Nothing

A Little Confusion to Stir with Your Coffee from Nothing by Brian Greene.

The importance of nothing:

"We have heard that when it arrived in Europe, zero was treated with suspicion. We don't think of the absence of sound as a type of sound, so why should the absence of numbers be a number, argued its detractors. It took centuries for zero to gain acceptance. It is certainly not like other numbers. To work with it requires some tough intellectual contortions, as mathemati­cian Ian Stewart explains.

"Nothing is more interesting than nothing, nothing is more puzzling than nothing, and nothing is more important than nothing. For mathematicians, nothing is one of their favorite topics, a veritable Pandora's box of curiosities and paradoxes. What lies at the heart of mathematics? You guessed it: nothing.

"Word games like this are almost irresistible when you talk about nothing, but in the case of math this is cheat­ing slightly. What lies at the heart of math is related to nothing, but isn't quite the same thing. 'Nothing' is ­well, nothing. A void. Total absence of thingness. Zero, however, is definitely a thing. It is a number. It is, in fact, the number you get when you count your oranges and you haven't got any. And zero has caused mathematicians more heartache, and given them more joy, than any other number.

Zero, as a symbol, is part of the wonderful invention of 'place notation.' Early notations for numbers were weird and wonderful, a good example being Roman numerals, in which the number 1,998 comes out as MCMXCVIII ­one thousand (M) plus one hundred less than a thousand (CM) plus ten less than a hundred (XC) plus five (V) plus one plus one plus one (III). Try doing arithmetic with that lot. So the symbols were used to record numbers, while calculations were done using the abacus, piling up stones in rows in the sand or moving beads on wires.

Ancient Egyptian symbol for zero (bottom left) also the symbol for "beautiful."

"At some point, somebody got the bright idea of representing the state of a row of beads by a symbol­ -- not our current 1, 2, 3, 4, 5, 6, 7, 8, 9, but something fairly similar. The symbol 9 would represent nine beads in any row -- nine thousands, nine hundreds, nine tens, nine units. The symbol's shape didn't tell you which, any more than the number of beads on a wire of the abacus did. The distinction was found in the position of the symbol, which corresponded to the position of the wire. In the notation 1,998, for instance, the first 9 means nine hundred and the second ninety.

"The idea of place notation made it rather important to have a symbol for an empty row of beads. Without it, you couldn't tell the difference between 14, 104, 140 and 1,400. So in the beginning the symbol for zero was inti­mately associated with the concept of emptiness, rather than being a number in its own right. But by the 7th century, that had started to change. The Indian astrono­mer Brahmagupta explained that multiplying a number by 0 produced 0 and that subtracting 0 from a number left the number intact. By using 0 in arithmetic on the same footing as the other numbers, he showed that 0 had genuine numberhood.

"Pandora's box was now wide open, and what burst forth was -- nothing. And what a glorious, untamed, infuriating nothing it was.

"The results obtained by doing arithmetic with zero were often curious, so curious sometimes that they had to be forbidden. Addition had the same effect as sub­traction: the number stayed the same. Linguistic purists may object that leaving something unchanged hardly amounts to addition, but mathematicians generally prefer convenience to linguistic purity. Multiplication by zero, as Brahmagupta said, always yielded zero. It was with div­ision that the serious trouble set in.

"Dividing 0 by a non-zero number is easy: the result is 0. Why? Because 0 divided by 7, say, should be 'what­ever number gives 0 when multiplied by 7,' and 0 is the only thing that fits the bill. But what is 1 divided by 0? It must be 'whatever number gives 1 when multiplied by 0.' Unfortunately, any number multiplied by 0 gives 0 not 1, so there's no such number. Division by zero is therefore forbidden, which is why calculators put up an error message if you try it.

"Instead of forbidding fractions like 1 divided by 0, it is possible to release yet another irritant from Pandora's mathematical box -- by defining 1 divided by 0 to be 'infinity.' Infinity is even weirder than zero; its use should always be accompanied by a government warning: 'Infin­ity can seriously damage your calculations.' Whatever infinity may be, it isn't a number in the usual sense. So mostly it's best to avoid things like 1 divided by 0.

"Sorry: Pandora's curse is not so easily evaded. What about 0 divided by 0? Now the problem is not an absence of suitable candidates, but an embarrassment of them. Again, 0 divided by 0 should mean 'whatever number gives 0 when multiplied by 0.' But since this is true whatever number you use to divide' by, unless you're very careful, you can fall into many logical traps -- the simplest such being the 'proof' that 1 = 2 because both equal 0 when they are divided by 0. So 0 divided by 0 is also forbidden."