Egyptian mathematics

Summary: Ancient Egyptians were adept at engineering and geometry and deeply dependent on accurate measurements of the annual Nile flood.

Our knowledge of Egyptian mathematics (3000-1000 b.c.e.) is based on hieroglyphic writings found on stone or as script (hieratic and demotic) in multiple papyri. Preserved in tombs and temples in the Nile valley, a papyrus is a narrow scroll of paper, about 15 feet in length, made by interweaving tiny strips of a water reed called papu. The key documents are the Moscow, Rhind, Rollin, and Harris papyri. These works are generally thought to be textbooks used by scribes to learn mathematics and solve problems.

In ancient Egypt, mathematics was used for many purposes necessary to everyday life: measuring time, drawing straight lines, measuring and recording the level of the Nile floodings, calculating land areas, and managing money and taxes. The Egyptians were also one ancient culture that came closest to determining the true length of Earth’s year with mathematics. Perhaps most well known to the modern world are the fantastic tombs, pyramids, and other architectural marvels constructed using mathematics. Though their knowledge ranged from arithmetic calculations to algebraic rules to geometrical formulas to numerical ideas, historians consider the Egyptians’ mathematical achievements to be somewhat less advanced compared to the Babylonians.

Egyptian Number System

Egyptian numbers are written using a simple grouping system whose symbols denote powers of 10. Their symbols included a vertical staff (10⁰), heel bone (10¹), scroll (10²), lotus flower (10³), pointing finger (10⁴), tadpole (10⁵), and astonished man (10⁶):

Using these symbols, a number was expressed additively. For example, the base-10 number 4501 was represented by a visual collection of 4 lotus flowers, 5 scrolls, and 1 vertical staff. As no place-value system is involved, these symbols can be written in any order or arrangement visually they equal a numerical value as a group. Though able to represent large values of numbers with these symbols, the Egyptians’ lack of place values deterred their ability to calculate proficiently using algorithms.

Again represented by hieroglyphic symbols, Egyptian fractions were restricted to unit fractions (numerator of 1) except for the special fraction 2/3. For example, the unit fraction 1/3 was represented by an ellipse (or dot) placed visually over 3 vertical staffs. The Egyptians had no symbol for zero as a place holder but such was not really needed because of their simple grouping system and use of distinct symbols for each power of 10.

Egyptian Arithmetic

Addition and subtraction are quite easy using the Egyptian numbers, involving only the union or removal of the grouped symbols. In addition, a symbol that appeared 10 times was replaced by the next higher level symbol; for example, 10 vertical staffs could be replaced by 1 heel bone. Similarly, in subtraction, a symbol could be traded in for 10 of the next lesser symbol if such was necessary. For example, to perform 23—8, a heel bone could be traded for 10 vertical staffs so that 8 vertical staffs could be taken from the 13 vertical staffs.

Egyptian multiplication involved repeated addition, using a doubling process along with a counter. For example, to multiply 23×13, their process (in modern notation) would look like the following, with the counter on the right:

23 1*

46 2

92 4*

184 8*

Using the starred counters (1+4+8=13), the product is obtained by adding the associated numbers (23+92+184=299). The key to this multiplication is the distributive process, since

Thus, base two notation also is the underlying principle, since

These processes of duplation and mediation (doubling and halving) remained as standard algorithms in Western mathematics until the 1500s.

Division required an inversion of the multiplication process. For example, to divide 299 by 23, the Egyptian scribe determined what number times 23 would produce 299, using a process like the following (in modern notation):

23* 1

46 2

92* 4

184* 8

Using the starred sums, 23+92+184=299, the desired factor (or quotient) is obtained by adding the associated numbers, or 1+4+8=13. The division process becomes complicated when no combination of the starred numbers equals the desired sum (for example, 300 divided by 23), requiring the use of unit fractions:

23* 1

46 2

92* 4

184* 8

1* 1/23

For more difficult divisions (for example, 301 divided by 23), considerable creativity was needed.

