Maths in the Time of the Pharaohs

The Great Pyramid of Cheops at Giza (image from Wikimedia Commons).

Why would the Ancient Egyptians have any interest in or need for mathematics? There are many reasons. They had a well-organised and developed civilisation extending over millennia. Science and maths must have played important or even essential roles in this culture. They needed measurement for land surveying and for designing irrigation canals, arithmetic for accounting and taxation and geometry for construction of large grain silos and, of course, for pyramid building.

The Egyptians had to tackle numerous tasks involving arithmetic and geometry. Specific problems included the division of food and other supplies amongst families, distribution of plots of land to farmers and allocation of tasks to workers and soldiers, Their large grain silos were cylindrical in form and their capacity required to be calculated. Of course, with their preoccupation with pyramids, they needed to know how to deduce the volume from the base-length and height or slant–height in order to estimate the quantity of blocks and bricks required to complete them.

The Rhind Mathematical Papyrus

The Rhind Papyrus is the most important source of information on Ancient Egyptian mathematics. It was found in Thebes (within modern-day Luxor). It was written by the scribe Ahmes (A’h-mosé) around 1650 BC and is often referred to as the Ahmes Papyrus. Ahmes wrote the papyrus in hieratic script, and described it as a copy of a much older document, dating back perhaps 1000 years. The papyrus contains about 80 problems, with solutions, covering all the topics mentioned in this article.

Jean-François Champollion deciphered hieroglyphs in 1822 by analysing the inscriptions at the temples of Karnak and Luxor. Hieroglyphs were the formal writing system in Ancient Egypt. The later hieratic script, derived from hieroglyphic writing, was a cursive writing system used to write the Egyptian language. It was primarily written in ink with a reed pen on papyrus.

Papyrus was much less durable than the baked clay tablets of Mesopotamia. As a result, we have a much more detailed knowledge of Babylonian or Mesopotamian mathematics than of the mathematics in the time of the Pharaohs. The main part of the Rhind Papyrus is now in the British Museum.

Egyptian Numerals

The number system of Ancient Egypt was decimal-based, with a new system being introduced for each power of 10. Units were written as strokes, tens as horseshoes, hundreds as ropes, thousands as lilies, and so on. and so on. Thus, ${ 1\ 2\ 3\ ... }$ were written as ${|\ ||\ |||\ \dots}$ and ${10\ 20\ 30\ ... }$ as ${\cap,\ \cap\cap,\ \cap\cap\cap,\ ...}$ . The number 32 would be written “ ${ ||\,\cap\cap\cap}$ ” or “ ${\cap\cap\cap ||}$ ”.

Unit Fractions

The basic idea of the Egyptians for dividing quantities was “the ${n}$ th part”. Only unit fractions were used. There was no way to write ${\frac{3}{5}}$ or ${\frac{13}{27}}$ . Unit fractions were denoted by a bar-like symbol above the denominator. Thus, ${\frac{1}{12}}$ was written ${ \overline{ ||\,\cap }}$ . The only exception was the fraction ${\frac{2}{3}}$ , which played a central role in their computing methods. It had a special symbol ${ \overline{ |^| }}$ .

The fifth part of 20 is 4, but what is the fifth part of 2? It is not a unit fraction, so it would be broken up into a sum of unit fractions:

$\displaystyle \frac{2}{5} \rightarrow \frac{1}{3} + \frac{1}{15} \,.$

Fractions were expressed as sums of unit fractions (excepting 2/3) in descending order, that is, with increasing denominators. For example

$\displaystyle \frac{7}{12} \rightarrow \frac{6+1}{12} \rightarrow \frac{6}{12}+\frac{1}{12} \rightarrow \frac{1}{2}+\frac{1}{12} \,.$

The repetition of a fraction, like ${\frac{2}{7} \rightarrow \frac{1}{7}+\frac{1}{7}}$ , was not permitted. Instead, the scribe would write something like ${\frac{2}{7} \rightarrow \frac{1}{4}+\frac{1}{28}}$ .

