It’s that time of year again – March 14th – that’s right, it’s **PI DAY**! PI is an irrational number. No, that doesn’t mean it’s unreasonable. It means it’s not rational – that is PI can not be represented as a ratio of two integers. In other words, PI can not be represented as a fraction (where the numerator and denominator are integers). However, * PI can be approximated *as a fraction or a decimal number. Many know PI as

**3.14**or

**3.1415**or even

**3.141592653**(

*knowing this many digits promotes one to dork status – but that’s alright by me*). No matter how many digits, these are just approximates of PI, often well within given or acceptable tolerances. Computers have found decimal approximations of PI to over a trillion decimal places (I kid you not!). There are many very interesting properties associated with PI. One that I find the most intriguing is the following summation:

So multiplying the right side of the equation by 4 will result in an approximation of PI. Although PI can not be represented as a fraction, it can be found by infinitely adding and subtracting fractions (

*this is called an infinite summation)*. Even index terms are added and odd index terms are subtracted. Each term is just the reciprocal of 2n-1 (e.g. 1/(2n-1)) where n is any natural number. Initially, the summation does not show much promise. The sum of the first 12 terms is approximately 2.8952 – not very close to PI. However the sum of the first 200 terms is approximately 3.12159 – much closer! I have created a quick program that computes the approximate of PI based on the number of summations you input. Try 1000, 100000, 1000000 (a million) and 1000000000 (that’s a billion – it takes a few seconds) to see how the approximation converges towards PI.

**Have fun with PI and HAPPY PI DAY!**

Tags: Infinite Sum, PI, PI Day, Summation