       Re: 0^0 = 1?

• To: mathgroup at smc.vnet.net
• Subject: [mg95644] Re: 0^0 = 1?
• From: Dave Seaman <dseaman at no.such.host>
• Date: Fri, 23 Jan 2009 05:09:45 -0500 (EST)
• Organization: Information Technology at Purdue
• References: <gl7211\$c8r\$1@smc.vnet.net> <gl9mua\$ajr\$1@smc.vnet.net>

```On Thu, 22 Jan 2009 11:56:58 +0000 (UTC), dh wrote:

> Hi,

> 0^0 means the limit if both base and exponent go to zero.

No, that is not how 0^0 is defined.  Does 2+2 mean the limit as both
summands go to 2?  The value may happen to be the same in that case, but
that is not how 2+2 is defined.

The value of x^y for cardinal numbers x and y is the cardinality of the
set of mappings from y into x.  In the case where x and y are the empty
set, there is exactly one such mapping.  Hence, 0^0 = 1.

It's a theorem of ZF (as stated in Suppes, _Axiomatic_Set_Theory_) that
m^0 = 1 for every cardinal number m.

Another way is to notice that 0^0 represents an empty product, whose
value is the identity element in the monoid of the integers (or the
reals).

In:= Product[0,{k,0}]

Out= 1

One might also consider the series expansion for Exp, which reduces to

1 = 0^0/0! + (lots of terms that all reduce to zero).

Having x^y be discontinuous at (0,0) does not "cause problems" any more
than having the Sign function be discontinuous at 0 causes problems.
Anyone who works with limits should be aware that you can't just blindly
assume continuity when evaluating limits.  You have to consider the
actual definition of the limit.

--
Dave Seaman
Third Circuit ignores precedent in Mumia Abu-Jamal ruling.
<http://www.indybay.org/newsitems/2008/03/29/18489281.php>

```

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