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class="firstHeading mw-first-heading"><span class="mw-page-title-main">Ancient Egyptian multiplication</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 14 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-14" 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">14 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%A7%D9%84%D8%AC%D8%AF%D8%A7%D8%A1_%D8%A8%D8%A7%D8%B3%D8%AA%D8%B9%D9%85%D8%A7%D9%84_%D8%A7%D9%84%D8%B7%D8%B1%D9%8A%D9%82%D8%A9_%D8%A7%D9%84%D9%85%D8%B5%D8%B1%D9%8A%D8%A9_%D8%A7%D9%84%D9%82%D8%AF%D9%8A%D9%85%D8%A9" title="الجداء باستعمال الطريقة المصرية القديمة – Arabic" lang="ar" hreflang="ar" data-title="الجداء باستعمال الطريقة المصرية القديمة" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Multiplicaci%C3%B3n_por_duplicaci%C3%B3n" title="Multiplicación por duplicación – Asturian" lang="ast" hreflang="ast" data-title="Multiplicación por duplicación" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Multiplicaci%C3%B3_per_duplicaci%C3%B3" title="Multiplicació per duplicació – Catalan" lang="ca" hreflang="ca" data-title="Multiplicació per duplicació" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Russische_Bauernmultiplikation" title="Russische Bauernmultiplikation – German" lang="de" hreflang="de" data-title="Russische Bauernmultiplikation" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A0%CE%BF%CE%BB%CE%BB%CE%B1%CF%80%CE%BB%CE%B1%CF%83%CE%B9%CE%B1%CF%83%CE%BC%CF%8C%CF%82_%CF%84%CF%89%CE%BD_%CF%87%CF%89%CF%81%CE%B9%CE%BA%CF%8E%CE%BD" title="Πολλαπλασιασμός των χωρικών – Greek" lang="el" hreflang="el" data-title="Πολλαπλασιασμός των χωρικών" 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/Multiplicaci%C3%B3n_por_duplicaci%C3%B3n" title="Multiplicación por duplicación – Spanish" lang="es" hreflang="es" data-title="Multiplicación por duplicación" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Technique_de_multiplication_dite_russe" title="Technique de multiplication dite russe – French" lang="fr" hreflang="fr" data-title="Technique de multiplication dite russe" 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%B3%A0%EB%8C%80_%EC%9D%B4%EC%A7%91%ED%8A%B8_%EA%B3%B1%EC%85%88%EB%B2%95" title="고대 이집트 곱셈법 – Korean" lang="ko" hreflang="ko" data-title="고대 이집트 곱셈법" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%8E%E0%B4%A4%E0%B5%8D%E0%B4%AF%E0%B5%8B%E0%B4%AA%E0%B5%8D%E0%B4%AF%E0%B5%BB_%E0%B4%97%E0%B5%81%E0%B4%A3%E0%B4%BF%E0%B4%A4%E0%B4%82" title="എത്യോപ്യൻ ഗുണിതം – Malayalam" lang="ml" hreflang="ml" data-title="എത്യോപ്യൻ ഗുണിതം" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Multiplica%C3%A7%C3%A3o_por_duplica%C3%A7%C3%A3o" title="Multiplicação por duplicação – Portuguese" lang="pt" hreflang="pt" data-title="Multiplicação por duplicação" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%94%D1%80%D0%B5%D0%B2%D0%BD%D0%B5%D0%B5%D0%B3%D0%B8%D0%BF%D0%B5%D1%82%D1%81%D0%BA%D0%BE%D0%B5_%D1%83%D0%BC%D0%BD%D0%BE%D0%B6%D0%B5%D0%BD%D0%B8%D0%B5" title="Древнеегипетское умножение – Russian" lang="ru" hreflang="ru" data-title="Древнеегипетское умножение" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B6%B4%E0%B7%94%E0%B6%BB%E0%B7%8F%E0%B6%AD%E0%B6%B1_%E0%B6%8A%E0%B6%A2%E0%B7%92%E0%B6%B4%E0%B7%8A%E0%B6%AD%E0%B7%92%E0%B6%BA%E0%B7%8F%E0%B6%B1%E0%B7%94_%E0%B6%9C%E0%B7%94%E0%B6%AB_%E0%B6%9A%E0%B7%92%E0%B6%BB%E0%B7%93%E0%B6%B8" title="පුරාතන ඊජිප්තියානු ගුණ කිරීම – Sinhala" lang="si" hreflang="si" data-title="පුරාතන ඊජිප්තියානු ගුණ කිරීම" data-language-autonym="සිංහල" data-language-local-name="Sinhala" class="interlanguage-link-target"><span>සිංහල</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%95%D0%B3%D0%B8%D0%BF%D0%B0%D1%82%D1%81%D0%BA%D0%BE_%D0%BC%D0%BD%D0%BE%D0%B6%D0%B5%D1%9A%D0%B5" title="Египатско множење – Serbian" lang="sr" hreflang="sr" data-title="Египатско множење" 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-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%84%D0%B3%D0%B8%D0%BF%D0%B5%D1%82%D1%81%D1%8C%D0%BA%D0%B8%D0%B9_%D0%BC%D0%B5%D1%82%D0%BE%D0%B4_%D0%BC%D0%BD%D0%BE%D0%B6%D0%B5%D0%BD%D0%BD%D1%8F" title="Єгипетський метод множення – Ukrainian" lang="uk" hreflang="uk" data-title="Єгипетський