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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  1. S
    Sadanand Patwardhan
    Mar 18, 2015 at 2:18 pm
    **any number, no matter how large, could be expressed as a perfect ratio of two natural numbers** Please tell us professor how could Pythagoras do his RATIOS without Positional Number System, which Greeks did not know.
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      JJ
      Mar 18, 2015 at 9:21 pm
      My question is broader than theories or subjects: if India and Indians were indeed that advanced in sciences then why has India always lagged behind China and European civilizations and economies? At least going by writings of scholars and even Babur the so called Golden Sparrow was little more than an arid brown wasteland - where is the evidence of that intellectual excellence? How could such towering intellects translate into nothing? Why did India not build weapon systems to protect itself from invaders and why are there no records of its ancient magnificence (line the Great Wall or the Pyramids). What happened to that knowledge - it was obviously not applied for welfare of mes. Did India just meditate on it?
      Reply
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        Anil Suri
        Mar 18, 2015 at 5:48 am
        Baudhayana merely mentions the 'Pythagorean' theorem. It was known to the Harappans from about 3500 BCE. Archaeological evidence has been found dating back to around 3500 BCE suggesting that the Harappans could accurately calculate the hypotenuse for a right-triangle, making them the earliest to know this theorem, that too in full detail. A few centuries later, the Babylonians got an inkling of this theorem too, but they could not accurately calculate the hypotenuse - they merely had sets of 'Pythagorean numbers' (ie {3,4,5} etc) obtained by crude measurement. Please see the book, 'From Sumer to Meluhha' edited by the Wisconsin Uni archaeologist and great authority on the Indus Valley Civilisation, Prof. JM Kenoyer for details.
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          Gauhar Vatsyayan
          Mar 18, 2015 at 2:04 pm
          Did I read it correctly. Is it really from a JNU scholar. Nice work
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          1. S
            Shashank
            Mar 19, 2015 at 1:45 am
            thoroughly enjo it, waiting for another such article
            Reply
            1. S
              Srilata
              Mar 18, 2015 at 10:57 pm
              In our youth (I am 71), we often asked this question. Later, a physicist had an answer that I could accept. It seems that after Buddhism was ousted, the caste distinctions became very rigid. The thinkers thought but that was not translated to the craftsmen. equipment was not invented beyond what was in practice. Sciences and mathematics were stultified. This went on until the East India Company the French came with their modern weapons. It was for this reason that Ram Mohan Ray asked for schools that taught modern sciences and mathematics, and not the Sanskrit, Persian or Arabic texts.
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                gary george
                Mar 18, 2015 at 3:40 am
                Greeks have so much in common with the Indians. Question is, who influenced whom?
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                  Giridharan Velamore
                  Mar 18, 2015 at 9:32 am
                  It's a wonderful article that throws significant light in the development of the concept of zero. I have read somewhere that the question, 'Should we represent a sign/ figure to denote nothingness' was one of the greatest questions. The development of zero is indeed amazing and considering that it was done couple of millennium ago is so great.
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                    politicalonion
                    Mar 18, 2015 at 3:12 pm
                    Greeks did not know, does not mean, Pythagoras did not know positional number system. If he could 'know' the theorem from India, he could have easily known other things that came with it from India.
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                      politicalonion
                      Mar 18, 2015 at 3:09 pm
                      If Pythagoras could borrow his theorem from India, he could have borrowed positional number system too. Some people would just keep on lying. Pity.Truth is the new secular, Chew it.
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                        politicalonion
                        Mar 18, 2015 at 1:28 pm
                        Pleasant surprise. Excellently written, factual, nor a bundle of lies denouncing ancient India neither a jingoistic flight of imagination, making up stories of ancient glories of India.
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                          Indyadhanush
                          Mar 19, 2015 at 2:41 am
                          Actually the Church had banned the Hindu aka Arabic Numericals as " THE WORK OF THE DEVIL "........ It took another 300 to 400 years before the Hindu Numericals were accepted and called Arabic numericals..... Westerners, lots of Muslims ( and Hindus too ) in India still think that the Arabs invented the numericals....
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                            Srivatsa
                            Mar 18, 2015 at 9:14 am
                            Good one! It's true that India contributed heavily to mathematical theory. However, you might want to edit the language to clarify uncertainty in statements which are unproven - to distinguish them from known facts (few leads from other comments below).
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                              Tomar
                              Mar 18, 2015 at 4:16 am
                              The beginning of the article is based on several myths. It is not known whether Pythagoras had a proof for his theorem. It is also not proved that Hippacus discovered irrational numbers and whether he was killed for that. Similarly, it is not clear if Sulbhashatras have a proof for pythagoras theorem. several ancient civilizations knew of pythagoras theorem. Overall, one can only say that many different people from all different parts of the world contributed in their own way to the mive edifice of science and math that we see today.
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                                S.Karyampudi
                                Jul 14, 2015 at 12:19 am
                                In a small paragraph it is difficult to answer all your questions. Indian civilization is a very ancient one. It had produced many intellectuals in many fields. All the knowledge is still in Sanskrit language books. Many were removed from the country to far of lands . You are from Seattle, you may very well understand about the first Nation people and what happened.
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                                  N.s Parameswaran
                                  Mar 18, 2015 at 9:31 am
                                  try teaching these concepts in india today. try to incorporate these facts in indian text books. you will be immediately hounded out by the MMPS (Macualite-marxist-pseudo-secular) crowd for saffronising the education system. This is the tragedy of "SECULAR" India.
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                                    Raja
                                    Mar 18, 2015 at 7:49 am
                                    This article is basically a recycling of issues that are well-known. The author does not present anything interesting.
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                                      Prashant Kulkarni
                                      Mar 19, 2015 at 6:28 am
                                      Not sure what is reason the occasion for this article...but certainly good to have it detailed account of the well-known fact of Indian intellectual tradition... I cover similar Indological articles in my blog below. Prashant Blog:htpps:ppkya.wordpress
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                                        Raj
                                        Mar 18, 2015 at 9:57 am
                                        While well written, please also note that other civilizations have contributed to numericals and the pythagoras theorem equally. I think Mesopotamia also had the concept of zero, probably prior or at the same time as the Indians. So, we all simultaneously came up with these innovations. It's best to avoid nationalism in common human history - likewise, it would not be good if the contributions of Indians is downpla (as it has become a fad in India) to an extent where every innovation is umed to be Western and everything Indian is ridiculed. Indians in history had their place and their contributions, like the rest of the world, is undeniable.
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                                          Rajesh
                                          Mar 18, 2015 at 8:20 am
                                          Surprised this article has come from a professor from JNU. Usual theme JNU is that we learnt every thing from invading aryan race. Even though it has been proven otherwise. Anyway a good starting for JNU. Though he have to watch out for vultures in JNu.
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                                            Girish Rao
                                            Mar 18, 2015 at 8:44 am
                                            Hopefully these become part of our text books and students are made aware and the feel proud of their ancestors and its rich culture.
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