Why can't you have zero to the power of zero?
2 Answers
This is a really good question. In general, and in most situations, mathematicians define
But that is the short answer. This question has been debated since the time of Euler (i.e. hundreds of years.)
We know that any nonzero number raised to the
And that zero raised to a nonzero number equals
Sometime
Two source I used are:
http://mathforum.org/dr.math/faq/faq.0.to.0.power.html
Well, you kind of could have
The problem (if it is a problem) is that they don't agree on what the definition should be.
Consideration 1:
For any number
This is actually a definition of what the zero exponent mean. It's a definition chosen for good reasons. (And it doesn't "break" arithmetic.)
Here's one of the good reasons: defining
For example,
So what about
Cancellation (reducing the fraction) gives us
So, maybe we should use the same rule to define
But . . .
Consideration 2
For any positive exponent,
So if it's true for positive exponants, maybe we should extend it to the
Consideration 3
We have looked at the expressions:
Now look at the expression
graph{y=x^x [-1.307, 3.018, -0.06, 2.103]}
One of the things you may notice about this, is that when
In some fields in mathematics, this is good reason to define
Final notes
Definition is important and powerful, but cannot be used carelessly. I mentioned "breaking arithmetic". Any attempt to define division so that division by
Last note: the definitions of