Recall from my last post what we are trying to accomplish: by assuming that is a rational number, we are going to define an unpossible function! So, without further ado:
Suppose , where and are positive integers. Define the function like this:
(In case you’ve forgotten, , pronounced “n factorial,” is the product of all the numbers from 1 to .) “OK… but… what is ?” I hear you ask. Good question. The short answer is, it doesn’t matter: can be any positive integer. We will show a bunch of things that are true about no matter what is. Later, we will see that we get a contradiction only for values of which are “big enough.” But that’s OK; since everything we prove up to that point will be true no matter what is, we can pick a value of which is as big as we like.
Let’s explore some properties of . First, it’s easy to see that . It’s not too hard to see that as well (remembering that , of course, which means that ):
So has zeros at and . But more is true: in fact, is symmetric (a mirror reflection of itself) around the line . That is,
Let’s prove this:
“I don’t see what’s so unpossible about so far,” you say? Patience! (Of course, it isn’t really itself which is the problem; the problem is our insistence that is actually defined in terms of the “numerator” and “denominator” of …)
Next time, we’ll see that the derivatives of also have some special properties.