## Tuesday, January 13, 2009

### Pi

π is the most famous irrational number known to mankind. It has been studied for over 3000 years and is one of the most important and most useful of mathematic entities. To sixty digits the value of π is...

3.1415926535897932384626433832795028841971693993751058209749445923078164...

I actually have the digits above memorized, a feat that took me awhile. If you like digits of π, here are some websites (here and here).

So, you might of heard π as 3.14 or 3.14159 but even 100, 1000, or even a million digits can't officially equal π. This is because π is an irrational number. An irrational number is a number that cannot be expressed as a ratio of two rational numbers. In simpler terms, an irrational number can't equal 3/7 or 231/576 or in general x/y, with x and y being rational numbers. Another example of an irrational number is the famed square root of 2 which equals approximately 1.414... Obviously 2π/2=π but π is irrational so the rule survives. Literally, π is the circumference of a circle whose diameter equals one. Therefore, the area and the circumference of any circle can be found by the formulae πr^2 and 2πr respectively (with r being the radius of the given circle). Some early methods of calculating π where by calculating the perimeter of polygons with n sides. As n increased the approximations would get more accurate because the more sides a polygon has the closer to a circle it will be. For example, in 480 A.D. Zu Chongzhi calculated an approximation of π with a polygon with an incredible 12288 sides!! This put π between 3.1415926 and 3.1415927, the most accurate approximation of pi for the next 900 years.

Today, more than a trillion (1,000,000,000,000) digits of π have be calculated using crazy formulae including infinite sums and factorials that extend to infinity.