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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
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
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 √pis constructible.
In 1836 Pierre Wantzel showed that if pis 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 pis 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 pcan 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 pis.