A few days ago, over the breakfast table, my 9-year-old tells me:
Dad, I told a boy at school that any number to the power zero is one. He said that was idiotic.
Indeed, I had told Ben earlier that any (nonzero) number to the power zero is one. I think I had even tried to explain the reason to him, but I wasn’t surprised that the reason hadn’t stuck. So I felt I had to explain it again. And then yesterday, in the math puzzle thread, that boy-at-school’s more polite uncle (OK, I’m making up the family relationship) commented:
So I’m a lawyer and therefore math dumb, but I don’t understand how you raise a number to a square root power, or to any non-whole number power for that matter. So 2^2 is the same as 2 x 2. 2^3 is the same as 2 x 2 x 2. What is 2^1.4?
Here’s the thing: As Leopold Kronecker supposedly observed in the 19th century, God created the positive integers (and maybe zero), and all else is the work of man. Less metaphysically, when we talk about 2+3, or 2×3, or 6/2, or 3-2, we’re talking about things that obviously correspond to real world phenomena: combining piles of things, or splitting piles of things. When we exponentiate integers, we can also easily conceptualize this as corresponding to an obvious operation on integers: 3^4 is four threes multiplied together. We might throw in the rational numbers to this real-world math as well.
But, seriously, doesn’t it seem a bit idiotic to talk with a straight face about multiplying zero threes together? And even beyond that, negative five — just what is that? Does anyone actually have negative five of anything? (I don’t mean owing five of something, which it turns out can be conveniently represented as having negative five, I mean actually having negative five of something.)
Here’s how I explained things to my 9-year-old: Let’s look at 1234.56. What does the 1 stand for? “Thousands.” And 10 to the power three is ... “One thousand.” How about the 2? “Hundreds.” And 10 to the power two is ... “One hundred!” (Always good to have some confidence-building questions there.) The 3? “Tens.” And 10 the power one is ... “Ten.”
Now at this point, we get to the 4, in the ones place and it sure looks very convenient just to go from 10^3 to 10^2 to 10^1 to 10^0 — the thousands to hundreds to tens to ones. It’s useful to just say that 10^0 = 1. And then since the 5 after the decimal is tenths, you’d want to say that 10^-1 = 0.1, and then that 10^-2 = 0.01, and so on.
There’s another way of thinking about this, I told my son: We know that a^b x a^c = a^(b+c), for instance 2^2 x 2^3 = 2^5 (4 x 8 = 32); always good to see that with familiar, observable integers. If that’s so, then if c = 0, a^b x a^0 = a^(b+0), which means that a^0 = a^b/a^b = 1. And you can do the same with negative powers, too; adapting the formula I gave at the start of the paragraph to negative numbers, we see that it fits well to say that a^(-b) equals 1/a^b.
How about my commenter? Well, let’s say this: We know that (a^b)^c = a^(b*c). Thus, for instance, (2^2)^3 = 4^3 = 64 = 2^6 = 2^(2*3). We can even easily show that this is true, in a way that makes it intuitively clear for the sorts of numbers for which exponentiation is intuitively clear. If that’s so, then what’s 9^(1/2)? Well, (9^1/2)^2 = 9^(1/2*2) = 9, so (9^1/2) = square root of 9. Raising something to the power 1/2 is taking the square root. Raising it to the power 1/10 is taking the tenth root. (1024^0.1 = the tenth root of 1024 = 2.) Raising a number to the power 1.4 is multiplying it by the square of its fifth root (i.e., raising it to the power 1+2/5), though it’s not clear that conceptualizing it this way is that helpful.
So once we get past some basic operations involving the positive integers, zero, and the positive rational numbers, which correspond pretty directly to the real world, most other things — negative numbers, raising something to the power 0, raising it to a negative power, raising it to a fractional power, and so on — have properties that mathematicians have agreed on because they’re useful.
We make some mathematical statements because they are “true” in the sense of corresponding to the observable world; the commutativity of addition of positive integers, which is to say (x + y = y + x). But other mathematical statements we make because that creates a more coherent way of dealing with certain sets of problems (though this coherence will often also help us deal with the observable world).
It’s not so much idiotic, I’d tell my son’s classmate, as postmodern. In fact, young man, it all depends on what “is” is. (Well, maybe I wouldn’t tell him exactly that, because then I might have to explain the reference, and that might give me a bad reputation around school.)
If by “a number to the power zero is one” you mean something like what we normally mean by “two plus three is five,” which is to say “there’s a real-world phenomenon that clearly corresponds to raising something to the power zero, and the result of that is one,” then that’s not really so. If you mean “when you multiple zero copies of a number by itself, you get one,” then that’s not really so, either. But if you mean “it really is very convenient to treat a number to the power zero as being one, and because of that pretty much all mathematicians — and those students of theirs who paid attention — define the operation in a way that the result is one,” then a number to the power zero indeed “is” one.
Idiotic? Postmodern? Sensible? Maybe all three.