What Do Exponents Mean?

(New Question of the Week)

We recently had a long discussion about a very common question from a somewhat different perspective: What do exponents (zero, negative, fractional, …) actually mean? The hard part, in the end, was to decide what “mean” means. What does it mean to define something in math? I will pick out the main thread of the discussion (which involved three Math Doctors at various times), editing heavily at times and omitting some interesting side trails, to avoid making this too long.

Here is the initial question, from Sarah:

What does raising a number to the power of zero mean? I know why it’s equal to 1, but I don’t know what it means. 2^3 means multiplying 2 by itself 3 times, but how can you multiply a number by itself 0 times? And what about fractional and negative fractional indices? What does 2^(-1/2) mean? I tried looking up a bit but what I get is mainly how to work them out and why, not what they actually mean.

Doctor Fenton took the question first:

The way I like to think about this is that as you observe, exponents are used to indicate the number of factors in a product of repeated terms: a^3 = a*a*a; and so on.  Using this definition, we can derive some properties of exponents.   Let m and n represent positive integers.  Then

(1) a^m*aⁿ = a^m+n 

(2) if m > n,  a^m/aⁿ = a^m-n

(3) (a^m)ⁿ = a^mn .

But look at the second property, which requires that m > n.  Suppose m=n?  Then the equation becomes

a^m/a^m= a^m-m  = a⁰ .

As you observe, the right side a^m-m  or a⁰ doesn’t make sense from the factor-counting point of view: you can’t make sense of 0 factors (well, you might be willing to accept that any product may be considered to contain a factor of 1, e.g. a*a = 1*a*a, so the “product” 1 contains 0 factors of a, in which case a⁰ would make sense).  But even if you don’t want to regard 1 as a product, in the equation

a^m/a^m= a^m-m  = a⁰

the left side makes sense (if a ≠ 0): it equals 1.  So you have an equation which always makes sense when a ≠ 0 and m > n > 0

a^m/aⁿ = a^m-n.

However, when m=n, the left side still makes sense, while the right side doesn’t have any meaning.  You can look at this in two ways: throw up your hands and say the equation doesn’t make sense in this case; or you can say that since a^m-m doesn’t have a meaning, you can give it a meaning – the meaning of the left side, which does make sense.  This means that we DEFINE a⁰ to be 1, and now the second property

a^m/aⁿ = a^m-n

is valid for m ≥ n .

If you now accept that a⁰ = 1, then for m=0, n > 0, the second exponent property would say that

a⁰/aⁿ = a^(0-n), or 1/aⁿ = a^-n .

Again, the left side makes sense while the right side doesn’t, so we can use the left side to DEFINE the right side.

But if 0 < m < n, then in the quotient a^m/aⁿ, n of the factors of a in the numerator cancel n of the factors of a in the denominator, leaving n-m factors of a in the denominator.  Let n-m=k, or n=m+k, so that

a^n-m =a^k.

If we try to apply the second exponent property in this situation, then

a^m/aⁿ = 1/a^n-m  and  1/a^n-m = a^-(m-n) =a^-k . 

We have now extended the second property to all integer exponents.  You can similarly use the third property to define fractional exponents: a^p/q = q-th root of a^p.

This is a typical presentation of the concepts, nicely tailored to Sarah’s specific issue, about the meaning of a zero or negative exponent: we give it a meaning by defining it. But this didn’t really answer Sarah’s question, in her mind:

Thanks for your detailed reply. But you do not define, for example 2^3 as 8, you DEFINE it as 2 multiplied by itself for 3 times. It is equal to 8, yes, but you do not define it as such. So why would you define the zeroth and fractional powers by their answer?

What is “meaning”? What is a “definition”? Is it enough to say what the values are, or is there something deeper?

I stepped in, giving a number of references to our archive where we have discussed exponents in a variety of ways, hoping that one of these (especially the last, which focuses on the idea of definition) would help:

We’ve discussed this sort of question many times, so you should look at Ask Dr. Math for various views of it, to see if any help you. Here is a starting point for the general concept:

Decimal Exponents

Here are some more pages specifically on 0, negative, and fractional exponents, followed by two more that put the ideas together:

How Can a Number Raised to the Zero Power Be One?

Numbers Raised to the Negative Power

Fractional Exponents

Are Negative Exponents Like Other Exponents?

