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.