What is a Dedekind cut? And why is it relevant in math?
In: Mathematics
Let’s start with the integers. The integers are all the non-fractional numbers, positive and negative:
…-3, -2, -1, 0, 1, 2, 3…
From the integers, we can create the rationals. The rationals are all numbers that are ratios of integers, like 1/2, or 5/6, or -7/3. It’s easy to link the two because the numerator and denominator are both integers. One way of saying this is we can *construct* the rationals from the integers.
But what about numbers that *can’t* be expressed as a ratio of integers? Like the square root of 2, or pi? We call these *ir*rational numbers. When you combine the irrationals and rationals together, we have an overarching set of numbers called *real* numbers.
One thing mathematicians like to do is root complex ideas in simpler ones. Just like we rooted the construction of the rationals in the integers. Well mathematicians wanted to do this with the real numbers and Dedkind cuts is one way they did this.
A Dedkind cut is just that, a cut. It cuts the rational numbers into two groups that we will call A and B. All of the elements of A are less than all of the elements of B. Imagine taking the number line and cutting it at some place. Everything to the left of the cut is in A and everything to the right of the cut is B. We will call the place of the cut C.
Based on how they are defined, B can have a smallest element and, if it does, that element is where the cut takes place, C.
A, however, doesn’t have a biggest element. This may seem odd, since everything in A is less than everything in B. But we can see how this can be. Let’s say we make our cut at the number 2. Everything in A is less than 2, but A doesn’t have a largest number. How? Well, let’s take 1, 1 is in A because it’s less than 2.
Now let’s average those two numbers: 1 + 2 = 3/2. 3/2 is still less than 2.
Average again. 3/2 + 2 = 7/4. Still less than two.
And you can keep doing this, forever, always getting closer, but never reaching, 2.
Now, I said that B *can* have a smallest element. But it doesn’t have to. When B has a smallest element, that smallest element will be a rational number and we say that this Dedekind cut represents, or corresponds, to that rational number.
But when B doesn’t have a smallest element, then it turns out that A and B both get closer and closer and closer to a number that isn’t rational. They get closer and closer to a single, unique irrational number. And in this case we say that the Dedekind cut represents/corresponds to that irrational number.
So we can use Dedekind cuts to represent both rational and irrational numbers which, when combined, means we can use Dedekind cuts to represent all the reals. And since Dedekind cuts are constructed from just the rational numbers themselves, we’ve found a way to construct the real numbers from the rational numbers, linking them together.
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