Monthly Archives: July 2011

Approximating pi

Every 14 March (or 3/14 in US format) for the last few years I’ve noticed an increasing amount of joviality and excitement about the constant \pi (\approx 3.141592653\ldots).  I assumed for a while that this was one of those internet-originating fads like Talk Like a Pirate Day (19 September) or the related Talk Like a Pilot Day (19 May): a bit of fun but (the adoption of Talk Like a Pirate Day as a religious holiday by the Church of the Flying Spaghetti Monster notwithstanding) essentially a bit of annual silliness.  But it turns out that Pi Day has a slightly more serious heritage: Wikipedia tells us that it was founded in 1988 by Larry Shaw, a physicist working at the Exploratorium, an amazing interactive science museum in San Francisco.  A ceremony is held at 1:59pm on 14 March, culminating in the participants singing happy birthday to Albert Einstein (born on that day in 1879).  In 2009, the US House of Representatives passed a non-binding resolution recognising the date as National Pi Day.

I’m in two minds over whether this sort of thing has a net positive effect on public understanding of mathematics, or whether (like those tedious corporate PR-driven “magic equation” stories that keep turning up in newspapers, and on which Ben Goldacre has eloquently grumbled several times) it merely serves to trivialise the subject and present mathematicians as just interested in numbers and hard sums.  I suppose it depends on how it’s presented: the situation wasn’t entirely helped by the recent news items about a supposed battle between the mathematical establishment and a small group of mavericks and renegades over whether \pi or \tau=2\pi is the more natural constant to use in most circumstances.  But by and large I think Pi Day is probably a good thing: it gets people thinking about mathematics and helps inspire some children to go and find out more (in particular this is the intention of the people behind the whole thing, and also explicitly stated as such in the resolution passed by the House of Representatives).

But today, 22 July, is Pi Approximation Day: a well-known rational approximation to \pi is given by 22/7 (=3.\overline{142857}).  Actually, 22/7 gives a slightly better approximation to the value of \pi than 3.14; an even better one is given by \frac{355}{113} \approx 3.1415929203\ldots but that hasn’t easily translated into a date since the reign of the Roman Emperor Constantius II.

In my EC119 Mathematical Analysis exam this year I set a question requiring the students to calculate the Taylor-MacLaurin series for the function \arcsin x near x=0 up to fifth order and use this (together with the fact that \sin\frac{\pi}{6}=\frac{1}{2}) to obtain a rational approximation to the value of \pi.

I thought this was an interesting bit of a question, but relatively few of them attempted it (which, I guess, goes to show that examiners and students have different ideas about what constitutes an “interesting” exam question).  Anyway, the fifth-order Taylor-MacLaurin expansion for \arcsin x is x = x + \frac12\cdot\frac{x^3}{3} + \frac12\cdot\frac34\cdot\frac{x^5}{5} and so by setting x=\frac12 and multiplying the result by 6 we should get a rational approximation to \pi.  As it happens, we get \frac{2009}{640} = 3.1390625, which is clearly getting there but isn’t as good as 3.14, 22/7 or 355/133.  And although we can get a much better approximation by calculating more terms of the Taylor-MacLaurin series, it’s clear that this isn’t a particularly efficient way of doing things.

Fortunately, there are many better ways of calculating rational or decimal approximations to \pi.  In 1706, the English mathematician John Machin (c.1686-1751) published the identity \frac{\pi}{4} = 4\arctan\frac{1}{5} - \arctan\frac{1}{239} and used it (together with the MacLaurin series for \arctan x) to calculate the first hundred decimal places of \pi.

The Indian mathematician Srinivasa Ramanujan (1887-1920) discovered the formula \frac{1}{\pi} = \frac{2\sqrt{2}}{9801} \sum^\infty_{k=0} \frac{(4k)!(1103+26390k)}{(k!)^4 396^{4k}} which converges extremely rapidly.  The best currently known formula (which I thank Arthur Clune for alerting me to) was discovered by the Ukrainian/American mathematicians Gregory and David Chudnovsky in 1989, based on Ramanujan’s work.  This formula \frac{1}{\pi} = 12 \sum^\infty_{k=0} \frac{(-1)^k (6k)! (13591409 + 545140134k)}{(3k)!(k!)^3 640320^{3k + 3/2}}, converges extremely quickly, with each additional term adding another fourteen or fifteen decimal places of accuracy.  They used it in August 2010 to calculate \pi to 5\times 10^{12} decimal places. (5\times 10^{12} is 5 billion or 5 trillion depending on whether you’re using the long or short scales; being English I tend to use the former, and experience a faint sense of guilt on the increasingly common occasions I find myself using the latter.)

An apparently much less accurate approximation to \pi turns up in the Bible.  1 Kings 7:23 says:

And he made a molten sea, ten cubits from the one brim to the other: it was round all about, and his height was five cubits: and a line of thirty cubits did compass it round about.

This verse, repeated almost verbatim in 2 Chronicles 4:2, implies the approximation \pi\approx3.  The other famous approximation to \pi turns up in an infamous 1897 bill considered (but ultimately not passed) by the state legislature of Indiana.  This includes various mathematical observations by the physician and amateur mathematician Edwin J Goodwin (c.1825-1902), such as

Furthermore, it has revealed the ratio of the chord and arc of ninety degrees, which is as seven to eight, and also the ratio of the diagonal and one side of a square which is as ten to seven, disclosing the fourth important fact, that the ratio of the diameter and circumference is as five-fourths to four

This indicates not only a value of 3.2 (=\frac{4}{1.25}) for \pi but also a value of 1.\overline{428571} for \sqrt2.  The bill also includes Goodwin’s “solutions” to the problems of squaring the circle, trisecting an angle and doubling the cube (three ancient unsolved problems known since the middle of the 19th century to be formally unsolvable).  The bill was passed unanimously by the House but due to the efforts of Clarence Waldo, a Professor of Mathematics at Purdue University, enough members of the Senate were persuaded of the errors in Goodwin’s work that the bill was stalled indefinitely in committee and never made it onto the statute book.

