Milü

Milü (Chinese: 密率; pinyin: mì lǜ; "close ratio"), also known as Zulü (Zu's ratio), is the name given to an approximation to π (pi) found by Chinese mathematician and astronomer, Zǔ Chōngzhī (祖沖之), born 429 AD. Using Liu Hui's algorithm (which is based on the areas of regular polygons approximating a circle), Zu famously 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.

Fractional approximations to π.
Milü
Chinese密率

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 1/3748629. The next rational number (ordered by size of denominator) that is a better rational approximation of π is 52163/16604, 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 86953/27678. For 8, 102928/32763 is needed.

The accuracy of Milü to the true value of π can be explained using the continued fraction expansion of π, the first few terms of which are . A property of continued fractions is that truncating the expansion of a given number at any point will give the "best rational approximation" to the number. To obtain Milü, truncate the continued fraction expansion of π immediately before the term 292; that is, π is approximated by the finite continued fraction , which is equivalent to Milü. Since 292 is an unusually large term in a continued fraction expansion, this convergent will be very close to the true value of π.[1]

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.[2] Zu Chongzhi's approximation π ≈ 355/113 can be obtained with He Chengtian's method.[3]

See also

References
  1. W., Weisstein, Eric. "Pi Continued Fraction". mathworld.wolfram.com. Retrieved 2017-09-03.
  2. Martzloff, Jean-Claude (2006). A History of Chinese Mathematics. Springer. p. 281.
  3. Wu Wenjun ed Grand Series of History of Chinese Mathematics. vol 4, p. 125
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