# 圓周率

${\displaystyle \mathbb {N} \subset \mathbb {Z} \subset \mathbb {Q} \subset \mathbb {R} \subset \mathbb {C} }$

1650年，約翰·沃利斯搵到${\displaystyle {\frac {\pi }{2}}={\frac {2}{1}}\cdot {\frac {2}{3}}\cdot {\frac {4}{3}}\cdot {\frac {4}{5}}\cdot {\frac {6}{5}}\cdot {\frac {6}{7}}\cdot {\frac {8}{7}}\cdot {\frac {8}{9}}\cdots }$

1674年，萊布尼茲搵到${\displaystyle {\frac {\pi }{4}}=1-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+{\frac {1}{9}}-{\frac {1}{11}}+{\frac {1}{13}}-\cdots }$

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。[4]

## 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。

Monte Carlo方法，用隨機嘅方法嚟搵π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${\displaystyle \pi }$嘅近似值[6]。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π大約就係[7]

${\displaystyle \pi \approx {\frac {2n\ell }{xt}}}$

## 參考

1. 髀算經
2. 楊炯。天賦
3. 60進制下60個小數位嘅嗰圓周率
4. 引用錯誤 無效嘅<ref>標籤；無文字提供畀叫做Arndt嘅參照
5. Arndt & Haenel 2006, p. 39
6. Ramaley, J.F. (October 1969). "Buffon's Noodle Problem". The American Mathematical Monthly. 76 (8): 916–918. doi:10.2307/2317945. JSTOR 2317945.
7. Arndt & Haenel 2006, pp. 39–40Posamentier & Lehmann 2004, p. 105