Posted on: 3 August 2012

A History of Zero
By J J O'Connor and E F Robertson
School of Mathematics and Statistics, University of St Andrews, Scotland

One of the commonest questions which the readers of this archive ask is: Who discovered zero? Why then have we not written an article on zero as one of the first in the archive? The reason is basically because of the difficulty of answering the question in a satisfactory form. If someone had come up with the concept of zero which everyone then saw as a brilliant innovation to enter mathematics from that time on, the question would have a satisfactory answer even if we did not know which genius invented it. The historical record, however, shows quite a different path towards the concept. Zero makes shadowy appearances only to vanish again almost as if mathematicians were searching for it yet did not recognise its fundamental significance even when they saw it.

The scene now moves to India where it is fair to say the numerals and number system was born which have evolved into the highly sophisticated ones we use today. Of course that is not to say that the Indian system did not owe something to earlier systems and many historians of mathematics believe that the Indian use of zero evolved from its use by Greek astronomers. As well as some historians who seem to want to play down the contribution of the Indians in a most unreasonable way, there are also those who make claims about the Indian invention of zero which seem to go far too far. For example Mukherjee in [6] claims:-

... the mathematical conception of zero ... was also present in the spiritual form from 17 000 years back in India.

What is certain is that by around 650AD the use of zero as a number came into Indian mathematics. The Indians also used a place-value system and zero was used to denote an empty place. In fact there is evidence of an empty place holder in positional numbers from as early as 200AD in India but some historians dismiss these as later forgeries. Let us examine this latter use first since it continues the development described above.

In around 500AD Aryabhata devised a number system which has no zero yet was a positional system. He used the word "kha" for position and it would be used later as the name for zero. There is evidence that a dot had been used in earlier Indian manuscripts to denote an empty place in positional notation. It is interesting that the same documents sometimes also used a dot to denote an unknown where we might use x. Later Indian mathematicians had names for zero in positional numbers yet had no symbol for it. The first record of the Indian use of zero which is dated and agreed by all to be genuine was written in 876.

We have an inscription on a stone tablet which contains a date which translates to 876. The inscription concerns the town of Gwalior, 400 km south of Delhi, where they planted a garden 187 by 270 hastas which would produce enough flowers to allow 50 garlands per day to be given to the local temple. Both of the numbers 270 and 50 are denoted almost as they appear today although the 0 is smaller and slightly raised.

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This pair of researchers write very well. For those who are interested in the topic, I recommend you search out on the net everything they've written...

When it comes to the 'history of Indian Mathematics'... I would rely on Rohini Chowdhury... more than anyone else!

No sabĂ­a que el cero lo inventasen los indios. Pues es un gran invento.

(An anecdote narrated by a friend years ago) Once R.K. Laxman (yes, the very same indomitable "common man" cartoonist of the Times of India) called upon Mr. Bertrand Russell in London some time in the late sixties. During the course of their meeting Russell, with a twinkle in his eyes, commented that India had contributed nothing to philosophy. Seeing the consternation his remark had caused, he quickly added, "India gave Zero to the world!"

The Indian zero seems to be more of a place holder in the number counting system. I am not sure if "operations" using zero as a number were known to Indians. In other words, zero was not recognised as a number in itself by Indians. Even as a number, zero has a peculiarity that it cannot be the denominator of any fraction or division by zero is not a permissible mathematical operation. It can be added to, subtracted from or can be raised to but cannot be a divisor. Also use of zero as the equaliser in an equation is unique. Is there any evidence that the Indian mathematicians treated zero as a number?

Shekhar, A brief article : "...... the Indians were the first to see that zero can be used for something beyond nothing - at different places in a number, it adds different values. For example, 76 is different from 706, 7006, 760 etc. Indian zero alluded to in the question was a multi faceted mathematical object: a symbol, a number, a magnitude, a direction seprator and a place holder, all in one operating with a fully established positional number system. Such a zero occured only twice in history- the indian zero which is now the universal zero and the Mayan zero which occupied in solitary isolation in central america around the beginning of commaon Era. Brahmgupta (598 AD - 660 AD) was the first to give the rules of operation of zero. A + 0 = A, where A is any quantity. A - 0 = A, A x 0 = 0, A / 0 = 0 He was wrong regarding the last formula. This mistake was corrected by Bhaskara (1114 AD - 1185 AD), who in his famous book Leelavati, claimed that division of a quantity by zero is an infinite quantity or immutable God."