Numbers & Functions

1 Counting Isn't Enough

CHAPTER 1 Counting Isn’t Enough Here is something nobody tells you when you’re little: the numbers you learned first were never the whole story. They were just the beginning — the part we needed before we needed anything else.

Let’s start where everyone starts.

1.1 The numbers you already trust

You can count. One sheep, two sheep, three sheep. These are the counting numbers: 1, 2, 3, 4, 5, . . .

They go on forever, and there’s something satisfying about them. They feel solid. You can hold up that many fingers. You can put that many cookies on a plate. Mathematicians call them the natural numbers, because there’s nothing artificial about them — they’re the numbers that show up the moment you have stuff and you want to know how much stuff.

For a long time in human history, this was basically all the numbers there were.

But here’s the thing about numbers: we don’t just count with them. We do things with them. We add. We subtract. We multiply. We divide. And the moment you start doing things with numbers, you discover that your nice solid collection of counting numbers has holes in it.

Let me show you what I mean. This is going to be the secret pattern of the whole chapter, so watch for it.

1.2 When subtraction breaks

Adding two counting numbers is no problem. 3 + 5 = 8. Still a counting number. You can add counting numbers all day and you’ll never leave the family of counting numbers. Nice.

Now subtract.

5 −3 = 2.

Fine. Still a counting number.

3 −5 = ?

Uh oh.

If all you have is counting numbers, this question has no answer. You’re being asked to take 5 cookies away from a plate that only has 3 cookies on it. With actual cookies, you’d just say “you can’t do that.” And for thousands of years, that’s exactly what people said. The problem was considered nonsense. But think about how strange that is. We have a perfectly good operation — subtraction — and it just. . . refuses to work sometimes. It works for 5 −3 but not for 3 −5. That’s annoying. An operation that only sometimes works is like a light switch that only sometimes turns on the light.

So mathematicians did something bold. Instead of saying “you can’t do that,” they said: What if we invented the numbers that would make it work?

And that’s where the negative numbers come from. We decided that 3 −5 should have an answer, and we gave that answer a name: −2. We extended our collection. Now we have: . . . , −3, −2, −1, 0, 1, 2, 3, . . .

This bigger family is called the integers. (Notice 0 snuck in there too — the number for “none,” which is also a surprisingly modern invention, but that’s a story for another day.)

Here’s the part I really want you to feel. We didn’t discover negative numbers lying around in nature. Nobody ever held up negative-two fingers. We built them, on purpose, because we wanted subtraction to always work. The integers exist because we refused to accept an operation that broke.

That’s the pattern. Hold onto it.

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