## A little irrationality for Pi Day

Those of you who are sufficiently geeky will probably already know that today, March 14, is “Pi Day”. Those of you who are regular readers of this blog will not be surprised to discover that we’re treating this as an excuse to present a neat wee piece of mathematics and a side helping of eccentricity…

One of the facts that almost everybody comes across at some point during their school maths education is that $\pi$ is irrational: it cannot be written as the ratio of two integers. What most of us don’t get to see is a proof of this result — which is a shame, because although proving it for yourself might be hard, there are plenty of “elementary” proofs which don’t need much mathematical knowledge to follow.

Here’s one of the simplest. It’s a slightly expanded version of this one given by Helmut Richter, itself a version of a classic proof by Ivan M. Niven. It proceeds by contradiction: we assume that $\pi$ can be written as a rational number, and we show that this leads inescapably to a contradiction so our starting assumption has to be wrong. The defining properties of $\pi$ that it uses are that $x = \pi$ is the first positive solution of $\sin(x) = 0$, and that $\cos(\pi) = -1$.

Assume that $\pi = \dfrac{a}{b}$ for some positive integers $a$ and $b$. For any natural number $n$, we may now define two functions:

$f_n(x) = \dfrac{x^n(a-bx)^n}{n!} \qquad \mbox{and} \qquad F_n(x) = \displaystyle\sum_{j=0}^n(-1)^jf^{(2j)}_n(x),$

where $f^{(2j)}_n(x)$ is the 2j-th derivative of $f_n(x)$ with respect to $x$.

The following statements are now straightforward to prove.

(i) The function $n!f_n(x)$ is a polynomial in $x$ with integer coefficients. The highest power of $x$ appearing is $x^{2n}$ and the lowest is $x^n$.

(ii) $f_n(x) = f_n(\pi-x)$. [Recall that $\pi = a/b$ by assumption.]

(iii) For $0 \leq x \leq \pi$, $0 \leq f_n(x) \leq \dfrac{\pi^na^n}{n!}$.

(iv) For $0 \leq j < n$, $f^{(j)}_n(0) = 0$ and $f^{(j)}_n(\pi) = 0$. [This follows from (i) and (ii).]

(v) For $j \geq n$, $f^{(j)}_n(0)$ and $f^{(j)}_n(\pi)$ are both integers. [This follows from (i) and (ii); note that having differentiated $n$ times we have “brought down” a prefactor of $n!$ as required.]

(vi) From (iv) and (v), $F_n(0)$ and $F_n(\pi)$ are sums of integers and both are therefore integers themselves.

(vii) By direct differentiation of the series and a bit of cancellation, $F_n(x) + F^{(2)}_n(x) = f_n(x)$. [Note that the final term in the series vanishes as $f_n$ is a polynomial of order $2n$.]

(viii) Using (vii), it’s easy to show that $\dfrac{\mathrm{d}}{\mathrm{d}x}\left[ F_n'(x)\sin(x) -F_n(x)\cos(x) \right] = f_n(x)\sin(x).$

We now consider the integral

$I = \displaystyle\int_0^{\pi}f_n(x)\sin(x)\mathrm{d}x.$

It is clear that $I > 0$ since from (iii) and the properties of $\sin(x)$ the integrand is everywhere positive. From (viii), we have

$\displaystyle\int_0^{\pi}f_n(x)\sin(x)\mathrm{d}x = \left[ F_n'(x)\sin(x) -F_n(x)\cos(x) \right]_0^{\pi} = F_n(\pi)+F_n(0),$

which we know from (vi) must be an integer.

But we can also obtain an upper bound on the magnitude of this integral:

$\left|\displaystyle\int_0^{\pi}f_n(x)\sin(x)\mathrm{d}x\right| \leq \displaystyle\int_0^{\pi}|f_n(x)|\mathrm{d}x \leq \displaystyle\int_0^{\pi}\dfrac{\pi^na^n}{n!}\mathrm{d}x \quad \mbox{using (iii) above},$

and thus we obtain the bound

$\left|\displaystyle\int_0^{\pi}f_n(x)\sin(x)\mathrm{d}x\right| \leq \dfrac{\pi^{n+1}a^n}{n!}.$

But we are free to take $n$ to be as large as we like. We can therefore certainly take $n$ to be large enough that $\dfrac{\pi^{n+1}a^n}{n!} < 1$.

But now the integral $I$ must be a non-zero integer with magnitude less than one… This is a contradiction, so we’re forced to conclude that $\pi$ cannot be written in the form $\pi = \dfrac{a}{b}$ for any natural numbers $a$ and $b$. QED 🙂

Sadly, $\pi$ has been the source of a great deal of irrationality in other senses as well. For whatever reason, the claim that $\pi$ is irrational prompts a certain type of person to set out to prove the mathematicians wrong (nothing wrong with that impulse in itself, of course), and a disturbing number of them waste huge portions of their lives trying to prove that $\pi = \dfrac{25}{8}$, $\pi = \dfrac{355}{133}$ and the likes, generally with logic that would disgrace a Dan Brown book. The tendency has dropped off in recent years as the internet has given such people a wide choice of other ideas to obsess about, but until recently dealing with excitable letters from these cranks was one of the  hazards of being a professional mathematician.

A co-founder of the London Mathematical Society, Augustus De Morgan, made a hobby of collecting and correcting what he called “paradoxers”, and a fair chunk of his monumental A Budget of Paradoxes (the closest colloquial translation of this title is probably “The Bumper Book of Bampots”) is devoted to them. How persuasive he was can be judged by the titles of the series of pamphlets written by one James Smith, a Liverpool businessman who had convinced himself that $\pi = \dfrac{25}{8}$. They start fairly mildly with The Problem of squaring the circle solved; or, the circumference and area of the circle discovered. A few years later the emphasis has changed, with A Nut to crack for the readers of Professor De Morgan’s ‘Budget of Paradoxes’. By the end, we have The British Association in Jeopardy, and Professor De Morgan in the Pillory without hope of escape — but $\pi$ remains solidly anchored at 3.125. (Meanwhile De Morgan was not exactly dampening the flames by describing Smith in print as “the ablest head at unreasoning, and the greatest hand at writing it, of all who have tried in our day to attach their names to an error”.) It’s almost comforting to know that the respectful spirit of the RateMyProfessors website is not a modern invention.

The spat between Smith and De Morgan pales into civility, though, compared with the one that $\pi$ provoked between the eminent mathematician John Wallis and the still more eminent philosopher Thomas Hobbes. (Technically this concerned not the irrationality but the transcendentality of $\pi$, but it involves closely-related fallacies.) After a few initial skirmishes, courtesies were being exchanged in pamphlets with titles like Six Lessons to the Professors of Mathematics; Due Correction to Mr Hobbes in School Discipline for not saying his Lessons right; and Marks of the Absurd Geometry, Rural Language, Scottish Church Politics, and Barbarisms of John Wallis. (More details can be found in the Martin Gardner book from which I filched these titles.) $\pi$ certainly doesn’t seem to bring out the best in people…

Enough of that. To end on a happier note, here’s one of Wallis’s more constructive contributions to the theory of $\pi$: the wonderful infinite product formula

$\dfrac{\pi}{2} = \dfrac{2\times2\times4\times4\times6\times6\times8\times8\times\dots}{1\times3\times3\times5\times5\times7\times7\times9\times\dots}.$

Happy Pi Day!

(DP)