метод множення" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span 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<div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Multiplication algorithm</div> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, <b>ancient Egyptian multiplication</b> (also known as <b>Egyptian multiplication</b>, <b>Ethiopian multiplication</b>, <b>Russian multiplication</b>, or <b>peasant multiplication</b>), one of two <a href="/wiki/Multiplication" title="Multiplication">multiplication</a> methods used by scribes, is a systematic method for multiplying two numbers that does not require the <a href="/wiki/Multiplication_table" title="Multiplication table">multiplication table</a>, only the ability to multiply and <a href="/wiki/Division_by_2" class="mw-redirect" title="Division by 2">divide by 2</a>, and to <a href="/wiki/Addition" title="Addition">add</a>. It decomposes one of the <a href="/wiki/Multiplicand" class="mw-redirect" title="Multiplicand">multiplicands</a> (preferably the smaller) into a set of numbers of <a href="/wiki/Powers_of_two" class="mw-redirect" title="Powers of two">powers of two</a> and then creates a table of doublings of the second multiplicand by every value of the set which is summed up to give result of multiplication. </p><p>This method may be called <b>mediation and duplation</b>, where <a href="/wiki/Division_by_two" title="Division by two">mediation</a> means halving one number and duplation means doubling the other number. It is still used in some areas.<sup id="cite_ref-neugebaue_1-0" class="reference"><a href="#cite_note-neugebaue-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>The second Egyptian multiplication and division technique was known from the <a href="/wiki/Hieratic" title="Hieratic">hieratic</a> <a href="/wiki/Moscow_Mathematical_Papyrus" title="Moscow Mathematical Papyrus">Moscow</a> and <a href="/wiki/Rhind_Mathematical_Papyrus" title="Rhind Mathematical Papyrus">Rhind Mathematical Papyri</a> written in the seventeenth century B.C. by the scribe <a href="/wiki/Ahmes" title="Ahmes">Ahmes</a>.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>Although in ancient Egypt the concept of <a href="/wiki/Binary_number" title="Binary number">base 2</a> did not exist, the algorithm is essentially the same algorithm as <a href="/wiki/Long_multiplication" class="mw-redirect" title="Long multiplication">long multiplication</a> after the multiplier and multiplicand are converted to <a href="/wiki/Binary_number" title="Binary number">binary</a>. The method as interpreted by conversion to binary is therefore still in wide use today as implemented by <a href="/wiki/Binary_multiplier" title="Binary multiplier">binary multiplier circuits</a> in modern computer processors.<sup id="cite_ref-neugebaue_1-1" class="reference"><a href="#cite_note-neugebaue-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Method">Method</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ancient_Egyptian_multiplication&action=edit&section=1" title="Edit section: Method"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Ancient_Egypt" title="Ancient Egypt">ancient Egyptians</a> had laid out tables of a great number of powers of two, rather than recalculating them each time. The decomposition of a number thus consists of finding the powers of two which make it up. The Egyptians knew empirically that a given power of two would only appear once in a number. For the decomposition, they proceeded methodically; they would initially find the largest power of two less than or equal to the number in question, <a href="/wiki/Subtraction" title="Subtraction">subtract</a> it out and repeat until nothing remained. (The Egyptians did not make use of the number <a href="/wiki/Zero" class="mw-redirect" title="Zero">zero</a> in mathematics.) </p><p>After the decomposition of the first multiplicand, the person would construct a table of powers of two times the second multiplicand (generally the smaller) from one up to the largest power of two found during the decomposition. </p><p>The result is obtained by adding the numbers from the second column for which the corresponding power of two makes up part of the decomposition of the first multiplicand.