_processor" title="Superscalar processor">Superscalar</a></li></ul></li> <li><a href="/wiki/Task_parallelism" title="Task parallelism">Task</a> <ul><li><a href="/wiki/Thread_(computing)" title="Thread (computing)">Thread</a></li> <li><a href="/wiki/Process_(computing)" title="Process (computing)">Process</a></li></ul></li> <li><a href="/wiki/Data_parallelism" title="Data parallelism">Data</a> <ul><li><a href="/wiki/Vector_processor" title="Vector processor">Vector</a></li></ul></li> <li><a href="/wiki/Memory-level_parallelism" title="Memory-level parallelism">Memory</a></li> <li><a href="/wiki/Distributed_architecture" class="mw-redirect" title="Distributed architecture">Distributed</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Multithreading_(computer_architecture)" title="Multithreading (computer architecture)">Multithreading</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Temporal_multithreading" title="Temporal multithreading">Temporal</a></li> <li><a href="/wiki/Simultaneous_multithreading" title="Simultaneous multithreading">Simultaneous</a> <ul><li><a href="/wiki/Hyper-threading" title="Hyper-threading">Hyperthreading</a></li> <li><a href="/wiki/Simultaneous_and_heterogeneous_multithreading" title="Simultaneous and heterogeneous multithreading">Simultaneous and heterogenous</a></li></ul></li> <li><a href="/wiki/Speculative_multithreading" title="Speculative multithreading">Speculative</a></li> <li><a href="/wiki/Preemption_(computing)" title="Preemption (computing)">Preemptive</a></li> <li><a href="/wiki/Cooperative_multitasking" title="Cooperative multitasking">Cooperative</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Flynn%27s_taxonomy" title="Flynn's taxonomy">Flynn's taxonomy</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Single_instruction,_single_data" title="Single instruction, single data">SISD</a></li> <li><a href="/wiki/Single_instruction,_multiple_data" title="Single instruction, multiple data">SIMD</a> <ul><li><a href="/wiki/Single_instruction,_multiple_threads" title="Single instruction, multiple threads">Array processing (SIMT)</a></li> <li><a href="/wiki/Flynn%27s_taxonomy#Pipelined_processor" title="Flynn's taxonomy">Pipelined processing</a></li> <li><a href="/wiki/Flynn%27s_taxonomy#Associative_processor" title="Flynn's taxonomy">Associative processing</a></li> <li><a href="/wiki/SWAR" title="SWAR">SWAR</a></li></ul></li> <li><a href="/wiki/Multiple_instruction,_single_data" title="Multiple instruction, single data">MISD</a></li> <li><a href="/wiki/Multiple_instruction,_multiple_data" title="Multiple instruction, multiple data">MIMD</a> <ul><li><a href="/wiki/Single_program,_multiple_data" title="Single program, multiple data">SPMD</a></li></ul></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Computer_performance" title="Computer performance">Processor<br />performance</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Transistor_count" title="Transistor count">Transistor count</a></li> <li><a href="/wiki/Instructions_per_cycle" title="Instructions per cycle">Instructions per cycle</a> (IPC) <ul><li><a href="/wiki/Cycles_per_instruction" title="Cycles per instruction">Cycles per instruction</a> (CPI)</li></ul></li> <li><a href="/wiki/Instructions_per_second" title="Instructions per second">Instructions per second</a> (IPS)</li> <li><a href="/wiki/FLOPS" class="mw-redirect" title="FLOPS">Floating-point operations per second</a> (FLOPS)</li> <li><a href="/wiki/Transactions_per_second" title="Transactions per second">Transactions per second</a> (TPS)</li> <li><a href="/wiki/SUPS" title="SUPS">Synaptic updates per second</a> (SUPS)</li> <li><a href="/wiki/Performance_per_watt" title="Performance per watt">Performance per watt</a> (PPW)</li> <li><a href="/wiki/Cache_performance_measurement_and_metric" title="Cache performance measurement and metric">Cache performance metrics</a></li> <li><a href="/wiki/Computer_performance_by_orders_of_magnitude" title="Computer performance by orders of magnitude">Computer performance by orders of magnitude</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Processor_(computing)" title="Processor (computing)">Types</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Central_processing_unit" title="Central processing unit">Central processing unit</a> (CPU)</li> <li><a href="/wiki/Graphics_processing_unit" title="Graphics processing unit">Graphics processing unit</a> (GPU) <ul><li><a href="/wiki/General-purpose_computing_on_graphics_processing_units" title="General-purpose computing on graphics processing units">GPGPU</a></li></ul></li> <li><a href="/wiki/Vector_processor" title="Vector processor">Vector</a></li> <li><a href="/wiki/Barrel_processor" title="Barrel processor">Barrel</a></li> <li><a