[Math Lair] Egyptian Mathematics

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"It was among the Egyptians that geometry is generally held to have been discovered. It owed its discovery to the practice of land measurement. For the Egyptians had to perform such measurements because the overflow of the Nile would cause the boundary of each person's land to disappear."
Proclus (5th century A.D.)

The story of mathematics starts at least 5,000 years ago in Ancient Egypt. Tradition holds that the first geometers were Egyptian. As the annual flooding of the Nile river wiped out all boundary markers, it was necessary to figure out each year where one property ended and another began.

Building the pyramids would also require mathematical knowledge. Some of the pyramids also contain interesting relationships which, although they are usually accidents or coincidences, are interesting to note. For example, the ratio of the length of one side of the Great Pyramid of Cheops at Giza is approximately π/2. The Greek historian Herodotus wrote that the pyramid was built so that the area of each lateral face would equal the area of a square with side length equal to the pyramid's height, in which case the ratio above would automatically approximate π. [Title of the Rhind Mathematical Papyrus]

The advent of an agricultural society in Ancient Egypt led to a class of priests and scribes who spent some of their time working on mathematics. These scribes would have wrote on papyrus. In some cases papyrus can be well-preserved in the dry Egyptian climate, and so some of the papyri have survived to this day. From a mathematical perspective, the most important are the Moscow Papyrus and the Rhind Papyrus.

The Rhind Papyrus is the work of the scribe Ahmes, and dates to around 1650 B.C. Ahmes is the earliest mathematician whom we know by name. The papyrus deals with many different problems that the ancient Egyptians would have encountered.

The papyrus is able to give us a fairly good picture of the state of Egyptian mathematics. The measurement of figures and solids plays an important part of this papyrus. There are no theorems as we would call them; everything is stated in the form of a problem, and not in general terms but using actual numbers (for example: "measure a rectangle the sides of which contain two and ten units of length").

There is also a section of arithmetical problems, which is headed "Directions for knowing all dark things". The first part contains directions for expressing as unit fractions numbers of the form 2/(2x+1). The ancient Egyptians, with the exception of 2/3, only dealt with unit fractions; see my Egyptian Fractions page for more details.

Ahmes also writes about multiplication, which he accomplished by repeated adding, and he also posed some algebra problems, which is interesting considering that the ancient Greeks paid little attention to algebra.

One interesting problem in the papyrus is problem 79, which is discussed further at St. Ives Riddle.

The Moscow Papyrus is a somewhat shorter papyrus written around 1850 B.C. The Moscow Papyrus contains several interesting geometry problems. One such problem is the fourteenth problem, which determines the volume of a truncated pyramid with a square base. The method used for doing so dovetails with the formula that we know today, namely

V = ⅓h(a² + ab + b²)
This is interesting because the formula is quite complicated and not easy to either deduce or discover empirically.

Here are a few other highlights:

The Egyptian Calendar: The Egyptian calendar appears to date from 4,241 B.C., although it is possible that it dates from 1,460 years later. It consisted of 365 days; as the mean solar year is about 365.2422 days, the Egyptian calendar would gradually fall out of synch with the seasons over a cycle of around 1,460 years.

Egyptian Numerals: The Egyptians wrote numbers using a base-10, non-place-value system. On a mace dating to 3100 B.C., numbers in the millions are recorded.

You may also be interested in Egyptian Fractions.

Math history timeline

On our sister site All Fun and Games, you can view interesting facts about Egypt, including a few ancient Egypt math facts.

Sources used (see bibliography page for titles corresponding to numbers): 35.