<sup id="cite_ref-neugebaue_1-2" class="reference"><a href="#cite_note-neugebaue-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Example">Example</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ancient_Egyptian_multiplication&action=edit&section=2" title="Edit section: Example"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>25 × 7 = ? </p><p>Decomposition of the number 25: </p> <dl><dd><table> <tbody><tr> <td>The largest power of two less than or equal to 25</td> <td>is 16:</td> <td style="text-align:center;"><b>25 − 16</b></td> <td><b>= 9</b>. </td></tr> <tr> <td>The largest power of two less than or equal to 9</td> <td>is 8:</td> <td style="text-align:center;"><b>9 − 8</b></td> <td><b>= 1</b>. </td></tr> <tr> <td>The largest power of two less than or equal to 1</td> <td>is 1:</td> <td style="text-align:center;"><b>1 − 1</b></td> <td><b>= 0</b>. </td></tr> <tr> <td colspan="3">25 is thus the sum of: 16, 8 and 1. </td></tr></tbody></table></dd></dl> <p>The largest power of two is 16 and the second multiplicand is 7. </p> <table style="text-align:right;width:8em;"> <tbody><tr> <td>1</td> <td>7 </td></tr> <tr> <td>2</td> <td>14 </td></tr> <tr> <td>4</td> <td>28 </td></tr> <tr> <td>8</td> <td>56 </td></tr> <tr> <td>16</td> <td>112 </td></tr></tbody></table> <p>As 25 = 16 + 8 + 1, the corresponding multiples of 7 are added to get 25 × 7 = 112 + 56 + 7 = 175. </p> <div class="mw-heading mw-heading2"><h2 id="Russian_peasant_multiplication">Russian peasant multiplication</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ancient_Egyptian_multiplication&action=edit&section=3" title="Edit section: Russian peasant multiplication"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In the Russian peasant method, the powers of two in the decomposition of the multiplicand are found by writing it on the left and progressively halving the left column, discarding any remainder, until the value is 1 (or −1, in which case the eventual sum is negated), while doubling the right column as before. Lines with <a href="/wiki/Parity_(mathematics)" title="Parity (mathematics)">even</a> numbers on the left column are struck out, and the remaining numbers on the right are added together.<sup id="cite_ref-peasant_3-0" class="reference"><a href="#cite_note-peasant-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Example_2">Example</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ancient_Egyptian_multiplication&action=edit&section=4" title="Edit section: Example"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>238 × 13 = ? </p> <table style="text-align:right;border-collapse:collapse;"> <tbody><tr> <td><b>13</b></td> <td></td> <td><b>238</b></td> <td> </td></tr> <tr> <td><b>6</b></td> <td>  (remainder discarded)</td> <td><b>476</b></td> <td> </td></tr> <tr> <td><b>3</b></td> <td></td> <td><b>952</b></td> <td> </td></tr> <tr> <td><b>1</b></td> <td>  (remainder discarded)</td> <td><b>1904</b></td> <td> </td></tr> <tr> <td style="width:4em;">  </td> <td>  </td> <td style="width:4em;">  </td></tr></tbody></table> <table style="text-align:right;border-collapse:collapse;"> <tbody><tr> <td><b>13</b></td> <td><b>238</b> </td></tr> <tr> <td><s>6</s></td> <td><s>476</s> </td></tr> <tr> <td><b>3</b></td> <td><b>952</b> </td></tr> <tr> <td><b>1</b></td> <td>+<b>1904</b> </td></tr> <tr> <td></td> <td><hr /> </td></tr> <tr> <td></td> <td><b>3094</b> </td></tr> <tr> <td style="width:4em;">  </td> <td style="width:4em;">  </td></tr></tbody></table> <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=Ancient_Egyptian_multiplication&action=edit&section=5" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Egyptian_fraction" title="Egyptian fraction">Egyptian fraction</a></li> <li><a href="/wiki/Egyptian_mathematics" class="mw-redirect" title="Egyptian mathematics">Egyptian mathematics</a></li> <li><a href="/wiki/Multiplication_algorithm" title="Multiplication algorithm">Multiplication algorithms</a></li> <li><a href="/wiki/Binary_numeral_system" class="mw-redirect" title="Binary