To aid in their computations, the Egyptians created tables for doubling and halving numbers, complemented by special 2/ⁿ tables that would help avoid odd-number situations. For example, the Rhind papyrus had a 2/ⁿ table for the odd numbers 5-101.

Egyptian Algebra

Though without an algebraic notation, the Egyptians solved numerous types of algebraic equations, known as “aha” calculations. The majority of their problems were linear equations with one unknown (called the “heap”). Their solution process involved the method of false position, where an initial guess is made, examined, and then adjusted to obtain the correct solution. This same process is now fundamental to the area of numerical analysis and is used extensively for scientific computing using computers.

Consider this Egyptian problem, “Heap and a seventh of the heap together give 19.” In modern notation, the associated linear equation is x + ^x/7 = 19, while their step-by-step solution was the following:

Make a guess for heap, for example, 7

The processes of multiplication and division, as well as the law of associativity, play very important roles:

The majority of the Egyptians’ “aha” problems created practical situations requiring the use of ratios and proportions, such as determining feed mixtures or combinations of grains to make bread. In some instances, the Egyptians did use special hieroglyphic symbols as part of their algebraic work, including “plus” (legs walking left to right), “minus” (legs walking right to left) and other ideograms for “equals” and the “heap.”

Egyptian Geometry

The Egyptians’ geometry was rooted in an algebraic perspective, devoid of any evidence of generalization or proof. Approximately one-fourth of the problems found in the papyri are geometrical focusing on practical measurements, such as the calculation of land areas, or volumes of storage containers. Similar to the Babylonians, the Egyptians used prescriptive formulas. For example, they viewed a circle’s area as equal to that of a square erected on 8/9 of the diameter. That is,

implying their value of π approximated 3.160493827.

Historians agree that the Egyptians knew key formulas for computing the area of a triangle, the volume of a cylinder, some curvilinear areas, and even the volume of the frustum of a square-based pyramid. These formulas were apparently put to great use by the Egyptians in their accurate construction of the pyramids, feats that required a solid understanding of ratios, proportions, dihedral angles, and even astronomy. No evidence suggests the Egyptians knew of the relationships described by the Pythagorean theorem. Some of their geometrical prescriptions were also incorrect. For example, the area of a general quadrilateral (with ordered side lengths a, b, c, d) was calculated by the formula

which is correct only if the quadrilateral is a rectangle or square.

Signs of Advanced Mathematical Thinking

Egyptian mathematics was utilitarian in its direct ties to the solution of practical problems. Also, because their numeration system involved simple grouping with no place values, it is not reasonable to expect that the Egyptians had explored ideas such as factors, powers, and reciprocals. This limitation perhaps explains why no record has been found of tables involving Pythagorean triples. Nonetheless, they did apparently use some number tricks; when multiplying a number by 10, they merely replaced each hieroglyphic symbol by the symbol representing the next higher power of 10 (that is, replacing each vertical staff with a heel bone, each heel bone with a scroll, and so forth).

Problem 79 in the Rhind Papyrus suggests that the Egyptians did some recreational mathematics that had no real-world applications. The problem states, “7 houses, 49 cats, 343 mice, 2401 ears of spelt, 16,807 hekats.” Historians assume that the scribe was creating a problem involving seven houses, each with seven cats, each of which eats seven mice, each of which had eaten seven ears of grain, each of which had sprouted seven grains of barley”…wanting to know the total number of houses, cats, mice, ears of spelt, and grains. Mathematically, the solution of this problem would require some knowledge of powers of 7 and geometric progressions.

Bibliography

Aaboe, Asger. Episodes From the Early History of Mathematics. Washington, DC: Mathematical Association of America, 1975.

Friberg, Jöran. Unexpected Links Between Egyptian and Babylonian Mathematics. Singapore: World Scientific Publishing, 2005.

Katz, Victor J., ed. The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook. Princeton, NJ: Princeton University Press, 2007.

Van der Waerden, B. L. Science Awakening. Oxford, England: Oxford University Press, 1985.

. Geometry and Algebra in Ancient Civilizations. Berlin: Springer, 1983.