To the 0th power

On twitter I got involved in a discussion on exponents. At first it started with square roots being the same as ‘to the power of 0.5’. The first tweet commented on a blog that stated suprise about the fact that students did not know why the square root of a number was equivalent to ‘to the 0.5th power’. Of course this can be readily proved by using the definition of sum of exponents. I certainly agree with the fact that a lot of students (and teachers?) don’t know why some rules apply. But to me, this is closely related to all the rules that we use. So later on the discussion became more general, and was about when you assume certain properties and when you don’t assume them. For example. proving $g^0=1$ can be done by using the rule for exponents $g^a/g^b=g^{a-b}$ and so if $a=b$ then $g^0=1$ . But you could also say that, just like $g^a/g^b=g^{a-b}$ , $g^0=1$ could be assumed when proven. If you would state: ‘but I want $g^0=1$ to be proven from the most basic rules and assumptions’ this seems a fair request.

However, then I would say it is not necessary to assume the rule $g^a/g^b=g^{a-b}$ but that ${g^a} \cdot {g^b} = g^{a+b}$ is enough. Because let $a=0$ then ${g^0} \cdot {g^b}$ which is (under this law) $g^{0+b}=g^b$ . So ${g^0} \cdot {g^b}= g^b$ and so $g^0$ must be 1.

Proofwiki has an excellent list of the (inter-)dependency of exponent laws. I know it perhaps is nitpicking if I say ‘assuming less is better’ but some of the rules and laws we commonly use are not always that trivial. For students this is a nice way to approach it. It even has a link to a great question by a student whether we have to define or prove $g^0=1$ . Actually I don’t really care whether you prove it or not, but I do think some ‘feel’ for assumptions, axioms, dependencies etc. is useful for everyone. This was the only point I was trying to make.

A special case involves $0^0$ which is indeterminate.

@sndrclsn (who also made this nice picture) tweeted about his great movie bij James Tanton on “zero to the zero”-th power.

It sparked many more musing, like plotting $x^y$ in Wolfram and in Google. But there are other views, like this one stating that $0^0=1$ and also the Math Forum has some interesting quotes.