numeral system">Binary numeral system</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ancient_Egyptian_multiplication&action=edit&section=6" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-neugebaue-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-neugebaue_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-neugebaue_1-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-neugebaue_1-2"><sup><i><b>c</b></i></sup></a></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="CITEREFNeugebauer1969" class="citation book cs1"><a href="/wiki/Otto_E._Neugebauer" title="Otto E. Neugebauer">Neugebauer, Otto</a> (1969) [1957]. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=JVhTtVA2zr8C"><i>The Exact Sciences in Antiquity</i></a> (2 ed.). <a href="/wiki/Dover_Publications" title="Dover Publications">Dover Publications</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-22332-2" title="Special:BookSources/978-0-486-22332-2"><bdi>978-0-486-22332-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Exact+Sciences+in+Antiquity&rft.edition=2&rft.pub=Dover+Publications&rft.date=1969&rft.isbn=978-0-486-22332-2&rft.aulast=Neugebauer&rft.aufirst=Otto&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DJVhTtVA2zr8C&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAncient+Egyptian+multiplication" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><a href="/wiki/Battiscombe_Gunn" title="Battiscombe Gunn">Gunn, Battiscombe George</a>. Review of The Rhind Mathematical Papyrus by T. E. Peet. The Journal of Egyptian Archaeology 12 London, (1926): 123–137.</span> </li> <li id="cite_note-peasant-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-peasant_3-0">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.cut-the-knot.org/Curriculum/Algebra/PeasantMultiplication.shtml">Cut the Knot - Peasant Multiplication</a></span> </li> </ol></div> <div class="mw-heading mw-heading3"><h3 id="Other_sources">Other sources</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ancient_Egyptian_multiplication&action=edit&section=7" title="Edit section: Other sources"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Boyer, Carl B. (1968) A History of Mathematics. New York: John Wiley.</li> <li>Brown, Kevin S. (1995) The Akhmin Papyrus 1995 --- Egyptian Unit Fractions.</li> <li>Bruckheimer, Maxim, and Y. Salomon (1977) "Some Comments on R. J. Gillings' Analysis of the 2/n Table in the Rhind Papyrus," Historia Mathematica 4: 445–52.</li> <li>Bruins, Evert M. (1953) Fontes matheseos: hoofdpunten van het prae-Griekse en Griekse wiskundig denken. Leiden: E. J. Brill.</li> <li>------- (1957) "Platon et la table égyptienne 2/n," Janus 46: 253–63.</li> <li>Bruins, Evert M (1981) "Egyptian Arithmetic," Janus 68: 33–52.</li> <li>------- (1981) "Reducible and Trivial Decompositions Concerning Egyptian Arithmetics," Janus 68: 281–97.</li> <li>Burton, David M. (2003) History of Mathematics: An Introduction. Boston Wm. C. Brown.</li> <li>Chace, Arnold Buffum, et al. (1927) The Rhind Mathematical Papyrus. Oberlin: Mathematical Association of America.</li> <li>Cooke, Roger (1997) The History of Mathematics. A Brief Course. New York, John Wiley & Sons.</li> <li>Couchoud, Sylvia. "Mathématiques égyptiennes". Recherches sur les connaissances mathématiques de l'Egypte pharaonique., Paris, Le Léopard d'Or, 1993.</li> <li>Daressy, Georges. "Akhmim Wood Tablets", Le Caire Imprimerie de l'Institut Francais d'Archeologie Orientale, 1901, 95–96.</li> <li>Eves, Howard (1961) An Introduction to the History of Mathematics. New York, Holt, Rinehard & Winston.</li> <li>Fowler, David H. (1999) The mathematics of Plato's Academy: a new reconstruction. Oxford Univ. Press.</li> <li>Gardiner, Alan H. (1957) Egyptian Grammar being an Introduction to the Study of Hieroglyphs. Oxford University Press.</li> <li>Gardner, Milo (2002) "The Egyptian Mathematical Leather Roll, Attested Short Term and Long Term" in History of the Mathematical Sciences, Ivor Grattan-Guinness, B.C. Yadav (eds), New Delhi, Hindustan Book Agency:119-34.</li> <li>-------- "Mathematical Roll of Egypt" in Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures. Springer, Nov. 2005.