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Re: More Fun with Zero!by John M. Dlugosz (Monsignor) 
on Jul 24, 2001 at 00:31 UTC ( #99157=note: print w/ replies, xml )  Need Help?? 
If you multiply an inteterminte form by zero, it's like Unknown OR True being True no matter what the left side was. Anything times zero is zero, so NotSure * 0 is certainly zero. So... Log 0 times 0 is zero, whatever we decide about the Log of zero. antilog of 0 is 1. So figuring out by the normal numeric methods, we get a result of 1 for 0**0. Ah, but what is an antilog? We're back to 1**0 there. So, it's a tautology, but consistent! Consistancy is the real point. Take a graph of y=42**x for all real values of x. When x is 0.1, you get one and a half. For 0.01, 1.04. For 0.0001, you get 1.0004. Likewise for negative values of x. If you graph it, you see a big V pointing right at (0,1). If you zoom in, you find that out of all the infinite (second level infinite, yet!) points on the curve, *one* is missing. Yuck. Now calculus deals with that all the time. You can't do it directly, but you can sneak up on it, finding the "limit" as x approaches zero. That's one of the reasons calculus was invented. So, it's definitly true that the limit as x approaches 0 of 42**x is zero. What does that buy us? Like in programming, special cases are a pain. If in real work you tacidly assume you mean the limit, the work becomes a lot easier, and you get the right (useful?) answer anyway. Like I pointed out at the top, this produces consistant results when used in larger systems. So for all intents, n**0 is defined to be 1. Now, do the same thing varying n. Draw the family of curves, and 42**x gives a V. 18**x gives a V with a different angle. .00000001 gives a nice V, too. All the curves in the family point to (0,1). What happens when n hits zero? BOOM! Again, take the limit and you get n**0 is 1 as n approaches arbitrarily close to zero.
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