href="/wiki/Stream_processing" title="Stream processing">Stream</a></li> <li><a href="/wiki/Tile_processor" title="Tile processor">Tile processor</a></li> <li><a href="/wiki/Coprocessor" title="Coprocessor">Coprocessor</a></li> <li><a href="/wiki/Programmable_Array_Logic" title="Programmable Array Logic">PAL</a></li> <li><a href="/wiki/Application-specific_integrated_circuit" title="Application-specific integrated circuit">ASIC</a></li> <li><a href="/wiki/Field-programmable_gate_array" title="Field-programmable gate array">FPGA</a></li> <li><a href="/wiki/Field-programmable_object_array" title="Field-programmable object array">FPOA</a></li> <li><a href="/wiki/Complex_programmable_logic_device" title="Complex programmable logic device">CPLD</a></li> <li><a href="/wiki/Multi-chip_module" title="Multi-chip module">Multi-chip module</a> (MCM)</li> <li><a href="/wiki/System_in_a_package" title="System in a package">System in a package</a> (SiP)</li> <li><a href="/wiki/Package_on_a_package" title="Package on a package">Package on a package</a> (PoP)</li></ul> </div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">By application</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Embedded_system" title="Embedded system">Embedded system</a></li> <li><a href="/wiki/Microprocessor" title="Microprocessor">Microprocessor</a></li> <li><a href="/wiki/Microcontroller" title="Microcontroller">Microcontroller</a></li> <li><a href="/wiki/Mobile_processor" title="Mobile processor">Mobile</a></li> <li><a href="/wiki/Ultra-low-voltage_processor" title="Ultra-low-voltage processor">Ultra-low-voltage</a></li> <li><a href="/wiki/Application-specific_instruction_set_processor" title="Application-specific instruction set processor">ASIP</a></li> <li><a href="/wiki/Soft_microprocessor" title="Soft microprocessor">Soft microprocessor</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Systems<br />on chip</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/System_on_a_chip" title="System on a chip">System on a chip</a> (SoC)</li> <li><a href="/wiki/Multiprocessor_system_on_a_chip" class="mw-redirect" title="Multiprocessor system on a chip">Multiprocessor</a> (MPSoC)</li> <li><a href="/wiki/Cypress_PSoC" title="Cypress PSoC">Cypress PSoC</a></li> <li><a href="/wiki/Network_on_a_chip" title="Network on a chip">Network on a chip</a> (NoC)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Hardware_acceleration" title="Hardware acceleration">Hardware<br />accelerators</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Coprocessor" title="Coprocessor">Coprocessor</a></li> <li><a href="/wiki/AI_accelerator" title="AI accelerator">AI accelerator</a></li> <li><a href="/wiki/Graphics_processing_unit" title="Graphics processing unit">Graphics processing unit</a> (GPU)</li> <li><a href="/wiki/Image_processor" title="Image processor">Image processor</a></li> <li><a href="/wiki/Vision_processing_unit" title="Vision processing unit">Vision processing unit</a> (VPU)</li> <li><a href="/wiki/Physics_processing_unit" title="Physics processing unit">Physics processing unit</a> (PPU)</li> <li><a href="/wiki/Digital_signal_processor" title="Digital signal processor">Digital signal processor</a> (DSP)</li> <li><a href="/wiki/Tensor_Processing_Unit" title="Tensor Processing Unit">Tensor Processing Unit</a> (TPU)</li> <li><a href="/wiki/Secure_cryptoprocessor" title="Secure cryptoprocessor">Secure cryptoprocessor</a></li> <li><a href="/wiki/Network_processor" title="Network processor">Network processor</a></li> <li><a href="/wiki/Baseband_processor" title="Baseband processor">Baseband processor</a></li></ul> </div></td></tr></tbody></table><div> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Word_(computer_architecture)" title="Word (computer architecture)">Word size</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/1-bit_computing" title="1-bit computing">1-bit</a></li> <li><a href="/wiki/4-bit_computing" title="4-bit computing">4-bit</a></li> <li><a href="/wiki/8-bit_computing" title="8-bit computing">8-bit</a></li> <li><a href="/wiki/12-bit_computing" title="12-bit computing">12-bit</a></li> <li><a href="/wiki/Apollo_Guidance_Computer" title="Apollo Guidance Computer">15-bit</a></li> <li><a href="/wiki/16-bit_computing" title="16-bit computing">16-bit</a></li> <li><a href="/wiki/24-bit_computing" title="24-bit computing">24-bit</a></li> <li><a href="/wiki/32-bit_computing" title="32-bit computing">32-bit</a></li> <li><a href="/wiki/48-bit_computing" title="48-bit computing">48-bit</a></li> <li><a href="/wiki/64-bit_computing" title="64-bit computing">64-bit</a></li> <li><a href="/wiki/128-bit_computing" title="128-bit computing">128-bit</a></li> <li><a href="/wiki/256-bit_computing" title="256-bit