[Edu-sig] Update from Urner [why 0**0 == 1?]

[Kirby Urner]

Thanks for the discussion.

An explanation of why 0^0 is undefined in standard arithmetic
is that anything-except-0 to the 0th power is 1, while 0 to
the anything-except-0th power is 0.  So when you try to deal
with the exception (0^0), you've got two rules which converge
to different answers at the limit i.e.

    lim x^0 = 1    versus   lim 0^x = 0
    x->0                    x->0

To take an excerpt from math-teach, you don't want 0^0 to be
defined because:

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Subject: Re: "0^0"
From: Soroban <afetrmath2@aol.com>
Date: 14 Jul 02 22:10:35 -0400 (EDT)

I use the following approach.

Remember the problem with "zero power"?
   What does 5^0 mean?  And why does it equal 1?

I remind them that: x^6/x^2 = x^4.  (The Division Rule:
   "With like bases, subtract exponents."

So when first confronted with: x^4/x^4, we may invoke the
   Division Rule and get x^(4-4) = x^0, which seems to be
   meaningless.
But wait!  We know that x^4/x^4 = 1, either by "cancelling"
   or from "anything divided by itself is 1".
   [Okay - there's ONE exception to that!]

Hence, the answer must be 1.
   And x^0 (as strange as it looks) must also equal 1.

Now, where or why does 0^0 show up at all?
   It must have come from a division problem.
   For example:  0^4/0^4
   And what is the answer?

Using the Division Rule, the answer is 0^0.

Using arithmetic, 0^4 = 0, so the fraction becomes 0/0.
   Why is this called "indeterminate"?  Why isn't it 1?

How do we CHECK a division problem?
   For example: does 12/3 = 4?
   How do we check?  We multiply the quotient (4) by the
   divisor (3) and we get the dividend (12).  Correct!

Very well, I claim that 0/0 = 7.  Care to check it?
   Someone else claims that 0/0 = 1.  It too checks out.
   The same for 0/0 = -137, etc.
It turns out that 0/0 equals ANY NUMBER.  There are TOO MANY
   answers (the value is indeterminate, can't be determiined).
   That renders the "number" 0^0 useless to us.

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[Full text in Jul 15 posting under:
http://www.mathforum.com/epigone/math-teach/brandghaichox ]

Sounds to me like there are good arguments on both sides, but
I personally am tilting towards 0^0 should produce a ValueError
(or whatever exception), as it does in Mathematica (but not
in MathCad, which says it's 1).  In the real world, we have to
conclude that there's no universal agreement on how best to
evaluate 0^0.  Interesting.

Kirby