Perhaps the most fundamental contribution of ancient India to the progress of civilisation is the decimal system of numeration including the invention of the number zero. This system uses 9 digits and a symbol for zero to denote all integral numbers, by assigning a place value to the digits. This system was used in Vedas and Valmiki Ramayana. Mohanjodaro and Harappa civilisations (3000 B.C.) also used this system.

If zero merely signified a magniutude or a direction seperatorI(i.e. separting those above the zero level from those below the zero level), the Egyptian zero, nfr, dating back atleast four thousand years, amply served these purposes. The ancient Egyptians (5000 B.C.) had a system based on 10, but they didn't use positional notation. Thus to represent 673, they would draw six snares, seven heel bones and three vertical strokes.

If zero was merely a place holder symbol, indicating the absence of a magnitude at a specified place position (such as, for example, the zero in 10 indicated the absence of any 'tens' in one hundred and one), then such a zero was already present in the babylonian number system long before the first recorded occurence of the Indian zero. Babylonians in Mesopotamia (3000 B.C.) had a sexagesimal system using base 60. Greeks and Romans had a cumbersome system (try to write 2376 in Roman numerals).

If zero was represented by just an empty space within a well defined postional number system, such zero was present chinese mathematics a few centuries before the Indian zero.

Many civilisations had some concept of "zero" as nothing - for example, if you have two cows and they both die, you are left with nothing.

However,

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.

The ancient Indians represented zero as a circle with a dot inside. The word 'zero' comes from the Arabic "al-sifr". Sifr in turn is a transiliteration of the Sanskrit word "

*soonya*" meaning of void or empty, which became later the term for zero. This and the decimal number system fascinated Arab scholars who came to India. Arab mathematician

*Al-Khowarizmi* (790 AD - 850 AD) wrote

*Hisab-al-Jabr wa-al-Muqabala* (Calculation of Integration and Equation) which made Indian numbers popular. "Soonya" became "al-sifr" or "

*sifr*". The impact of this book can be judged by the fact that "al-jabr" became "Algebra" of today.

An Italian

*Leonardo Fibonacci* (1170 AD - 1230 AD) took this number system to Europe. The Arabic "sifr" was called "zephirum" in Latin, and acquired many local names in Europe including "cypher". In the beginning, the merchants used to Roman numbers found the decimal system a new idea, and referred to these numbers as "

*infidel numbers*", as the Arabs were called infidels because they had invaded the holy land of Palestine.

However, nowadays this system is called

*Hindu-Arabic System*. This positional system of representing integers revolutionised the mathematical calculations and also helped in Astronomy and accurate navigation. The use of positional system to indicate fractions was introduced around 1579 AD by Francois Viete. The dot for a decimal point came to be used a few years later, but did not become popular until its use by Napier.

* ***A history of Zero**
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 first thing to say about zero is that there are two uses of zero which are both extremely important but are somewhat different. One use is as an empty place indicator in our place-value number system. Hence in a number like 2106 the zero is used so that the positions of the 2 and 1 are correct. Clearly 216 means something quite different. The second use of zero is as a number itself in the form we use it as 0. There are also different aspects of zero within these two uses, namely the concept, the notation, and the name. (Our name "zero" derives ultimately from the Arabic *sifr* which also gives us the word "cipher".)

Neither of the above uses has an easily described history. It just did not happen that someone invented the ideas, and then everyone started to use them. Also it is fair to say that the number zero is far from an intuitive concept. Mathematical problems started as 'real' problems rather than abstract problems. Numbers in early historical times were thought of much more concretely than the abstract concepts which are our numbers today. There are giant mental leaps from 5 horses to 5 "things" and then to the abstract idea of "five". If ancient peoples solved a problem about how many horses a farmer needed then the problem was not going to have 0 or -23 as an answer.

