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.

What is pi?

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"

An attempt at a complete list. This is one overworked letter. :)

  1. The 16th letter of the Greek alphabet.
  2. Ancient Greek term for the circumference of a circle.
  3. Modern term, as N-wing mentioned, for the irrational ratio of a circle's diameter to its circumference. This ratio pops up in many other places too.
  4. N-wing states population percentage in statistics. I'll take his word for it.
  5. Product series when capitalized. As in, Π(1<x<5) of (x-1), x in Z =(2-1)(3-1)(4-1)
  6. π(x) is the number of primes below a number x - for example, π(100) = 25 (this, btw, approaches x/ln(x))

Given its general popularity as a concept meaning product, I'm sure there are other uses in various sciences. Will keep on hunting. Just had to add #6, my personal area of obsession.

15,000? HA! here's the first 58,688 decimal places of pi:

3.141592653589793238462643383279 5028841971 6939937510 5820974944
5923078164 0628620899 8628034825 3421170679 8214808651 3282306647 0938446095 5058223172
5359408128 4811174502 8410270193 8521105559 6446229489 5493038196 4428810975 6659334461
2847564823 3786783165 2712019091 4564856692 3460348610 4543266482 1339360726 0249141273
7245870066 0631558817 4881520920 9628292540 9171536436 7892590360 0113305305 4882046652
1384146951 9415116094 3305727036 5759591953 0921861173 8193261179 3105118548 0744623799
6274956735 1885752724 8912279381 8301194912 9833673362 4406566430 8602139494 6395224737
1907021798 6094370277 0539217176 2931767523 8467481846 7669405132 0005681271 4526356082
7785771342 7577896091 7363717872 1468440901 2249534301 4654958537 1050792279 6892589235
4201995611 2129021960 8640344181 5981362977 4771309960 5187072113 4999999837 2978049951
0597317328 1609631859 5024459455 3469083026 4252230825 3344685035 2619311881 7101000313
7838752886 5875332083 8142061717 7669147303 5982534904 2875546873 1159562863 8823537875
9375195778 1857780532 1712268066 1300192787 6611195909 2164201989 3809525720 1065485863
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0167113900 9848824012 8583616035 6370766010 4710181942 9555961989 4676783744 9448255379
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2011204698 8609209339 0853657732 2239

Editor's note: Pi to 100 million decimal places

Pi is a programming (better a specification) language. It was developed in 1984 and is used at some German universities (for example at University of Potsdam).
It was made to develop specifications. Specifcations are provisional programs. They are not for making real programs which can be shipped, but to formulate a given problem into an algorithm, implement this algorithm, test it and show it to client to verify, that that is what he wanted. Then you can start making the real program basing on this specification.
The given Pi-Compiler (can be downloaded from http://www.cs.uni-potsdam.de/~owilhelm/Manuskripte/Pi/pi.zip ) translates the given specification into Haskell (in fact the compiler is buggy so there is a pre-compiler which translates the given code, so that the compiler understands it and a post-compiler, so that the Haskell-code is working).

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

The import is empty as we only import one specification, but this is also needed to be exported
import

Now follow the common parameters
common parameters
type Logic
operation true : -> Logic operation false : -> Logic constructor true : -> Logic constructor false : -> Logic

Now the body
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)

A constructor is the basis on which a type is constructed.

Please Note: Most of the modules are German so it can be hard for non-german speakers to understand

The world's longest mnemonic for pi is a short story by Michael Keith entitled Cadaeic Cadenza. This record was previously held by Poe, E., Near a Raven (encoding the first 740 digits of pi) by the same author.

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.

Pi

~Wislawa Szymborska~

The admirable number pi:
Three point one four one.
All the following digits are also initial,
Five nine two because it never ends.
It can't be comprehended six five three five at a glance,
Eight nine by calculation,
Seven nine or imagination,
Not even three two three eight by wit, that is, by comparison
Four six to anything else
Two six four three in the world.
The longest snake on earth calls it quits at about forty feet.
Likewise, snakes of myth and legend, though they may hold out a bit longer.
The pageant of digits comprising the number pi
Doesn't stop at the page's edge.
It goes on across the table, through the air,
Over a wall, a leaf, a bird's nest, clouds, straight into the sky,
Through all the bottomless, bloated heavens.
Oh how brief - a mouse tail, a pigtail - is the tail of a comet!
How feeble the star's ray, bent by bumping up against space!
While here we have two three fifteen three hundred nineteen
My phone number your shirt size the year
Nineteen hundred and seventy-three the sixth floor
The number of inhabitants sixty-five cents
Hip measurement two fingers a charade, a code,
In which we find hail to thee, blithe spirit, bird thou never wert
Alongside ladies and gentlemen, no cause for alarm,
As well as heaven and earth shall pass away,
But not the number pi, oh no, nothing doing,
It keeps right on with its rather remarkable five,
Its uncommonly fine eight,
Its far from final seven,
Nudging, always nudging a sluggish eternity
To continue.
starting with the following function:
f(x)= -1     x < 0 
    = 0      x = 0
    = 1      x > 0 
notice that f(x) is odd and as such, it's fourier series expansion will only have sin() coeficients:
     +inf
f(x)= Σ an * sin(2*n*Pi*x/L)
     n=1
where

