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

如果個圓直徑係一，佢個圓周就等如圓周率($\pi$)。

手寫體嘅 $\pi$

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

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

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。

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 $\pi$，數值大約係$3.1415927$，呢個係目前公認嘅定義。

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

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

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$\pi$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π大約就係：

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

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$\pi /4$。

睇埋

• E (數學常數)
• 黃金比例

參考

來源

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