What is pi?
The simplest definition of pi is the ratio of the circumference of any circle to its diameter. It can be proved that pi is a constant.
So, for a circle of radius r, pi = C/(2r)
where C is the circumference of the circle.
It can be proved that pi is an irrational number, that is it cannot be expressed as a fraction.
[Strictly, it cannot be expressed by any p/q: {p,q} in ZZ, q!=0]
Since pi is irrational it can never be exactly evaluated by any finite decimal. Thus, pi can only ever be approximated by a value of arbitrarily many decimal places.
Whilst there have been many approximation formulae discovered, an efficient approximation of pi was found by Leonard Euler in the 18th century to be:
pi^2/6 = sum_(i=1)^oo 1/i^2 ->pi approx 3.1415926535897932384626433...
[NB: It can also be proved that pi is a transcendental number. That is it cannot be the root of any polynomial equation with real coefficients.]
How is pi used in real life?
The practical uses of pi are too numerous to set out here. I'll list a few basic examples below.
(i) As can be seen from the definition above, using pi we can find the circumference of a circle of radius r which is 2pir
(ii) The area of a circle of radius r is pir^2
(iii) The volume of a sphere of radius r is 4/3pir^3
There are a vast number of instances involving pi in the physical
world as well as many other applications in pure mathematics.
