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<!doctype html> <html lang="en" class="historyandterminology mathworldcontributors"> <head> <title>Egyptian Mathematical Leather Roll -- from Wolfram MathWorld</title> <meta name="DC.Title" content="Egyptian Mathematical Leather Roll" /> <meta name="DC.Creator" content="Weisstein, Eric W." /> <meta name="DC.Description" content="The Egyptian Mathematical Leather Roll (EMLR), dates to the Middle Kingdom, and was purchased in Egypt in 1858 by Henry Rhind, near the time when the Rhind papyrus was purchased. While the Rhind papyrus dates to 1650 BC, no specific date has been determined for the EMLR. Both the EMLR and the Rhind papyrus have been in the British Museum since 1864, donated by the estate of Henry Rhind. The EMLR was not unrolled until 1927. It consists of 26 unit fraction series each of which is an..." /> <meta name="description" content="The Egyptian Mathematical Leather Roll (EMLR), dates to the Middle Kingdom, and was purchased in Egypt in 1858 by Henry Rhind, near the time when the Rhind papyrus was purchased. While the Rhind papyrus dates to 1650 BC, no specific date has been determined for the EMLR. Both the EMLR and the Rhind papyrus have been in the British Museum since 1864, donated by the estate of Henry Rhind. The EMLR was not unrolled until 1927. It consists of 26 unit fraction series each of which is an..." /> <meta name="DC.Date.Created" scheme="W3CDTF" content="2004-11-02" /> <meta name="DC.Subject" scheme="MathWorld" content="Mathematics:History and Terminology:History" /> <meta name="DC.Subject" scheme="MathWorld" content="Mathematics:History and Terminology:Disciplinary Terminology:Aeronautical Terminology" /> <meta name="DC.Subject" scheme="MathWorld" content="Mathematics:History and Terminology:Disciplinary Terminology:Culinary Terminology" /> <meta name="DC.Subject" scheme="MathWorld" content="Mathematics:MathWorld Contributors:Gardner" /> <meta name="DC.Subject" scheme="MSC_2000" content="01A" /> <meta name="DC.Rights" content="Copyright 1999-2025 Wolfram Research, Inc. 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While the Rhind papyrus dates to 1650 BC, no specific date has been determined for the EMLR. Both the EMLR and the Rhind papyrus have been in the British Museum since 1864, donated by the estate of Henry Rhind. The EMLR was not unrolled until 1927. It consists of 26 unit fraction series each of which is an..."> <meta name="twitter:card" content="summary_large_image"> <meta name="twitter:site" content="@WolframResearch"> <meta name="twitter:title" content="Egyptian Mathematical Leather Roll -- from Wolfram MathWorld"> <meta name="twitter:description" content="The Egyptian Mathematical Leather Roll (EMLR), dates to the Middle Kingdom, and was purchased in Egypt in 1858 by Henry Rhind, near the time when the Rhind papyrus was purchased. While the Rhind papyrus dates to 1650 BC, no specific date has been determined for the EMLR. Both the EMLR and the Rhind papyrus have been in the British Museum since 1864, donated by the estate of Henry Rhind. The EMLR was not unrolled until 1927. It consists of 26 unit fraction series each of which is an..."> <meta name="twitter:image:src" content="https://mathworld.wolfram.com/images/socialmedia/share/ogimage_EgyptianMathematicalLeatherRoll.png"> <link rel="canonical" href="https://mathworld.wolfram.com/EgyptianMathematicalLeatherRoll.html" /> <meta http-equiv="x-ua-compatible" content="ie=edge"> <meta name="viewport" content="width=device-width, initial-scale=1"> <meta charset="utf-8"> <script async src="/common/javascript/analytics.js"></script> <script async src="//www.wolframcdn.com/consent/cookie-consent.js"></script> <script async src="/common/javascript/wal/latest/walLoad.js"></script> <link rel="stylesheet" href="/css/styles.css"> <link