Ahmes Biography

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World of Mathematics on Ahmes

Relatively few mathematical writings have survived from the time the rulers of Egypt built the pyramids. This is largely because writing at that time was done on laminated sheets of the papyrus plant, and such materials readily decompose under adverse environmental conditions.

One of the documents that has survived is known as the Rhind papyrus, named for the Scottish antique dealer, Henry Rhind, who purchased it in 1858. This document, about one foot high and 18 feet long, is now held by the British Museum (except for a few fragments held by the Brooklyn Museum). The document is also known as the Ahmes papyrus, after the individual who authored it. The papyrus is believed to date from about 1650 BC, but according to information contained in it, the material contained in it is derived from an earlier version from the Middle Kingdom (about 2000 to 1800 BC).

Although other mathematical writings from that time have been found, including the Moscow papyrus (now held in Moscow) and fragments of others, the Ahmes papyrus is one of the longest. These mathematical papyri are believed to have been written by ecclesiastical and government scribes.

The opening lines of the papyrus, written in the phonetic hieratic script (a simplification of pictorial hieroglyphics) that was used for everyday writing, reads "Directions for Obtaining the Knowledge of All Dark Things." The actual text appears to record the types of problems that business and administrative clerks frequently had to solve.

Because the document does not actually give any rules for solving arithmetical problems, there has been speculation that the Ahmes papyrus may have been an advanced arithmetical text for students, written in much the same tone as modern texts that leave proofs as an exercise for the reader. Others have speculated that it was the work of a student.

In the text, all fractions were decomposed into what are known as unit fractions. The author includes a table that expresses fractions with numerator 2 and odd denominators from 5 to 101 as sums of fractions with numerator 1. For example, Ahmes writes 2/5 = 1/3 + 1/15.

To express a fraction like 7/29 as a sum of unit fractions, Ahmes first notes that 7 = 2 + 2 + 2 + 1, and then proceeds to convert each 2/29 to a sum of fractions with numerator 1. In this way, he comes up with the following result: 7/29 = 1/6 + 1/24 + 1/58 + 1/87 + 1/232.

In the papyrus, Ahmes' sixty-third problem reads "Directions for dividing 700 breads among four people, 2/3 for one, 1/2 for the second, 1/3 for the third, 1/4 for the fourth." (In its modern representation, this problem would read 2x/3 + x/2 + x/3 + x/4 = 700, where one is to solve for x). Ahmes' solution was as follows: "Add 2/3 + 1/2 + 1/3 + 1/4. This gives 1 + 1/2 + 1/4. Divide 1 by (1 + 1/2 + 1/4). This gives 1/2 + 1/14. Now find (1/2 + 1/14) * 700. This is 400."

In Ahmes' papyrus, there is only minimal use of symbols. Addition and subtraction are represented by the legs of a man coming and going, respectively.

Some have speculated that one of the reasons that the Egyptians never developed advanced arithmetical or algebraic methods was that, although the method of fractions proved adequate for carrying out basic arithmetical operations, use of the method required extensive and complicated manipulations.

One of the exercises in Ahmes' papyrus, most certainly intended for students, simply reads: "seven houses, 49 cats, 343 mice, 2401 ears of spelt, 1607 hekats." Modern interpreters believe that the author was describing a problem in which each of seven houses contained seven cats, and that each cat would eat seven mice, and each mouse seven ears of grain, with each ear producing seven measures of grain. The statement and solution of the problem, which called for a calculation of the total number of houses, cats, mice, ears of spelt, and measures of grain, has not a little in common with our nursery rhyme that begins "As I was going to St. Ives, I met a man with seven wives," which some have interpreted as an indication that Ahmes had something of a sense of humor.