Ask A Scientist Mathematics Archive

name         Ronak S.
status       student
age          16

Question -   Why is it that (infinity^0) still remains infinite, when
theory tells us that any number raised to 0 is 1? If so how can the above
conjecture be proved?
----------------
Ronak,

The quantity (infinity^0) is what mathematicians call an "indeterminate
form".  It is not "infinite" as you suggest.  It is grouped in a class of
mathematical expressions that are not defined and require other tools and
ways to investigate situations that lead to these indeterminate forms.
Other examples of indeterminate forms include (0/0), (0^0), (1^infinity).

Here is a web site that has some elementary details about the definition of
indeterminate forms and their evaluations:

http://www.sosmath.com/calculus/indforms/intro/intro.html

Regards,

Todd Clark, Office of Science
U.S. Department of Energy
=====================================================
Ronak,
Infinity is not a number.  Infinity is a limit.  When you use limits,
unexpected things can happen.  It is possible for infinity^0 to be infinite.
It is possible for infinity^0 to be zero.  It is not actually defined.  A
simpler example appears with division.

What is infinity divided by infinity?  Most people would say it equals one.
(infinity)x(infinity)=(infinity), so (infinity)=(infinity)/(infinity).
Depending on how you figure it out, you can get infinity divided by infinity
to equal just about anything.

Dr. Ken Mellendorf
Physics Instructor
Illinois Central College
=====================================================