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Re: Re: More Fun with Zero!

by spudzeppelin (Pilgrim)
 on Jul 23, 2001 at 21:13 UTC ( #99082=note: print w/replies, xml ) Need Help??

in reply to Re: More Fun with Zero!
in thread More Fun with Zero!

Now you got me thinking. I have always been taught (as most of us have) that n^0 is 1. Now, why, do you suppose, that n multiplied by itself zero times, would it equal one?

It's a matter of consistency: remember, when you take two values which are exponents of the same base and multiply them together, you get that base, to the sum of the exponents. So, what is

((n)^1) * ((n)^-1)     ?

It is     n * (1/n) = (n/n) = 1.

This behaves politely for all real n != 0. But, it still remains 1 for all n->0 from each side, so it's what our friend in analysis call (and what tilly alluded to above) a removable singularity. That is, there is definitely a "hole" there, but for all-intents-and-purposes we can "plug" it when doing calculus.

Spud Zeppelin * spud@spudzeppelin.com

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Re: Re: Re: More Fun with Zero!
by hding (Chaplain) on Jul 23, 2001 at 22:45 UTC

The essence of the difficulty in this case is that defining a "consistent" value of the function f(x,y)=x^y isn't possible. By definition, x^y is defined as e^(y log x). (Which, incidentally, is itself not well-defined; one has to make a branch cut for log to have it make sense. If you don't know what that means, just assume that we've done it, as it doesn't really impact this discussion very much.) Note that the limit as x->0 (with y <> 0 fixed) is 0 and the limit as y->0 (with x <> 0 fixed) is 1. So we can't fill in a value at (0,0) that is going to make the function continuous there.

However, as others have noted, one can make a sensible convention 0^0=1 -- sensible in the sense that it makes doing certain things easier. There are other contexts in mathematics where one makes similar conventions (without sacrificing rigor, of course) in order to simplify notation or statements of results and so on. (E.g. in measure theory one might use the convention that 0 * infinity = 0.)

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