Milü

The name Milü (Chinese: 密率; pinyin: mì lǜ; "close 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). 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.

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, we need 102928/32763.

{\begin{aligned}\pi &\approx 3.141\ 592\ 653\ 5\dots \\\\{\frac {355}{113}}&\approx 3.141\ 592\ 920\ 3\dots \\\\{\frac {52163}{16604}}&\approx 3.141\ 592\ 387\ 4\dots \\\\{\frac {86953}{27678}}&\approx 3.141\ 592\ 600\ 6\dots \end{aligned}} 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 $[3;7,15,1,292,1,1,...]$ . 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 $[3;7,15,1]$ , 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.

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/π113/355.

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. Zu Chongzhi's approximation π355/113 can be obtained with He Chengtian's method.