Look at your hands and—most likely—you have a finite number of fingers on them. That is, you don’t have an endless number of fingers on any hand. On average, a human has ten fingers and ten toes. But, what exactly is “ten?”—not the word, but the quantity. Is there something deeper to the abstract concept of “ten,” an idealized number form that we just approximate when sticking out all ten of our fingers at once? We are going to explore what exactly “numbers” mean, and how different
“levels” of numbers are layered on top of each other like shortcake.
Look to your fingers and count them. One, two, three, four, and so on, until you get to the maximum number of fingers on your hands. Notice something interesting here? You counted them in the same way you would count a quantity of apples, or books, or something that looked “concrete” and “indivisible.” When referring to how many fingers we have, we treat the finger as a single unit and count the overall quantity of fingers. It seems straightforward and almost natural, in a way. It is like the type of counting that you’ve learned from the start, and has been etched into the deepest recesses of your brain.
That’s why they’re given an amusingly fitting name—the “natural numbers” or, one, two, three, four, five, and so on. Do we stop? No! The natural numbers comprise a “set,” which is just a convenient name for a collection of unique objects. This collection can be finite or infinite. When talking about the natural numbers, we can think of a bottomless bucket that contains all of those straightforward-looking numbers from one and so forth. If we want to pull out a number from this bucket, whether it’s two or 200,000,000,000,001, we will find it. That’s simply because there is no such thing as the “biggest natural number!” Here’s a quick proof for you: think of the biggest number you can, big enough to serve as a candidate for “biggest natural number.” Got something? Good. Now add 1 to it. Whoops, it looks like you found an even bigger number!
But, cleverly, you say, “What about infinity?” Well, infinity isn’t a “number,” but there are different levels of infinity, which we will get to soon enough. Some “infinities” are, in a sense, more “infinite” than others.
What kind of basic arithmetic can we do with just our fingers? Well, we can add them: 1 + 1 = 2 and 4 + 5 = 9—maybe 2 + 2 = 5, if you’re currently studying Orwell. Given two natural numbers, adding them together will always pop out another natural number as a sum. Neat, huh? What about multiplication? It’s basically just adding a number to itself a certain amount of times. Three times three is just the number three added to itself three times, and so we know it’s nine. Reducing multiplication to these addition sequences makes it clear that multiplying two natural numbers pops out another natural number. In fancy terms, we say that the set of natural numbers is closed under addition and multiplication.
Just as a quick remark for the perceptive: do we consider zero to be a natural number? Yes and no. This has been the subject of harsh disputes for decades—I wish I were joking. For our purposes, it doesn’t matter either way, and it’s convenient to think of zero as one whenever you’ll need it.
There’s other “natural” sorts of operations that the natural numbers aren’t closed under, though. If we have five fingers and put down three then we’re left with two fingers, but how do we work out the idea of having three fingers and putting down five? Can we even find a natural number that corresponds to three subtract five? We can’t. So, we just extend the numbers we can use and this is where we begin to think of things a bit more abstractly. Abandon the idea of numbers being tied explicitly to quantities and past this point of no return, numbers will just become more and more abstract.
Now, we can introduce the integers! We grab our set of natural numbers (and concretely stick the zero in here without dispute), and extend them in the “opposite direction” to include negative numbers. So now we have a bucket that has … -3, -2, -1, 0, 1, 2, 3, … and so on. It’s convenient to start thinking about an infinite line here. This line will be useful the further we go into this story, but for now, we can picture each of these integers being represented by equally-spaced notches on the line, extending off in both directions. Now we have an answer to our question of three subtract five. We just represent it as (3) plus (-5). What does that look like? On the line, I’ll walk three steps to the right and five steps to the left. That’ll leave me off at two steps to the left. So we’ve got 3 – 5 = -2!
It’s peculiar to think that we can just “extend a number system” like that, but it’s a fairly useful trick. Can we go further than the integers? We can add, multiply, and now subtract them. Division is still a problem, though. If we divide three by four, we won’t get anything in the set of integers. So let’s extend our set again and include any numbers that we can represent as a fraction of two integers (just make sure the denominator isn’t a zero). Now we can divide in this nice new set we call the “rational numbers”. But what’s cool about them is this is where the granularity of our sets and lines begins to break a bit. Let’s look at the distance between any two numbers on our integer line, like 0 and 1. If we want to take half this distance, we get 1/2. Half of that is 1/4. Half of that is 1/8. As we continue, we keep getting smaller and smaller intervals, all of which we can easily represent as integer fractions. But just like how we didn’t have a biggest number, we can keep dividing these intervals to get smaller and smaller numbers. So not only does our set of rationals go off into infinity in both positive and negative directions, but between every rational number there is an infinite number of other rational numbers!
That takes care of addition, subtraction, multiplication, and division! What else is left? Of course, numbers that cannot be represented as a fraction of two integers, of course. Wait, what? Those exist? They do indeed, and two of the most famous ones are pi and the square root of 2, which humans have tried for centuries to approximate after mathematically proving that no integer fraction representations exist. We call them “irrational numbers” for that reason, and what’s surprising is that, between any two rational numbers, no matter how close they may be, there is also an infinite amount of irrational numbers between them!
These infinities are piling up. We can extend our number set to include these irrationals, and define our new set to be “anything we can uniquely represent with decimals.” So this can include anything from “1.00000 …” to “3.141592654 …”. While it’s not a perfect definition by any means, it’s a rough idea of the set of “real numbers,” which are closed under the four major algebraic operations and contain rationals and irrationals.
Let’s look back at how we pictured “infinity.” With the natural numbers, we basically thought of infinity as “reaching something really, really big.” With the integers, we thought of infinity as the same but in two directions—positive and negative infinity, extending into the horizon. With the rationals, we knew they extended into positive and negative infinity, but there was also an infinite amount of rationals between each rational.
The real numbers are on another level of infinity. They extend into infinity in both positive and negative directions. But if you look at any two real numbers on a line, no matter how close together, you will find an infinite number of other real numbers—rational and irrational—between them. And if you were to choose two numbers from the infinity between your two numbers, you would find that even between these two new numbers, there exists an infinite amount of real numbers. Try it—pick any decimal number you can think of. Then divide it by any other larger decimal number and you’ll get something smaller.
This infinity is so deep—”an infinity of infinities”—that we call the set of real numbers something special: an “uncountable set.” The natural numbers and integers are both countable, as you can find a one-to-one way to “map” each number in these sets to a subset of the natural numbers. With some clever thinking through using the fact that rational numbers use integers as their fractions, it’s been proven that even they are countable—but, the real numbers are not.
Two interesting ideas arise out of this. The first is, that the unique infinity of the real numbers has lent itself to theorems that shake the foundations of mathematics. The second is that there’s another mystery left: what’s the square root of negative one? It certainly isn’t in the set of real numbers, so there are still some gaps!
However, a look into these questions will be for another time. For the time being, make sure your fingers are still intact.