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What is the smallest number? This question was once deemed unimportant, however now is puzzling many hunderds of mathematicians. Here there work is summed up. For further reading, consult New Scientist (26 Jan 2002) or go get a degree in pure mathematics.
How small is small?
is quite small,
is very very small,
so what must equal???
Archimedes was one of the first to think of this 'number'. He called it a fluxion. Newton also came across is when he formed The Calculus, and this concept of reducing a distance soooo small that it is practically zero, yet still can be used to divide, is fundemental to any high-school calculus.
BUT... If you have a number, can't you simply divide it in two to make a smaller number? In this case, the infinitesimal couldn't possibly exist, as you could always divide it.
HANG ON... Whoever said a number had to exist to be used? Scientists and matematicians currently think this weird number is an imaginary, or complex, number. This means it does not 'exist'. If you think of numbers as a line.... eg. -3 -2 -1 0 1 2 3 that goes on forever, then these imaginary numbers are not on this line, they are off to the side a bit. Weird, huh?
So there you have it - The infinitesimal. Do you think that it exists? If you come up with a solution, send it to a university - You will have solved one of the great holes in mathematics.