MY CHILDREN are at an age when they bring home maths problems from school involving the calculation of areas and circumferences of circles. Inevitably, I am often asked to check whether a particular problem has been solved correctly. This has set me to thinking about the term pi, its meaning and its history.
Pi has fascinated mathematicians for thousands of years and certain aspects of its meaning remain under active investigation even today. It is defined as the ratio of the circumference (c) of a circle to the diameter (d) of the circle, i.e. pi = c/d.
The formula for the area of a circle is: area = pir2 (r=radius, d=2r). When you divide the circumference of a circle by the diameter, you get a number beginning with 3.14159 that continues on from there with infinite length and a completely random chain of digits.
Pi was present at the birth of mathematics, some 4,000 years ago, long before it even had a name. In 1748, Leonhard Euler, a Swiss mathematician, settled on the Greek letter pi to denote pi. Euler also introduced the symbol i for the square route of -1 and oc for infinity. Old as it is, pi remains elusive and it can never be fully written out in a decimal number system. In other words, when you divide c by d it never ends no matter show many decimal places you go to.
An Egyptian scroll dating back to 1650BC contains the earliest known estimate of pi. The Rhind scroll details 84 problems and their solutions. One of the problems assumes that the area of a circle with a diameter of nine units is the same as that of a square with sides of eight units. The assumption is incorrect, but it is a very good approximation. If you use the correct formula for the area of a circle (pir2) you can work out that the scroll implicitly sets pi equal to about 3.1605.
For about 1,000 years, pi remained a rough tool for measuring circular plots of land. In the third century BC, the most brilliant of the ancient Greek mathematicians, Archimedes, devised a way to calculate the value of pi with real precision. His method can be understood in a simple way as follows. Draw a circle with a diameter of one. The circumference of this circle is equal to pi x 1 = pi.
Now draw within the circle, and circumscribe about it, a regular polygon (sides of equal length) of any number of sides. The circumference of the circle lies physically between the two polygons and therefore pi lies, arithmetically, between the two perimeters. The polygon perimeters serve as upper and lower bounds to the true value of pi. The greater the number of sides on the polygons, the closer the bounds move to pi.
Archimedes used polygons of 96 sides and showed that the value of pi lies between 3 10/70 and 3 10/71.
The same technique was used by the Dutch mathematician Ludolph van Ceulen to calculate pi to 20 decimal places. He later extended this to 35 decimal places, a feat that so impressed the Germans that they still refer to pi as the Ludolphine number. The last three digits of Van Ceulen's pi calculation - 288 - were engraved on his tombstone in St Peter's Church in Leiden.
A 35 decimal place pi would seem to be accurate enough for any calculation. For example, imagine a gigantic circle with its centre at the Earth and extending out to a radius which reaches the star Sirius, about 8.8 light years away. If you calculate this circle's circumference using the 35 digit pi, your answer will be in error by no more than a thousandth part of a millimetre. Nevertheless, many mathematicians have spent years extending the estimate of pi, digit by digit.
IN 1873 William Shanks, a British school headmaster, after 20 years of calculations by hand, published pi to the 707th decimal. Shanks held the record for 72 years until 1945 when it was noticed that he had mistakenly written a five rather than a four in the 528th place. Off course, every digit after this was therefore wrong.
Ultimately, mathematicians "would like to know, once and for all, whether pi is a totally random sequence of numbers or a chain with a subtle pattern. The founding father of computer science, John von Neuman, examined the first 500,000 digits of pi and found that they are completely random. The current record for working out the decimal expansion of pi (by computer) is held by Yasuniasa Kanada, of the University of Tokyo, who calculated pi to 6 billion decimal places in 1997.
Whether pi is a completely random number is still not known for sure, but it certainly appears to be a normal number, i.e. all digits occur equally often.
We are all familiar with the phrase attempting to square the circle. The phrase means attempting to bring about some impossible reconciliation. The phrase has its origin in the well known and ancient mathematical puzzle of trying to draw, using only a compass and ruler, a square having exactly the same area as a circle. Let me emphasise this is an impossible task, lest you become obsessed with the problem and end up like Edwin J. Goodwin.
Goodwin was an Indiana physician who became obsessed with the ancient problem of squaring the circle. His obsession eventually led him to conclude that every geometer since Euclid had miscalculated the value of pi. He deduced that the ratio of the diameter and circumference is as five fourths to four, which means that pi works out as 3.2 exactly.
Emboldened by his discovery, Goodwin introduced a Bill Introducing a New Mathematical Truth to the Indiana State Legislature. The State Committee of Education recommended that the bill be passed. The reasoning was that the State and the author could earn a lot of money when everyone else had to pay royalties for the use of this new and correct value of pi. Eventually, a mathematics professor, C.A. Waldo, proved that Goodwin was wrong and the Bill was dropped.
Why this endless fascination with pi? Pi is a fundamental relationship in mathematics and geometry. We are introduced to the concept as children and it engages us both rationally and aesthetically. Then, the fact that such a fundamental number completely escapes exact expression in our mathematical language, vividly engages our imagination. Of course, for those of us with a pre disposition to accept a literal translation of the Bible, the problem of the exact value of pi is solved. In the First Book of Kings in the Old Testament it states and he made a molten sea, 10 cubits from one brim to the other: it was round "all about and a line of 30 cubits did encompass it round about. Pi therefore equals 3.0.