Fun with Maths - Part 2 - Squaring the circle

Fun with Maths - Part 2 - Squaring the circle - by P.K.Odendaal - August 2012. 

There are numbers which are not natural or rational numbers, and are called transcendental numbers. These are numbers which cannot be expressed exactly in our number system, having infinite decimals, like p and e - with the result that we will never know what the circumference of a circle is equal to exactly, if we have its radius. The formula is easy C (circumference)=2*p*R (Radius).

There are two ways of trying to establish it.

Let us try the geometric one first.

If we take a circle and approximate the curve with straight lines we do the following:

1.     We draw three spokes from the circle centre, each 120 degrees apart, and we then join the end points of these lines and it gives us three lines approximating the circumference of the circle. We know from Pythagoras that the length of each of these lines is R*√3 - and here we start to run into problems because √3 is a transcendental number. If we had a calculator, which we do not have in these articles, we would have found that p=2.6

2.     Next we are going to take four spokes, and have four lines, each with length R*√2 - and this is our next problem, because √2 is a transcendental number. If we had a calculator, we would have found that p=2.83

3.     We might get lucky one day. Let's take six spokes. The length of one line so generated is exactly equal to R. Yes we have struck luck, but p=3 in this case and we are still 5% from the truth. Without a calculator, this is our last try.

Square roots are constructible as shown by Euclid in 300 B.C., so the key to the problem is whether √p is constructible. In 1836 Pierre Wantzel showed that if p is a transcendental number, then it is not a constructible number. The final piece of the puzzle arrived in 1882 when the Lindemann-Weierstrass theorem implied that p is indeed a transcendental number, and so squaring the circle is impossible.

In 1914 Ramanujan brought us closer to the truth with algebra. He established that p can be expressed as a rational number namely ⁴√2143/22 - i.e. the fourth root of the ratio given, which, in this instance, is correct to eight decimals - indeed remarkable.

If you wish to express it in rational numbers, here are a few : 3 ; 22÷7 ;   333÷106 ;  355÷113 ;  103993÷33102  . Eric Weisstein has calculated its value to 100 million terms in 2003.
 

It seems that we will never know what the exact value of p is.