Questions about Zero - Undefined or Indeterminate?

From: Norman Rogers
Subject: Nothing about nothing
Date: Wed, 16 Aug 95 22:56:55 EDT

1/	Is 0,0,0,... a geometric sequence ?

2/	What is the value of 0/0 ? (is it really undefined or are there an
	infinite number of values) 

3/	What is the value of 0^0 (^ means exponent)? Can it be considered to
	be the limit of a^0 as a approaches 0 ?

Any thoughts on these would be appreciated. Thanx

Norman Rogers

Date: 809188843
From: Doctor Ken
Subject: Re: Nothing about nothing

Hello!

>1/	Is 0,0,0,... a geometric sequence ?
>
Well, sort of.  You could say that it's a geometric sequence with common
ratio 4, or whatever, but I wouldn't.  The reason is that you can't find
the common ratio by looking at the sequence, dividing one term by the 
previous term.  So I guess I'd say no.  What you might say is that this 
is a degenerate case of a geometric sequence.

>2/	What is the value of 0/0 ? (is it really undefined or are there an
>	infinite number of values) 
>
There's a special word for stuff like this, where you could conceivably
give it any number of values.  That word is "indeterminate."  It's not the
same as undefined.  It essentially means that if it pops up somewhere,
you don't know what its value will be in your case.  For instance, if
you have the limit as x->0 of x/x and of 7x/x, the expression will have
a value of 1 in the first case and 7 in the second case.  Indeterminate.

>3/	What is the value of 0^0 (^ means exponent)? Can it be considered to
>	be the limit of a^0 as a approaches 0 ?
>
0^0 is indeed indeterminate.  It turns out that you could make
it have any value between 0 and 1, inclusive.  You could have 0 if it's the
limit as a->0 of 0^a, you could have 1 if it's the limit as a->0 of a^0, 
and for x in between 0 and 1, (and this is the neat part from Dr. Shimimoto)
look at the expression (x^n)^(1/n).  This just equals x for all positive
values of n.  As n->Infinity, this fraction goes to 0^0, but if it's 
just x the whole time, the limit of the expression as it goes to 0^0 is x.
So we could make it anything in between 0 and 1, so it's got to be 
indeterminate.

-Doctor Ken,  The Geometry Forum