</li> <li>Gillings, Richard J. (1962) "The Egyptian Mathematical Leather Roll," Australian Journal of Science 24: 339–44. Reprinted in his (1972) Mathematics in the Time of the Pharaohs. MIT Press. Reprinted by Dover Publications, 1982.</li> <li>-------- (1974) "The Recto of the Rhind Mathematical Papyrus: How Did the Ancient Egyptian Scribe Prepare It?" Archive for History of Exact Sciences 12: 291–98.</li> <li>-------- (1979) "The Recto of the RMP and the EMLR," Historia Mathematica, Toronto 6 (1979), 442–447.</li> <li>-------- (1981) "The Egyptian Mathematical Leather Role–Line 8. How Did the Scribe Do it?" Historia Mathematica: 456–57.</li> <li>Glanville, S.R.K. "The Mathematical Leather Roll in the British Museum" Journal of Egyptian Archaeology 13, London (1927): 232–8</li> <li>Griffith, Francis Llewelyn. The Petrie Papyri. Hieratic Papyri from Kahun and Gurob (Principally of the Middle Kingdom), Vols. 1, 2. Bernard Quaritch, London, 1898.</li> <li><a href="/wiki/Battiscombe_Gunn" title="Battiscombe Gunn">Gunn, Battiscombe George</a>. Review of The Rhind Mathematical Papyrus by T. E. Peet. The Journal of Egyptian Archaeology 12 London, (1926): 123–137.</li> <li><a href="/wiki/Friedrich_Hultsch" title="Friedrich Hultsch">Hultsch, F.</a> Die Elemente der Aegyptischen Theihungsrechmun 8, Übersicht über die Lehre von den Zerlegangen, (1895):167-71.</li> <li><a href="/wiki/Annette_Imhausen" title="Annette Imhausen">Imhausen, Annette</a>. "Egyptian Mathematical Texts and their Contexts", Science in Context 16, Cambridge (UK), (2003): 367–389.</li> <li>Joseph, George Gheverghese. The Crest of the Peacock/the non-European Roots of Mathematics, Princeton, Princeton University Press, 2000</li> <li><a href="/wiki/Victor_Klee" title="Victor Klee">Klee, Victor</a>, and <a href="/wiki/Stan_Wagon" title="Stan Wagon">Wagon, Stan</a>. Old and New Unsolved Problems in Plane Geometry and Number Theory, Mathematical Association of America, 1991.</li> <li><a href="/wiki/Wilbur_Knorr" title="Wilbur Knorr">Knorr, Wilbur R.</a> "Techniques of Fractions in Ancient Egypt and Greece". Historia Mathematica 9 Berlin, (1982): 133–171.</li> <li>Legon, John A.R. "A Kahun Mathematical Fragment". Discussions in Egyptology, 24 Oxford, (1992).</li> <li>Lüneburg, H. (1993) "Zerlgung von Bruchen in Stammbruche" Leonardi Pisani Liber Abbaci oder Lesevergnügen eines Mathematikers, Wissenschaftsverlag, Mannheim: 81=85.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNeugebauer1969" class="citation book cs1"><a href="/wiki/Otto_E._Neugebauer" title="Otto E. Neugebauer">Neugebauer, Otto</a> (1969) [1957]. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=JVhTtVA2zr8C"><i>The Exact Sciences in Antiquity</i></a> (2 ed.). <a href="/wiki/Dover_Publications" title="Dover Publications">Dover Publications</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-22332-2" title="Special:BookSources/978-0-486-22332-2"><bdi>978-0-486-22332-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Exact+Sciences+in+Antiquity&rft.edition=2&rft.pub=Dover+Publications&rft.date=1969&rft.isbn=978-0-486-22332-2&rft.aulast=Neugebauer&rft.aufirst=Otto&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DJVhTtVA2zr8C&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAncient+Egyptian+multiplication" class="Z3988"></span></li> <li><a href="/wiki/Gay_Robins" title="Gay Robins">Robins, Gay</a> and <a href="/wiki/Charles_Shute_(academic)" title="Charles Shute (academic)">Charles Shute</a>, <i>The Rhind Mathematical Papyrus: an Ancient Egyptian Text</i>, London, British Museum Press, 1987.</li> <li>Roero, C. S. "Egyptian mathematics" Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences" I. Grattan-Guinness (ed), London, (1994): 30–45.</li> <li>Sarton, George. Introduction to the History of Science, Vol I, New York, Williams & Son, 1927</li> <li>Scott, A. and Hall, H.R., "Laboratory Notes: Egyptian Mathematical Leather Roll of the Seventeenth Century BC", <i><a href="/wiki/British_Museum_Quarterly" title="British Museum Quarterly">British Museum Quarterly</a></i>, Vol 2, London, (1927): 56.