Egyptian algebra

In the history of mathematics, Egyptian algebra, as that term is used in this article, refers to algebra as it was developed and used in Ancient Egypt. Ancient Egyptian mathematics as discussed here spans a time period ranging from ca. 3000 BC to ca. 300 BC.

We only have a limited number of resources (problems) from ancient Egypt that concern algebra. Problems of an algebraic nature appear in both the Moscow Mathematical Papyrus (MMP) and in the Rhind Mathematical Papyrus (RMP) as well as several other sources.^[1]

Fractions

The mathematical writings show that the scribes used (least) common multiples to turn problems with fractions into problems using integers. The multiplicative factors were often recorded in red ink and are referred to as Red auxiliary numbers.^[1]

Aha problems, linear equations and false position

Aha in hieroglyphs

Aha problems involve finding unknown quantities (referred to as Aha) if the sum of the quantity and part(s) of it are given. The Rhind Mathematical Papyrus also contains four of these type of problems. Problems 1, 19, and 25 of the Moscow Papyrus are Aha problems. For instance problem 19 asks one to calculate a quantity taken 1 and ½ times and added to 4 to make 10.^[1] In other words, in modern mathematical notation we are asked to solve the linear equation:

$3/2 \times x + 4 = 10.\$

Solving these Aha problems involves a technique called method of false position. The technique is also called the method of false assumption. The scribe would substitute an initial guess of the answer into the problem. The solution using the false assumption would be proportional to the actual answer, and the scribe would find the answer by using this ratio.^[1]

Pefsu problems

Many of the practical problems contained in the Moscow Mathematical Papyrus are pefsu problems: 10 of the 25 problems. A pefsu measures the strength of the beer made from a heqat of grain

$\mbox{pefsu} = \frac{\mbox{number loaves of bread (or jugs of beer)}}{\mbox{number of heqats of grain}}.$

A higher pefsu number means weaker bread or beer. The pefsu number is mention in many offering lists. For example problem 8 translates as:

(1) Example of calculating 100 loaves of bread of pefsu 20

(2) If someone says to you: “You have 100 loaves of bread of pefsu 20

(3) to be exchanged for beer of pefsu 4

(4) like 1/2 1/4 malt-date beer

(5) First calculate the grain required for the 100 loaves of the bread of pefsu 20

(6) The result is 5 heqat. Then reckon what you need for a des-jug of beer like the beer called 1/2 1/4 malt-date beer

(7) The result is 1/2 of the heqat measure needed for des-jug of beer made from Upper-Egyptian grain.

(8) Calculate 1/2 of 5 heqat, the result will be 2¹⁄₂

(9) Take this 2¹⁄₂ four times

(10) The result is 10. Then you say to him:

(11) Behold! The beer quantity is found to be correct.^[1]

Geometrical progressions

The use of the Horus eye fractions shows some (rudimentary) knowledge of geometrical progression.^[1] One unit was written as 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + 1/64. But the last copy of 1/64 was written as 5 ro, thereby writing 1 = 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + (5 ro). These fractions were further used to write fractions in terms of $1 / 2^k$ terms plus a remainder specified in terms of ro as shown in for instance the Akhmim wooden tablets.^[2]

Arithmetical progressions

Knowledge of arithmetic progressions is also evident from the mathematical sources.^[1]

References

1. Clagett, Marshall. 1999. Ancient Egyptian Science: A Source Book. Volume 3: Ancient Egyptian Mathematics. Memoirs of the American Philosophical Society 232. Philadelphia: American Philosophical Society. ISBN 0-87169-232-5
2. Vymazalova, H. "The Wooden Tablets from Cairo: The Use of the Grain Unit HK3T in Ancient Egypt." Archiv Orientalai, Charles U., Prague, pp. 27–42, 2002.