Powers of Infinity
Name: Ronak S.
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?
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:
Todd Clark, Office of Science
U.S. Department of Energy
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
Illinois Central College
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Update: June 2012