# What is pi? How do we use it in life?

Apr 28, 2018

See below.

#### Explanation:

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 $r$, $\pi = \frac{C}{2 r}$
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 $\frac{p}{q} : \left\{p , q\right\} \in \mathbb{Z} , q \ne 0$]

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:

${\pi}^{2} / 6 = {\sum}_{i = 1}^{\infty} \frac{1}{i} ^ 2 \to \pi \approx 3.1415926535897932384626433 \ldots$

[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 $2 \pi r$

(ii) The area of a circle of radius $r$ is $\pi {r}^{2}$

(iii) The volume of a sphere of radius $r$ is $\frac{4}{3} \pi {r}^{3}$

There are a vast number of instances involving $\pi$ in the physical
world as well as many other applications in pure mathematics.