Centuries-old homework
Have a spot of fun putting yourself in the place of Babylonian schoolchildren by trying to solve a couple of ancient math problems.
The full 360
Perhaps best known for their use of base-60, rediscovered tablets have allowed historians to discover a lot about Babylonian numerals.
Dams and drills
You probably know a bit about the Egyptians, but do you know about their legacy? It’s not all mummies and pyramids. Take a fresh look at the Egyptians.
Related programmes
The classical Greeks believed that mathematics was invented in Egypt but there is disappointingly little evidence of the Egyptians’ mathematical attainments. Dr June Barrow-Green examines the evidence that exists.
The Egyptian number system is not too difficult to follow since integers were written according to a decimal system, with different symbols being used to represent the powers of ten: 1, 10, 100, … up to 1,000,000. In hieroglyphic notation these symbols were written additively with each symbol being repeated as often as necessary, eg the number 472 was expressed by writing the symbol for 100 four times, the symbol for 10 seven times, and the symbol for 1 twice. In hieratic, each number from 1 to 9 had a specific symbol, as did each multiple of 10, each multiple of 100, and so on. Thus in hieratic a given number, such as 472 was expressed by putting the symbol for two next to that for seventy and putting both of these symbols next to the symbol for four hundred. Although a zero is not necessary in such a system, the Egyptians did have a symbol for zero but it only occurs in papyri dealing with architecture and accounting.
Egyptian calculations were fundamentally additive. The most frequent operations were doubling and halving. Multiplication is reduced to repeated additions, and division, because it is the inverse of multiplication, is seen in terms of what one number must be multiplied by in order to get another, e.g. a problem such a 100 divided by 13 would be given as multiply 13 so as to get 100.
The most remarkable feature of Egyptian mathematics is its use of fractions. All fractions, with the lone exception of 2/3, are reduced to sums of what we call unit fractions, that is fractions with numerator 1, eg 1/2, 1/7, 1/34. Like integers, unit fractions are written additively, so that:
In hieroglyphic fractions are written with an elongated oval above the whole number, and in hieratic fractions are written with a dot over the whole number. The exception, 2/3, had its own special symbol. The reduction to sums of unit fractions was made possible by tables which gave the decompositions for fractions of the form 2/n, e.g:
The Rhind papyrus contains a 2/n table giving the decompositions for all odd n from 5 to 101.
Many of the problems are quite simple and do not go beyond a linear equation with one unknown. They deal with everyday concerns, such as the strength of bread and of different kinds of beer, the feeding of animals and the storage of grain, although often the numbers involved mean that the problems do not have any basis in reality. There are also geometrical problems, mostly related to measuring, and they too are conceived in a practical setting—finding the volume of a granary was particularly popular. From these problems we know that the Egyptians had formulae for the area of a triangle and of a circle, a value for the constant we call pi of 256/81 (3.1605), and formulae for solid volumes, such as the cube, the cylinder and, remarkably, the truncated pyramid. We are still waiting to find an Egyptian formula for an ordinary pyramid!
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Content last updated: 13/01/2005
