11:12 PM

Last time we I outlined the proof that the square root of 2 is not a rational number. Now I will discuss why this is a problem, and define the real numbers as a solution.

Consider a set of rational numbers, for example all rational numbers less than 1. If there exists a number such that every number in the set is less than or equal to that number, that number is an upper bound for the set. For example, 2 is an upper bound of the set of rational numbers less than 1. Of course, 1 is also an upper bound. If a set has any upper bound it has lots of upper bounds.

So instead of talking about any upper bound for a set, we want to talk about the least upper bound. That is, the smallest number that is an upper bound. If we choose a smaller number, numbers in the set will be greater than the number we choose. If we choose a larger number, then there exists a number smaller than that which is still an upper bound. So the least upper bound sits right in the middle, and could be thought of as the "best" upper bound.

The least upper bound for the set of rational numbers less than 1 is obviously 1. The problem comes when we try to find a least upper bound for the set of rational numbers less than the square root of 2. Since the square root of 2 is not a rational number, it cannot be the least upper bound. Any rational number greater than the square root of 2 is an upper bound, but if you try to claim that any particular rational number is the least upper bound, it is always possible to find a smaller upper bound. The only conclusion is that there is no rational least upper bound for this set.

An informal interpretation of this result is to say, okay, the square root of 2 is not a rational number. Why don't we use the "best" rational approximation of the square root of 2 instead? Leaving aside the fact that approximations aren't really a mathematical way of thinking, the least upper bound result shows that there is no best approximation, because you can always find a better one.

Essentially, this means that the rational numbers have holes. It's true that given any two rational numbers, you can always find a rational number between them, so the rational numbers are very close together. However, there are still numbers we would like to use (like the square root of 2) which just don't exist in the set of rational numbers, and there's no rational number we can use as a substitute for these missing numbers.

This is where the real numbers come in. The real numbers are defined as an extension of the rational numbers with the property that any set of real numbers with an upper bound has a least upper bound. The least upper bound of the set of rational numbers less than the square root of 2 is the square root of 2. Therefore, the square root of 2 is a real number.

Now that the real numbers have been defined, I will post some thoughts about them next time.

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