history of mathematics - National STEM Centre

22
The Babylonians
Activity 2.10 Quadratic equations
b Find out how many iterations you need to reach the approximation 1; 24,51,10
for V2.
2 Experiment with the algorithm to see how you can find other square roots. What
difference does it make when you start with other values, such as 1 and 5, or 2 and
2\ for V5?
3 How does this algorithm compare to other square-root algorithms that you
know?
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4 Use this algorithm to find a formula for approximating to A/a 2 + h .
Quadratic problems involving one unknown
The existing mathematical tablets from the Old Babylonian period fall broadly into
two categories: mathematical tables, and problems. You have seen examples of
tables in the section about number notation: these include multiplication tables,
reciprocal tables, and the table of squares. These tablets consist solely of tables of
numbers; about four hundred such tablets have been found.
Problem tablets, by contrast, are rarer - only a hundred or so of them have been
found. Below are two examples. Notice that, although the words 'side' and 'square'
are used, the problem is not really geometrical, unless Babylonian geometry was
quite different from ours.
The Babylonians probably used the terms 'side' and 'area' to mean the unknown
quantity and its square, rather than geometric, quantities. This is similar, perhaps, to
the way in which you think of x squared without necessarily imagining a square. Yet
it is possible that the terms 'side' and 'area' trace back to an earlier geometric use.
This is in contrast to the algebra of the Arab mathematicians for whom there was a
strong association between computational procedures and geometric representation.
This association remained until the time of Descartes in western mathematics.
I have added up the area and the side of my square: 0; 45. You write down
1, the coefficient. You break off half of 1. 0; 30 and 0; 30 you multiply: 0;
15. You add 0; 15 to 0; 45: l.This is the square of 1. From 1 you subtract 0;
30, which you multiplied. 0; 30 is the side of the square.
In that example, and in the next, the problem is given in the first sentence; the rest is
its solution. Here is the second problem.
I have subtracted the side of my square from the area: 14,30. You write
down l,the coefficient. You break off half of 1. 0; 30 and 0; 30 you multiply.
You add 0; 15 to 14,30. Result 14,30; 15. This is the square of 29; 30. You
add 0; 30, which you multiplied, to 29; 30. Result 30, the side of the square.
1 Verify that the results 0;30 and 30 solve the problems quoted above.
2 Express the first problem in modern algebraic notation.