</li> <li><a href="/wiki/James_Joseph_Sylvester" title="James Joseph Sylvester">Sylvester, J. J.</a> "On a Point in the Theory of Vulgar Fractions": American Journal of Mathematics, 3 Baltimore (1880): 332–335, 388–389.</li> <li>Vogel, Kurt. "Erweitert die Lederolle unserer Kenntniss ägyptischer Mathematik Archiv für Geschichte der Mathematik, V 2, Julius Schuster, Berlin (1929): 386-407</li> <li>van der Waerden, Bartel Leendert. Science Awakening, New York, 1963</li> <li>Hana Vymazalova, The Wooden Tablets from Cairo:The Use of the Grain Unit HK3T in Ancient Egypt, Archiv Orientalai, Charles U Prague, 2002.</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=Ancient_Egyptian_multiplication&action=edit&section=8" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://rmprectotable.blogspot.com/">RMP 2/n table</a></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20130625181118/http://weekly.ahram.org.eg/2007/844/heritage.htm">The Ahmes code</a></li> <li><a rel="nofollow" class="external text" href="http://emlr.blogspot.com">Egyptian Mathematical Leather Roll</a></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20120913011126/http://planetmath.org/encyclopedia/FirstLCMMethodRedAuxiliaryNumbers.html">The first LCM method Red Auxiliary numbers</a></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20120606142257/http://planetmath.org/encyclopedia/RationalNumbers.html">Egyptian fraction</a></li> <li><a rel="nofollow" class="external text" href="http://mathforum.org/kb/message.jspa?messageID=6579539&tstart=0">Math forum and two ways to calculate 2/7</a></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20100117021426/http://ahmespapyrus.blogspot.com/2009/01/ahmes-papyrus-new-and-old.html">New and Old classifications of Ahmes Papyrus</a></li> <li><a rel="nofollow" class="external text" href="http://mathforum.org/dr.math/faq/faq.peasant.html">Russian Peasant Multiplication</a></li> <li><a rel="nofollow" class="external text" href="http://www.lafstern.org/matt/col3.pdf">The Russian Peasant Algorithm (pdf file)</a></li> <li><a rel="nofollow" class="external text" href="http://www.cut-the-knot.org/Curriculum/Algebra/PeasantMultiplication.shtml">Peasant Multiplication</a> from <a href="/wiki/Cut-the-knot" class="mw-redirect" title="Cut-the-knot">cut-the-knot</a></li> <li><a rel="nofollow" class="external text" href="http://demonstrations.wolfram.com/EgyptianMultiplication/">Egyptian Multiplication</a> by Ken Caviness, <a href="/wiki/The_Wolfram_Demonstrations_Project" class="mw-redirect" title="The Wolfram Demonstrations Project">The Wolfram Demonstrations Project</a>.</li> <li><a rel="nofollow" class="external text" href="http://thedailywtf.com/Articles/Programming-Praxis-Russian-Peasant-Multiplication.aspx">Russian Peasant Multiplication</a> at <a href="/wiki/The_Daily_WTF" title="The Daily WTF">The Daily WTF</a></li> <li><a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=Ih1ZWE3pe9o">Michael S. Schneider explains how the Ancient Egyptians (and Chinese) and modern computers multiply and divide</a></li> <li><a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=HJ_PP5rqLg0&feature=youtu.be">Russian Multiplication - Numberphile</a></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul 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test">Baillie–PSW</a></li> <li><a href="/wiki/Elliptic_curve_primality" title="Elliptic curve primality">Elliptic curve</a></li> <li><a href="/wiki/Pocklington_primality_test" title="Pocklington primality test">Pocklington</a></li> <li><a href="/wiki/Fermat_primality_test" title="Fermat primality test">Fermat</a></li> <li><a href="/wiki/Lucas_primality_test" title="Lucas primality test">Lucas</a></li> <li><i><a href="/wiki/Lucas%E2%80%93Lehmer_primality_test" title="Lucas–Lehmer primality test">Lucas–Lehmer</a></i></li> <li><i><a href="/wiki/Lucas%E2%80%93Lehmer%E2%80%93Riesel_test" title="Lucas–Lehmer–Riesel test">Lucas–Lehmer–Riesel</a></i></li> <li><i><a href="/wiki/Proth%27s_theorem" title="Proth's theorem">Proth's theorem</a></i></li> <li><i><a href="/wiki/P%C3%A9pin%27s_test" title="Pépin's test">Pépin's</a></i></li> <li><a