# Milü

Fractional approximations to π.

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.

{\displaystyle {\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 ${\displaystyle [3;7,15,1,292,1,1,...]}$. 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 ${\displaystyle [3;7,15,1]}$, 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.[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/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.[2] Zu Chongzhi's approximation π355/113 can be obtained with He Chengtian's method.[3]