Flexistentialism Welcome Guest | Login or Register

Update: My computer has been dragged to its knees by the C program “Roy Williams’ pi to arbitrary precision”:http://www.cacr.caltech.edu/~roy/upi/pi.txt which uses Machin’s formula, π/4 = 4*arctan(1/5) – arctan(1/239). It has spent the last two hours working on calculating π to 1,000,000 digits, and I don’t know how much longer it will take. 10,000 digits took only a few minutes. However, it felt good to know that while we were eating Pizza πs and Key Lime πs, the computer was celebrating in its own special way. Pictures of the π pizza and π pie are forthcoming.

As many of you may know, today is March 14th, or 3/14, or… 3.14 ~= π! Today is the day to celebrate the goodness that is the number π (aka, Pi).

The most common definition of π is the following familiar equation: π = A/r^2 where A is the area of a circle, and r is the radius of said circle.

You may have seen it written like this: A = πr^2 in school, when using it to calculate the area of circles.

The number is very old, and was known in its significance by the Egyptians and the Babylonians, though their calculations weren’t as accurate as ours. The Babylonians considered π equivalent to 3 1/8 = 3.125, and the Egyptians had the more difficult to use, and slightly less accurate approximation of 4*(8/9)^2 ~= 3.160484.

π is an infinite decimal, meaning that as you calculate π, you will never find a repeating string of digits. The number never stops, you can continue to add digits forever. This isn’t just shown by experiment, in 1768 Johann Lambert proved mathematically that a repeating pattern can never exist.

π shows up in some amazing places, like probability, and the “famous five” equation that combines 5 of the most unique and key constants in mathematics: e^(i*π) + 1 = 0

e is the base of the natural log, and another infinite decimal, which is approximately equal to 2.71828, and is found all over, especially when studying growth or decay (including compound interest at the bank), the statistical “bell curve” functions, and the shape of suspended wires, or the shape of arch’s like St. Louis’ own Gateway Arch). E is just about as amazing as π, but we’ll talk more about it later, since today is π’s day.

i is the so called “imaginary number” and is equal to the square root of -1. The astute among you will remember that it is impossible to take the square root of a negative number, because any number multiplied by itself equals a positive number. The explanation for i then is that it isn’t a real number. The “real number set” is defined most generally as all the numbers you are probably familiar with. Negative numbers, zero, positive numbers. Number like π and e are considered real numbers, as are fractions. Basically, imagine an infinitely long number line, with zero in the middle. Any number that can be found on that line is a “real” number. You won’t find i on that line anywhere, because there is another number line, perpendicular to the real numbers, that contains i. It is a difficult concept, but is crucial to mathematics. i is the most clearly defined imaginary number, and is used to build other imaginary numbers, like 3i, or πi.

1 and 0 you are familiar with, but they are still crucial to mathematics, despite being some of the very first numbers you learn as a child. In fact, the reason you learn them early on, is that you find them everywhere, from the simplest calculations to the most complex number theory. 1 and 0, indicating the most basic form of “somethingness” and “nothingness” respectively, are key to mathematics.

People have calculated π out to billions of digits. Here are a bunch of digits of π:

bq. 3.141592653589793238462643383279502884197169399375105820974944
592307816406286208998628034825342117067982148086513282306647
093844609550582231725359408128481117450284102701938521105559
644622948954930381964428810975665933446128475648233786783165
271201909145648566923460348610454326648213393607260249141273
724587006606315588174881520920962829254091715364367892590360
011330530548820466521384146951941511609433057270365759591953

All in all, π is a pretty cool thing. To celebrate π, Dennis will be bringing over beer, and we’ll make a pizza π, and watch the movie π, and maybe eat a key lime π, and maybe even calculate some digits of π. Cool, huh?