Numbers Through the Ages

Number plays an essential role in our culture as indeed it does in any recognisable form of society. However far we go back in history, we can be certain that number has played its part in ways of thought and in human reaction to the world. This book is essentially about number in some of its many aspects.

Our counting system is based on the number ten. Why just ten? Probably because we have ten fingers. Finger counting is widespread among both primitive and civilised peoples (see Figure 1).

In Chapter 2 the discussion of the counting systems of primitive tribes was confined largely to what is known of their present or recent state. The history of these counting systems is not known in any detail. Hypotheses about the distribution of these systems, interesting though they are, cannot truly be classified as ‘history’. They are speculative and uncertain. However, for the Sumerian and Babylonian number system, there are written sources going back to around 3000 BC. The gradual development of this system from 3000 BC until around 1700 BC (the time of Hammurabi¹) can therefore be studied. After 1700 BC the Babylonian system remained the same until the time of Christ.

The discussion so far has been concerned almost entirely with the history of spoken numbers. This chapter is concerned solely with the history of numerals, i.e. written numbers. It describes numeral systems which have been used in different parts of the world and traces the familiar decimal numbers back to their origin in the Indian subcontinent.

In spoken English certain simple fractions like 1/2 or 1/4 or 3/4 have special names. We do not say ‘one-fourth’ but one quarter, and 1/2 is pronounced as one half. The French have a special name tiers for 1/3 Fractions of this kind, which serve the purposes of everyday life, may be called natural fractions.

In a positional system like our own, calculations with pencil or pen and paper are easy, but in a non-positional notation like that of the Romans they are much more difficult. To see this more clearly, consider the multiplication of 325 by 47. Using our numerals, the calculation is simple: Now try to multiply CCCXXV by XLVII. In the former system, one had to multiply every single figure contained in 325 with every figure in 47 (altogether 6 simple multiplications), to write the partial results in the right places, and to add them. If one tries to do the same thing with CCCXXV and XLVII, the first problem that arises is that XLVII cannot be decomposed into parts X + L + V + I + I, because the notation XL is subtractive. One can try writing XXXX instead of XL and try computing the product CCCXXV.XXXXVII by a method similar to today’s, multiplying every single component C -or- X or V contained in the first factor by every component X or V or I of the second factor. This method, however, will involve 42 single multiplications, followed by the addition of the results. A very cumbersome method!