The spread of Hindu-Arabic numerals in the tradition of European practical mathematics

by Raffaele Danna (University of Cambridge)

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0, 1, 2, 3, 4, 5, 6, 7, 8, 9

The ten digits we use to represent numbers are everywhere in our modern world. But they reached a widespread diffusion in the ‘west’ only at a relatively late stage. The positional numeral system was central for the development of the scientific revolution, but – contrary to what one might expect – their spread in Europe was not driven just by scientists, but also by practitioners.

How did these numbers reach the almost universal diffusion we see today? What were the causes and broad consequences of their introduction?

As a matter of fact, for a very long time the ‘west’ did not know the numbers we now use every day. People had to rely on Roman numerals and the corresponding reckoning tools (such as counting boards).

Arabic numbers, or more precisely Hindu-Arabic numbers, were invented sometime in fifth century India. From India they spread westwards, together with the spread of Islam, reaching the Mediterranean around the eighth century.

Europe picked up these numbers from the Arabic civilisation, and that is the reason why we call them ‘Arabic’. But it took a long time before Europeans widely adopted Arabic numbers in their practice. This was due to difficult relationships with Islam, but also to the low levels of literacy and numeracy in Europe at the time, together with a more general cultural backwardness in comparison with the Arabic civilisation.

Starting from the eleventh century, Europe experienced an economic renaissance that reached its peak in the thirteenth century. With the development of international trade, several key financial and organisational innovations were introduced. This is the moment when the first international companies appear, together with the earliest examples of banking and international finance.

This new economic complexity raised the need for a higher level of computing power, especially to solve calculations of interest and exchange rates. It is at this stage that merchant-bankers, who were already literate and numerate, realised that Hindu-Arabic numerals suited their needs better than Roman ones. Arithmetic with Hindu-Arabic numerals became part of the required training for merchant-bankers.

By the late thirteenth century, we see the first examples of practical arithmetic texts published in central Italy, the cradle of early finance and banking. From here, the publication of these manuals slowly spread to the rest of Europe, with a dramatic acceleration in the sixteenth century driven by the introduction of the printing press.

A detailed reconstruction of these traditions, comprising more than 1,280 manuals, makes it possible to study the main characteristics of such spread. It was a movement from the south to the north of Europe, with late adopters – such as the north of Germany and England – taking up such texts only in the second half of the sixteenth century.

The spread of these texts allows us to reconstruct a slow process of transmission of practical mathematics throughout Europe. The use of such knowledge transformed economic practices, together with several other fields, such as visual arts, architecture, shipbuilding, surveying and engineering.

During the seventeenth century, this practical mathematics combined with the academic understanding of astronomy, reaching a new synthesis in the scientific revolution. Following the story of the adoption of Hindu-Arabic numerals allows us to appreciate that the scientific revolution was also indebted to more than three centuries of mathematical experimentation carried out by European practitioners.