Signing in

The London Mathematical Society is one of the main UK learned societies for mathematics (others include the Institute of Mathematics and its Applications and the Royal Statistical Society).  Founded in 1865, its list of presidents includes many of the most eminent names in British mathematics over the last century and a half: De Morgan, Cayley, Sylvester, Rayleigh, Kelvin, Hardy, Littlewood, Atiyah, Zeeman and many others.  Its membership is largely composed of research mathematicians working in academia, and its emphasis has historically been more on pure mathematics than its applications.  Indeed, this was one of the factors which led (in the 1960s) to the founding of the IMA, although since then the LMS has gradually adopted a more balanced position and now caters better for the interests of applied mathematicians.

I joined the LMS, as many people do, while still a graduate student.  There’s a formal application process whereby you must be nominated by a current member; like most people, I asked my PhD supervisor.  I sent off the form together with a cheque for my first year’s subscription fees (there’s a reduced rate for graduate students) and received a letter a month or so later informing me that I had been duly “elected” to membership at the most recent meeting.

The LMS is pleasingly down-to-earth as learned societies go.  It has no complicated hierarchy of membership grades, whereas some other societies (the IMA or the Institute of Physics, for example) have a plethora: Associates, Members, Fellows, Corporate Membership, Chartered Status, and so forth, each carrying slightly different privileges, responsibilities and subscription fees.  The LMS has only a single grade of membership (well, technically, students are classed as “associate members” but apart from the reduced subscription rate, I was never able to discern any practical difference).  It confers no postnominal letters or academic dress – not that there’s anything wrong with either of those, indeed I’m also a member of another society which happily does both.

What it does do, and well, is everything you’d expect from a learned society: it holds regular meetings throughout the UK, it funds and otherwise supports conferences, it publishes several peer-refereed journals and (together with the RSS and the IMA) it lobbies the government, seeking to influence public policy on mathematics and science in the interests of its members.

Pretty much the only formal, ceremonial aspect of the society takes place at the beginning of a society meeting.  Although I’ve been a member since 1999, I’ve only made it to a few meetings in that time and so I’d never witnessed this formal bit until a couple of weeks ago, at the most recent West Midlands regional meeting at the University of Birmingham.  (A couple of years ago I went to another meeting in Leicester but my train was late so I missed the formal proceedings, although I got there in time for the first talk.)

A brief account of the previous meeting is displayed, and the members presents asked to confirm this.  Next, a list is displayed of prospective members from whom application forms have been received, and the assembled throng is invited to approve their election.  Something similar happens at Oxford and Cambridge graduation ceremonies, where the senior members present (essentially, anyone with an MA) are given the opportunity to veto the award of a given degree, by calling “non placet” (“it does not please”) at the appropriate juncture.  In practice, I believe this hasn’t happened in living memory, and I strongly suspect the same goes for election to the LMS.  As we were looking at the handful of names (complete with educational qualifications and institutional affiliations) it occurred to me that sometime in 1999, a similar group of people must have muttered cordial assent to the election of one Nicholas James Jackson BA MSc (PhD student, University of Warwick).

Next came the bit I’d actually been waiting for, the nearest the LMS has to a formal ceremony.  The president invited any subscribing member present, who had not already done so, to come forward and sign their name in the members’ book.  This is a leatherbound tome, about 3-4 inches thick, containing the signatures of most members of the society since its foundation in 1865.  I scribbled my own name in the next available space (about halfway through – there’s plenty of room for future expansion), and shook hands with the president, who formally admitted me as a member of the society.  I returned to my seat as about four other members were also admitted.

I took the opportunity afterwards to leaf through the earlier pages of the book, and spotted the signatures of such eminent mathematicians as A Cayley, J J Sylvester and G H Hardy.  (I suspect that this is the only time my name will find itself in such august company.)  I also noticed the signature, sometime during the 1960s, of Maurice Dodson, who in the early 1990s, during my final undergraduate year at the University of York, introduced me to algebraic topology, the subject which I ended up specialising in as a graduate student a couple of years later.

While very little fuss is made of this ceremony, and even though some members never get around to signing the book (it took me twelve years and this is certainly not a record), I think it all serves a valid purpose: to remind us that we are just the latest successors to, and participants in, a tradition of inquiry and scholarship that underpins the foundations of our civilisation.  In a little over a week’s time, I’ll be attending the graduation ceremony of some of my own students, an occasion which is perhaps a little more grandiose but which ultimately serves the same purpose.

After this formal (if somewhat low-key) part of the meeting had concluded, there followed three talks on various aspects of mathematics: Miles Reid (Warwick) spoke about graded rings, Shaun Stevens (East Anglia) spoke about representations of p-adic groups and the local Langlands conjectures, and Catharina Stroppel (Bonn) spoke about algebraic categorification; the first two topics are at present beyond my ken, while the latter is a subject I’m trying to learn about at the moment, and although I didn’t understand very much of the content of the talk, it gave me a few more fragments of insight into some of the ideas in question, as well as an indication of the sort of problems people are currently trying to solve.

So, in all, a worthwhile use of a Tuesday afternoon.