Milü

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.

$\begin{align}\pi & \approx 3.141\ 592\ 653\ 5\dots \\ \\ \frac{355}{113} & \approx 3.141\ 592\ 920\ 3\dots \\ \\ \frac{52163}{16604} & \approx 3.141\ 592\ 387\ 4\dots \\ \\ \frac{86953}{27678} & \approx 3.141\ 592\ 600\ 6\dots\end{align}$

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

See also

• Continued fraction expansions of π
• History of numerical approximations of π
• Pi Approximation Day

External links

• Fractional Approximations of Pi