# Pi

The mathematical constant Pi occurs widely in mathematics and science. The number expresses the ratio of any circle’s circumference to its diameter. It also appears in higher level mathematics.

For practical purposes there are two equations that regularly crop up involving Pi.

Suppose we have a circle of radius r

The length of the circumference of that circle is always 2*Pi*r.

The area enclosed by the circumference is always Pi*r*r.

These two formula form the basis of numerous practical calculations. Ancient mathematicians used these calculations to calculate a value for Pi. Their methods are very easy to reproduce today.

The first method illustrates a connection between pi and rotary motion. When a cartwheel completes one full turn it has moved forward by a distance equal to the the circumference of the wheel. The distance travelled on the ground can then be divided by twice the wheel radius to give an estimate for Pi.

A second method involves drawing a large circle on graph paper. The area of the circle can then be estimated by counting squares on the graph paper. Then the estimated area of the circle can be divided by the square of the radius to give an estimate for Pi.

Using practical methods like these the ancient Babylonians calculated Pi to be 25/8. The Egyptians thought it was 256/81 while the Indians reckoned on 339/108. Remarkably, although we can now calculate Pi to much greater precision the ratio ratio of 22/7 is sufficient for.practical purposes.

Early mathematicians suspected that Pi had some peculiar properties.

They believed but could not prove that Pi was irrational which meant that it could never be expressed as fraction in the form a/b. Muhammad ibn Musa al Khwarizmi suspected this in the eighth century. Johann Lamert proved it using continued fractions in 1768. More recent proofs by Ivan Niven and Mary Cartwright rely upon a knowledge of trigonometry and integral calculus.

Mathematicians also suspected that Pi was transcendental. This meant that they thought that Pi could not be written as a solution to a polynomial equation with rational coefficients. The proof, that Pi was transcendental, was published by the German mathematician Ferdinand von Lindemann in 1882. von Lindemann’s proof also showed that the ancient problem of squaring the circle was impossible.

Many mathematical insights concerning Pi derive from its relationship with angle. When angles are measured in radians (there are 2*Pi radians in a ciircle) it becomes possible to develop series expansions for the trigonometric functions that involve Pi.

The most intriguing relationship that involves Pi is the formula

e P.i + 1 = 0

The equation is attributed to Leonhard Euler and is often regarded as the most e1egant equation in mathematics. Three mathematical constants,e , Pi and i are related using the basic mathematical operators of exponentiation, multiplication and addition.

There are many facets to Pi that are yet to be discovered.