**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.