href="/wiki/Quadratic_Frobenius_test" title="Quadratic Frobenius test">Quadratic Frobenius</a></li> <li><a href="/wiki/Solovay%E2%80%93Strassen_primality_test" title="Solovay–Strassen primality test">Solovay–Strassen</a></li> <li><a href="/wiki/Miller%E2%80%93Rabin_primality_test" title="Miller–Rabin primality test">Miller–Rabin</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Generating_primes" class="mw-redirect" title="Generating primes">Prime-generating</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/Sieve_of_Atkin" title="Sieve of Atkin">Sieve of Atkin</a></li> <li><a href="/wiki/Sieve_of_Eratosthenes" title="Sieve of Eratosthenes">Sieve of Eratosthenes</a></li> <li><a href="/wiki/Sieve_of_Pritchard" title="Sieve of Pritchard">Sieve of Pritchard</a></li> <li><a href="/wiki/Sieve_of_Sundaram" title="Sieve of Sundaram">Sieve of Sundaram</a></li> <li><a href="/wiki/Wheel_factorization" title="Wheel factorization">Wheel factorization</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Integer_factorization" title="Integer factorization">Integer factorization</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/Continued_fraction_factorization" title="Continued fraction factorization">Continued fraction (CFRAC)</a></li> <li><a href="/wiki/Dixon%27s_factorization_method" title="Dixon's factorization method">Dixon's</a></li> <li><a href="/wiki/Lenstra_elliptic-curve_factorization" title="Lenstra elliptic-curve factorization">Lenstra elliptic curve (ECM)</a></li> <li><a href="/wiki/Euler%27s_factorization_method" title="Euler's factorization method">Euler's</a></li> <li><a href="/wiki/Pollard%27s_rho_algorithm" title="Pollard's rho algorithm">Pollard's rho</a></li> <li><a href="/wiki/Pollard%27s_p_%E2%88%92_1_algorithm" title="Pollard's p − 1 algorithm"><i>p</i> − 1</a></li> <li><a href="/wiki/Williams%27s_p_%2B_1_algorithm" title="Williams's p + 1 algorithm"><i>p</i> + 1</a></li> <li><a href="/wiki/Quadratic_sieve" title="Quadratic sieve">Quadratic sieve (QS)</a></li> <li><a href="/wiki/General_number_field_sieve" title="General number field sieve">General number field sieve (GNFS)</a></li> <li><i><a href="/wiki/Special_number_field_sieve" title="Special number field sieve">Special number field sieve (SNFS)</a></i></li> <li><a href="/wiki/Rational_sieve" title="Rational sieve">Rational sieve</a></li> <li><a href="/wiki/Fermat%27s_factorization_method" title="Fermat's factorization method">Fermat's</a></li> <li><a href="/wiki/Shanks%27s_square_forms_factorization" title="Shanks's square forms factorization">Shanks's square forms</a></li> <li><a href="/wiki/Trial_division" title="Trial division">Trial division</a></li> <li><a href="/wiki/Shor%27s_algorithm" title="Shor's algorithm">Shor's</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Multiplication_algorithm" title="Multiplication algorithm">Multiplication</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 class="mw-selflink selflink">Ancient Egyptian</a></li> <li><a href="/wiki/Long_multiplication" class="mw-redirect" title="Long multiplication">Long</a></li> <li><a href="/wiki/Karatsuba_algorithm" title="Karatsuba algorithm">Karatsuba</a></li> <li><a href="/wiki/Toom%E2%80%93Cook_multiplication" title="Toom–Cook multiplication">Toom–Cook</a></li> <li><a href="/wiki/Sch%C3%B6nhage%E2%80%93Strassen_algorithm" title="Schönhage–Strassen algorithm">Schönhage–Strassen</a></li> <li><a href="/wiki/F%C3%BCrer%27s_algorithm" class="mw-redirect" title="Fürer's algorithm">Fürer's</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Euclidean_division" title="Euclidean division">Euclidean</a> <a href="/wiki/Division_algorithm" title="Division algorithm">division</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/Binary_division" class="mw-redirect" title="Binary division">Binary</a></li> <li><a href="/wiki/Chunking_(division)" title="Chunking (division)">Chunking</a></li> <li><a href="/wiki/Fourier_division" title="Fourier division">Fourier</a></li> <li><a href="/wiki/Goldschmidt_division" class="mw-redirect" title="Goldschmidt division">Goldschmidt</a></li> <li><a href="/wiki/Newton%E2%80%93Raphson_division" class="mw-redirect" title="Newton–Raphson