computing">256-bit</a></li> <li><a href="/wiki/512-bit_computing" title="512-bit computing">512-bit</a></li> <li><a href="/wiki/Bit_slicing" title="Bit slicing">bit slicing</a></li> <li><a href="/wiki/Word_(computer_architecture)#Table_of_word_sizes" title="Word (computer architecture)">others</a> <ul><li><a href="/wiki/Word_(computer_architecture)#Variable-word_architectures" title="Word (computer architecture)">variable</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Core count</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Single-core" title="Single-core">Single-core</a></li> <li><a href="/wiki/Multi-core_processor" title="Multi-core processor">Multi-core</a></li> <li><a href="/wiki/Manycore_processor" title="Manycore processor">Manycore</a></li> <li><a href="/wiki/Heterogeneous_computing" title="Heterogeneous computing">Heterogeneous architecture</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Components</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Central_processing_unit" title="Central processing unit">Core</a></li> <li><a href="/wiki/Cache_(computing)" title="Cache (computing)">Cache</a> <ul><li><a href="/wiki/CPU_cache" title="CPU cache">CPU cache</a></li> <li><a href="/wiki/Scratchpad_memory" title="Scratchpad memory">Scratchpad memory</a></li> <li><a href="/wiki/Data_cache" class="mw-redirect" title="Data cache">Data cache</a></li> <li><a href="/wiki/Instruction_cache" class="mw-redirect" title="Instruction cache">Instruction cache</a></li> <li><a href="/wiki/Cache_replacement_policies" title="Cache replacement policies">replacement policies</a></li> <li><a href="/wiki/Cache_coherence" title="Cache coherence">coherence</a></li></ul></li> <li><a href="/wiki/Bus_(computing)" title="Bus (computing)">Bus</a></li> <li><a href="/wiki/Clock_rate" title="Clock rate">Clock rate</a></li> <li><a href="/wiki/Clock_signal" title="Clock signal">Clock signal</a></li> <li><a href="/wiki/FIFO_(computing_and_electronics)" title="FIFO (computing and electronics)">FIFO</a></li></ul> </div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Execution_unit" title="Execution unit">Functional<br />units</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Arithmetic_logic_unit" title="Arithmetic logic unit">Arithmetic logic unit</a> (ALU)</li> <li><a href="/wiki/Address_generation_unit" title="Address generation unit">Address generation unit</a> (AGU)</li> <li><a href="/wiki/Floating-point_unit" title="Floating-point unit">Floating-point unit</a> (FPU)</li> <li><a href="/wiki/Memory_management_unit" title="Memory management unit">Memory management unit</a> (MMU) <ul><li><a href="/wiki/Load%E2%80%93store_unit" title="Load–store unit">Load–store unit</a></li> <li><a href="/wiki/Translation_lookaside_buffer" title="Translation lookaside buffer">Translation lookaside buffer</a> (TLB)</li></ul></li> <li><a href="/wiki/Branch_predictor" title="Branch predictor">Branch predictor</a></li> <li><a href="/wiki/Branch_target_predictor" title="Branch target predictor">Branch target predictor</a></li> <li><a href="/wiki/Memory_controller" title="Memory controller">Integrated memory controller</a> (IMC) <ul><li><a href="/wiki/Memory_management_unit" title="Memory management unit">Memory management unit</a></li></ul></li> <li><a href="/wiki/Instruction_decoder" class="mw-redirect" title="Instruction decoder">Instruction decoder</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Logic_gate" title="Logic gate">Logic</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Combinational_logic" title="Combinational logic">Combinational</a></li> <li><a href="/wiki/Sequential_logic" title="Sequential logic">Sequential</a></li> <li><a href="/wiki/Glue_logic" title="Glue logic">Glue</a></li> <li><a href="/wiki/Logic_gate" title="Logic gate">Logic gate</a> <ul><li><a href="/wiki/Quantum_logic_gate" title="Quantum logic gate">Quantum</a></li> <li><a href="/wiki/Gate_array" title="Gate array">Array</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Hardware_register" title="Hardware register">Registers</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Processor_register" title="Processor register">Processor register</a></li> <li><a href="/wiki/Status_register" title="Status register">Status register</a></li> <li><a href="/wiki/Stack_register" title="Stack register">Stack register</a></li> <li><a href="/wiki/Register_file" title="Register file">Register file</a></li> <li><a href="/wiki/Memory_buffer_register" title="Memory buffer register">Memory buffer</a></li> <li><a href="/wiki/Memory_address_register" title="Memory address register">Memory address register</a></li> <li><a href="/wiki/Program_counter" title="Program counter">Program