The Integer Hat Shop

Say you’re going shopping for some new hats to add to your wardrobe. You walk into the hat shop and pick up a nice baseball cap. But you don’t just want one—you want three. Or maybe you want five. Point is, you will walk out of the store with a specific number of hats. That number won’t be 1.5, or the square root of 7: it will be a positive integer.

Why the word “integer?” Why use such a term?

Math is full of not-quite-English words that make many people cringe.

But the words aren’t meant to confuse people. No one decided to use the word “integer” just to make it difficult for an eighth grader to do his homework (even though it can seem that way). The words are not used make math seem complicated or exclusive.

They’re shortcuts—used so that a concept doesn’t need to be explained every time it’s invoked. Every language, including English, uses shortcuts in this way. “Cookie” is a shortcut that refers to a round, bite-size, crumbly baked dessert. “Quack” is a shortcut that refers to the sound that ducks make. Some ideas don’t require shortcuts, like the sound that zebras make. But we would make a word for it if we referred to it a lot.

Math isn’t (usually) talking about cookies, zebras or ducks. It needs its own shortcuts – that is, its own special words.

Take the word integer. Integer is a shortcut term.

You can think of integers as the units used to count whole objects. Those objects can’t be broken into pieces. For example, 3 is an integer. You can buy 3 baseball caps: that’s one baseball cap, plus one baseball cap, plus one baseball cap. But you can’t buy 1.5 baseball caps: that would be one baseball cap, plus a half of a baseball cap. That doesn’t make sense. If you have to rip up a baseball cap to get the number of hats you want, that number is not an integer. So 1.5 is not an integer.

What about the “positive” part? To understand positive integers, look no further than their rivals, the negative integers.

Imagine a baseball cap made out of some fantastical antimatter. It looks kind of like a regular baseball cap, but it is different in one important way. When it comes in contact with a regular baseball cap, both vanish! They cancel each other out.

Regular hat.


In this way, the negative number -2 is the “antimatter” of the positive number 2. When you add them together, both vanish. You are left with 0.

You can add up hats and anti-hats, just like you can add positive and negative integers. This kind of concrete idea can really help when you’re doing arithmetic. Let’s say you add 4 anti-hats to 5 regular hats. Then 4 anti-hats will vanish right along with 4 of the regular hats. That leaves 1 regular hat in your hands. It is a concrete way to understand the equation (-4) + 5 = 1.

What about 6 regular hats added to 10 anti-hats? All cancel out except for 4 anti-hats. That is like saying 6 + (-10) = -4. Positive integers correspond to baseball caps, and negative integers correspond to anti-caps.

Zero is another interesting concept. It doesn’t fit neatly into either the “positive” or “negative” category. If you have zero hats and zero anti-hats, do you have a positive number of hats? Do you have a negative number? You don’t really have either. Zero gets its own category: still an integer, but neither positive nor negative.


Zero hats.

I can’t speak for all people, but for me, this kind of analogy is the best way to understand a new math concept.

It is unsatisfying to learn a technique, memorize it, and repeat it without understanding. But with a picture in your mind, your approach changes immediately. I can mentally pick up a hat and move it around, smash together hats and anti-hats, and toss around an ephemeral zero. Once I’ve done that enough, I can skip the hats and do the same things with integers.

This is the process I have gone through for almost every math concept I have ever learned. To understand integers, I thought of solid objects like hats. To gain an intuition about angles, I thought about toothpicks, connected, at the corners, by balls of clay. To learn about derivatives, I thought about curvy shapes, speeds, and snapshots.

Stereo_wiggle_3DMath isn’t supposed to be scary. It comes from real, basic ideas: ideas that make sense to the human mind, like numbers, shapes, and speeds. When these ideas are singled out for abstract discussion, they need their own special vocabulary to go with them. “Integer” means something more specific than a word like “amount.” So it often pays to use it.

Math ideas are precise, so to express them, we need precise terminology. That’s it. That’s how math language develops and why we use it.

So next time you hear a math term that seems intimidating, keep in mind: it’s probably a lot simpler than you think.

The University of Iowa