rel="preload" href="//www.wolframcdn.com/fonts/source-sans-pro/1.0/global.css" as="style" onload="this.onload=null;this.rel='stylesheet'"> <noscript><link rel="stylesheet" href="//www.wolframcdn.com/fonts/source-sans-pro/1.0/global.css"></noscript> </head> <body id="topics"> <main 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History and Terminology </a> <a href="/topics/NumberTheory.html" id="sidebar-numbertheory"> Number Theory </a> <a href="/topics/ProbabilityandStatistics.html" id="sidebar-probabilityandstatistics"> Probability and Statistics </a> <a href="/topics/RecreationalMathematics.html" id="sidebar-recreationalmathematics"> Recreational Mathematics </a> <a href="/topics/Topology.html" id="sidebar-topology"> Topology </a> </nav> <nav class="secondary-nav"> <a href="/letters/"> Alphabetical Index </a> <a href="/whatsnew/"> New in MathWorld </a> </nav> </section> <section id="content">  <nav class="breadcrumbs"><ul class="breadcrumb"> <li> <a href="/topics/HistoryandTerminology.html">History and Terminology</a> </li> <li> <a href="/topics/History.html">History</a> </li> </ul><ul class="breadcrumb"> <li> <a href="/topics/HistoryandTerminology.html">History and Terminology</a> </li> <li> <a href="/topics/DisciplinaryTerminology.html">Disciplinary Terminology</a> </li> <li> <a href="/topics/AeronauticalTerminology.html">Aeronautical Terminology</a> </li> </ul><ul class="breadcrumb"> <li> <a href="/topics/HistoryandTerminology.html">History and Terminology</a> </li> <li> <a href="/topics/DisciplinaryTerminology.html">Disciplinary Terminology</a> </li> <li> <a href="/topics/CulinaryTerminology.html">Culinary Terminology</a> </li> </ul><ul class="breadcrumb"> <li> <a href="/topics/MathWorldContributors.html">MathWorld Contributors</a> </li> <li> <a href="/topics/Gardner.html">Gardner</a> </li> </ul><a class="show-more">More...</a><a class="display-n show-less">Less...</a></nav>   <h1>Egyptian Mathematical Leather Roll</h1>  <hr class="margin-t-1-8 margin-b-3-4">   <div class="entry-content"> <p> The Egyptian Mathematical Leather Roll (EMLR), dates to the Middle Kingdom, and was purchased in Egypt in 1858 by Henry Rhind, near the time when the <a href="/RhindPapyrus.html">Rhind papyrus</a> was purchased. While the <a href="/RhindPapyrus.html">Rhind papyrus</a> dates to 1650 BC, no specific date has been determined for the EMLR. Both the EMLR and the Rhind papyrus have been in the British Museum since 1864, donated by the estate of Henry Rhind. The EMLR was not unrolled until 1927. </p> <p> It consists of 26 <a href="/UnitFraction.html">unit fraction</a> series each of which is an expression of a rational number of the form <img src="/images/equations/EgyptianMathematicalLeatherRoll/Inline1.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="31" height="21" alt="1/p" /> or <img src="/images/equations/EgyptianMathematicalLeatherRoll/Inline2.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="54" height="21" alt="1/(pq)" /> into an <a href="/EgyptianFraction.html">Egyptian fraction</a>. Five methods were listed to generally convert any <img src="/images/equations/EgyptianMathematicalLeatherRoll/Inline3.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="31" height="21" alt="1/p" /> or <img src="/images/equations/EgyptianMathematicalLeatherRoll/Inline4.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="54" height="21" alt="1/(pq)" /> to a concise and exact unit fraction series. Four methods have been confirmed as being additive, with three of them being identities and the fourth based on a remainder (Boyer and Merzbacher 1991). </p> <p> For the last 75 years, only the first four methods have been stressed as fairly representing the central theme in the EMLR. However, in 2002, a connection to the <a href="/RhindPapyrus.html">Rhind papyrus</a> <img src="/images/equations/EgyptianMathematicalLeatherRoll/Inline5.