# 圓周率

如果個圓直徑係一，佢個圓周就等如圓周率( WIKINIUM\_MATH\_PH\_28 )。
手寫體嘅 WIKINIUM\_MATH\_PH\_30
**圓周率**，一般用 **π** 表示，係一個喺數學同物理學普遍存在嘅常數，大約等於 3.14159。佢嘅定義係平面幾何（或者歐幾里得幾何）入面圓形嘅圓周同直徑嘅比例，亦等於圓形嘅面積同半徑平方嘅比例。佢係精確計算圓周長、圓面積、球體積等幾何量嘅關鍵。喺分析學上， WIKINIUM\_MATH\_PH\_32 可以定義為最細嘅 WIKINIUM\_MATH\_PH\_34 令到 WIKINIUM\_MATH\_PH\_36 WIKINIUM\_MATH\_PH\_38 。

圓周率係一個無理數，唔可以用分數準確表示

圓周率亦係一個超越數，冇辦法用有理係數嘅多項式嚟表達。

古代最初估計圓周率係 WIKINIUM\_MATH\_PH\_40 ，正所謂「周三徑一」。後尾有人發現有理數 WIKINIUM\_MATH\_PH\_42 可以當做圓周率嘅近似值，叫做**約率**。中國南北朝數學家祖沖之發現有理數 WIKINIUM\_MATH\_PH\_44 （ WIKINIUM\_MATH\_PH\_46 ）更加接近（只係大咗 WIKINIUM\_MATH\_PH\_48 或接近千萬分之一），所以叫做**密率**。

日本數學家三上義夫為咗記念祖沖之嘅成就，提議將呢個近似值叫做**祖率**。喺一般應用， WIKINIUM\_MATH\_PH\_50 或約率 WIKINIUM\_MATH\_PH\_52 就已經夠數，但係工程學成日用 WIKINIUM\_MATH\_PH\_54 （5位有效數字）或者 WIKINIUM\_MATH\_PH\_56 （6位有效數字）。至於密率 WIKINIUM\_MATH\_PH\_58 就係一個易記啲、精確到7位有效數字嘅分數。

1650年，約翰·沃利斯搵到 WIKINIUM\_MATH\_PH\_60

1674年，萊布尼茲搵到 WIKINIUM\_MATH\_PH\_62

巴比倫人用嘅六十進制圓周率係

3.8,29,44,0,47,25,53,7,24,57,
36,17,43,4,29,7,1,3,41,17,
52,36,12,14,36,44,51,5,15,33,
7,23,59,9,13,48,22,12,21,45,
22,56,47,39,44,28,37,58,23,21,
11,56,33,22,4,42,31,6,6,4。

## 定義

因為任何圓形都係相似形，所以圓周同直徑嘅比例係一個常數，叫做**圓周率**，符號係 WIKINIUM\_MATH\_PH\_64 ，數值大約係 WIKINIUM\_MATH\_PH\_66 ，呢個係目前公認嘅定義。

無論圓嘅大細點樣，比值 WIKINIUM\_MATH\_PH\_68 為恆值。如果個圓嘅直徑變為頭先嘅二倍，佢嘅周長亦都變為二倍，比值 WIKINIUM\_MATH\_PH\_70 唔變。而家 WIKINIUM\_MATH\_PH\_72 比值定義隱性咁樣用咗歐幾里得幾何入面比值啲定理，雖然個圓嘅定義可以擴展到任意曲面（即唔係歐幾里得幾何），但呢啲圓都唔再符合定律 WIKINIUM\_MATH\_PH\_74 。

另外有啲數學家建議用圓周÷半徑嚟定義**圓周率**，符號係 WIKINIUM\_MATH\_PH\_76 ，數值大約係 WIKINIUM\_MATH\_PH\_78 ，即係 WIKINIUM\_MATH\_PH\_80 。

## 搵圓周率數值嘅現代方法

### Monte Carlo方法

Buffon嘅針，a針同b針係隨機掉嘅。隨機擺啲點落去外接住四份一個圓嘅正方形入面。Monte Carlo方法，用隨機嘅方法嚟搵 WIKINIUM\_MATH\_PH\_82 嘅近似值。
Monte Carlo方法即係重覆好多次隨機嘅過程去搵答案，可以用嚟搵 WIKINIUM\_MATH\_PH\_83 嘅近似值。Buffon嘅針係一個例子：一個平面上面畫咗一柞平行線，每條相隔t個單位，隨機掉n次每條長度係l個單位嘅針落去，記錄啲針同平行線相交嘅次數係x，噉 WIKINIUM\_MATH\_PH\_85 大約就係：

 WIKINIUM\_MATH\_PH\_86

另一個方法係係畫一個正方形外接住一個圓，隨機擺啲點落去，喺圓入面嘅點占嘅比例大約係 WIKINIUM\_MATH\_PH\_88 。

## 睇埋

* E (數學常數)
* 黃金比例

## 參考

### 來源

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