division">Newton-Raphson</a></li> <li><a href="/wiki/Long_division" title="Long division">Long</a></li> <li><a href="/wiki/Short_division" title="Short division">Short</a></li> <li><a href="/wiki/SRT_division" class="mw-redirect" title="SRT division">SRT</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Discrete_logarithm" title="Discrete logarithm">Discrete logarithm</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/Baby-step_giant-step" title="Baby-step giant-step">Baby-step giant-step</a></li> <li><a href="/wiki/Pollard%27s_rho_algorithm_for_logarithms" title="Pollard's rho algorithm for logarithms">Pollard rho</a></li> <li><a href="/wiki/Pollard%27s_kangaroo_algorithm" title="Pollard's kangaroo algorithm">Pollard kangaroo</a></li> <li><a href="/wiki/Pohlig%E2%80%93Hellman_algorithm" title="Pohlig–Hellman algorithm">Pohlig–Hellman</a></li> <li><a href="/wiki/Index_calculus_algorithm" title="Index calculus algorithm">Index calculus</a></li> <li><a href="/wiki/Function_field_sieve" title="Function field sieve">Function field sieve</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Greatest_common_divisor" title="Greatest common divisor">Greatest common divisor</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/Binary_GCD_algorithm" title="Binary GCD algorithm">Binary</a></li> <li><a href="/wiki/Euclidean_algorithm" title="Euclidean algorithm">Euclidean</a></li> <li><a href="/wiki/Extended_Euclidean_algorithm" title="Extended Euclidean algorithm">Extended Euclidean</a></li> <li><a href="/wiki/Lehmer%27s_GCD_algorithm" title="Lehmer's GCD algorithm">Lehmer's</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Quadratic_residue" title="Quadratic residue">Modular square root</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/Cipolla%27s_algorithm" title="Cipolla's algorithm">Cipolla</a></li> <li><a href="/wiki/Pocklington%27s_algorithm" title="Pocklington's algorithm">Pocklington's</a></li> <li><a href="/wiki/Tonelli%E2%80%93Shanks_algorithm" title="Tonelli–Shanks algorithm">Tonelli–Shanks</a></li> <li><a href="/wiki/Berlekamp%E2%80%93Rabin_algorithm" title="Berlekamp–Rabin algorithm">Berlekamp</a></li> <li><a href="/wiki/Kunerth%27s_algorithm" title="Kunerth's algorithm">Kunerth</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other algorithms</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/Chakravala_method" title="Chakravala method">Chakravala</a></li> <li><a href="/wiki/Cornacchia%27s_algorithm" title="Cornacchia's algorithm">Cornacchia</a></li> <li><a href="/wiki/Exponentiation_by_squaring" title="Exponentiation by squaring">Exponentiation by squaring</a></li> <li><a href="/wiki/Integer_square_root" title="Integer square root">Integer square root</a></li> <li><a href="/wiki/Integer_relation_algorithm" title="Integer relation algorithm">Integer relation</a> (<a href="/wiki/Lenstra%E2%80%93Lenstra%E2%80%93Lov%C3%A1sz_lattice_basis_reduction_algorithm" title="Lenstra–Lenstra–Lovász lattice basis reduction algorithm">LLL</a>; <a href="/wiki/Korkine%E2%80%93Zolotarev_lattice_basis_reduction_algorithm" title="Korkine–Zolotarev lattice basis reduction algorithm">KZ</a>)</li> <li><a href="/wiki/Modular_exponentiation" title="Modular exponentiation">Modular exponentiation</a></li> <li><a href="/wiki/Montgomery_reduction" class="mw-redirect" title="Montgomery reduction">Montgomery reduction</a></li> <li><a href="/wiki/Schoof%27s_algorithm" title="Schoof's algorithm">Schoof</a></li> <li><a href="/wiki/Trachtenberg_system" title="Trachtenberg system">Trachtenberg system</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow hlist" colspan="2"><div> <ul><li><i>Italics</i> indicate that algorithm is for numbers of special forms</li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Ancient_Egyptian_multiplication&oldid=1253902303">https://en.wikipedia.org/w/index.php?title=Ancient_Egyptian_multiplication&oldid=1253902303</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Egyptian_mathematics" title="Category:Egyptian mathematics">Egyptian mathematics</a></li><li><a href="/wiki/Category:Multiplication" 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