In Search of πAuthor(s): John SykesSource: Mathematics in School, Vol. 29, No. 3 (May, 2000), pp. 32-33Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30212346 .

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In search of

11 3.141 592 653 589.......

by John Sykes

Having read 'The Joy of n' by David Blatner I copied out n to a thousand places on to pieces of card with which I circumnavigated my classroom. The interest this produced in my pupils was fascinating with many wishing to know how they could obtain n to so many decimal places. When I informed them that I had merely copied them from the front of Blatner's book they admired my stamina if not my mathematical skills. So during the following term different classes tried different ways of calculating n. A variety of skills are tested. Basic numeracy and fractions can be combined with long division or calculator use. Algebraic manipulation is necessary to obtain n from circumferences and areas. Random numbers and plotting points gives rise to an estimate for n. The ideas of infinite series can be introduced. Higher level skills of iterations and analytical, as well as numerical, integration allow senior pupils alternative approaches. A practical approach can be made with a simple pendulum.

Having completed some work on percentage pie charts to revise for Key Stage Three tests the class made further use of their pie chart templates in two ways. First, a piece of string was wrapped around the circumference and its length measured. A wiser pupil used some lateral thinking and rolled the template along the edge of some paper and measured the distance covered. Most found the circumference to be 31.4 cm and because the diameter of the template is 10 cm, n was soon found to be 3.14.

The next use of the template was to draw around it on a piece of card. 5% sectors were marked and the circle cut out and divided into the twenty sectors. The pieces were arranged in the familiar form shown in the diagram.

This was taken to be a parallelogram and its area calculated to be approximately 78 cm2. Using the equation: Area = n r2 the value of n was found to be 3.12.

The processes so far had included revision of the equations for the circumference and area of a circle and their algebraic manipulation to make n the subject.

Following this lesson the class's homework was based on an historical survey. This gave an interesting background and provided some basic calculator use. The first example came from around 1500 BC with the ancient Egyptians who calculated the area of a circle in terms of its diameter d using

Area= d - 2

256 A simple exercise in algebra produced a value of = 256 3.160 to 3 decimal places. 81

Next was a conversion from fractions to decimals. In the third century BC Archimedes calculated upper and lower bounds for the area of circles using polygons and found that

10 1 3- < i< 3-

71 7 i.e. 3.140 845 070 4 < n < 3.142 857 142 9

1 22 Interestingly 3 7 or 7 is still used by many as an approximation to i.

32 Mathematics in School, May 2000

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Eight hundred years later the Chinese found n to be 16

3 = 3.141 592 92 113

which is remarkably accurate.

I then introduced the idea of an infinite series with Francoise Viete's method of 1593 in which

2

1/2 1/2 +

1/2i1/2

i 2f1 + 1/2 1/2 +

1/2 1/2

It proved a good exercise in the use of a calculator to evaluate this expression (to give 3.121 445 152 ...).

Another series by John Wallis dated 1655 is

n 22 4 4 6 6 8 8 -----X--X--X--X--X--X--X--... 2 1 3 3 5 5 7 7 9

This is easier to calculate but even with twenty terms gives a poor value for n. Blatner lists many other infinite series to evaluate n.

Curiously, the Greek letter n was not introduced until 1706 by William Jones.

With a sixth form group who were preparing for the UCLES Pure 3 module we used the iteration

Xn+1 = Xn + sin xn which on a spreadsheet gives

3.00000000000000

3.14112000805987

3.14159265357220

3.14159265358979

3.14159265358979

What is surprising about this iteration is the speed with which it converges. This could also be done with a higher level GCSE group, providing the opportunity to introduce radians!

A further mathematics group working on the Pure 4 module showed that the integral

0.5 0.5

fi dx_=[sin-'x] _ o 1-x2 0 6

can be evaluated using the trapezium rule with, say, 10 strips to give n = 6 x 0.523 759 = 3.142 554.

Another, more accurate method, is to use Simpson's rule with 1 1

S1 dx=[tan-'x] _

4 0 1 +x2 0 4

which, with 10 strips, gives n to be approximately

3.141 592 614.

The first practical method that could be used is the simple pendulum for which the time period T is given by

T= 2n gI

where 1 is the length of the pendulum. The value of n will depend not only on the accuracy of the measurement of the period but on the value of g (9.81 cm s-2 is not very helpful).

A different approach is the probability experiment using random numbers. Axes are drawn from 0 to 1000 and a square and arc of a circle are drawn as shown:

1000

y

1000 0 x

With a calculator pairs of random numbers are generated which are then plotted as coordinates, e.g. 0.735 and 0.248 would be plotted as (735, 248). When sufficient points are plotted the total number of points is N and the number falling inside or on the quarter circle is n. Then the probability of a point being in or on the quarter circle is

n 0.25 x n x 10002

N 10002 hence a value for n can be obtained using

4n R =

N It would be interesting to know what other methods

readers have used to evaluate n or what techniques they may have employed to find n to more than 10 decimal places with a calculator or 15 decimal places with a spreadsheet.

Reference Blatner, D. 1997 The Joy of iT, Allen Lane, Harmondsworth.

Keywords: Pi; Shape and Space; History.

Author John Sykes, Head of Mathematics, Sedbergh School, Cumbria LA10 5HG.

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Mathematics in School, May 2000 33

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