# A Common Book of

3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679...

The mysterious and wonderful is reduced to a gargle that helps computing machines clear their throats.

-- Philip J. Davis

In recent years, the computation of the expansion of has assumed the role of a standard test of computer integrity.

-- David H. Bailey

It requires a mere 39 digits of in order to compute the circumference of a circle of radius (an upper bound on the distance travelled by a particle moving at the speed of light for 20 billion years, and as such an upper bound for the radius of the universe) with an error of less than meters (a lower bound for the radius of a hydrogen atom).

-- Jonathan and Peter Borwein

The number has been the subject of a great deal of mathematical (and popular) folklore. It's been worshipped, maligned, and misunderstood. Overestimated, underestimated, and legislated. Of interest to scholars, crackpots, and everyday people.

Pretty amazing accomplishments for a number!

The next few pages will attempt to teach you a few facts about .

You will find here, among other things, a brief history of extended precison approximations of , including Archimedes' method for estimating , a page full of "oh, wow!" formulas used to estimate over the centuries, and a brief look at a modern algorithm used to compute . I also have a list of references for further reading and a list of other pages devoted to pi on the Web.

Before we begin, it might not hurt to remind you that is defined as the (constant) ratio of the circumference to the diameter in any circle. In other words, the circumference and diameter of every circle are known to be related by .

It's not hard to see (using only elementary geometry) that is bigger than 3 but less than 4.

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Neal Carothers - carother@bgnet.bgsu.edu