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The magic of Pi (π)

David Galton
DOI: http://dx.doi.org/10.1093/qjmed/hcp008 439-440 First published online: 18 February 2009

Pi can be taken as a symbol for one's education. The deeper one goes into it the more mysterious it becomes and the more ramifications one can find. You can go as far with pi as your abilities can take you. Pi is first met in elementary geometry at school and it occurs in one of the first equations that you can solve; that is to find the circumference or areas of circles. For pi you substitute the fraction 22/7 and the answer comes out right enough to satisfy the school master. In the sixth form when doing finite and infinite series you learn that there does not appear to be an exact number for pi. Although it is a simple ratio of the circumference of a circle to its diameter, the decimal places appear to go on forever. This is not too unfamiliar because you have learnt that the decimal places for such fractions as 1/3 or 1/9 go on as 0.333… or 0.111… recurring. But pi cannot even be expressed as a definite fraction. Now if you consider a number line scaled from 0 to 5 Units you would imagine that pi would occupy a definite position lying somewhere between 3.1415 and 3.1416 and that it could be defined accurately. Taking the half-way point at 3.14155 might be it. No—so you go on taking smaller and smaller intervals and using each time the half-way point to be pi, accumulating more and more decimal points but you never seem to arrive at the exact number. In 1966, pi was calculated on an IBM 7030 computer to 500 000 decimal places. In 1994, the calculation was run out to a billion decimal places; there were no recurring motifs and leads one to believe that there is no finite number of digits for pi. The numbers on such a line appear to be continuous and there are no gaps between them; so you can go on dividing ever smaller intervals to infinity.

At university I was astonished to be shown the relationship of pi to Euler's exponential e and the imaginary number (i), Embedded Image. The tutor produced it like a magician pulling a rabbit out of a top hat: Embedded Image Incidentally this has been voted by some mathematicians to be the most beautiful equation of their whole subject.

Why on earth should there be such a simple connection between pi and e? It has nothing to do with circles. Euler's e like pi is an irrational number (i.e. cannot be written in the form of a fraction a/b) and its value cannot be given exactly; to five decimal places it is 2.71828. And why is the Embedded Image involved? I have always felt uneasy about Embedded Image. Is it just a fudge factor to make it possible to find the square roots of minus numbers? I know exactly what the number means in two books or three apples; and also the Embedded Image apples means something. But what does the Embedded Image apples mean? Physicists tell you Embedded Image is a number reflecting ‘real’ life too; and it is a most useful quantity to study rotating vectors, alternating currents or analysis of other wave forms.

Does pi come into medicine? We all use the normal (or Gaussian) distribution to work out the student's t-test for the probability values of our observations. This bell-shaped curve approaches the horizontal axis without ever quite reaching it. Yet, the total area under the curve can be accurately determined by a method of advanced calculus involving double integrals (which is beyond me); and it works out to involve the square root of pi. This Gaussian integral (or probability integral) of the standard normal distribution can be expressed as:

The area under the curve Embedded Image

Such a strange way to come across pi again makes it look quite scary. Does pi really have magical properties?

Going to 2000 BC there is a clay tablet written in cuneiform script that is on display at the current exhibition on Babylon in the British Museum. The tablet shows you how to calculate the areas of squares, rectangles and circles and was probably used as architectural exercises for students. The tablet states that the perimeter of a regular hexagon approximates to the circumference of a circumscribed circle; and that to calculate the area of the circle you need to employ a ‘circle ratio’ (the symbol pi was not used until the 18th century). The ‘circle ratio’ came out as 31/8, a slight underestimation of the true value. The Babylonians were perhaps hampered by using 60 as a base system for counting rather than 10; which lingers on to this day with 60 min to the hour and 360° to turn a full circle.

This idea of calculating pi by using polygons to work out the circumference of a circle to any desired degree of accuracy was taken up by Archimedes (287–212 BC) of Syracuse. In his book On the Measurements of Circles instead of just using a hexagon to circumscribe the circle like the Babylonians, he used regular polygons of n sides inscribed within a circle as smaller than the circumference, whereas the perimeter of a similar polygon circumscribed outside the circle is taken as greater than the circumference. By making n sufficiently large, the two perimeters of the polygons will approach the true circumference of the circle, one from above, the other from below. Archimedes started with the hexagon and progressively doubled the number of sides to a polygon of 96 sides. The diameter of the circle was easy to measure; so he obtained a value for pi, expressed as an inequality: Embedded Image

As far as I understand Euclid (∼300 BC) contributed nothing original to pi. His Elements systematize the mathematical knowledge available to him. But he is not the father of geometry; he is more the father of mathematical rigour and logical thinking. The first four books of the Elements often turn up on the school syllabus and Book 3 deals with the properties of circles. All his theorems can be derived from his five basic postulates or axioms but nothing new is added to the ‘circle ratio’.

After the Greeks, pi took a rest for a while. Fibonacci and Galileo worked on it half-heartedly. But gradually ways of calculating pi by using series were discovered. For a giant like Newton (1642–1727) calculating pi was a chicken feed and in his Method of Fluxions he shows in four lines how to get a value of pi to 16 decimal places. The series he uses converges on pi much more quickly than the first of these series for pi co-discovered by his rival Leibniz (1646–1716). Newton devotes only a short paragraph to his series, apologizing for such triviality with a ‘by the way’. He later writes that ‘I am ashamed to tell you to how many places I carried these computations’. His real business was to work on the big problems of the Universal Laws of Gravitation and the nature of light. The history of calculating pi comes somewhat to an end with Euler (1701–83). He produced formulas for calculating pi by the dozens. To take just one example: the series of inverse squares Embedded Image had baffled mathematicians for decades. No one had been able to find its sum until Euler showed it to be π2/6. He went on to discover many more series and expressions that summed to terms involving pi. Although Euler had finished off one chapter in the history of pi he opened up a new one. What sort of number is pi? We know it is irrational, but is it transcendental as well? In other words, can pi be the root of an algebraic equation of finite degree with rational coefficients? It took another 100 years to settle this question but the proof was not easy. Lindemann's proof of 1882 runs to 13 pages of tough mathematics (not read by me) showing that pi is indeed transcendental.

So in the end this little number of pi turns out to be irrational, transcendental and its decimal places may run to infinity. All big concepts for something we met and used unthinkingly in our early school days.

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