Disappointed by Definitions: Where’s the Deduction?

The key idea, as discussed especially in the last, is that we are extending the definition step by step from a simple idea of repeated multiplication to new ideas that have different forms, but are no less definitions. For example, we define x^-n as 1/xⁿ, using the original definition for a positive integer exponent to create a definition for a negative integer exponent.

Please read all of these, and see if at least some of them help.

Since I hadn’t really added much, Sarah still wanted something more:

I’ve read all the links you sent (I had already read some of them), and I really enjoyed it, especially your last discussion.

I can see it a bit differently now, but I still don’t fully agree with defining, for example n^0 as 1. Defining n^(1/2) as the square root of n makes sense, because that is exactly what it means, but the fact that n^0 (or anything to the power of 0, or rather except 0, for that matter) is equal to 1, doesn’t mean that that is its definition. By defining n^0 as 1, it doesn’t fully explain what it means to raise n to the zeroth power, in my opinion. As for negative exponents, I do agree that n^(-1/2)  is equal to 1/sqrt(n), but I wouldn’t use that as its definition. As I said, you wouldn’t define 4^2 as 16. 4^2 is equal to 16, but that does not make 16 the definition of 4^2.

So, a very basic question, but how would you define a^n, for example, if not “a multiplied by itself for n times”?

But the links did give me a few things to think about, and I do see it more clearly now. It’s probably because when you’re young, you’re taught some definitions that have to change when you get older as more advanced concepts are introduced.

After a little more back-and-forth with me and Doctor Fenton, I wrote,

I think a big part of the difficulty here is what “definition” means. A definition of an operation is any statement from which we can determine the result of that operation. In saying, “I don’t see how by defining n^0 as 1, that explains what you’re doing in raising n to the zeroth power”, I think you want a definition to tell you “what is going on under the hood” (to use an American expression describing how something works, by comparison to knowing how the engine of a car runs), as if there is some real, deep meaning to the operation. But really, there isn’t. An operation is just a function that takes two numbers as input, and produces an output. Defining the output for any input defines that operation. Nothing deeper has to be there. We can explain why we choose to define an operation as we do (as the various links I gave you do), but that is all the deeper explanation we need.

The explanations you prefer are not formal definitions, but introductions that make something feel reasonable. They are good; but definitions don’t have to be like that, and usually aren’t.

Sarah wrote back,

So what you’re saying is that defining a^n as a multiplied by itself for n times is a basic definition, not a formal one? And you’re saying that an operation doesn’t need to make sense as long as it gives a reasonable output that can be explained? Is that what you mean when you say “as if there is some real, deep meaning to the operation. But really, there isn’t.”?

So how would you define 3^3? Would you define it as 27, since it’s the output of the operation you performed? (Also, this is something else, but I wonder why they’re called “operations”)

What I don’t get from what you said is this: how is the output of an operation a good definition? The output of an operation, as I see it, doesn’t DEFINE what you did, it just gives the final answer, but it does not EXPLAIN what you are doing. I define a “definition” as an explanation that shows something’s exact meaning.

At this point, Doctor Rick joined in with some helpful distinctions:

Hi, Sarah. I’d like to add something to this discussion.

You ask if we just define 3^3 as 27, since we’ve said we define n^0 as 1. No, we don’t; if we did, we’d need to have an infinite number of definitions, even just for integer exponents: 3^0 = 1, 3^1 = 3, 3^2 = 9, 3^3 = 27, 3^4 = 81, … . That obviously doesn’t make sense.

There’s a way to formalize the informal definition of positive-integer powers that Dr. Peterson has discussed. Here’s a recursive definition of x^n:

x^0 = 1

x^n = x^(n-1) * x   (for any positive integer n)

It works this way. Once we have the starting point (x^0 = 1), we can find x^1:

x^1 = x^(1-1) * x = x^0 * x = 1 * x = x

And then we can find x^2:

x^2 = x^(2-1) * x = x^1 * x = x * x

And so on, for any non-negative integer x you want — you just keep applying the second part of the recursive definition as many times as you need. (“Recursive” means “going back again”, doing the same thing over and over.) Thus, with two lines, I’ve defined x^n for any non-negative integer n (and any number x).

Now, we’d need to extend this definition to cover negative integer exponents, and then rational exponents, and finally real exponents, as Dr. Peterson has outlined. But my point now is to show you what a formal defintion of non-negative-integer powers might look like.

