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Sorry for my English.When b is a positive integer, is defined to be the product of a multiplied by itself b times.
I do not understand why we agree with the axiom : .
The question is, what is the most natural way to extend this definition to the case when b=0? Here are several ways to see that the definition is the only reasonable one:
Therefore, should be the number of ways of writing no numbers, each of which is from 1 to a. There is exactly one way of doing this, namely, don't write any numbers at all!
(This reason is more compelling if you make it more mathematically precise, using the fact that is the number of functions from a b-element set B to an a-element set A, and when b=0 the set B is the empty set, and there is exactly one function from the empty set into A, namely, the empty function).
The above reasons all illustrate why defining to be 1 is the only reasonable definition.
There's one other point worth mentioning: some of the reasons above are less compelling when a=0. For instance, in the first reason, we need to have , and if a is non-zero we can divide by to deduce that . However, if a=0 we no longer get a reason for to be 1.
Some of the reasons are still compelling, and, especially if we are in a context where only integer exponents are being considered, we still normally define to be 1.
However, if we define a two-variable function , then this function does not have a well-defined limit as (x,y) -> (0,0). We can define if we like, but the limit still won't exist. In other words, if A and B each approach zero, there's no guarantee as to what (if anything) approaches. It need not approach our definition of .
That's why, in calculus, is often called an indeterminate form. If one is working in situations where the exponent can continuously vary, it is usually better to leave undefined to avoid making mistakes. However, if one is working in situations in which the exponent is always integral, is usually defined to be 1.
These complications are only for . When a is nonzero, is always defined to be 1, for the reasons given above.
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