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="54" height="21" alt="2/(pq)" /> series rule was published, so there is also a fifth method using the rule </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/EgyptianMathematicalLeatherRoll/NumberedEquation1.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="110" height="43" alt=" 1/(pq)=1/A×A/(pq) " /></td><td align="right" width="3"> <div id="eqn1" class="eqnum"> (1) </div> </td></tr> </table> </div> <p> for <img src="/images/equations/EgyptianMathematicalLeatherRoll/Inline6.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="42" height="21" alt="A=5" />, 7, 25, as used implicitly in four of the EMLR's 26 series. </p> <p> As an example of method five consider <img src="/images/equations/EgyptianMathematicalLeatherRoll/Inline7.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="131" height="21" alt="1/8=1/A×A/8" />. </p> <div class="table-responsive-noborders"> <table align="center" width="100%" cellpadding="0" cellspacing="0" style="padding-left: 50px" border="0"> <tr style=""><td align="right" width=""><img src="/images/equations/EgyptianMathematicalLeatherRoll/Inline8.svg" class="displayformula" style="max-width:100%;max-height:100%;" width="12" height="26" border="0" alt="1/8" /></td><td align="center" width="14"><img src="/images/equations/EgyptianMathematicalLeatherRoll/Inline9.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EgyptianMathematicalLeatherRoll/Inline10.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="52" height="26" alt="1/(25)×(25)/8" /></td><td align="right" width="10"> <div id="eqn2" class="eqnum"> (2) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/EgyptianMathematicalLeatherRoll/Inline11.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/EgyptianMathematicalLeatherRoll/Inline12.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EgyptianMathematicalLeatherRoll/Inline13.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="46" height="26" alt="1/5×(25)/(40)" /></td><td align="right" width="10"> <div id="eqn3" class="eqnum"> (3) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/EgyptianMathematicalLeatherRoll/Inline14.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/EgyptianMathematicalLeatherRoll/Inline15.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EgyptianMathematicalLeatherRoll/Inline16.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="84" height="26" alt="1/5×(3/5+1/(40))" /></td><td align="right" width="10"> <div id="eqn4" class="eqnum"> (4) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/EgyptianMathematicalLeatherRoll/Inline17.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/EgyptianMathematicalLeatherRoll/Inline18.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EgyptianMathematicalLeatherRoll/Inline19.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="116" height="26" alt="1/5×(1/5+2/5+1/(40))," /></td><td align="right" width="10"> <div id="eqn5" class="eqnum"> (5) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/EgyptianMathematicalLeatherRoll/Inline20.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/EgyptianMathematicalLeatherRoll/Inline21.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EgyptianMathematicalLeatherRoll/Inline22.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="146" height="26" alt="1/5×(1/5+1/3+1/(15)+1/(40))" /></td><td align="right" width="10"> <div id="eqn6" class="eqnum"> (6) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/EgyptianMathematicalLeatherRoll/Inline23.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/EgyptianMathematicalLeatherRoll/Inline24.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/EgyptianMathematicalLeatherRoll/Inline25.