Notice too that what I’ve done here is not simply to define 3^3 — which is a number — but to define the operation of exponentiation (for a restricted set of exponents). I’ve defined the operation by presenting an “algorithm” by which the result of the operation could be found, in principle, for any pair of numbers x and n (again, so far n must be a non-negative integer).

You asked why exponentiation is called an operation? Like addition, subtraction, multiplication, or division, it takes a pair of numbers and “operates” on them — does something with them — to produce a result (a single number). Perhaps you have some other sense of “why” in mind, but that’s the basic meaning.

Note that 3^3 is not an operation, just as 2+2 is not an operation; rather, addition is an operation. So defining 3^3 as 27 does not define the operation of exponentiation; but my two-part recursive definition does define the operation — again, in the sense that it allows me to find the result of the operation for any pair of numbers.

It’s possible that you’re confused about why we need to define x^0 specifically as the number 1. It may have seemed like we’re saying each power must be defined as a number, since we do that for x^0. But what’s special about x^0 is that it is a starting point for the recursive definition. We’ve got to start somewhere! Once that one power of x has been defined, the recursive definition allows us to define every integer power of x greater than 0.

Does this help at all?

In the course of subsequent discussion, Sarah said,

Also, I don’t agree with defining an operation so that it allows you to find its result. Shouldn’t defining something explain the meaning of something. Why does it necessarily allow you to find a result?

To which Doctor Rick replied,

Can you tell me what “meaning” means? I suspect that you’re looking for something that isn’t really part of math at all. As Dr. Peterson said, there is no “real, deep meaning to the operation. … An operation is just a function that takes two numbers as input, and produces an output. Defining the output for any input defines that operation.”

We look for meaning in relation to ourselves and the world we live in. We find meaning when math is applied to the world; and we usually start with an application in order to get our minds around a new concept — like addition, where we might start by using it to count blocks. However, math itself is just what it is, no more and no less. The result of an operation is the essence of what the operation is; if you can find the result of the operation on any pair of numbers (in its domain), then you fully understand the operation. So that’s a good definition. (And this definition would play a part in our extension of the definition of exponentiation to rational exponents.)

This is not to say that a definition of an operation will always enable you to find the result. For example, I can define the square root of a number x as the (non-negative) number y such that y^2 = x. This in itself does not enable me to find the square root of 2 as a decimal fraction; in fact, we can’t find that number exactly. However, we can know that it exists (if x is non-negative), and we can approximate it by many methods, all of which trace back in some way to the definition I have given.

Sarah wrote back:

I find it quite difficult to explain “meaning”, I see something’s meaning and its definition as being the same thing. You say “We find meaning when math is applied to the world”, but I’ve never seen exponents used in real life, especially the zeroth and fractional exponents. This sounds interesting!

The conversation moved from here into a discussion of applications of various kinds of exponents, which I will omit for the sake of space (except to comment that compound interest commonly involves fractional exponents, and scientific notation uses negative and zero exponents). But first, Doctor Rick continued with the issue of meaning:

I too would normally regard “definition” and “meaning” as synonyms. However, we presented you with a definition, and you weren’t satisfied with that, saying, “The output of an operation, as I see it, doesn’t DEFINE what you did, it just gives the final answer, but it does not EXPLAIN what you are doing. I define a ‘definition’ as an explanation that shows something’s exact meaning.”

We do use the word “meaning” in more than one way. When someone is searching for “the meaning of life,” she is usually not looking for a dictionary definition of the word, or for a scientific or legal definition of the distinction between a living being and an inanimate object. She is asking questions like “Why am I here? Is there a purpose for which I was created, or is it all just random?” Those have to do with “deeper meaning”. I don’t think you’re looking for something that deep, but I imagine that what you wanted was an answer to some “why?” question. It can be difficult to figure out what someone means by “why”, too — what sort of answer would satisfy the person.

I hope that you’re getting the idea now — as you say, you’re beginning to see why one would define x^0 = 1, etc. Perhaps that’s all you needed — not something that goes outside of mathematics itself.

Sarah concluded (for now):

Yes, I agree the word ‘meaning’ can be used in different ways.

And just because I see why you define exponents as such, it doesn’t mean I fully agree with it, but I see the sense in your reasoning. It will take some time for me to accept it – I’m very hard to convince!