2014-03-18

From the first year of secondary school to postgraduate studies at universities around the world, there is nowhere to hide from Π. It is a nightmare for maths-hating teenagers and a never-ending mystery for professors who teach it in the daytime and dream about it at night. Since Archimedes tried to ‘square the circle’ in third century B.C.E. Sicily, at that time a part of the Greek Empire, mathematicians have wondered about this number. Nowadays, supercomputers can calculate Π to 1.4 trillion places, although we only need the first thirty-two digits to work out the size of the universe. So, why does Π continue to excite scientists and bore school kids more than two thousand years after Archimedes first played with the idea of this strange number?

The Ancient Greeks thought in lines and squares. They tried to calculate the area of a circle by working out what it could look like as a square. In his book, Archimedes described how he drew a larger rectangular hexagon outside a circle and a slightly smaller one inside it. He then halved the length of the sides until each polygon had 96 sides. He did this because he wanted to get closer to what a circle looked like. When each polygon had 96 sides, he calculated that the value of Π was between 31⁄7 or about 3.1429, and 310⁄71, around 3.1416. Archimedes was very nearly right – but only very nearly, not exactly. He also showed that the area of a circle was equal to Π multiplied by the square of the radius of that circle. We can write this as πr^{2}. Of course, if we cannot know the exact value of Π, we cannot know the exact area of a circle.

For more than 1500 years, nobody after Archimedes seemed very interested in Π. The first man to take up the subject again was Madhava, who lived from about 1340 to 1425 near the town of Cochin in southern India. He calculated Π arithmetically (not geometrically) as:

Π ÷ 4 = 1 – 1/3 + 1/5 – 1/7 + 1/9 … + (-1)ⁿ/2n + 1 ….

What Madhava did was very special because he showed Π as an infinite series, one that could never end. (Don’t forget that the supercomputers are still calculating the value of Π after 1.4 trillion places!) In other words, Π is a number that can never have an exact value. So, Madhava moved mathematics from the finite to the infinite.

About three and a half centuries later on another continent, the Swiss mathematician, Lambert, showed in 1761 that Π would never repeat the same pattern of numbers and was, therefore, an irrational number. Obviously, then, we could never write it as a fraction

Then, in 1882, the great German mathematician, Lindemann proved that Π was a transcendental number, which means that we cannot write an equation with values of x and whole numbers to show its value. So, we cannot use algebra to show the value of Π. This is different from Lambert’s finding that Π is irrational. For example, √2 is an irrational number because it will never repeat the same pattern of numbers as decimals, but it can be written in algebra as: *x*2 - 2 = 0. If Π is transcendental, then we can solve the Greek problem of trying to square the circle by saying that it can never be done!

So, Π is an infinite series if we write it in arithmetic, an irrational number and a transcendental one too. But we do not know its value and the more that mathematicians find out about numbers, the less we seem to know about it. As Ian Stewart says, the search for Π is like the drunk man who loses his keys and tries to find them in the light of a lamppost when all the time, they are lying in the dark just outside that circle of light. Perhaps one day, every mathematician hopes, she will step away from the place where the answer to Π is expected and find it in the dark … perhaps.

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