S. ANANTHANARAYANAN
March 14, denoted as 3/14 in places such as the US, is celebrated because the number, pi, or π, and known to most as 22/7 has the value 3.14. The day was even more special in the year 2015, because the date read as 3/14/15, which gives the value of pi to four decimal places and happens only once in a century. Four centuries ago, March 14, 1592, gave the value of pi, 3.141592, to six decimal places! But even in our own century, 9.26:53 a.m. on March 14, 2015, had the value of pi to 10 digits.
But apart from fitting into the calendar in this way, the number pi has features that make it quite a star, among mathematical things. The purpose, of naming a day after the number pi, and celebrating the day, by eating pies, for instance, one hopes, is to use fun and food to draw attention to different qualities of this important number.
The first property of the number, of course, is that it is the ratio of the diameter of a circle to the circumference [c=2πr]. The fact that it is also the ratio of the square of the radius to the area of the circle [area = πr2] would suggest how this ratio arises. But at first, it was worked out simply as the constant ratio that masons and carpenters all over the world had noticed in the course of their work.
It is perhaps Archimedes who first used a mathematical method to work out the ratio without recourse to actual measurement of circles. This way of working things out is called the analytical method. One way of doing this is to divide the circle into strips, as show in the picture. While each strip could be considered to have some width, its height can be worked out from its distance from the centre. With the help of the radius and the width and height, we can work out the area of each strip. And then,adding together the area of all the strips leads us somewhat near the area of the circle. Now, if we make the strips narrower and narrower, our estimate of the area would keep getting more accurate. We can similarly, work out almost the circumference of the circle, by adding up all the slanting ends of the strips, as the strips get narrower and narrower. Archimedes actually did something like this, only he worked out the area, or the sides, of a polygon, which is a many-sided shape, fitted inside and outside the circle, and then increasing the number of sides, to make the shape more nearly like the circle.
In later centuries, the methods were refined and we have exact formulas. The Leibniz (1646-1716) formula, for instance, says: π/4 = 1-1/3 + 1/5 – 1/7 + 1/9 – 1/11+ 1/13 -…….. to infinite terms. We can notice 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. A problem with this formula is that the approach to the correct value is slow, requiring many terms before the value is good enough. And there are better formulas, and we have very accurate computations of the value of pi indeed.
There is even an ‘experimental method’ – of dropping a needle on to a grid of parallel lines and counting the number of times the pin falls across 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 a huge number of times, which can be automated, can thus generate an accurate value for π.
Is the value important?
The value, beyond a few decimal places, is of no practical value, except in large surveys or astronomy, where more decimal places are needed. But there is great interest in the fact that the exact value of π can never, ever, be evaluated. This is because the correct value is an infinite series of non-repeating digits after the decimal point. This amounts to saying that the value cannot be expressed as a ratio of two integers, like 7/3, or 13/5 or even 22/7. All fractions like this, where one whole number divides another, can be evaluated as a terminating decimal number, or as a recurring decimal, where the division keeps giving the same remainder or the same series of remainders. Examples of recurring decimals are 1/3= 0.3333333… or 31/99= 0.31313131…
Numbers like this, which can be expressed as a ratio, are called rational numbers, while those which cannot are called irrational numbers.
Random progression
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 the pattern. Irrational numbers are thus a source of randomness, and a portion of a computer-generated value, running into millions of digits, could be used to generate a code. It would be a code that is very difficult to break, unless the eavesdropper knows at which stage of the progression the portion selected 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, based on 8, or the hexadecimal, based on 16, which computers use. The properties, in fact, represent basic features of circles, lines, and angles, and even geometries in more than three dimensions, and the study of the number, pi, is of fundamental importance.
(The writer is a retired civil servant and science writer. He can be contacted at response@simplescience.in)
