In the

history of mathematics The history of mathematics deals with the origin of discoveries in mathematics and the History of mathematical notation, mathematical methods and notation of the past. Before the modern age and the worldwide spread of knowledge, written examples ...

, Egyptian algebra, as that term is used in this article, refers to

algebra Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...

as it was developed and used in

ancient Egypt Ancient Egypt () was a cradle of civilization concentrated along the lower reaches of the Nile River in Northeast Africa. It emerged from prehistoric Egypt around 3150BC (according to conventional Egyptian chronology), when Upper and Lower E ...

. Ancient Egyptian mathematics as discussed here spans a time period ranging from 3000 BCE to 300 BCE. There are limited surviving examples of ancient Egyptian algebraic problems. They appear in the

Moscow Mathematical Papyrus The Moscow Mathematical Papyrus, also named the Golenishchev Mathematical Papyrus after its first non-Egyptian owner, Egyptologist Vladimir Golenishchev, is an ancient Egyptian mathematical papyrus containing several problems in arithmetic, ge ...

(MMP) and in the

Rhind Mathematical Papyrus The Rhind Mathematical Papyrus (RMP; also designated as papyrus British Museum 10057, pBM 10058, and Brooklyn Museum 37.1784Ea-b) is one of the best known examples of ancient Egyptian mathematics. It is one of two well-known mathematical papyri ...

(RMP), among others.

Fractions

Known mathematical texts show that 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.

Aha problems, linear equations and false position

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 types of problems. Problems 1, 19, and 25 of the Moscow Papyrus are Aha problems. Problem 19 asks one to calculate a quantity taken 1 and one-half times and added to 4 to make 10. In modern mathematical notation, this

linear equation In mathematics, a linear equation is an equation that may be put in the form a_1x_1+\ldots+a_nx_n+b=0, where x_1,\ldots,x_n are the variables (or unknowns), and b,a_1,\ldots,a_n are the coefficients, which are often real numbers. The coeffici ...

is represented: :

\frac 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.

Pefsu problems

10 of the 25 problems of the practical problems contained in the Moscow Mathematical Papyrus are pefsu problems. A pefsu measures the strength of the

beer Beer is an alcoholic beverage produced by the brewing and fermentation of starches from cereal grain—most commonly malted barley, although wheat, maize (corn), rice, and oats are also used. The grain is mashed to convert starch in the ...

made from a heqat of grain. :

\mbox = \frac.

A higher pefsu number means weaker bread or beer. The pefsu number is mentioned 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 2 : (9) Take this 2 four times : (10) The result is 10. Then you say to him: : (11) Behold! The beer quantity is found to be correct.

Geometrical progressions

The use of the Horus eye fractions shows some (rudimentary) knowledge of geometrical progression. 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.

Arithmetical progressions

Knowledge of

arithmetic progression An arithmetic progression or arithmetic sequence is a sequence of numbers such that the difference from any succeeding term to its preceding term remains constant throughout the sequence. The constant difference is called common difference of that ...

s is also evident from the mathematical sources.

References

{{Ancient Egypt topics Egyptian mathematics History of algebra