Zero - Charles Seife [27]
Sometime around the fifth century AD, Indian mathematicians changed their style of numbering; they moved from a Greek-like system to a Babylonian-style one. An important difference between the new Indian number system and the Babylonian style was that Indian numbers were base-10 instead of base-60. Our numbers evolved from the symbols that the Indians used; by rights they should be called Indian numerals rather than Arabic ones (Figure 14).
Nobody knows when the Indians made the switch to a Babylonian-style place-value number system. The earliest reference to the Hindu numerals comes from a Syrian bishop who wrote, in 662, of how the Indians did calculations “by means of nine signs.” Nine—not ten. Zero was evidently not among them. But it’s hard to tell for sure. It is fairly clear that the Hindu numerals were around before the bishop wrote about them; there is evidence that zero appeared in some variants of the Indian system by that time, though the bishop hadn’t heard about it. In any case, a symbol for zero—the placeholder in the base-10 numbering system—was certainly in use by the ninth century. By then Indian mathematicians had already made a giant leap.
The Indians had borrowed little of Greek geometry. They apparently didn’t have a deep interest in the plane figures that the Greeks loved so much. They never worried about whether the diagonal of a square was rational or irrational, nor did they investigate the conic sections as Archimedes had. But they did learn how to play with numbers.
The Indian system of numbering allowed them to use fancy tricks to add, subtract, multiply, and divide numbers without using an abacus to help them. Thanks to their place-number system, they could add and subtract large numbers in roughly the same way we do today. With training, a person could multiply with Indian numerals faster than an abacist could tally. Contests between the abacists and the so-called algorists who used Indian numerals were the medieval equivalents of the Kasparov versus Deep Blue chess match (Figure 15). Like Deep Blue, the algorists would win in the end.
Figure 14: The evolution of our numerals
Though the Indian number system was useful for everyday tasks like addition and multiplication, the true impact of Indian numbers was considerably deeper. Numbers had finally become distinct from geometry; numbers were used to do more than merely measure objects. Unlike the Greeks, the Indians did not see squares in square numbers or the areas of rectangles when they multiplied two different values. Instead, they saw the interplay of numerals—numbers stripped of their geometric significance. This was the birth of what we now know as algebra. Though this mind-set prevented the Indians from contributing much to geometry, it had another, unexpected effect. It freed the Indians from the shortcomings of the Greek system of thought—and their rejection of zero.
Figure 15: An algorist versus an abacist
Once numbers shed their geometric significance, mathematicians no longer had to worry about mathematical operations making geometric sense. You can’t remove a three-acre swath from a two-acre field, but nothing prevents you from subtracting three from two. Nowadays we recognize that 2 – 3 = –1: negative one. However, this was not at all obvious to the ancients. Many times they solved equations only to get a negative result and concluded that their answer had no meaning. After all, if you are thinking in geometric terms, what is a negative area? It simply didn’t make any sense to the Greeks.
To the Indians, negative numbers made perfect sense. Indeed, it was in India (and in China) that negative numbers first appeared. Brahmagupta, an Indian mathematician of the seventh century, gave rules for dividing numbers by each other, and he included the negatives. “Positive divided by positive, or negative by negative, is affirmative,” he wrote. “Positive