One might think that once a place-value number system came into existence then the 0 as an empty place indicator is a necessary idea, yet the Babylonians had a place-value number system without this feature for over 1000 years. Moreover there is absolutely no evidence that the Babylonians felt that there was any problem with the ambiguity which existed. Remarkably, original texts survive from the era of Babylonian mathematics. The Babylonians wrote on tablets of unbaked clay, using cuneiform writing. The symbols were pressed into soft clay tablets with the slanted edge of a stylus and so had a wedge-shaped appearance (and hence the name cuneiform). Many tablets from around 1700 BC survive and we can read the original texts. Of course their notation for numbers was quite different from ours (and not based on 10 but on 60) but to translate into our notation they would not distinguish between 2106 and 216 (the context would have to show which was intended). It was not until around 400 BC that the Babylonians put two wedge symbols into the place where we would put zero to indicate which was meant, 216 or 21 '' 6.

The two wedges were not the only notation used, however, and on a tablet found at Kish, an ancient Mesopotamian city located east of Babylon in what is today south-central Iraq, a different notation is used. This tablet, thought to date from around 700 BC, uses three hooks to denote an empty place in the positional notation. Other tablets dated from around the same time use a single hook for an empty place. There is one common feature to this use of different marks to denote an empty position. This is the fact that it never occured at the end of the digits but always between two digits. So although we find 21 '' 6 we never find 216 ''. One has to assume that the older feeling that the context was sufficient to indicate which was intended still applied in these cases.

If this reference to context appears silly then it is worth noting that we still use context to interpret numbers today. If I take a bus to a nearby town and ask what the fare is then I know that the answer "It's three fifty" means three pounds fifty pence. Yet if the same answer is given to the question about the cost of a flight from Edinburgh to New York then I know that three hundred and fifty pounds is what is intended.

We can see from this that the early use of zero to denote an empty place is not really the use of zero as a number at all, merely the use of some type of punctuation mark so that the numbers had the correct interpretation.

Ptolemy in the *Almagest* written around 130 AD uses the Babylonian sexagesimal system together with the empty place holder O. By this time Ptolemy is using the symbol both between digits and at the end of a number and one might be tempted to believe that at least zero as an empty place holder had firmly arrived. This, however, is far from what happened. Only a few exceptional astronomers used the notation and it would fall out of use several more times before finally establishing itself. The idea of the zero place (certainly not thought of as a number by Ptolemy who still considered it as a sort of punctuation mark) makes its next appearance in Indian mathematics. 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.

Brahmagupta attempted to give the rules for arithmetic involving zero and negative numbers in the seventh century. He explained that given a number then if you subtract it from itself you obtain zero. He gave the following rules for addition which involve zero:- *The sum of zero and a negative number is negative, the sum of a positive number and zero is positive, the sum of zero and zero is zero.*

Subtraction is a little harder:-

*A negative number subtracted from zero is positive, a positive number subtracted from zero is negative, zero subtracted from a negative number is negative, zero subtracted from a positive number is positive, zero subtracted from zero is zero.*

Brahmagupta then says that any number when multiplied by zero is zero but struggles when it comes to division:- *A positive or negative number when divided by zero is a fraction with the zero as denominator. Zero divided by a negative or positive number is either zero or is expressed as a fraction with zero as numerator and the finite quantity as denominator. Zero divided by zero is zero.*

Really Brahmagupta is saying very little when he suggests that *n* divided by zero is *n*/0. Clearly he is struggling here. He is certainly wrong when he then claims that zero divided by zero is zero. However it is a brilliant attempt from the first person that we know who tried to extend arithmetic to negative numbers and zero. *... a number multiplied by zero is zero, and a number remains the same when zero is subtracted from it.*

*A number remains unchanged when divided by zero.*

Bhaskara wrote over 500 years after Brahmagupta . Despite the passage of time he is still struggling to explain division by zero. He writes:- *A quantity divided by zero becomes a fraction the denominator of which is zero. This fraction is termed an infinite quantity. In this quantity consisting of that which has zero for its divisor, there is no alteration, though many may be inserted or extracted; as no change takes place in the infinite and immutable God when worlds are created or destroyed, though numerous orders of beings are absorbed or put forth.*

