# Egyptian Mathematics: The Ancient Egyptian Number System

**by**Caroline Seawright

**March 19, 2001**

In ancient Egypt mathematics was used for measuring time, straight lines, the level of the Nile floodings, calculating areas of land, counting money, working out taxes and cooking. Maths was even used in mythology - the Egyptians figured out the numbers of days in the year with their calendar. They were one of the ancient peoples who got it closest to the 'true year', though their mathematical skills. Maths was also used with fantastic results for building tombs, pyramids and other architectural marvels.

A part of the largest surviving mathematical scroll, the Rhind Papyrus (written in hieratic script), asks questions about the geometry of triangles. It is, in essence, a mathematical text book. The surviving parts of the papyrus show how the Egyptian engineers calculated the proportions of pyramids as well as other structures. Originally, this papyrus was five meters long and thirty three centimeters tall.

It is again to the Nile Valley that we must look for evidence of the early influence on Greek mathematics. With respect to geometry, the commentators are unanimous: the mathematician-priests of the Nile Valley knew no peer. The geometry of Pythagoras, Eudoxus, Plato, and Euclid was learned in Nile Valley temples. Four mathematical papyri still survive, most importantly the Rhind mathematical papyrus dating to 1832 B.C. Not only do these papyri show that the priests had mastered all the processes of arithmetic, including a theory of number, but had developed formulas enabling them to find solutions of problems with one and two unknowns, along with "think of a number problems." With all of this plus the arithmetic and geometric progressions they discovered, it is evident that by 1832 B.C., algebra was in place in the Nile Valley.

Problem no. 56 in the Rhind Papyrus gives an equation to find the angle of the slope of a pyramid's face, which in fact is its cotangent. With a cotangent, one automatically has a tangent by taking the inverse of the cotangent. Moreover, the means were present with pyramidal models to obtain sine and cosine values. Thus, trigonometry was also developed earliest in the Nile Valley. The advanced state of this math is confirmed by an architectural drawing even older than the Rhind Papyrus that shows that Nilotic engineers had learned to find the area under a curve more than 5,000 years ago. Finally, as Flinders Petrie found, the architects had several times built into their structures right triangles that obeyed the theorem: *a*^{2} + *b*^{2} = *c*^{2}, where *a* and *b* are the two sides and *c* is the hypotenuse. Since Pythagoras studied in the temples of the Nile Valley for 22 years it would not have surprised him to learn there was the source of the theorem that bears his name.

The Rhind Papyrus also asks questions like "From a certain amount of grain, how many loaves can be baked?" or "Given a ramp of length *x* and height *y*, how many bricks are needed?" These are typical examples of what Egyptian school students had to do in their mathematics class.

The papyrus was found in Waset (Thebes) in the ruins of a small building near the Rameseum. It is a copy made by the scribe Ahmose during the 15th Dynasty reign of the Hyksos Pharaoh, Apepi I. Ahmose states that his writings are similar to those of the time of Amenemnhat III (1842 - 1797 B.C.)

Egyptians knew addition, subtraction, some division and multiplication. They only multiplied and divided by two, so if they wanted to find *e x 5*, they would use *e x 2 + e x 2 + e*. *13 / 4* was done as *4 x 2 + 4 = 12*, *13 - 12 = 1*, and so the answer was 3 ¼.

Being only able to multiply and divide by two, Egyptian math was unwieldy. To get whole numbers like 32, the Egyptians would have to write: *10 + 10 + 10 + 1 + 1*. Although simple, the way the Egyptians wrote their maths made it long and repetitive.

The Egyptians were somewhat familiar with both roots and square roots. They could plot an arch by using offsets that were measured at regular intervals from a base line, and they could also find out areas. To find the area of a circle, the Egyptians used an area of a square on an 8/9 of the diameter, or (7/8) squared. They could also figure out the area of a triangle. They knew that the volume of a frustum of a square pyramid equalled (1/3) height (*a* squared + *ab* + *b* squared). They also knew that to make right angled triangles, they had to use the ratio of 3:4:5.

As for fractions, 'r' was used for the word 'part'. This means that r-10 is equivalent to our 1/10.

The Egyptian sign 'gs' was used for the word 'side' or 'half' ½. The word 'hsb' meant 'fraction', but it came to mean 'part-4' or ¼. 'rwy' meant 'two parts out of three' 2/3, and 'khmt rw' , though rare, was 'three parts out of four' 3/4.

Other fractions could be made from representing the different parts of the 'wdj3t', the Eye of Horus. It was split up into ½, ¼, 1/8, 1/16, 1/32 and 1/64. The 'wdj3t' (*udjat*) is split into parts because of the myth where Set, Horus' uncle, tore his eye from his head and ripped it to pieces. Thoth, later on, completed the eye, joining the parts together, giving it the name 'the sound eye'. The parts add up to 63/64, and the missing 1/64 was presumably the part filled by Thoth.

The Egyptians, though, had no concept for zero. The zero was invented independently both by the Indians (thanks to Ranjeev Ravi for pointing this out) and the Maya. The Indians used a space for zero, and the Maya used a symbol for zero in their calendars in the 3rd century AD. Eventually, the Indians came to use a dot for zero, which was picked up by the Arabs. Through the Arabs, the number zero reached European civilisation after 800 AD. The ancient Egyptians, as with the ancient Greeks and Romans, had no use for zero.

In their daily lives, the Egyptians who used mathematics most likely were priests and priestesses in charge of workers, surveyors, masons and engineers, tax collectors, shop keepers and at least some of the buyers, and cooks. The higher form of maths, of course, was done by those with the building-related jobs and the priests. Shop keepers, cooks and the lower classes probably only used the simple types of mathematics that we, today, use in our every day lives.

The Great Pyramid of Khufu from the Fourth Dynasty was a mathematical wonder:

It was laid out with geometric precision - a near-perfect square base, with sides of 230 meters that differ from each other by less than twenty centimeters, and faces that sloped upwards at an angle of 51º to reach an apex nearly 150 meters above the desert floor.

There are about 2,300,000 giant, heavy stone blocks (approx. 2.5 tons each) in the pyramid, which are placed so close together that a knife blade can not be inserted between them! The sides of the square base have an error of less than 1/14,000, while the right angles have an error of less than 1/27,000.

Some people have made certain discoveries about the Great Pyramid, using maths:

- When using the Egyptian cubit the perimeter is 365.24 - the amount of days in the year
- When doubling the perimeter, the answer is equal to one minute of one degree at the equator
- The apex to base slant is equal to 600th of a degree of latitude
- The height x 10 to the power of 9 gives approximately the distance from the earth to the sun
- The perimeter divided by 2 x the height of the pyramid is equal to
*pi*- 3.1416 - The weight of the pyramid x 10 to the power of 15 is equal to the approximate weight of the earth
- When the cross diagonals of the base are added together, the answer is equal to the amount of time (in years) that it takes for the earth's polar axis to go back to its original starting point - 25,286.6 years
- The measurements of the King's Chamber gives 2-5-3 and 3-4-5 which are basic Pythagorean triangles

Statistics can prove anything, so all of this should be taken with a grain of salt.

What the Egyptians did with their unwieldy mathematics system was, actually, pretty fantastic. Not only did they just use it in their day-to-day lives, but they built one of the Seven Wonders of the ancient world. Something that we, today, can not replicate despite our more complex mathematics and our modern technology!

**© Caroline 'Kunoichi' Seawright 2001 - present**

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