What does 0^0 equal? Why do mathematicians and high school teachers disagree?

Technically, 0^0 is an indeterminate form and has no specific solution. Accurate but unhelpful.

Practically, 0^0 highlights the issue that most of us don't have a good conceptual model for what exponents really do. How would you explain to a 10-year old why 3^0 = 1 beyond "it's necessary to make the algebra of powers work out".

I use an "expand-o-tron" analogy

http://betterexplained.com/articles/understanding-exponents-...

to wrap my head around what exponents are really doing: some amount of growth (base) for some amount of time (power). This gives you a "multiplier effect". So, 3^0 means "3x growth for 0 seconds" which, being 0 seconds, changes nothing -- the multiplier is 1. "0x growth for 0 seconds" is also 1, since it was never applied. "0x growth for .00001 seconds" is 0, since a miniscule amount of obliteration still obliterates you.

This can even be extended to understand, intuitively, why i^i is a real number (http://betterexplained.com/articles/intuitive-understanding-...).

That's addressed in the article under "cleverest student."

Fun digression late on a Wednesday :)

The indeterminate form seems the most correct based on the analysis of the limit of f(x,y) = x^y as x approached zero from different paths.

I had always thought of it more of an algebraic identity thing; x^n * x^m = x^(n+m). Obviously x^(n) = x^(n+0) = x^n * x^0 which can only be satisfied if x^0 = 1. But this article (and really, the wikipedia treatment that beej71 linked to) made me think more about it.

The high school teacher in the link is a B.S. in math education. They're usually reflexive Platonists, believing that math is out there, and we merely discover it. This is a result of the teaching of undergrad math as essentially a series of completed works, with little history attached to it. This teacher probably hasn't thought critically about why, say, 1/x^2 = x^(-2).

By contrast, a mathematician has a Ph.D. in math, and has had to do original research and look at some history of topics. Thus, the mathematician knows that math is not a finished product, but is under constant refinement.

Now, let's talk about negative exponents. Students are taught that 1/x^2 = x^(-2). High school teachers often don't understand why. It is so ingrained that even asking "Why?" seems almost grammatically incorrect.

The reason why is that we know the following things about exponents: 1) x^n = x* x* ...* x for n a positive integer 2) As a consequence of 1, x^m* x^n = x^(m+n) for m,n positive integers

Now, the question is not "what is x^n if n is negative?" (which is what a high school teacher might ask). Rather, the question is "Can we define (!) x^n for negative n in a way consistent with the item two above?" (mathematician's framing). And, of course, we can. If x^n = 1/x^(-n) for n negative, then item two works.

So, a high school teacher most likely thinks that the negative exponent rule is simply a rule, handed down from the Gods of math. A mathematician recognizes that it is a convention, and such a smooth convention that there is simply no better choice.

Now, about 0^0: The HS teacher asks "What is 0^0?" and is therefore under the impression that 0^0 is undefined because according to certain reasonings it could be 0 or it could be 1. Textbooks (not written by mathematicians) wouldn't correct this. The TI-86 gives a domain error when 0^0 is input. The mathematician asks "What value of 0^0 makes my preferred formulae continue working?" and thereby defines 0^0=1.