So Bhaskara tried to solve the problem by writing *n*/0 = ∞. At first sight we might be tempted to believe that Bhaskara has it correct, but of course he does not. If this were true then 0 times ∞ must be equal to every number *n*, so all numbers are equal. The Indian mathematicians could not bring themselves to the point of admitting that one could not divide by zero. Bhaskara did correctly state other properties of zero, however, such as 0^{2} = 0, and √0 = 0. Perhaps we should note at this point that there was another civilisation which developed a place-value number system with a zero. This was the Maya people who lived in central America, occupying the area which today is southern Mexico, Guatemala, and northern Belize. This was an old civilisation but flourished particularly between 250 and 900. We know that by 665 they used a place-value number system to base 20 with a symbol for zero. However their use of zero goes back further than this and was in use before they introduced the place-valued number system. This is a remarkable achievement but sadly did not influence other peoples.

You can see a separate article about Mayan mathematics .

The brilliant work of the Indian mathematicians was transmitted to the Islamic and Arabic mathematicians further west. It came at an early stage for al-Khwarizmi wrote *Al'Khwarizmi on the Hindu Art of Reckoning* which describes the Indian place-value system of numerals based on 1, 2, 3, 4, 5, 6, 7, 8, 9, and 0. This work was the first in what is now Iraq to use zero as a place holder in positional base notation. Ibn Ezra , in the 12^{th} century, wrote three treatises on numbers which helped to bring the Indian symbols and ideas of decimal fractions to the attention of some of the learned people in Europe. *The Book of the Number* describes the decimal system for integers with place values from left to right. In this work ibn Ezra uses zero which he calls galgal (meaning wheel or circle). Slightly later in the 12^{th} century al-Samawal was writing:- *If we subtract a positive number from zero the same negative number remains. ... if we subtract a negative number from zero the same positive number remains.*

Fibonacci was one of the main people to bring these new ideas about the number system to Europe. As the authors of [ 12 ] write:-

Of course there are still signs of the problems caused by zero. Recently many people throughout the world celebrated the new millennium on 1 January 2000. Of course they celebrated the passing of only 1999 years since when the calendar was set up no year zero was specified. Although one might forgive the original error, it is a little surprising that most people seemed unable to understand why the third millennium and the 21^{st} century begin on 1 January 2001. Zero is still causing problems!

* ***The Story of Indian Zero**
The story of zero is always been an interesting and proud story for any indian. Today I found the story behind zero which is almost unknown to me

here. I am not a very good learner of vedic mathematics but I could not resist to be a fan of them.

Vedic Mathematics has got most powerful algorithm at the ages of ancient india and rediscovered in 1911 by Sri Bharati Krsna Tirthaji (1884-1960).

**Some interesting notes on Zero Evolution.**

India: 458 A.D. (debated)
The final independent invention of the zero was in India. However, the time and the independence of this invention has been debated. Some say that Babylonian astronomy, with its zero, was passed on to Hindu astronomers but there is no absolute proof of this, so most scholars give the Hindus credit for coming up with zero on their own.

The reason the date of the Hindu zero is in question is because of how it came to be.

Most existing ancient Indian mathematical texts are really copies that are at most a few hundred years old. And these copies are copies of copies of copies passed through the ages. But the transcriptions are error free...can you imagine copying a math book without making any errors? Were the Hindus very good proofreaders? They had a trick.

Math problems were written in verse and could be easily memorised, chanted, or sung. Each word in the verse corresponded to a number. For example,

**viya dambar akasasa sunya yama rama veda**
sky (0) atmosphere (0) space (0) void (0) primordial couple (2) Rama (3) Veda (4)

0 0 0 0 2 3 4

Indian place notation moved from left to right with ones place coming first. So the phrase above translates to 4,230,000.

Using a vocabulary of symbolic words to note zero is known from the 458 AD cosmology text

*Lokavibhaga*. But as a more traditional numeral--a dot or an open circle--there is no record until 628, though it is recorded as if well-understood at that time so it's likely zero as a symbol was used before 628.

Which it probably was, considering that 30 years previously, an inscription of a date using a zero symbol in the Hindu manner was made in Cambodia.
A striking note about the Hindu zero is that, unlike the Babylonian and Mayan zero, the Hindu zero symbol came to be understood as meaning "nothing." This is probably because of the use of number words that preceded the symbolic zero.