[Izzy]

I dunno why, but today I realized I had a problem accepting that x⁰=1. (Maybe I'm just saying it wrong in my head, but something multiplying by itself zero times, and that equaling 1 doesn't click with me.) So I came home and looked up the proofs for it, the best so far involves that rule of exponents.

This makes total sense to me:

5^3 / 5^2 = 125 / 25 = 5

5^3 / 5^2 = 5 ^(3-2) = 5^1 = 5

5^3 / 5^3 = 125 / 125 = 1

5^3 / 5^3 = 5 ^(3-3) = 5^0 =

... and the result must be 1. So 5^0 =1.

But I can't help but wonder whether we're expressing it wrong.

Is there any sort of proof verifying that x^b / x^c = x^(b-c)? I'm sure there is, but typing that into Google gave me random weird things, and I'd like to see it for myself.

[Yoruichi-san]

Well, I'm not sure how 'fundamental' of a proof you want...I mean, all forms of math build off some fundamental assumptions (like the shortest distance b/w two points is a straight line), and then works up step by step...so without knowing exactly what step your at I don't know what kind of proof you'll accept...

But x^b/x^c=x^(b-c) seems pretty intuitive to me...if you have b # of x's in the numerator and c # of x's in the denominator, then the c # of x's in the denominator cancel out c # of x's in the numerator, leaving you with the difference, i.e. (b-c) # of x's in the numerator...

If you've accepted that dividing by a number is the same as multiplying by the inverse of that number...an easy 'proof' would be just to say:

1/x^c=x^(-c)

x^b/x^c=(x^b)(x^-c)=x^(b-c)

[Guest]

I dunno why, but today I realized I had a problem accepting that x⁰=1. (Maybe I'm just saying it wrong in my head, but something multiplying by itself zero times, and that equaling 1 doesn't click with me.) So I came home and looked up the proofs for it, the best so far involves that rule of exponents.

There is no proof. x⁰ is defined to be 1 (x != 0) because lim n->0 xⁿ = 1.

Is there any sort of proof verifying that x^b / x^c = x^(b-c)?

It is fairly straightforward to prove by induction....

[Guest]

Keep questioning, Izzy. Never stop questioning! This is wonderful.

What gives a group of x objects its "power"? Arbitrary definition?

What, then, does it mean to have none of that power? Does it mean that the number of objects in the group doesn't matter? Why?

Must a group that lacks any power act as a unit? Why do the individuals within it have no useful identity in this case?

The questioning can get pretty profound.

On the other hand, given that you seem to be mostly considering only integers, maybe you're more comfortable with infinity than with zero and are willing to take d3k3's approach but using: lim (n->infinity) x^(1/n)

[unreality]

all of the above, and also, a null multiplication is assumed to be 1. I forget the exact term let me look it up:

aha! It's Empty Product

maybe not an exact answer but it has to do with 0! = 1 and x^0 = 1 (x != 0 of course, 0^0 is effed up)

[Izzy]

...The result of multiplying no numbers is 1. That's totally messed up. I hate it when things are done for convenience.. Like particles having 'charge'. Bah.

[unreality]

actually it makes perfect sense. 1 is the focal multiplicative identity just like 0 is the focal additive identity (edit - but they are supported with lower-level axioms too)

Edited April 14, 2009 by unreality