Computation rules were used. For example,

$\displaystyle \frac{2}{3}\times \frac{1}{n} = \frac{1}{2n} + \frac{1}{6n} \,, \qquad\mbox{which yields}\qquad \frac{2}{3}\times\frac{1}{5} = \frac{1}{10}+\frac{1}{30} \,.$

Any fraction can be expressed in a binary form, ${q = 0.b_0 b_1 b_2 \dots }$ , or

$\displaystyle q = \frac{b_0}{2^0} + \frac{b_1}{2^1} + \frac{b_2}{2^2} + \cdots$

where ${b_k}$ is either ${0}$ or ${1}$ . Likewise, any integer can be expressed as ${n = \dots B_2 B_1 B_0}$ where ${B_k}$ is either ${0}$ or ${1}$ . Or as a sum of powers of 2:

$\displaystyle n = {B_0}{2^0} + {B_1}{2^1} +{B_2}{2^2} + \cdots$

Thus, any number is the sum of a subsequence of ${\{1, 2, 4, 8, \dots \}}$ . There is no record that the scribes knew this, but their method of multiplication depended on it so they my well have understood it at an intuitive level

Multiplication by Duplation

Multiplication was carried out by a successive sequence of duplications. Two operations were used:

Multiplication of numbers by 2.
Finding two-thirds of any number.

Egyptian arithmetic rested on these two simple-but-solid foundations. Prepared tables were probably used for the basic operations ${(+, -, \times, \div )}$ used in the calculations.

We illustrate the process of calculating 7 by 13 by repeated doublings. Multiply 7 by powers of 2, listing the powers and doublings in two columns. Then add the powers of two in the left column from the bottom upwards, omitting rows where necessary, to obtain 13. In this case, we need to omit the second row, since ${8 + 4 + 1 = 13}$ . Finally, add the values in the right column, excluding omitted rows. The result is ${13 \times 7 = 7 + 28 + 56 = 91}$ . The product has been obtained by “duplation” and addition.

The Pythagorean Theorem

The Pythagorean theorem was known in Mesopotamia more than a thousand years before Pythagoras. This is shown from the cuneiform tablet Plimpton 322, translated by Otto Neugebauer in 1945 [see That’s Maths reference below].

There is no evidence whatsoever that Ancient Egyptian mathematicians had any knowledge of the Pythagorean theorem. Indeed, no mention has been found in extant papyri even of the simple case of a triangle with sides 3, 4 and 5, althought it is reasonable to speculate that the Egyptian surveyors — known as rope-stretchers — who used knotted chords to measure lengths, would have stumbled upon this simple example.

Areas and Volumes

The Ahmes Papyrus includes several problems on calculating areas and volumes, giving us an idea of the extent of knowledge of geometry. The simple recipe for the area of a rectangle as length times breadth presented no problems. Likewise, the formula for the area of a triangle as half the base by the height was known.

Detail of Rhind Papyrus with calculation of the area of a circle.

For circles, the recipe for area was “subtract one ninth of the value of the diameter ${D=2r}$ and square the result”. This gave the formula

$\displaystyle A_\circ = \left(\frac{8 D}{9}\right)^2 = \left(\frac{16 r}{9}\right)^2 = \frac{256}{81} r^2 \,.$

This leads to an implicit value, ${\pi = \frac{256}{81}}$ , with an approximate value ${\pi \approx 3.16}$ , well within one per-cent of the true value.

Influences of Egyptian Mathematics

The extent to which Egyptian mathematics influenced that of Ancient Greece has long been debated. The Ancient Greeks wrote that geometry originated in Egypt. Pythagoras and Thales are believed to have studied in Egypt. The historian Herodotus, and other commentators, ascribed the origins of the subject to the Egyptians.

The Ancient Egyptians had no theorems or proofs but presented their findings in the form of problems with specific numerical values and with solutions worked out. They had a small number of general methods; for example, they could solve simple equations of first and second degree. But, in general, the state of development of the subject in Egypt was severely limited. Moreover, there is little evidence of the philosophical and speculative frame of mind that was such an inspiring force among the Greeks, whose mathematics reached impressive heights.

The axiomatic foundations of geometry and the systematic methods of proof are not found in Egyptian mathematics. These were essential aspects of what has been called “The Greek Miracle”.

Sources

${\bullet}$ Gillings, R.J., 1972: Mathematics in the Time of the Pharaohs. Dover, 288pp. ISBN: 0-987-4862-4315-3.

${\bullet}$ Joseph, George G., 2010: The Crest of the Peacock: Non-European Roots of Mathematics. Third Edition. Princeton University Press. Princeton University. ISBN: 978-0-6911-3526-7.

${\bullet}$ Wikipedia article Rhind Mathematical Papyrus. Link.

${\bullet}$ That’s Maths: Pythagorean Triples. Link.

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