counter</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Control_unit" title="Control unit">Control unit</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Hardwired_control_unit" class="mw-redirect" title="Hardwired control unit">Hardwired control unit</a></li> <li><a href="/wiki/Instruction_unit" title="Instruction unit">Instruction unit</a></li> <li><a href="/wiki/Data_buffer" title="Data buffer">Data buffer</a></li> <li><a href="/wiki/Write_buffer" title="Write buffer">Write buffer</a></li> <li><a href="/wiki/Microcode" title="Microcode">Microcode</a> <a href="/wiki/ROM_image" title="ROM image">ROM</a></li> <li><a href="/wiki/Counter_(digital)" title="Counter (digital)">Counter</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Datapath" title="Datapath">Datapath</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Multiplexer" title="Multiplexer">Multiplexer</a></li> <li><a href="/wiki/Demultiplexer" class="mw-redirect" title="Demultiplexer">Demultiplexer</a></li> <li><a class="mw-selflink selflink">Adder</a></li> <li><a href="/wiki/Binary_multiplier" title="Binary multiplier">Multiplier</a> <ul><li><a href="/wiki/CPU_multiplier" title="CPU multiplier">CPU</a></li></ul></li> <li><a href="/wiki/Binary_decoder" title="Binary decoder">Binary decoder</a> <ul><li><a href="/wiki/Address_decoder" title="Address decoder">Address decoder</a></li> <li><a href="/wiki/Sum-addressed_decoder" title="Sum-addressed decoder">Sum-addressed decoder</a></li></ul></li> <li><a href="/wiki/Barrel_shifter" title="Barrel shifter">Barrel shifter</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Electronic_circuit" title="Electronic circuit">Circuitry</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Integrated_circuit" title="Integrated circuit">Integrated circuit</a> <ul><li><a href="/wiki/Three-dimensional_integrated_circuit" title="Three-dimensional integrated circuit">3D</a></li> <li><a href="/wiki/Mixed-signal_integrated_circuit" title="Mixed-signal integrated circuit">Mixed-signal</a></li> <li><a href="/wiki/Power_management_integrated_circuit" title="Power management integrated circuit">Power management</a></li></ul></li> <li><a href="/wiki/Boolean_circuit" title="Boolean circuit">Boolean</a></li> <li><a href="/wiki/Circuit_(computer_science)" title="Circuit (computer science)">Digital</a></li> <li><a href="/wiki/Analogue_electronics" title="Analogue electronics">Analog</a></li> <li><a href="/wiki/Quantum_circuit" title="Quantum circuit">Quantum</a></li> <li><a href="/wiki/Switch#Electronic_switches" title="Switch">Switch</a></li></ul> </div></td></tr></tbody></table><div> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Power_management" title="Power management">Power<br />management</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Power_Management_Unit" title="Power Management Unit">PMU</a></li> <li><a href="/wiki/Advanced_Power_Management" title="Advanced Power Management">APM</a></li> <li><a href="/wiki/ACPI" title="ACPI">ACPI</a></li> <li><a href="/wiki/Dynamic_frequency_scaling" title="Dynamic frequency scaling">Dynamic frequency scaling</a></li> <li><a href="/wiki/Dynamic_voltage_scaling" title="Dynamic voltage scaling">Dynamic voltage scaling</a></li> <li><a href="/wiki/Clock_gating" title="Clock gating">Clock gating</a></li> <li><a href="/wiki/Performance_per_watt" title="Performance per watt">Performance per watt</a> (PPW)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/History_of_general-purpose_CPUs" title="History of general-purpose CPUs">History of general-purpose CPUs</a></li> <li><a href="/wiki/Microprocessor_chronology" title="Microprocessor chronology">Microprocessor chronology</a></li> <li><a href="/wiki/Processor_design" title="Processor design">Processor design</a></li> <li><a href="/wiki/Digital_electronics" title="Digital electronics">Digital electronics</a></li> <li><a href="/wiki/Hardware_security_module" title="Hardware security module">Hardware security module</a></li> <li><a href="/wiki/Semiconductor_device_fabrication" title="Semiconductor device fabrication">Semiconductor device fabrication</a></li> <li><a href="/wiki/Tick%E2%80%93tock_model" title="Tick–tock model">Tick–tock model</a></li> <li><a href="/wiki/Pin_grid_array" title="Pin grid array">Pin grid array</a></li> <li><a href="/wiki/Chip_carrier" title="Chip carrier">Chip carrier</a></li></ul> </div></td></tr></tbody></table></div> <style data-mw-deduplicate="TemplateStyles:r1130092004">.mw-parser-output .portal-bar{font-size:88%;font-weight:bold;display:flex;justify-content:center;align-items:baseline}.mw-parser-output .portal-bar-bordered{padding:0 2em;background-color:#fdfdfd;border:1px solid #a2a9b1;clear:both;margin:1em auto 0}.mw-parser-output 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