Babylonian ‘algebra’ from Crest of the Peacock

 

Discussion question: How could one state a general mathematical principle in a time before the development of algebra and algebraic notations? Is mathematics all about generalization and abstraction? Think about various areas of mathematical knowledge – number theory, geometries, calculus, graph theory – imagine how could you imagine stating general or abstract relationship without algebra?

In the time of Babylonians, there were no symbols for the variables and the mathematical operators. That is, they did not have the algebra of our time. Taking advantage of base 60 system, Babylonians had developed extensive tables of multiplications, and also tables of reciprocals (terminating sexagesimal fractions). These developments were believed to be started by the need in accounting matters on the trade. However we also found that they did have techniques to solve mathematical problems involving squares and cubes. Instead of symbols they used words and geometric shapes. They also had technique to find approximate square root and cube roots of some numbers. 

These findings from the Babylonian time show that it is possible to state some general mathematical principles. However it has limitations.

To state this for any positive values of a and A without using a symbol is possible. It would need several sentences. Suppose the word “product” is available or invented to represent the result of multiplying two numbers. Suppose also that we have the words “difference” and “sum” of two numbers; as well as “reciprocal”, “half” and “square root” of a number.

 Suppose we know the ‘difference’ between side of your square and side of my square. My side is bigger than yours. Suppose we also know the ‘product’ of side of my square and side of your square. Then, the following procedure calculates side of my square and side of your square.

Step (1): Half the ‘difference’, and calculate its ‘square’.
Step (2): Add the ‘product’ to the ‘square’ obtained in the step (1).
Step (3): Calculate the ‘square root’ of the result from step (2).
Step (4): Side of my square is the result of step (3) added to the half of the ‘difference’.
Step (5): Side of your square is the result of step (3) less the half of the ‘difference’.

 Without having a symbol for any mathematical operation means, further development will be slowed and limited. We can write the functional relationship between two variables x and y if they are related in a polynomial, reciprocal or root function of a friendly degree to that time, and also we have names to represent the coefficients. However, to describe in details about a family of quadratic functions by varying any one of the coefficients will require a good mastering of the language.