Burn Math Class: And Reinvent Mathematics for Yourself (2016)

2. The Infinite Power of the Infinite Magnifying Glass

Interlude 2: How to Get Something from Nothing

Ex Nihilo Redux: From an Abbreviation to an Idea?

Soon after we began our adventure, we introduced the concept of powers. Using the term “concept” is a bit of a stretch, since powers weren’t really an idea. They were just a meaningless abbreviation that we invented for the sake of convenience. Currently, the symbol (stuff)ⁿ is nothing more than a abbreviation for

which is just repeated multiplication. There’s nothing to know about powers that isn’t already true of multiplication, so powers don’t really have a life of their own. However, if you leave our private mathematical universe and spend some time stumbling around in the world, you’ll occasionally hear people talking about strange things like “negative powers” or “fractional powers” or “zeroth powers,” along with a variety of mysterious sentences like (stuff)⁰ = 1. Now, given how we defined powers, it’s not at all clear that (stuff)⁰ should be 1, because if we translate that abbreviation back into English, it says that (stuff)⁰ is (stuff) · (stuff) · · · (stuff), where there are 0 copies of (stuff) in a row. At first glance, it seems like this should be 0, not 1, right? We used x⁰ ≡ 1 as an abbreviation earlier, but its only motivation was aesthetics. It allowed us to write the expression for an arbitrary plus-times machine in a simpler way, but we had no reason to assume that the sentence x⁰ ≡ 1 made any sense beyond that.

Although we have no idea what a power like 0 or −1 or could possibly mean, we do have a bit of experience inventing things. So instead of simply trusting other people’s assertions about how x⁰ is 1 and so on, let’s see if we can invent a way to extend the idea of powers that makes sense to us. That is, let’s try to generalize our own definition of (stuff)ⁿ to situations where n could be any number, not just a whole number.

Whenever we try to extend a familiar definition to a new context, we immediately run into the fact that there are an infinite number of ways we could do this. However, even though there are lots of ways we could generalize the concept of powers, the vast majority of such generalizations would be sterile, boring, and useless. For example, we could extend our definition of (stuff)ⁿ by saying that (stuff)^# means the same thing it always meant when # is a positive whole number, but whenever # isn’t a positive whole number, then we define (stuff)^# to be 57. That’s not illegal, but it seems pretty boring. However, it’s consistent with our old definition, so its merit (or lack thereof) is to be judged only by the fact that it doesn’t help us at all, and we personally don’t think it’s interesting.

Faced with this infinite abundance of choices, how should we choose to generalize the concept of powers? Obviously, any generalization is useless unless it is useful. Let’s use this obvious fact to move beyond the familiar. We’ll define our generalization not directly but indirectly, as “whatever keeps the definition useful,” and as always, it’s up to us to say what we mean by “useful.” That is, we need to find some property or behavior of the abbreviation (stuff)ⁿ that we think is useful, or which makes it easier to deal with things that look like (stuff)ⁿ.

So let’s find a way of saying something that is useful about the familiar definition of (stuff)ⁿ. You might think, “Well, everything we would ever want to know about (stuff)ⁿ is contained in the definition itself, so if we’re looking to say something that’s useful about (stuff)ⁿ, why not just give its definition: (stuff)ⁿ ≡ (stuff) · (stuff) · · · (stuff)?” That’s all perfectly correct, but there’s one problem. The definition

doesn’t give us any hint about how we can define something like (stuff)⁻¹ or (stuff)^1/2, because this old definition relies on the idea of how many times (stuff) shows up. What could it possibly mean for (stuff) to show up half a time, or a negative number of times? That’s clearly not the way we want to think about things if we want to make sense of negative or fractional powers. Our goal is to choose some behavior of the familiar definition that makes just as much sense for powers that aren’t whole numbers. Here’s something more useful.

If n is some whole number (say, 5), then we can always write sentences like (stuff)⁵ = (stuff)²(stuff)³, because if we translate all the abbreviations on both sides back into English, then the left is saying “5 copies of stuff in a row” and the right is saying “2 copies of stuff in a row, then 3 more copies of stuff.” It’s clear that these are the same thing. For the same reason, whenever n and m are positive whole numbers, the sentence (stuff)^n+m = (stuff)ⁿ(stuff)^m is true, because both sides are just different ways of saying “n + m copies of stuff in a row.” That’s certainly useful, and it expresses the same idea that the original definition did, but it doesn’t necessarily require n and m to be whole numbers! So maybe we can use this as the raw material to build a more general concept of powers. Now comes the important part.

At this point, we just choose to say

I have no idea what (stuff)^# means when # isn’t a whole number. . .

But I really want to hold on to the sentence

(stuff)^{n + m} = (stuff)ⁿ(stuff)^m

So I’ll force (stuff)^# to mean:

whatever it has to mean in order to make that sentence keep being true.

