Why is π Irrational?

I've written about my (not super positive) feelings towards π day in the past. I don't want to rehash those arguments; if you're curious, you can read some of my thoughts here, here, and here.

At the same time, I feel less curmudgeonly these days, and if your idea of a good time is memorizing a sequence of numbers and reciting them or putting them to music, more power to you.

But if we're going to let π steal the limelight from other deserving irrational numbers, let's at least explore something more interesting than its decimal expansion. Here, I'd like to provide a little visual aid for exploring a proof of the irrationality of π.

π's irrationality undoubtedly contributes to some of its popularity: after all, every rational number's decimal expansion eventually terminates or repeats. This is why nobody is impressed if you can recite the decimal expansion for 1/3 = 0.33333333... from memory. The decimal expansion of π, on the other hand, lacks this sort of simple pattern.

Just because you don't find a repeating pattern in the first 100, 1000, or even million digits, however, doesn't mean that such a pattern can't exist. So looking at the decimal expansion of π out to an arbitrary number of digits won't suffice to prove its irrationality. For a valid proof, we need a more systematic approach.

There are many proofs that π is irrational, though none of them is elementary. At the very least, you need to know a bit of calculus. But even without the machinery that calculus provides, we can sketch a proof that hits on the main ideas.

Let's follow the argument accredited to mathematician Ivan Niven. Niven's proof proceeds by contradiction: let's assume that π IS rational, and follow that train of logic until it crashes head-first into an impossible conclusion.

Arguably the most confusing part of this proof is its starting point. If π is rational, then it can be written as a fraction a/b with a and b integers. The confusing part is the following leap: for any positive integer n, let's consider the (polynomial) function

As if that didn't seem like enough of a non-sequiter, Niven next asks us to consider the area under the curve from 0 to π of f(x) × sin(x).

Now we're at a point where a little technology can help. Here's a graph with the area Niven is referring to. You can adjust n and b; when you change b, the value of a will automatically update to give you the best approximation to π.

b
a =
π
Rectangle

So, why do we care about this area? Well, as you may have noticed, as you increase n, the area increases up to a point, but then begins to decrease, and will ultimately approach 0 as n tends towards infinity. This is because the factorial in the denominator of f grows more quickly than the numerator.

Put another way, since f(x) attains its maximum at the midpoint between 0 and a/b, we can compare the area under the curve to the area of the green rectangle on the graph, whose height is equal to f's maximum. From this picture, you can see that the are under the curve must be less than

,

and this (eventually) goes to 0 as n goes to infinity. This is the first key ingredient in the proof.

Okay. The area goes to zero. So what? Well, if you calculate these areas by hand for small values of n, you may notice something nice: the areas are always integers. To prove this fact for general n requires some calculus, but this is the second key ingredient. I'll spare you the details (here's where you'll need some calculus), but if you get stuck trying to verify it for yourself, Wikipedia has all the details. So does this nice article from Mind Your Decisions.

In any event, combine these two ingredients, and we're basically done! On the one hand, the area under the curve is always a positive integer, because f(x) and sin(x) are both positive from 0 to π. In other words, the area is always at least 1. But on the other hand, the area tends to 0, so for n sufficiently large, the area must be less than 1. No number can be both greater than and less than 1. Contradition: obtained!

I'm not saying this is a particularly intuitive proof. The jumping-off point isn't at all obvious. But sometimes this is how mathematics proceeds: one obtains a true result before understanding the intuition behind the truth. If you can follow along with this proof, I'd encourage you to start poking at it to try to develop your intuition. For example, why look at the area under f(x) × sin(x), rather than just the area under f(x)? What's the special property that binds π and sin(x) together?

Of course, π is more than just irrational: it turns out to be transcendental, an even stronger condition than irrationality. But that discussion will have to wait for another (pi) day.

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