  an - fourier series coeficients yet to be determined
  L - length of the interval


an (the fourier series coeficients) are calculated as:

an=2/L * integral(a, a+L, f(x)*sin(2*n*pi*x/L), x)


if we calculate the expansion from a=-1 to a+L=1 we get (L=2):

an=2/2*integral(-1,1,f(x)*sin(2*n*pi*x), x)

noticing that f(x) and sin() are odd functions, their product will be even, and as such we can say that

integral(-1,1, ..., x)=2*integral(0,1, ..., x)

then:

an =2*integral(-1,1,f(x)*sin(2*n*pi*x), x)
   = 2/(n*pi) * (1-(-1)^n)
   = 0           n even
     4/(n*pi)    n odd

f(x) then becomes:
      +inf
f(x) = Σ 2/(n*pi) * (1-(-1)^n) * sin(2*n*Pi*x/2)
      n=1
changing variables n=2k+1 (getting rid of the even terms (which are 0)):
      +inf
f(x) = Σ 4/((2k+1)*pi) * sin((2k+1)*Pi*x)
      k=0
if we get the value of f(x)at x=1/2 (which we know to be 1):
   +inf
1 = Σ 4/((2k+1)*pi) * sin(k*pi+Pi/2)
   k=0

   +inf
  = Σ 4/((2k+1)*pi) * (-1)^k
   k=0
divide by 4/pi:
      +inf
pi/4 = Σ (-1)^k/(2k+1)
      k=0

     = 1 - 1/3 + 1/5 - 1/7 + ....
just thought it'd be interesting to know how to actually get the result.

note though that this converges very slowly. Check the pi metanode for faster ways of calculating pi.
Here's yet another way to derive pi:

Set xo to 2*sqrt(3).
Set yo to 3.

Define xn as:

2 * xn-1 * yn-1
xn-1 + yn-1

Define yn as sqrt(xn * yn-1).

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.

An interesting point: To the best scientific guesses we have, the Universe .. Everything, ;-) .. shouldn't be more than a few Googol (10^100) kilometers wide, and it's probably a sphere or hypersphere. So you would use Pi to calculate its area or how much stuff is inside of it.

The thing is that to calculate anything to a Googol with significant digits, you only need 100 digits of Pi. So we could calculate the number of quarks (small atomic particles) in the entire Universe with, giving ourselves a thousand times more head-room than we think we need, 100,000 digits of Pi.

Here is the first 119,000 digits of Pi. Just imagine - with this number we could calculate perfectly the amount of atomic particles in Creation, if we had Its diameter (or some related precise measurement).

So, if this is the case, why do we have supercomputers working at this minute on Pi's digits well into the Billions? And why do people have it memorized to all those digits?

π

Released in 1998. Written and directed by Darren Aronofsky. Rated R for language and disturbing imagery.

CAST:

Max believes that everything in life can be reduced to underlying patterns, or its basic math (kind of like Lawrence Waterhouse). His theory is as follows:

  1. Mathematics is the language of nature;
  2. Everything around us can be represented and understood by numbers;
  3. If you graph these numbers, patterns emerge. Therefore: there are patterns everywhere in nature.

The project that he is currently working on is to find a pattern in the stock market. He has built a supercomputer called Euclid (a little in-joke, as Euclid, in Elements of Geometry, made certain observations that resulted in the discovery of π) to help him find this pattern.

Max's research with Euclid does not go unnoticed. A young woman named Marcy Dawson leaves cryptic messages on his answering machine, and arranges to bump into him "accidentally." She claims to be from a Wall Street firm that wants to hire him. While sitting in a diner, he meets a Hasidic Jew named Lenny Meyer, who tells him that the Torah is also numerical, and that people have been trying to find the underlying code of it as well.

As a result of these chance meetings, Max is a desirable commodity. This furthers his paranoia and heightens his stress, resulting in more migraines and hallucinations. At one point, he sees a human brain pulsing on the floor of a New York City subway car. It also drives his research. Unfortunately, Euclid keeps spitting out a 216 digit bug. This bug has been found before, by Sol. It is what caused his nervous breakdown. 216 is also believed, by Kabalists, to be the number of letters in the ultimate name of God. So everyone wants this number, which truly exists only in Max's mind.

Pi (?), n. [See Pica, Pie magpie, service-book.] (Print.)

A mass of type confusedly mixed or unsorted. [Written also pie.]

 

© Webster 1913


Pi, v. t. [imp. & p. p. Pied (?); p. pr. & vb. n. Pieing (?).] (Print.)

To put into a mixed and disordered condition, as type; to mix and disarrange the type of; as, to pi a form. [Written also pie.]

 

© Webster 1913


Pi (pI), n. [Gr. pi^.]

1.

A Greek letter (Π, π) corresponding to the Roman letter P.

2.

Specif.: (Math.) The letter Π, π, as used to denote the number or quotient approximately expressing the ratio of the circumference of a circle to its diameter; also, the quotient or the ratio itself. The value of the quotient pi, to eight decimal places, is 3.14159265. The quotient pi cannot be expressed as a root of an algebraic equation; and from this fact follows the impossibility of the quadrature of the circle by purely algebraic processes, or by the aid of a ruler and compass.

 

© Webster 1913

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