# Mathematicsmathsnumberhistory Greekromanindianarabitalianeuropeanphilosophy

A Big, Fat Zero.

The prodigious power of nothing.

A paradox that probably doesn’t really occur to people until someone with the warped, twisted mind of, say, ex-Monty Pythoner Terry Jones points it out.

Interviewed by the B.B.C. for his 2005 programme, “The Power of One” he said Europeans started using the zero in medieval times:

“”The numerals we use are from India. We think they are Arabic but they aren’t, The Arabs got it from the Indians and we got it from the Arabs. But don’t ask me when that was…I’m hopeless with dates. I’d never make an academic.”

Historians believe the concept was proposed by Indian astronomer Brahmagupta. His treatise was then translated into Arabic and forwarded in about 620AD.

The Romans, whose system was in use in much of Europe until the late Middle Ages, had no proper concept of nothing as a number.

“Roman numerals may have held the Romans back. Their system was so cumbersome and they never produced any mathematicians, unlike the Greeks.”

Their famous and long-lived numerals didn’t cover zero. Instead, they just gave it a name: “nulla”, or nothing. Other culture used other methods to indicate the absence a value – not one looked upon it as a bona fide number in itself. The Babylonians credited with the earliest system of writing in the west and with place-value system like that of modern times for over a millennium, simply used two wedges instead of our zero.

Even this took centuries to develop. Prior to about 400 B.C, they relied on the context of the number to tell them if 216 was intended or 2106. As
J J O’Connor and E F Robertson point out in A history of Zero:
“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.”

Here, the zero is clearly a punctuation mark verbally ensuring the numbers had the correct interpretation.

The seemingly-simple and basic ten-digit set of numerals, from which any other number could be synthesised, was like all evolutions,one which spanned an astonishing quantity land and milennia.

But, just thinking about the difficulties created by this ‘unnumber’ – which is what zero is, a symbol for something that isn’t in existence – it’s pretty easy to see why it took so long to be accepted.

One of the uses of zero is as an empty place A number, 2106 for instance the zero ensures the positions of the 2 and 1 are correct. Clearly, 216 is “something completely different”.

The Greeks Terry Jones mentioned took their maths very seriously. To them, however, numbers weren’t matters for philosophical discussion; they intended to make use of them. They found the area of a triangle and unknown quantities using geometric proofs, determined the value of pi. None of these studies encompassed or included the counting or mensuration of nothing!

If the area of of a farm had x acres of land on one side and y on another, neither x nor y could ever signify nothing, the second modern use of zero! This developed into fiercely-heated debate on the nature of numbers.Could numerical status be applied to a …nothingness.

The system of numeracy in India was also based on a place-value system and, here in C.E. 876, is found the first accepted use of zero as a number. An inscription on a stone tablet dates itself to 876. The inscription concerns the town of Gwalior, 400 km south of Delhi. Here a garden measuring 187 by 270 hastas was planted to allow 50 garlands per day to be given to the local temple. Both numbers, 270 and 50, are denoted almost as they appear today although the 0 is smaller and slightly raised.

An immediate problem arises when zero and negatives as numbers is how they interact in regard to the operations of arithmetic, addition, subtraction, multiplication and division. The difficulties were taken up by three India mathematicians over 500 years.

Brahmagupta stated

“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.”

while saying also

“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. “

Clear as mud so far? It gets worse because he says a number multiplied by zero is zero but even he ties up in in knots when tackling division:-

“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

Within a couple of centuries, Mahavira buffed up Brahmagupta, saying ” … a number multiplied by zero is zero, and a number remains the same when zero is subtracted from it. However, both he and Bashkara, some 300 years later, found division a Gordian knot, the latter bravely and confusingly asserting that

“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.”

None of the whiz-kids , it seems, could bear to face up to the fact that no number can be divided by nothing.

Then, given the opening of zero being established, a wonderland of weirdness popped out of the Pandora’s box: negative numbers, irrational numbers, decimals and imaginary numbers, all of which certain mathematicians (specifically Pythagoras), shall we say disquieting.
When Arabic numerals, with its base-ten system, zero became a necessity The number zero as a standalone entity truly became important with the advent of algebra, for often times in solving an algebraic equation, the number zero becomes a crucial element by itself, while still retaining its basic meaning the representation of nothing.

It was the Italian Fibonacci who grew up among the Arabian Moors of modern Algeria. Clearly recognizing the superiority of their methods of counting over the accepted well- obsolete Roman way, he wrote “Liber Abaci” (The Book of Calculating) and introduced the ten-digit oppositional system with zero we know and love (!) today.