Once, while I was still a child in school, I heard that
'minus times minus is plus'. How strange it seemed that
negatives could cancel out -- as though two wrongs could
make a right. math helper; I wondered if there could be something else,
still like arithmetic, but having yet another sign. Why
not make up some number things, I thought, that go not
just two ways, but three? I searched for days, making up
new little multiplication tables. Alas, each system ended
either with impossible arithmetic (e.g., with one and two
the same), with no signs at all, or with an extra sign.
Eventually, I gave up. If I had had the courage to persist,
as Gauss did, I might have discovered the arithmetic of
complex numbers, or, as Pauli did, the arithmetic of spin
matrices. But no one ever finds a three-signed imitation
of arithmetic because, it seems, it simply doesn't exist.
Try, for example free online math tutor, to make a new number system that's like the
ordinary one except that it skips some number -- say, 4. It
just won't work. Everything will go wrong. You'll have to
decide what 2 plus 2 is. If you say that this is 5, then 5
will have to be an even number, and so also must 7 and 9.
Then, what's 5 plus 5? Is it 8, or 9, or 10? You'll find that
to make the new system at all like arithmetic you'll have to
change the properties of all the other numbers. Then, when
you're done, you'll find that you have changed only those
numbers' names and not their properties at all. more examples on math forum.
