The story of shunya

How Indian mathematicians defined zero as a concept.

Written by Brishti Guha | Updated: March 18, 2015 12:15 am

In 6th century BC, a mathematician named Pythagoras proved that the square of the hypotenuse of any right-angled triangle is the sum of the squares of its two other sides. By doing so, he acquired everlasting fame. The immediate reception of the proof must have been gratifying for him as well; the Greek colony of Croton declared a 10-day holiday in celebration and had 100 oxen killed, treating the entire population of the colony to a sumptuous feast. Pythagoras had officially become an academic superstar.

We know far less about how the “Pythagorean” theorem was received 200 years before Pythagoras, in 8th century BC India, when the same result was pointed out by Baudhayana in his Sulba Sutra. Baudhayana showed that the square formed by the “diagonal” of a triangle has the combined area of the squares formed by the length and breadth of the triangle — the geometric analogue of the Pythagoras theorem.

Let’s skip 200 years forward to see what Pythagoras did after proving his theorem. He asserted that any number, no matter how large, could be expressed as a perfect ratio of two natural numbers (that is, he believed all numbers were rational). One day, a student (named Hippasus, according to most ancient commentators) made a major discovery. The square root of 2 could never be exactly expressed as a ratio! Yet, Pythagoras’s theorem had already proved that the square root of 2 had a real physical meaning: it was the length of the hypotenuse of an isosceles triangle whose other two sides were of length 1. Pythagoras was now presented with a major threat to his reputation: if he were to uphold the Pythagorean theorem, he must accept that his statement about all numbers being rational was wrong.

His way out of the dilemma was chillingly simple — he murdered his student. Poor Hippasus paid with his life for his intellectual curiosity.

Academics’ lives were relatively more peaceful in ancient India, as far as we know. There was a lack of any rigid preconceptions about the world of numbers. Baudhayana (and later Aryabhata) do not seem to have had the least trouble accepting that numbers could be irrational — both provided approximations for the square root of 2, and “pi”, without being disturbed by the fact that neither could be exactly expressed as a ratio of two natural numbers.

While the more learned among the ancient Greeks were busy protecting their academic reputations, we might presume that less illustrious Greeks — the ones who transacted their day-to-day business in the marketplace — had an easy time dealing with the numbers needed to settle their accounts. This wasn’t the case, though. Arithmetic was extremely difficult before the invention of our modern place-value number system. Just think of Roman numerals to understand why. With symbols for different numbers but no place-value system, there was no easy way of adding two numbers. This might not have mattered much if the numbers were small, but it became more of a handicap when dealing with large numbers. They performed sums by drawing geometrical figures in the sand and adding or subtracting areas of figures, not very efficient. What’s more, the Greeks did not have a zero. They were uncomfortable with the concept of a void. Nor did they have negative numbers, as it made no sense to subtract a larger area from a smaller one.

Thus begins our story of zero as a concept — a story that takes us to India of the 6th and 7th centuries AD, the era of the mathematician Brahmagupta. Even before Brahmagupta, other mathematicians had been using zero, but only as a symbol; they did not know how to perform arithmetical operations with it. Brahmagupta was the first to clearly define zero (as what remains when a number is subtracted from itself) and to explore all its properties. The zero, or shunya, could now be fully integrated into arithmetic and completed the place-value decimal system. Brahmagupta also invented negative numbers as a concept. Rather than treat numbers simply as abstract concepts, however, Brahmagupta was also able to give negative numbers practical significance by calling them “debts” — something that must have instantly resonated with lenders and borrowers.

Brahmagupta’s major work on mathematics, the Brahmasphutasiddhanta or The Opening of the Universe, was written in 628 AD. More than a century later, around 770 AD according to al-Biruni, Caliph al-Mansur of Baghdad heard about Brahmagupta through a visiting Indian scholar, Kanka, who brought with him a copy of the Brahmasphutasiddhanta and would commission an Arabic translation of his book. The Arabs then gradually became comfortable with the concept of zero, which they called sifr. However, zero remained unknown to Europe for another 400 years, until the Moors conquered Spain and brought zero with them. Accountants and businessmen all over Europe eagerly adopted it, finding a simple way of balancing their books by having their assets and liabilities sum to zero. But governments were not as keen — Florence banned it in 1299. One reason provided was that it would be easy for cheats to inflate figures simply by adding a zero at the end. Merchants, however, were not ready to give up zero so easily, and continued to use a secret symbol for it despite the ban. Zero, or sifr, thus became associated with secret codes — the origin of the modern term “cipher”.

There is probably no greater testament to the popularity of a number system than the fact that a secret code was devised in order to keep using it illegally. Brahmagupta could not have known how his number system, complete with zero and negative numbers, would become the number system, just as Baudhayana may not have anticipated how famous his result would become. Unfortunately for them, Baudhayana’s result is now known only as the Pythagorean theorem, while few people know Brahmagupta as the genius behind “Arabic numerals”. (Ironically, the Arab mathematician al-Khowarizmi, who became famous for Arabic numerals, referred to them as “Hindu numerals”.)

The writer is associate professor of economics, School of International Studies, JNU, Delhi

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