圓周率

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，${\displaystyle \pi }$ 可以定義為最細嘅 ${\displaystyle x>0}$ 令到 ${\displaystyle \sin(x)}$ ${\displaystyle =0}$。

數學嘅數

基本

${\displaystyle \mathbb {N} \subset \mathbb {Z} \subset \mathbb {Q} \subset \mathbb {R} \subset \mathbb {C} }$ 自然數 ${\displaystyle \mathbb {N} }$ 整數 ${\displaystyle \mathbb {Z} }$ 二進分數 有限小數 循環小數 有理數 ${\displaystyle \mathbb {Q} }$ 高斯整數 ${\displaystyle \mathbb {Z} [i]}$ 代數數 ${\displaystyle \mathbb {A} }$ 實數 ${\displaystyle \mathbb {R} }$ 複數 ${\displaystyle \mathbb {C} }$ 負數 分數 單位分數 無限小數 規矩數 無理數 超越數 二次無理數 虛數 艾森斯坦整數 ${\displaystyle \mathbb {Z} [\omega ]}$

延伸

雙複數 四元數 ${\displaystyle \mathbb {H} }$ 共四元數 八元數 ${\displaystyle \mathbb {O} }$ 超數 上超實數 超現實數 超複數 十六元數 ${\displaystyle \mathbb {S} }$ 複四元數 Tessarine 大實數 超實數 ${\displaystyle {}^{\star }\mathbb {R} }$

其他

對偶數 雙曲複數 序數 質數 同餘 可計算數 艾禮富數 公稱值 超限數 基數 P進數 規矩數 整數序列 數學常數 圓周率 π = 3.141592653… 自然對數嘅底 e = 2.718281828… 虛數單位 i = ${\displaystyle +{\sqrt {-1}}}$ 無窮大量 ∞

如果個圓直徑係一，佢個圓周就等如圓周率(${\displaystyle \pi }$)。

手寫體嘅 ${\displaystyle \pi }$

圓周率係一個無理數，唔可以用分數準確表示^[1]。

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 ${\displaystyle 3}$，正所謂「周三徑一」^[2][3]。後尾有人發現有理數 ${\displaystyle {\frac {22}{7}}}$ 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 ${\displaystyle {\frac {355}{113}}}$ （${\displaystyle 3.1415929203539823008849557522124\cdots \cdots \cdots }$）更加接近（只係大咗${\displaystyle 0.0000002667641890624223123\cdots \cdots \cdots }$ 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，${\displaystyle 3.14}$ 或約率 ${\displaystyle {\frac {22}{7}}}$ 就已經夠數，但係工程學成日用 ${\displaystyle 3.1416}$（5位有效數字）或者 ${\displaystyle 3.14159}$（6位有效數字）。至於密率 ${\displaystyle {\frac {355}{113}}}$ 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${\displaystyle {\frac {\pi }{2}}={\frac {2}{1}}\cdot {\frac {2}{3}}\cdot {\frac {4}{3}}\cdot {\frac {4}{5}}\cdot {\frac {6}{5}}\cdot {\frac {6}{7}}\cdot {\frac {8}{7}}\cdot {\frac {8}{9}}\cdots }$

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

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。^[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 ${\displaystyle \pi }$，數值大約係${\displaystyle 3.1415927}$，呢個係目前公認嘅定義。

無論圓嘅大細點樣，比值 ${\displaystyle {\frac {C}{d}}}$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 ${\displaystyle {\frac {C}{d}}}$ 唔變。而家 ${\displaystyle \pi }$ 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 ${\displaystyle \pi ={\frac {C}{d}}}$。^[5]

另外有啲數學家建議用圓周÷半徑嚟定義圓周率，符號係 ${\displaystyle \tau }$，數值大約係${\displaystyle 6.2831853}$，即係${\displaystyle 2\times \pi }$。

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]：

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

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${\displaystyle \pi /4}$^[8]。

睇埋

• E (數學常數)
• 黃金比例

參考

1. https://archive.org/stream/Pi_to_100000000_places/pi.txt
2. 《周髀算經》。
3. 楊炯。《渾天賦》。
4. 60進制下60個小數位嘅嗰圓周率
6. Arndt & Haenel 2006, p. 39
7. Ramaley, J.F. (October 1969). "Buffon's Noodle Problem". The American Mathematical Monthly. 76 (8): 916–918. doi:10.2307/2317945. JSTOR 2317945.
8. Arndt & Haenel 2006, pp. 39–40Posamentier & Lehmann 2004, p. 105

來源

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