By definition, Pi is the ratio of the circumference of a circle to its diameter. Pi is always the same number, no matter which circle you use to compute it.

Pi is a very old number. The Egyptians and the Babylonians knew about the existence of the constant ratio Pi, and they had figured out that it was a little bigger than 3. The Babylonians had an approximation of 3 1/8 (3.125), and the Egyptians calculated it to be approximately (4/3)^4 which equals 3.1604. Indian Jaina religious works (dating from 500 B.C. to 100 B.C.) used \sqrt{10} as the value of \pi (circumference = \sqrt{10} × Diameter)

The first mathematician to calculate pi with reasonable accuracy was Archimedes, around 250 B.C. Using the formula: A = pi*r^2 for the area of a circle, he approximated pi by considering regular polygons with many sides inscribed in and circumscribed around a circle. Since the area of the circle is between the areas of the inscribed and circumscribed polygons, you can use the areas of the polygons (which can be computed just using the Pythagorean Theorem) to get upper and lower bounds for the area of the circle. This was the first general method for calculating approximations to pi, and at least theoretically it could be used to get any degree of accuracy if you could just do the computations. Archimedes showed in this way that pi is between 3 1/7 and 3 10/71. The same method was used by the early seventeenth century with polygons with more and more sides to compute pi to 35 decimal places (Van Ceulen did the biggest 674 calculations.)

The modern symbol for Pi (\pi) was first used in our modern sense in 1706 by William Jones. Pi was chosen to represent the number 3.141592… because the letter (\pi) in Greek, pronounced like our letter ‘p’, stands for ‘perimeter’.

Pi is an infinite decimal. If you write Pi down in decimal form, the numbers to the right of the dot never repeat in a pattern. Many mathematicians tried to find a repeating pattern for Pi till Johann Lambert proved tin 1768 hat there cannot be any such repeating pattern. As a number that cannot be written as a repeating decimal or a finite decimal (you can never get to the end of it) Pi is irrational: it cannot be written as a fraction (the ratio of two integers).

When Newton and Leibnitz developed calculus in the late seventeenth century, more formulas were discovered that could be used to compute pi. For example, there is a formula for the arctangent function:

arctan(x) = x – x^3/3 + x^5/5 – x^7/7 + …

If you substitute x = 1 and notice that arctan(1) is pi/4 you get a formula for pi. It takes too many terms to get any accuracy. A more useful formula is Machin’s formula:

pi/4 = 4 arctan(1/5) – arctan(1/239)

This formula and similar ones were used to calculate approximations to pi to over 500 decimal places by the early eighteenth century (this was all hand calculation!) The modern computers have made it much simpler. In 1910 the great Indian mathematician Ramanujan discovered a formula that in 1985 was used to compute pi to 17 million digits. In daily life, we don’t have to remember so many digits. Approximations 22/7 and 355/113 are good enough.