Egyptian Fractions 2/n roll The Akhmim Wooden Tablet Egyptian Math Papyri

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The Akhmim Wooden Tablet INTRODUCTION

In the Babylonian base 60 system the division of its smallest numbers, by prime numbers, used a round off system. In base 60, the inverse of a prime number, 1/p, was written as a multiple of 2, 3 and 5. For example, 1/91 was usually rounded off
to 1/90, though scribes were generally free to select their own round off rules, meaning that 1/92 could have been selected. Though the Babylonians surely knew of arithmetic fixes to repair the very old 2, 3, and 5 denominator limitation. They, aparently, never replaced the convention, and therefore never culturally allowed any number, n, one of the fixes, to generally appear in denominators.

In the nearby Egyptian case, prior to 2,000 BC, the same type of cursive round off error is suggested by its binary fraction Horus-Eye system. 1/64, was rounded off, leaving an awkward system, when contrasted to Babylon's relative accurate system. However, the Egyptians developed a formal correction of its awkward Horus-Eye errors. The method was recorded in a wooden tablet called the Akhmim Wooden Tablet (AWT) also known as the Cairo wooden tablet.

AWT dates aproximately to 2000 BC. More accurately, sometime from the 12th dynasty to the 15th dynasty in the Egyptian Middle Kingdom. It is 46.5 x 26n cm and is housed in the Cairo Museum. Its contents include 27 servant names, and an unknown king's name (citing the 8th year of his reign), and for our interest, five division calculations, two of which were repeated several times. The document was reported in 1901, analyzed and published in 1906, by Georges Daressy who left out some divisions. In 2002, this was clarified by Hana Vymazalova in her re-translation.

The AWT explains the Egyptian shorthand division remainder system. In addition it touches on Egyptian arithmetic operations, especially division.

Acknowledgements:

We wish to thank Milo Gardner, Jr. of Sacramento for enlightening us about The Akhmim Wooden Tablet. The primary initial source of this material comes from letters to the author from Milo Gardner or http://akhmimwoodentablet.blogspot.com.

The Reader may also wish to see Gardner's contribution on the AWT to the Math World site, and Gardner's very informative Math Forum contribution.

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