pi......

圓周率，一般以π來表示，是一個在數學及物理學普遍存在的數學常數。它定義為圓形之周長與直徑之比。它也等於圓形之面積與半徑平方之比。是精確計算圓周長、圓面積、球體積等幾何形狀的關鍵。分析學上，π 可定義為是最小的 x > 0 使得 sin(x) = 0。

常用的 π 10進位近以值為3.1415926，另外還有由祖沖之給出的疏率：及密率：。

精確到小數點後第100位的圓周率值

π = 3. 1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679……

π的特性和相關方程
幾何：

若圓的半徑為 r，其圓周為 C = 2 π r
若圓的半徑為 r，其面積為 A = π r2
若橢圓的長、短兩幅分別為 a 和 b ，其面積為 A = π ab
若球體的半徑為 r，其體積為 V = (4/3) π r3
若球體的半徑為 r，其表面積為 A = 4 π r2
角度： 180 度相等於 π 弧度

代數
π 是個無理數，不可以是兩個整數之比，是由Johann Heinrich Lambert於1761年證明的。 1882年，Ferdinand Lindemann更證明了 π 是超越數，即不可能是某有理數多項式的根。

圓周率的超越性否定了化圓為方這古老尺規作圖問題的可能性，因所有尺規作圖只能得出代數數。


數學分析
(Leibniz 定理)
(Wallis乘積)
(歐拉)


(斯特林(Stirling)公式)
(歐拉(Euler)公式)
π 有個特別的連分數表達式：


π 本身的連分數表達式(簡寫)為 [3;7,15,1,292,1,1,1,2,1,3,1,14,2,1,1,2,...]，其近似部分給出的首三個漸近分數




第一個和第三個漸近分數即為疏率和密率的值。數學上可以證明，這樣得到的漸近分數，在分子或分母小於下一個漸進分數的分數中，其值是最接近精確值的近似值。

(另有 12 個表達式見於 [1] )


數論
兩個任意自然數是互質的機率是 6/π2。
一個任意整數沒有重複質因數的機率為 6/π2。
一個任意整數平均可用 π/4 個方法寫成兩個完全數之和。

機率論
取一枚長為l的針，再取一張白紙在上面畫上一些距離為2l的平行線。把針從一定高度釋放，讓其自由落體到紙面上。針與平行線相交的機率是圓周率的倒數（泊松針）。曾經有人以此方法來尋找 π 的值。

Dynamical Systems / Ergodic Theory

對[0, 1]中幾乎所有 x0，其中 xi 是 iterates of the Logistic map for r=4.

物理學
(海森堡測不準原理)

(相對論的場方程)


統計學
(The probability density function for the normal distribution.)

尚待解決的問題
關於 π 未解決的問題包括

它是否是一個 normal number，即 π 的十進位表達式是否包含所有的有限數列。對於二進位表達式，答案是肯定的，這是 Bailey 及 Crandall 於2000年從 Bailey-Borwein-Plouffe 方程的存在而引申出來的。
0,...,9是否以完全隨機的形出現在 π 的十進位表達式中。若然，則對於非十進位表達式，亦應有類似特質。
究竟是否所有0,...,9都會無限地出現在 π 的小數表達式中。
到底超級電腦計算出來的上億位的圓周率是否正確。

文化

背誦π的位數
世界記錄是100000位，原口証（en:Akira Haraguchi）於2006年10月3日背誦圓周率π至小數點後100000位。中文用諧音記憶的有「山巔一寺一壺酒」，就是3.14159。


π在數學外的用途
在Google公司2005年的一次公開募股中，集資額不是通常的整頭數，而是$14,159,265，這當然是由π小數點後的位數得來。（順便一提，谷歌公司2004年的首次公開募股，集資額為$2,718,281,828，與數學常數e有關）
排版軟體TeX從第三版之後的版本號為逐次增加一位小數，使之越來越接近π的值：3.1，3.14，……當前的最新版本號是3.141592
3月14日為圓周率日

http://hk.knowledge.yahoo.com/search/search_result?tab=0&p=Pi%3D%3F%3F+

What is pi ()? Who first used pi? How do you find its value? What is it for? How many digits is it?

By definition, pi is the ratio of the circumference of a circle to its diameter. Pi is always the same number, no matter which circle you use to compute it.
For the sake of usefulness people often need to approximate pi. For many purposes you can use 3.14159, which is really pretty good, but if you want a better approximation you can use a computer to get it. Here's pi to many more digits: 3.14159265358979323846.

The area of a circle is pi times the square of the length of the radius, or "pi r squared":


A = pi*r^2
A very brief history of pi

Pi is a very old number. We know that the Egyptians and the Babylonians knew about the existence of the constant ratio pi, although they didn't know its value nearly as well as we do today. They had figured out that it was a little bigger than 3; the Babylonians had an approximation of 3 1/8 (3.125), and the Egyptians had a somewhat worse approximation of 4*(8/9)^2 (about 3.160484), which is slightly less accurate and much harder to work with. For more, see A History of Pi by Petr Beckman (Dorset Press).
The modern symbol for pi [] was first used in our modern sense in 1706 by William Jones, who wrote:


There are various other ways of finding the Lengths or Areas of particular Curve Lines, or Planes, which may very much facilitate the Practice; as for instance, in the Circle, the Diameter is to the Circumference as 1 to (16/5 - 4/239) - 1/3(16/5^3 - 4/239^3) + ... = 3.14159... = (see A History of Mathematical Notation by Florian Cajori).
Pi (rather than some other Greek letter like Alpha or Omega) was chosen as the letter to represent the number 3.141592... because the letter [] in Greek, pronounced like our letter 'p', stands for 'perimeter'.

About Pi
Pi is an infinite decimal. Unlike numbers such as 3, 9.876, and 4.5, which have finitely many nonzero numbers to the right of the decimal place, pi has infinitely many numbers to the right of the decimal point.
If you write pi down in decimal form, the numbers to the right of the 0 never repeat in a pattern. Some infinite decimals do have patterns - for instance, the infinite decimal .3333333... has all 3's to the right of the decimal point, and in the number .123456789123456789123456789... the sequence 123456789 is repeated. However, although many mathematicians have tried to find it, no repeating pattern for pi has been discovered - in fact, in 1768 Johann Lambert proved that there cannot be any such repeating pattern.

As a number that cannot be written as a repeating decimal or a finite decimal (you can never get to the end of it) pi is irrational: it cannot be written as a fraction (the ratio of two integers).

Pi shows up in some unexpected places like probability, and the 'famous five' equation connecting the five most important numbers in mathematics, 0, 1, e, pi, and i: e^(i*pi) + 1 = 0.
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Pi to one MILLION decimal places (For Reference):
3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067982148086513282306647093844609550582231725359408128481117450284102701938521105559644622948954930381964428810975665933446128475648233786783165271201909145648566923460348610454326648213393607260249141273724587006606315588174881520920962829254091715364367892590360011330530548820466521384146951941511609.........

pi = 圓周率 = 圓周/直徑