Alex's Adventures in Numberland - Alex Bellos [48]
This type of enumeration is known as a ‘place-value’ system, which we discussed earlier. An abacus bead has a value dependent on which column it is in. Likewise, each number in the above list has a value dependent on its position in the list. Place-value systems, however, require the concept of a ‘place-holder’. For example, if a number has two dasa, no sata and three sahasra it cannot be written two, three since that refers to two dasa and three sata. A place-holder is needed to maintain the correct positions, to make it clear that there are no sata, and the Indians u a e word shunya – meaning ‘void’ – to refer to this place-holder. The number that is just two dasa and three sahasra would be written as two, shunya, three.
The Indians were not the first to introduce a place-holder. That honour probably went to the Babylonians, who wrote their number symbols in columns with a base 60 system. One column was for units, the next column was for 60s, the next for 3600s and so on. If a number had no value for that column, it was initially left blank. This, however, led to confusion, so they eventually introduced a symbol that denoted the absence of a value. This symbol, however, was used only as a marker.
After adopting shunya as a place-holder, the Indians took the idea and ran with it, upgrading shunya into a fully fledged number of its own: zero. Nowadays, we have no difficulty in understanding that zero is a number. But the idea was far from obvious. The Western civilizations, for example, failed to come up with it even after thousands of years of mathematical enquiry. Indeed, the scale of the conceptual leap achieved by India is illustrated by the fact that the classical world was staring zero in the face and still saw right through it. The abacus contained the concept of zero because it relied on place value. For example, when a Roman wanted to express 101, he would push a bead in the first column to signify 100, move no beads in the second column, indicating no tens, and push a bead in the third column to signify a single unit. The second, untouched column was expressing nothing. In calculations, the abacist knew he had to respect untouched columns just as he had to respect ones in which the beads were moved. But he never gave the value expressed by the untouched column a numerical name or symbol.
Zero took its first tentative steps as a bonafide number under the tutelage of Indian mathematicians such as Brahmagupta, who in the seventh century showed how shunya behaved towards its number siblings:
A debt minus shunya is a debt
A fortune minus shunya is a fortune
Shunya minus shunya is shunya
A debt subtracted from shunya is a fortune
A fortune subtracted from shunya is a debt
The product of shunya multiplied by a debt or fortune is shunya
The product of shunya multiplied by shunya is shunya
If ‘fortune’ is understood as a positive number, a, and ‘debt’ as a negative number, –a, Brahmagupta has written out the statements:
–a – 0 = –a
a – 0 = a
0 – 0 = 0
0 – (–a) = a
0 – a = –a
0× a = 0, 0
×– a = 0
0×0 = 0
Numbers had emerged as tools for counting, as abstractions that described amounts. But zero was not a counting number in the same way; understanding its value required a further level of abstraction. Yet the less that maths was tied to actual things, the more powerful it became.
Treating zero as a number meant that the place-value system that had made the abacus