Constructing the numbers

As children, we learn how to count on our fingers and our toes — one, two, three and so on.

Then, as we begin school, we are taught about the negative numbers, the violent creatures which cancel out our “natural” numbers, and the fractions, for those times when we only need half a minute. Then we learn about the irrational numbers, those mystical entities which stretch out toward the horizon and never end.

But at their core, what are the numbers really? How do we know a number such as pi — the ratio of a circle’s circumference to its diameter — really exists? Intuition lies to us if we listen too closely, so it is not enough merely to assume the real numbers exist because we can see them. We must define them to understand them.

It all begins with a single element. This lone quantity, the empty set which we assume exists, becomes the number zero, the basis for our cathedral of arithmetic.

Now we add a fearsome function into the cauldron to build something from this nothing that we’re given. This function has two special properties to make it work: One, that it can never give zero as a value, and two, that it’s one-to-one, meaning that if we know it spits out five, we must have given it four or six as an input but not both.

Because of this, the function is a bit like the gift that just keeps on giving.

We give it zero, and it gives us some new element, which we define as the number one. Then, we give it one, and it gives us some different element, which we define as the number two.

On and on, it goes, giving us more and more natural numbers until we have an infinity of them, all the counting numbers.

With these in hand, we can define what addition means.

To start off, we say that any natural number plus zero is itself, so that zero has the “identity” property. Then, any natural number plus another number can be defined by taking the second number and stepping back a rung on the ladder of the natural numbers but then applying the function.

In other words, if we wanted to add eight to a number, we could say this was the same as adding seven to the number but then applying the function once, or the same as adding zero to the number but then applying the function eight times, just like we’d expect.

From this we can go on to build the integers, the set of all these natural numbers with the negative numbers. We form pairs of two natural numbers A and B, writing (A,B), and think of this as the difference between A and B. Every natural number A corresponds to the pair (A,0), and its inverse “-A” corresponds to (0,A). To add, we simply add coordinate-wise.

But, the problem is that both (0,0) and (1,1) equivalently represent “zero,” so we have to throw these together into large storage bins called “equivalence classes,” which then become our integers.

To form the rationals, we repeat about the same process, thinking of (A,B) as A/B and defining the fractions as equivalence classes of these pairs. Now we have more fractions that we know how to deal with, with our rationals forming a dense speckle of numbers on the real number line.

But this isn’t all the real numbers. As the Greeks knew (and killed each other because of), the square root of two is irrational — it can never be represented as a fraction. So, we must perform one more procedure to get all the possible real numbers.

Unfortunately, a problem emerges: There are too many irrational numbers.

Paradoxically, there are exactly the same number of fractions as there are counting numbers, but there are many more real numbers than both combined. Because of this, the process from above will not work, so we need something else.

One way mathematicians deal with this is a “limiting” procedure. Because we have the fractions, we can begin to pick a sequence of numbers - ½, ?, ¼, ?, and so on that each step gets smaller, approaching zero step by step, inch by inch but never equaling zero. In fact, the terms in this “Cauchy” sequence are said to get arbitrarily close, meaning that if we go far enough out in the sequence, we can pick two terms whose distance is smaller than any number we pick.

With a little more work, we can pick a Cauchy sequence in the rationals such as this one that will approach the square root of two, even though its limit of square root of two is not in the set of the rationals. This is exactly how we find the reals, all the Cauchy sequences of the rationals together.

And just like that, we can build the real number line out of the zero element, making something serious emerge out of nothing.

­— sidfletc@indiana.edu