Understanding the Controversy: Simplifying 0^0 in Calculus

[Null_]

Homework Statement

It's a word problem, but I just need to simplify 0^0.

Homework Equations

n/a

The Attempt at a Solution

0^n = 0.
n^0 = 1.




I'm just going through a calculus book on my own this summer. It's not even a calculus question, but I've never come across this before. Which is it? 0 or 1? Why?

 

[Ryker]

I believe it's 0 (or not even defined, such as in poles of a polynom), because from what I can remember n^0 = 1 is by definition only true for those n that are different from 0.

 

[System]

0^0 is undefined.

 

[skeptic2]

From Wolfram Mathworld http://mathworld.wolfram.com/ExponentLaws.html

The definition 0^0=1 is sometimes used to simplify formulas, but it should be kept in mind that this equality is a definition and not a fundamental mathematical truth (Knuth 1992; Knuth 1997, p. 56).

 

[Null_]

Okay, thanks guys. Wolfram is awesome!

 

[LCKurtz]

Consider the binomial expansion:

$$1 = 1^2=(x+(1-x))^2 =\binom 2 0 x^0(1-x)^2+\binom 2 1 x^1(1-x) + \binom 2 2 x^2(1-x)^0$$

This doesn't work for x = 0 or 1 unless you define 0⁰ = 1. That is an example of why it is usually defined as 1. Also why 0! = 1 in the same example.