Take a moment to think about what we’re saying in that box. It’s not a complicated idea, but it is wildly different from the way we’re normally taught to think about mathematics in school. Yet as unfamiliar as this style of thinking might be, it is an infinitely more honest representation of real mathematical reasoning than any number-juggling calculation could ever be. This style of thought lies at the core of mathematical invention, and a surprising number of mathematical concepts are invented in exactly this way. Generalizing ideas like this is really nice for two reasons. First of all, it lets us import things we’re already familiar with into unfamiliar territory. That is, rather than simply surrendering to the sad fact that new things are unfamiliar, we can instead use a clever conceptual hack, and choose to define new things in a way that guarantees that we will already be familiar with them. That hack is simply to define things indirectly, not by what they are, but by how they behave. Second, instead of having to remember the meaning of strange things like (stuff)⁰ or (stuff)⁻¹ or (stuff)^1/2 we can figure out what they mean for ourselves! Let’s do that.

Why This Forces Zero Powers to Be 1

We don’t know what (stuff)^# means, but we’re forcing it to mean whatever it has to mean in order to let us write (stuff)^a+b = (stuff)^a(stuff)^b, where a and b are any numbers, not necessarily whole or positive. Let’s use this unusual way of thinking to see what (stuff)⁰ has to mean. Using the idea in the box above, we can write:

(stuff)^# = (stuff)^#+0 = (stuff)^#(stuff)0

This is telling us that (stuff)⁰ has to be whatever number doesn’t change things when you multiply by it. So I guess (stuff)⁰ has to be 1. Hey, that finally makes sense! Let’s write it down.

Our indirect definition forces it to be true that:

(stuff)⁰ = 1

Why This Forces Negative Powers to Be Handstands

We don’t know what (stuff)^−# means, but let’s try the same strategy as before. Notice that the sentence we used as the basis for our generalization only involves the addition of powers: (stuff)^a+b = (stuff)^a(stuff)^b. What about subtraction of powers, like (stuff)^a−b? Well, by writing a − b in the odd form (a) + (−b), we can trick the original sentence into letting us talk about subtraction in the language of addition. Using this idea, together with our recently acquired knowledge that (stuff)⁰ = 1, we can do this:

1 = (stuff)⁰ = (stuff)^#−# = (stuff)^#+(−#) = (stuff)^#(stuff)^−#

Now, if we divide both sides by (stuff)^#, we can build a sentence that tells us how to translate negative powers into the language of positive powers. Let’s write it down:

Our indirect definition forces it to be true that:

In textbooks, a term that looks like would usually be called the “reciprocal” of x. We won’t need a name for this concept very often, so it doesn’t really matter what we call it. Still, the term “reciprocal” is a bit unclear, so let’s use the term “handstand,” because is just x upside down.

Why This Forces Fractional Powers to Be n-Cube Side Lengths

Alright, we’ve unraveled zero powers and negative powers. What about powers that aren’t whole numbers, like ? First let’s assume that n itself is a whole number, and see what we get.

This is strange. We have an unfamiliar thing (namely, the power of stuff) and an unfamiliar thing (namely, stuff). Usually when we use the word “explanation,” we’re describing an unfamiliar thing in terms of one or more things that are more familiar. This does the opposite! It describes stuff in terms of a bunch of unfamiliar things: n copies of its 1/n power. Let’s say this more simply.

Our indirect definition forces it to be true that:

is any number that can say the following sentence without lying:

“Multiply me by myself n times and you get (stuff).”

In a sense, this is the opposite process of finding the “volume” of an n-dimensional cube. Earlier, when we invented the concept of area, we convinced ourselves that the n-dimensional volume of an n-dimensional cube should be ℓⁿ, where ℓ is the length of each edge. Now, the definition of we just obtained sort of seems to be talking about the volumes of n-dimensional cubes. . . but in reverse. It is not saying the usual thing:

From Lengths to n-Volumes: n^th powers

If you hand me the length ℓ of an n-dimensional cube, then the volume is ℓⁿ

Rather, it’s saying the opposite. Something like:

From n-Volumes to Lengths: (1/n)^th powers

If you hand me the volume V, then the length of each side is , whatever that is.

I assume this is where the funny terms “square root” and “cube root” came from. If all we knew was the area of a square (which we can abbreviate as A), then how might we figure out the length of its sides? Well, we may not know how to figure out an exact number for the length if A is something crazy like 9235, but that’s fine. Numbers aren’t the point. Ideas are. Here’s the idea: We know that the “side length” is whatever number turns into A when we multiply it by itself. That is, the side length is any number (?) that makes the sentence (?)² = A true. Which number is that? I don’t know, but that number is called . So even though we have no idea how to compute specific numbers for or , we know how these numbers behave and what they mean: if a square’s area is A, then its side lengths are . The same type of story works for cubes in the normal three-dimensional sense of the word, or even for those weird n-dimensional cubes we’ve talked about before (we can’t picture these, but again, that doesn’t matter). This is why is often called the “n^th root of (stuff).”

The word “root” isn’t really necessary: an n^th root of something is just a (1/n)^th power. Giving “roots” their own name (not to mention their own notation) often seems to give people the impression that the term “root” refers to a different (and more mysterious) concept than powers. But there’s nothing mysterious about them, and understanding the concept certainly doesn’t require you to know how to compute the arbitrary root of an arbitrary number! We’ll need to invent some more calculus before we figure out how to do that. But again, computing specific numbers isn’t the point. The point is the ideas and how they’re created. And in the case of powers, as is always the case in mathematics, the fundamental ideas are extremely simple.