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Putting Nothing in Context

Edward Bowers sent me a question from an unproduced script he wrote for the old sitcom Charles in Charge. The script was titled "When You're a sophomore." Here's his question, in context:

My Charles in Charge script opens with the kids doing some homework. They get to exponents, and the rule of thumb about tens to the power of whatever. An argument ensues when they get to ten to the zeroeth ("Zeroeth?"). Just where do they get off making that one instead of zero? Seems like no tens multiplied by nothing should be zero. :-)

BUDDY [as if loved ones had lost their lives over this issue]

O-oh, don't get me started.

This is another of those questions that I can answer although it is a little outside my training. A mathematician would likely give a more thorough or more precise answer, but this one is basically correct. If any mathematicians have comments, I would happily receive them, and I would ask that they accept my apologies for any lack of precision.

There are two issues here. The first is that zero and nothing are not exactly the same thing. The other has to do with context. I'll tackle them in that order.

In certain circumstances, zero and nothing can behave in ways that are equivalent, even indistinguishable. But, in other cases, they are very different. To more easily tell when they are different, substitute "not anything" for the word "nothing."

First, consider addition:


If I add zero to 5, I get 5.

If I do not add anything to 5, I get 5.

Now, consider multiplication:

If I multiply 5 by zero, I get zero.

If I do not multiply 5 by anything, I still have 5.

Note, especially, that in the case of multiplication "nothing" and "zero" have very different effects.

So, when you say 10⁰, you are saying that there are no tens to be multiplied. You are not saying to multiply by zero.

Now on to the subject of context.

What if there are no tens there at all? What is there? Nothing? That is certainly a reasonable response in some sense. But is it really practical to assume there is absolutely nothing? At the very least, "nothing" is a bit unwieldy in mathematical terms. To wit, it's hard to work out where "nothing" goes on a number line.

Even if you ignore that last paragraph, it seems unlikely that you have "nothing" if you are concerned about multiplying it by tens.

In other words, there must be some sort of context in which one is or is not multiplying tens; otherwise, what difference does it make?

Let's consider some contexts.

If I write 5 x 10², then I am multiplying 5 by ten and then by ten again. The answer is 500.

If I write 5 x 10¹, then I am multiplying 5 by ten. The answer is 50.

If I write 5 x 10⁰, then I am not multiplying 5 by any tens. The answer is 5.

This latter is equivalent to multiplying 5 by 1. It is not the same as multiplying 5 by 0.

So, when there isn't a context already defined, a default context of "1" is used ("1" is the "identity element" for multiplication-- remember that?). This means that when you say 10⁰, you are, by implication, saying 1 x 10⁰.

In fact, any number raised to the power of zero is equal to one for the same reason:

A x B⁰ = A

Here, A is not being multiplied by any B's. The answer is A.

posted by Ask Miss Science @ 5:36 PM