×
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT

Celebrating the significance of pi

Last Updated 24 March 2015, 05:06 IST

Pi – the simple ratio of the circumference of a circle to its diameter was
celebrated on March 14. This mathematical constant is found useful in formulae used in branches of science such as cosmology, number theory, statistics, fractals, thermodynamics, mechanics and electromagnetism. The ubiquity of pi makes it one of the most widely-known mathematical constants both inside and outside the scientific community.

March 14 is written as 3/14 in many parts of the world. As 3.14 is also the
commonly known value, apart from 22/7, of the number, pi, March 14 was observed as an International Pi Day. In fact, March 14, 2015 was even more special, because 3/14/15, gives the value of pi to four decimal places and this happens only once in a
century. An even more notable day was March 14, 1592, four centuries ago,
because 3.141592 is the value of pi to six decimal places. But, of the date in our own century, the newspaper, The Guardian, said, “Mass elation will peak at 9:26:53 am when the date and time will describe pi to 10 digits”.

While this coincidence of a date and a value does bring attention to the number pi, it is necessary to say that the number has features that make it quite remarkable, among mathematical constants.
What makes it special?

The first property of the number, of course, is that it is the ratio of the diameter of a circle to the circumference. The fact that it is also the ratio of the square of the radius to the area of the circle would suggest how this ratio arises. But initially, in Egypt, Babylon and ancient India, the value was worked out simply as the
constant ratio that masons and carpenters had noticed in the course of their work.

Although this value was approximate, it was practical. It was only in the third century BC that Archimedes worked out the ratio with the help of a mathematical method, without recourse to actual measurement of circles, using what we would call an analytical method. One direct way of doing this is by dividing the circle into parallel strips, as shown in the graph above. As each strip has a known width, the distance of any strip from the centre is known and with the help of simple geometry, the height of the strip is also calculated. The height and the width then, easily give us the area of each strip.

Many centuries later, when advancements evolved in the field of mathematics, the principles of these methods have been refined and we have exact formulae to work things out. The Liebniz formula, for instance, gives the value of pi as π/4 = 1-1/3+1/5-1/7+1/9–1/11+1/13 -…….. to infinite terms. A feature of this series is that every odd and positive term is slightly greater than the adjacent, negative term. The
value of the series hence gradually increases, to approach an exact value of Pi, the addition by the terms further down in the series being smaller and smaller. But the trouble with this formula is that the approach to the correct value is slow, and it takes many terms before the value is good enough.

There is even a laboratory method to work out the value of pi – of dropping a needle on to a grid of parallel lines and counting the number of times the pin fell over a line. It has been worked out that if the lines are drawn one unit apart and the length of the pin, k, is less than this unit, then the probability that the pin will cross a line is 2k/π. Throwing the pin,  many times (which can be automated), can thus generate an accurate value for π. The value, beyond a few decimal places, is of no practical value, except in large surveys or in astronomy, where more decimal places are needed. But there is great academic interest in the fact that the exact value of π can never be evaluated. This is because the correct value is an infinite series of non-repeating digits after the decimal point, never evaluating to an exact value.

Such a quality of a number amounts to saying that it can never be correctly expressed as a ratio of two integers, which are not multiples of each other, like 7/3, or 13/5 or even 22/7. All fractions like this, where one whole number divides another, can be evaluated as an exact decimal number, or as a recurring decimal, where the division keeps giving the same remainder or same series of remainders.

An important application of this kind of number, whose decimal expansion is infinite, is that the progression of digits in the decimal form is essentially random. This must be so, as else, there would be an endless repetition of patterns. Irrational numbers are thus a source of randomness, and a computer generated value, running into millions of digits, could be used as a code that it would be very difficult to break, unless the eavesdropper knew at which stage of the progression the code maker had started.

These properties of pi and other irrational numbers are the same even if we change the system of counting from the decimal, or on the base of 10, to another, like the binary, based on 2, or the Octal (8) or the hexadecimal (16), which computers use. The properties, in fact represent basic features of circles, lines and angles, and even geometries in more than three dimensions. Thus, the study of the number pi is of great and fundamental importance.

ADVERTISEMENT
(Published 23 March 2015, 18:00 IST)

Follow us on

ADVERTISEMENT
ADVERTISEMENT