Paul's Page of Pi

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Archimedes' Method

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Welcome

Welcome to Paul's Page of Pi. This site started as just a small page that I put up on the Internet in early 1996 to celebrate . Via the excuse of a PreCalculus term paper, I've expanded my page into an entire site. Most of the content of the site is focused on methods of calculating . I hope you find it amusing and useful.

Why Calculate Pi?

Pi is the constant ratio between the diameter of a circle and the circumference of that circle. Although a small minority of people still try to prove that pi is a rational, essentially all mathematicians agree that pi is an irrational number. Pi's irrationality can be demonstrated through a short proof. It is because of this irrationality that people are forced to calculate the digits of pi.

But why would anyone want to calculate beyond the first few digits? The ten digits (3.141592654) that are built into most scientific calculators are sufficient for nearly any real-world calculation; you can calculate the circumference of the Earth's orbit around the sun, and be off by less than 100 meters. Even extremely precise scientific work never requires more than twenty places. Clearly hundreds of thousands of digits of have no practical value. Mathematician Hermann Schubert offers an example to show the uselessness of hundreds of digits of :

Conceive a sphere constructed with the earth at its center, and imagine its surface to pass through Sirius, whis is 8.8 light years distant from the earth [that is, light, traveling at a velocity of 186,000 miles per second, takes 8.8 years to cover this distance]. Then imagine this enormous sphere to be so packed with microbes that in every cubic millimeter millions of millions of these diminuitive animalcula are present. Now conceive these microbes to be unpacked and so distributed singly along a straight line that every two microbes are as far distant from each other as Sirius from us, 8.8 light years. Conceive the long line thus fixed by all the microbes as the diameter of a circle, and imagine its circumference to be calculated by multiplying its diameter by to 100 decimal places. Then, in the case of a circle of this enormous magnitude even, the circumference so calculated would not vary from the real circumference by a millionth part of a millimeter. This example will suffice to show that the calculation of to 100 or 500 decimal places is wholly useless.

Some argue that by calculating many digits of mathematicians can empirically verify the theoretical ideas about . In my opinion, this argument is rather unconvincing. One theory that mathematicians were trying to prove empirically is that the statistical distribution of the digits of is equal. That is, the frequency of each digit (0..9) approaches 1/10 as the number of digits approaches infinity. As early as 1960 it was shown that each frequency is statistically equivalent for the first 16,000 digits of . Since then the digits of have passed this test every time it has been run. The other theory that mathematicians try to prove (or to disprove) is that is irrational. While this last idea has little mathematical validity, it does hint at what I believe is the real motivation for calculating .

Mathematician Petr Beckmann writes in A History of Pi: For the most part, I suspect, that driving force behind these calculations was the spirit that makes people go over Niagra Falls in a barrel or to top the world record of pole sitting by another 20 minutes.

Pi is fundamental to the way in which our universe functions; practically everything is dependent on at some basic level: light, sound, energy, gravity, electromagnetic fields, matter itself... In fact, is so central that it can be seen as a symbol of our universe. Humans like to think that we live in a rational world. We like to take the often chaotic cosmos and create order so that we can understand it. Notwithstanding, transcends rationality, and in doing so, it disturbs the order that we like to see. Symbolically, 's irrationality denotes an irrationality to the universe that we do not like. Pi represents an omniscience which we will never posses, but that we can nudge closer and closer to as we approach its true value. Calculating is a quest parallel to trying to fully understand our universe. It is for this reason that we wish to calculate to millions of places and beyond.