Π or π (or in English, "pi") is the 16th letter of the Greek alphabet.
The upper case form, Π , means:
The lower case form, π , means:
As the top of this e2node indicates, pi is also an Everythingite.
Pi is the number equal to the circumference divided by the diameter of a circle. It's an irrational number so most people just use 3.14 or 3.14159. Though many memorize a lot more (hundreds) of digits, nobody is stupid enough to try to multiply by that.
The circumference of a circle is πR2 (R=Radius) or just πD (D=diameter).The area of a circle is πR2.
See A brief history of Pi for, well, the history of pi.
Also the title of a poem which I first saw as an easter egg on Audiofind.com. Sadly I have no idea who the original poet was.
i knelt down next to you watched you laying there i put my hand on your forehead saw your eyes open it's time to wake up there are a lot of people who want to meet you
you looked back at me stared into my eyes i'll never forget what you said "you" "you are the builder" "the creator of things unseen"
15.000? HA! here's the first 119.000 decimal places of pi:
How to write a Pi-specification (please make yourself familiar with specifications before reading the following): As stated in specification it consists of the export, the import, the common parameters and the body. In the first three there are only the signatures of the operations and types ( in import and common parameters are these which are not in this module (specification) ). A signature is the first line of an operation (ex. in C int add (int x, int y) ). To show you how this all works I show you a short specification of the natural numbers + 0 ({0,1,...}):
this is what is exported.
cem NAT type view specification export type Nat constructor null : -> Nat constructor successor: Nat -> Nat operation null : -> Nat operation successor: Nat -> Nat operation add : Nat,Nat -> Nat operation one : -> Nat operation two : -> Nat
import
common parameters type Logic operation true : -> Logic operation false : -> Logic constructor true : -> Logic constructor false : -> Logic
body construction of type Nat is internal constructor null : -> Nat constructor successor: Nat -> Nat operation add : Nat,Nat -> Nat variables x,y : Nat equations add(x,null) = x; add(x,successor(y)) = successor(add(x,y)) operation one : -> Nat equations one = successor(null) operation two : -> Nat equations two = successor(one)
Please Note: Most of the modules are German so it can be hard for non-german speakers to understand
Cadaeic Cadenza encodes the first 3835 digits of pi, while simultaneously being an interesting read -- not a simple feat.
I think I should elaborate on what others have said about solving for pi. You can approximate pi/4 as accurately as your heart desires by the formula 1 - 1/3 + 1/5 - 1/7 + 1/9 ... always alternating adding and subtracting, and adding two to the denominator. Note that when you are on the adding phase you will always be slightly above pi/4, and on the subtracting phase you will always be just a little below pi/4. But if you just decide how many digits you need pi to, you can run the equation until the digits you will actually use stop changing, and are therefor accurate. Then multiply your result by 4 (because remember, this is solving for pi/4) and ta-da, pi. Here look, I wrote this C++ program to find pi:
#include "iostream.h" int main(int argc, char* argv) { long double pi=1; int denominator = 3; bool alt = 0; long double divisor; while (true) { divisor = 1 / (long double)denominator; cout << pi*4 << " "; if (alt == 0) { pi = pi - divisor; alt = 1; } else { pi = pi + divisor; alt = 0; } denominator += 2; } return 0; }
I'm no expert coder, so there is probably a way to get more digits in the output that I'm to stupid to know about, I'll update here and say so if I ever figure it out. -grin- But in the mean time, play with that.
f(x)= -1 x < 0 = 0 x = 0 = 1 x > 0
+inf f(x)= Σ an * sin(2*n*Pi*x/L) n=1
= 2/(n*pi) * (1-(-1)^n) = 0 n even 4/(n*pi) n odd
+inf f(x) = Σ 2/(n*pi) * (1-(-1)^n) * sin(2*n*Pi*x/2) n=1
+inf f(x) = Σ 4/((2k+1)*pi) * sin((2k+1)*Pi*x) k=0
+inf 1 = Σ 4/((2k+1)*pi) * sin(k*pi+Pi/2) k=0 +inf = Σ 4/((2k+1)*pi) * (-1)^k k=0
+inf pi/4 = Σ (-1)^k/(2k+1) k=0 = 1 - 1/3 + 1/5 - 1/7 + ....
The limit for both xn and yn as n approaches infinity will be pi.
When working with vectors, Π (uppercase) is also often used as a symbol for a plane.
For example:
Π: r.(4 5 1) = 7
is the equation of the plane Π
In number theory, the function pi(x) is defined to be the number of primes less than or equal to x. For example, pi(3)=2 since 2 and 3 are prime.
There have been many attempts to approximate the value of pi(x) for any given x. The logarithmic integral is one approximation of pi(x); indeed, in the limit (as x approaches infinity), the ratio pi(x)/Li(x) approaches one. This is one of the underlying results of the prime number theorem. Another famous approximation is given by Riemann and is based the logarithmic integral and the Mobius numbers:
∞ ----- \ \ u(n) / 1/n \ / -----*li | x | / n \ / ----- n=1
Where u(n) is the nth Mobius number. For most values of x, Riemann's estimate is among the best. But, whenever Li(x)<pi(x) (see Skewes number), Riemann's estimate is worse than many others.
For related information, see the Node of Prime, prime number and the Riemann zeta function. Note that since there are an infinite number of primes, pi(x) returns the value of every natural number at least once.
According to http://mathworld.wolfram.com/PrimeCountingFunction.html, the upper bound for pi(x) is (2x-6)/ln(x), where ln(x) is the natural logarithm of x.
Released in 1998. Written and directed by Darren Aronofsky. Rated R for language and disturbing imagery.
CAST:
Max Cohen, the protagonist, is a brilliant mathematician who makes his living by finding patterns.However, Max's brilliance comes at a steep price. He has debilitating migraine headaches, constant nose