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="130" height="26" alt="1/(25)+1/(15)+1/(75)+1/(200)," /></td><td align="right" width="10"> <div id="eqn7" class="eqnum"> (7) </div> </td></tr> </table> </div> <p> as listed in the EMLR. </p> <p> The <a href="/RhindPapyrus.html">Rhind papyrus</a> considered <img src="/images/equations/EgyptianMathematicalLeatherRoll/Inline26.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="82" height="21" alt="A=(p+1)" />, showing that the EMLR was a student test results paper, teaching the student to use several less optimal values for A when learning to convert <img src="/images/equations/EgyptianMathematicalLeatherRoll/Inline27.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="31" height="21" alt="1/p" /> and <img src="/images/equations/EgyptianMathematicalLeatherRoll/Inline28.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="54" height="21" alt="1/(pq)" />, as a foundation to learning <img src="/images/equations/EgyptianMathematicalLeatherRoll/Inline29.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="31" height="21" alt="2/p" /> and <img src="/images/equations/EgyptianMathematicalLeatherRoll/Inline30.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="54" height="21" alt="2/(pq)" /> conversion methods. </p> </div>  <hr class="margin-b-1-1-4"> <div class="c-777 entry-secondary-content">  <h2>See also</h2><a href="/AkhmimWoodenTablet.html">Akhmim Wooden Tablet</a>, <a href="/EgyptianFraction.html">Egyptian Fraction</a>, <a href="/RhindPapyrus.html">Rhind Papyrus</a>, <a href="/UnitFraction.html">Unit Fraction</a>     <p class="contributor"> <i>This entry contributed by <a target="_blank" href="/topics/Gardner.html">Milo Gardner</a></i> </p>   <h2>Explore with Wolfram|Alpha</h2> <div id="WAwidget"> <div class="WAwidget-wrapper"> <img alt="WolframAlpha" title="WolframAlpha" src="/images/wolframalpha/WA-logo.png" width="136" height="20"> <form name="wolframalpha" action="https://www.wolframalpha.com/input/" target="_blank"> <input type="text" name="i" class="search" placeholder="Solve your math problems and get step-by-step solutions" value=""> <button type="submit" title="Evaluate on WolframAlpha"></button> </form> </div> <div class="WAwidget-wrapper try"> <p class="text-align-r"> More things to try: </p> <ul> <li> <a target="_blank" href="http://www.wolframalpha.com/input/?i=history"> history </a> </li> <li><a target="_blank" href="https://www.wolframalpha.com/input/?i=0xff42ca">0xff42ca</a></li> <li><a target="_blank" href="https://www.wolframalpha.com/input/?i=edge+detect+Abraham+Lincoln+image+with+radius+x">edge detect Abraham Lincoln image with radius x</a></li> </ul> </div> </div>   <h2>References</h2><cite>Boyer, C. B. and Merzbacher, U. C. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0471543977/ref=nosim/ericstreasuretro">A History of Mathematics, 2nd ed.</a></i> New York: Wiley, 1991.</cite><cite>Gardner, M. "The Egyptian Mathematical Leather Roll, Attested Short Term and Long Term." In <i><a href="http://www.amazon.co.uk/exec/obidos/ASIN/8185931453/ref=nosim/mathworld-21">History of the Mathematical Sciences</a></i> (Ed. I. Grattan-Guiness and B. S. Yadav). Hindustan Book Agency, pp. 119-134, 2002.</cite><cite>Gillings, R. <i><a href="http://www.amazon.com/exec/obidos/ASIN/048624315X/ref=nosim/ericstreasuretro">Mathematics in the Time of the Pharaohs.</a></i> Boston, MA: MIT Press, pp. 89-103, 1972.</cite><cite>Glanville, S. R. K. "The Mathematical Leather Roll in the British Museum." <i>J. Egyptian Arch.</i> <b>13</b>, 232-238, 1927.