# Why do mathematicians need to define a zero exponent?

I think a zero exponent can be defined logically from various premises;

1. We can define an exponent as the number of times the base "appears" in a multiplication process. Thus: $2^3 = 2*2*2$, $2^1 = 2$, $2^0 = ( )$ doesn't appear at all.

Here's the thing, I know that we can literally define $2^{(something)}$ to be anything and as long as we are consistent with our definition, we can show that any result follows directly from our rules and definitions.

1. But mathematicians observed (and defined) a very nice property of exponents, namely; $b^n * b^m = b^{n+m}$ Thus, if we we want to be consistent with this definition AND WE WANTED TO INCLUDE THE NUMBER ZERO IN THIS, and we put n=0, then we have that $b^0 * b^m = b^m$ And since we have already defined this process $Anything * 1 = anything .....(*)$ Then it should follow that in order to be consistent with our rules, bexp 0 has to be defined as 1. (ofcourse if (*) was defined such as $anything * 5 = anything$, then we'd have to say that $b^0 = 5$).

My question is, however, why do we need to check the case when $n=0$? Why is it important/helpful? What problems will we face if we simply ignored $0$? I mean, the number $0$ wasn't even defined before? ofcourse the number $0$ alone is very useful in other contexts, but why here?

Lastly, fractional exponents are defined in such a way that they "act" like roots; thus making writing and dealing with roots easier, negative exponents make quotients easier and so on.. Is there some situation where we would for example get (something raised to the zero) as a result and therefore we would have to KNOW what that must mean?

I'm writing from a mobile version of the website so I can't use the Math symbols. Sorry about that. Thank you.

• To make negative exponents meaningful. – Bernard Feb 14 '17 at 18:53
• It's the obvious choice if you want $\exp$ to be continuous everywhere on $\Bbb R$ (or even just $\Bbb R_{\ge 0}$). – Bobbie D Feb 14 '17 at 18:57
• "I'm writing from a mobile version of the website so I can't use the Math symbols" I'm pretty sure you can notwithstanding. – MathematicsStudent1122 Feb 14 '17 at 20:15
• To make continual exponents over the reals meaningful, which is to make continuous growth functions calculable. – fleablood Feb 14 '17 at 22:21

You're absolutely right that it is not compulsory to define extended versions of $a^b$. However, it is very often convenient to extend notational devices to situations where the original sense does not directly apply. And, naturally, we'd like the extensions to behave as much like the originals as possible, so that our acquired intuition and reflexes for manipulation of the notation/idea can likewise extend to a larger class of situations. In particular, as you observe, the goal is not just to define things, but to set notational conventions compatible with previous more limited use. (Otherwise we're pranking ourselves by setting ourselves up for silly errors.)
First I'd say that $b^0=1$ is a natural generalization of $b^1=b,$ $b^2=b*b,$ etc. To see this, note we could write $b^1 = 1*b,$ $b^2 = 1*b*b,$ etc. Since $1$ is the multiplicative identity, $1$ is the natural answer to the seemingly-nonsensical question "what if I multiply $b$ zero times".
Secondly, as you correctly note in point $2$, the left hand side of the equation $b^{m+n} = b^m b^n$ makes sense when $m$ or $n$ is zero (or, for that matter when one of them is negative but still $m+n>0$) and the definitions $b^{0} = 1,$ $b^{-1}=1/b,$ $b^{-2}=1/b^2$ make the right hand side equal the left hand side in these cases.
Being that we have a nice natural extension of exponentiation to the integers, why not define it that way? You ask whether you ever need to take something to the zero-th power. Well, in a literal sense that's asking if I ever need to take an arbitrary number and turn it into $1,$ so yeah that's not that useful. However since the definition's natural it often happens that we can and want to interpret the expression $b^0$ when it comes up. Here's a couple of examples: