Every one knew how laborious the usual Method is of attaining to Arts and Sciences; whereas by his Contrivance, the most ignorant Person at a reasonable Charge, and with a little bodily Labour, may write Books in Philosophy, Poetry, Politicks, Law, Mathematicks and Theology, without the least Assistance from Genius or Study.

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The Evolution of Numbers

Posted on Tuesday 13 February 2007

Counting in groups of ten, a practice evidently suggested by the number of fingers on two hands has been practiced by many cultures for millennia . But the act of writing them down using a system of place notation system which can expand without limit is only comparatively recent innovation.

Here then is the story of 1 to 9 and of 0 and 10.

In the last few centuries before the Christian era, by our reckoning, India emerged from a dark age that had endured since the fall of the Indus valley civilization fifteen hundred years earlier. It was at this time that the written word started to reappear, especially in the form of edicts and inscriptions left by Ashok , the great emperor of the Mauryan Empire. These words were written in a script known as Brahmi and in amongst its letters we find symbols to express numeric quantities which look like this:


Even in this embryonic form it is possible to see the outlines of their future shapes, but it is important to realise that they did not as yet comprise a fully developed place-notation system, something which requires the symbol zero. Instead Brahmi used special symbols to represent 10, 20, 30, 100, 1000 and so on. The inclusion of zero or "nothing" as a numeral occurred some time around 600 AD and it transformed the Indian counting system into one that allowed numbers to expand without end. It could achieve this remarkable feat economically and without cumbersome notation or need to invent more and more symbols, a feature that all previous systems lacked. In computer parlance, the new positional system was really scalable .

As a slight diversion it is worth looking at how the Greeks represented numbers at the time. Many of us are familiar with Roman numerals but what system did the Greeks use? All of the famous classical mathematicians were Greeks, right?
The Sand Reckoner

Greek mathematical notation was not positional; it utilized many symbols and was cumbersome to work with.



The "M" is a myriad , and represents 10,000. The Greek work is murious (uncountable, pl. murioi ). The Romans converted to this to myriad .
It has been argued that the reason why this innovation occurred in India rather than the West was largely because of a peculiarly Indian fascination with astronomically huge numbers .
The traditional Indian cosmology states that the universe undergoes cyclic periods of birth, development and decay, lasting 4.32×109 years, each of these periods is called a Kalpa or ``day of Brahma''. During each Kalpa the universe develops by natural means and processes, and by natural means and processes it decays; the destruction of the universe is as certain as the death of a mouse (and equally important). Each Kalpa is divided into 1000 ``great ages'', and each great age into 4 ages; during each age humankind deteriorates gradually (the present age will terminate in 426,902 years). These is no final purpose towards which the universe moves, there is no progress, only endless repetition. We do not know how the universe began, perhaps Brahma laid it as an egg and hatched it; perhaps it is but an error or a joke of the Maker.

This description of the universe is remarkable for the enormous numbers it uses. The currently accepted age of the universe is about 1018 seconds and this corresponds to about 7 Kalpas+335 great ages. A unique feature of Indian cosmology is that no other ancient cosmology manipulates such time periods.

In the Surya Siddanta it is stated that the stars revolved around the cosmic mountain Meru at whose summit dwell the gods. The Earth is a sphere divided into four continents. the planets move by the action of a cosmic wind and, in fact, the Vedic conception of nature attributes all motion to such a wind. It was noted that the planets do not move in perfect circles and this was attributed to ``weather forms'' whose hands were tied to the planets by ``cords of wind''
The Brahmi script went through a continuous evolution, spawning numerous variants, the most important of which was the Devanagari (or sometimes simply Nagari) script. With Devanagari numerals, the 1 was rotated by 90 degrees and had developed a serif-like loop at the top. The 2 and 3 took on their familiar shapes due to shortcuts taken by scribes, who chose to link the parallel bars rather than lifting their pens.



Knowledge of Indian numerals spread quickly to the West. As early as 662, Severus Sebokht, a Nestorian bishop who lived in Keneshra on the Euphrates river, wrote:
I will omit all discussion of the science of the Indians, ... , of their subtle discoveries in astronomy, discoveries that are more ingenious than those of the Greeks and the Babylonians, and of their valuable methods of calculation which surpass description. I wish only to say that this computation is done by means of nine signs. If those who believe, because they speak Greek, that they have arrived at the limits of science, would read the Indian texts, they would be convinced, even if a little late in the day, that there are others who know something of value.
However, it had to wait until the Arab conquests before the Indian numerals began to be adopted widely and even then only very gradually. In the 11th century, the Muslim mathematician and astronomer al-Biruni referring to Indian numerals wrote:
Whilst we use letters for calculation according to their numerical value, the Indians do not use letters at all for arithmetic. And just as the shape of the letters that they use for writing is different in different regions of their country, so the numerical symbols vary.
While the Devanagari numerals already look quite familiar to Western eyes, in the process of adoption by the Arabs led to a stylistic split between East and West. The Western Arabs of Morocco and Andalusia continued to use numerals that quite closely resembled their Devanagari forebears, even as late as the 14th century:



However in the East, the numerals evolved quite rapidly in a different direction.

This example comes from a work dating from 969.


But only 120 years later they looked like this.



And this is what they look like today in modern Arabic


On closer examination, it can be seen that the numbers 2, 3 and 7 have become rotated by 90 degrees but the other figures have not. One explanation for this is that Arab scribes who write from right to left do so by turning the paper 90 degrees so that the right hand edge is at the top. Lines are then laid down by writing them from top to bottom in columns. It's thought that some scribes less familiar with the Indian signs failed to rotate them correctly.

From Spain and North Africa, the Devanagari numerals passed practically without modification to Europe and the rest is, so to speak, history...

Margarita philosophica by Gregor Reisch (early 16th century)
Pythagoras thinks: Hmm, me thinks this referee dame is unfairly prejudiced.

But while migration of the Indian numerals westward was to have a dramatic effect on later developments, it would be wrong to think that this was the only direction of their movement.The first millennium AD was India's Golden Age, a time when India's power and prestige were at their zenith and its culture was being transmitted to all of its neighbours, both East and West.

This was the time of Greater India .


Buddha statue on the upper terrace of Borobudur Stupa, Java, Indonesia

So here then is a brief survey of some of the other paths taken.

Tibetan


Burmese



Cham


Western (Cambodia)
Eastern (Vietnam)


Khmer



Thai


Javanese



Further examples can be found at the excellent Omniglot .