Mathematical Irrationality for Dummies

On one of my restless wanderings around the Internet, I came upon a collection of proofs of the statement that the square root of 2 is an irrational number. That is, you can’t express ##\sqrt{2}## as the ratio of two integers. There were no less than 28 proofs on that page. Some were old, some new. Some followed an algebraic path, while others were geometric. A few proofs generalized the statement to the square root of any integer, not just the root of 2. Such a root has to be either an integer or an irrational number, but never the ratio of two integers such as ##\frac{m}{n}##. I felt as if I had stumbled into some old and musty workshop full of many strange instruments, the exact workings of which I could but dimly fathom.

Proof #1 in the collection was by Julius Dedekind, who first clarified the meaning of irrational numbers and their place among all the other numbers. In common with Dedekind’s proof, the overwhelming majority start by assuming that you can indeed find a rational number ##\frac{m}{n}## that is the root of a given integer ##p## (where ##m## and ##n## are also integers). From this point, some methods end up showing that your assumption must be false if it is true, which convinces you that you can’t find the ##m## and ##n## that you assumed in the first place. In other versions, you start by assuming that you have the smallest ##m## and ##n## that square to give you the integer in question (having first divided ##m## and ##n## down to remove all common factors). Then you realize that you can still find ever smaller ##m##s and ##n##s with the same property. So if you do happen to find your ##m## and ##n## someday, you will be sucked into a monstrous vortex, a seething maelstrom, a swirling infinitude of ever-smaller ##m##s and ##n##s. Here be dragons.

Now, I am not mathematically gifted, particularly when it comes to the abstract properties of numbers. But I found myself hooked by this whole idea of irrational versus rational roots. I have this neurological quirk that I am quite hopeless at thinking about the properties of whole numbers and their factors. As a child, I disliked thinking about things like GCDs and LCMs. I still do. However, I am reasonably comfortable with continuous quantities and their functions. So my mind balked at having to do a preliminary division to reduce the ratio ##\frac{m}{n}## to its “smallest form”. And I desperately wanted to escape from that proof-by-contradiction and the “contrapositive” reasoning – i.e. the “ain’t not” line of argument.

I am not alone in my discomfort with that “ain’t”. There are online discussions where the OP raises just this point, and one reply goes like this: “Irrational numbers are defined as the ones that ain’t rational, so you can’t help saying ain’t!”. Despite this, I wanted a way to explain this stuff clearly to myself, without sacrificing too much rigor. I wanted to see the phenomenology in my mind’s eye without needing a pencil and paper, along with a derivation that would build on that picture. So I spent days manipulating pages of expressions and equations, only to find that they simplified back, without warning, into my starting point of ##p=(m/n)^2##. Many quadratic equations yielded roots that were either trivial and not helpful or complicated-looking but not helpful. In the end, I did converge on a (to me) appealing visual picture that also led to a workmanlike QED.

The denominator n in ##p=(m/n)^2## can be viewed as dividing the X-axis into steps of 1/n, with n steps per unit. Starting from X=1, we can reach X=2 in ##n## steps, or we can just stop at ##k## steps, which would be ##X=1+k/n##. For any finite n, this walk never ever lands on a point where ##p=(1+k/n)^2## is an integer. Even if ##n## is arbitrarily large, you would perhaps miss an integer-valued Y by a whisker, but you would never actually land on one. The diagram shows examples of just missing 2 and 3 on the Y axis, as we walk from 1 towards 2 on the X-axis. It is only when we reach an integer X, i.e. a perfect square Y, that we ever see an integer on the Y-axis.

Mathematically speaking, when is ##(1+k/n)^2## an integer, and when not?

Well, let ##(1+k/n)^2 = k^2/n^2+2k/n+1## be equal to some integer ##z_0##, so that:
##k^2/n^2+2k/n+1=z_0##.

But we can just as well drop the “1” term and check if the first two terms add up to some integer ##z##:
##k^2/n^2+2k/n-z=0##,
i.e. ##k^2+2kn-zn^2=0##.

For this choice of n and z, we can solve this quadratic equation for k and get:
##k=n(\sqrt{z+1}-1)##

So if we want to land on an integer on the Y axis, then ##z+1## must be the root of an integer. Furthermore, once such a ##z## is selected, we find that ##k## is a multiple of ##n##.

But if ##k## is a multiple of ##n##, then our X value, ##1+k/n##, is also an integer and ##p=(1+k/n)^2## is a perfect square. So if we are looking for the square root of, say, two, then we have to look outside the set ##\{\sqrt{p}=k/n, k\in\mathbb{Z}, n\in\mathbb{Z}\}##. And it is these “outsider” numbers that Dedekind called irrational numbers.

Now that I have cut my teeth on some basic number theory, maybe I am ready for something that’s a bit harder. Proving the Riemann Hypothesis, perhaps? Watch this space…