</cite>   <h2>Cite this as:</h2> <p> <a href="/topics/Gardner.html">Gardner, Milo</a>. "Egyptian Mathematical Leather Roll." From <a href="/"><i>MathWorld</i></a>--A Wolfram Web Resource, created by <a href="/about/author.html">Eric W. Weisstein</a>. <a href="https://mathworld.wolfram.com/EgyptianMathematicalLeatherRoll.html">https://mathworld.wolfram.com/EgyptianMathematicalLeatherRoll.html</a> </p>  <h2>Subject classifications</h2><nav class="breadcrumbs"><ul class="breadcrumb"> <li> <a href="/topics/HistoryandTerminology.html">History and Terminology</a> </li> <li> <a href="/topics/History.html">History</a> </li> </ul><ul class="breadcrumb"> <li> <a href="/topics/HistoryandTerminology.html">History and Terminology</a> </li> <li> <a href="/topics/DisciplinaryTerminology.html">Disciplinary Terminology</a> </li> <li> <a href="/topics/AeronauticalTerminology.html">Aeronautical Terminology</a> </li> </ul><ul class="breadcrumb"> <li> <a href="/topics/HistoryandTerminology.html">History and Terminology</a> </li> <li> <a href="/topics/DisciplinaryTerminology.html">Disciplinary Terminology</a> </li> <li> <a href="/topics/CulinaryTerminology.html">Culinary Terminology</a> </li> </ul><ul class="breadcrumb"> <li> <a href="/topics/MathWorldContributors.html">MathWorld Contributors</a> </li> <li> <a href="/topics/Gardner.html">Gardner</a> </li> </ul><a class="show-more">More...</a><a class="display-n show-less">Less...</a></nav>  </div> </section> </section>  </div> </main> <aside id="bottom"> <style> #bottom { padding-bottom: 65px; } #acknowledgment { display:none; } .attribution { font-size: .75rem; font-style: italic; } footer ul li:not(:last-of-type)::after { background: #a3a3a3; margin-left: .3rem; margin-right: .1rem; } @media all and (max-width: 900px) { .attribution { font-size: 12px; } } @media (max-width: 600px) { footer { max-width: 360px; } footer ul { max-width: 360px; } footer ul:nth-child(1) li:nth-child(2):after { content: ""; height: 11px; } footer ul:nth-child(1) li:nth-child(3):after { content: ""; height: 0px; } } </style> <footer> <ul> <li><a href="/about/">About MathWorld</a></li> <li><a href="/classroom/">MathWorld Classroom</a></li> <li><a href="/contact/">Contribute</a></li> <li><a href="https://www.amazon.com/exec/obidos/ASIN/1420072218/ref=nosim/weisstein-20" target="_blank">MathWorld Book</a></li> <li class="display-n display-ib__600"><a href="https://www.wolfram.com" target="_blank">wolfram.com</a></li> </ul> <ul> <li class="display-n__600"><a href="/whatsnew/">13,246 Entries</a></li> <li class="display-n__600"><a href="/whatsnew/">Last Updated: Fri Feb 21 2025</a></li>  <li><a href="https://www.wolfram.com" target="_blank">©1999–2025 Wolfram Research, Inc.</a></li> <li><a href="https://www.wolfram.com/legal/terms/mathworld.html" target="_blank">Terms of Use</a></li> </ul> <ul class="wolfram"> <li class="display-n__600 display-n__900"><a href="https://www.wolfram.com" target="_blank" aria-label="Wolfram"><img src="/images/footer/wolfram-logo.png" alt="Wolfram" title="Wolfram" width="121" height="28"></a></li> <li class="display-n__600"><a href="https://www.wolfram.com" target="_blank">wolfram.com</a></li> <li class="display-n__600"><a href="https://www.wolfram.com/education/" target="_blank">Wolfram for Education</a></li> <li class="attribution">Created, developed and nurtured by Eric Weisstein at Wolfram Research</li> </ul> </footer> <section id="acknowledgment"> <i>Created, developed and nurtured by Eric Weisstein at Wolfram Research</i> </section> </aside> <script type="text/javascript" src="/scripts/scripts.js"></script> <script src="/common/js/c2c/1.0/WolframC2C.js"></script> <script src="/common/js/c2c/1.0/WolframC2CGui.js"></script> <script src="/common/js/c2c/1.0/WolframC2CDefault.js"></script> <link rel="stylesheet" href="/common/js/c2c/1.0/WolframC2CGui.css.en"> <style> .wolfram-c2c-wrapper { padding: 0px !important; border: 0px; } .wolfram-c2c-wrapper:active { border: 0px; } .wolfram-c2c-wrapper:hover { border: 0px; } </style> <script> let c2cWrittings = new WolframC2CDefault({'triggerClass':'mathworld-c2c_above', 'uniqueIdPrefix': 'mathworld-c2c_above-'}); </script> <style> #IPstripe-outer { background: #47a2af; } #IPstripe-outer:hover { background: #0095aa; } </style> <div id="IPstripe-wrap"></div> <script src="/common/stripe/stripe.en.js"></script> </body> </html>