Date: Sat, 16 Dec 2000 02:58:05 -0500

From: Cale Gibbard

Under addition, 0 has no effect. a+0=a However, under multiplication, not 0, but 1, is the value that does this: 1a=a, since multiplication can be thought of as simply repeated addition. Multiplying a*b (a,b elem N) gives: a+a+..+a (b times) and adding 0 to the front of this results in the same value. Under multiplication the value 1 holds this special role. Exponentiating a^b (a, b elem N) gives: a*a*a*...*a (b times) and multiplying by 1 in front gives the s ame value. so:

0+a+a+a+a...+a (b times) = a*b

1*a*a*a*a...*a (b times) = a^b

Hopefully this pattern should make it clear why raising things to the exponent 0 gives 1. There are 0 factors a in the expansion, but that doesn't affect the fact that 1*a^0=1 (1 not multiplied by a is 1). We wouldn't want to mess that up by defining a^0=0 since then 1*a^0=0, that is, not multiplying 1 by a would give 0. If you like that - you're free to build your own branch of mathematics from that arithmetic, but don't blame me if you end up empty handed, or with no hands at all for that matter.

64485484 |