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Fractional approximations to π.

The name Milü (Chinese: 密率; pinyin: mì lǜ; "detailed (approximation) ratio"), also known as Zulü (Zu's ratio), is given to an approximation to π (pi) found by Chinese mathematician and astronomer Zǔ Chōngzhī (祖沖之). He computed π to be between 3.1415926 and 3.1415927 and gave two rational approximations of π, 22/7 and 355/113, naming them respectively Yuelü 约率 (approximate ratio) and Milü.

355/113 is the best rational approximation of π with a denominator of four digits or fewer, being accurate to 6 decimal places. It is within 0.000009% of the value of π, or in terms of common fractions overestimates π by less than 1/3 748 629. The next rational number (ordered by size of denominator) that is a better rational approximation of π is 52 163/16 604, still only correct to 6 decimal places and hardly closer to π than 355/113. To be accurate to 7 decimal places, one needs to go as far as 86 953/27 678. For 8, we need 102 928/32 763.

An easy mnemonic helps memorize this useful fraction by writing down each of the first three odd numbers twice: 1 1 3 3 5 5, then dividing the decimal number represented by the last 3 digits by the decimal number given by the first three digits. Alternatively, 1 / π ≈ 113 / 355.

Zu's contemporary calendarist and mathematician He Chengtian (何承天) invented a fraction interpolation method called "harmonization of the divisor of the day" to obtain a closer approximation by iteratively adding the numerators and denominators of a "weak" fraction and a "strong" fraction.[1] Zu Chongzhi's approximation π355/113 can be obtained with He Chengtian's method[2]

See also[edit]


  1. ^ Martzloff, Jean-Claude (2006). A History of Chinese Mathematics. Springer. p. 281. 
  2. ^ Wu Wenjun ed Grand Series of History of Chinese Mathematics vol 4 p125

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