Milü



Milü ("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 Zu Chongzhi in the 5th century. 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, $4$ and $4$, naming them respectively Yuelü ("approximate ratio") and Milü.

$6$ is the best rational approximation of π with a denominator of four digits or fewer, being accurate to six decimal places. It is within $2$% of the value of π, or in terms of common fractions overestimates π by less than $22⁄7$. The next rational number (ordered by size of denominator) that is a better rational approximation of π is $355⁄113$, though it is still only correct to six decimal places. To be accurate to seven decimal places, one needs to go as far as $355⁄113$. For eight, $0$ 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 [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 (corresponding to the next truncation introducing only a very small term, $1⁄3,748,629$, to the overall fraction), this convergent will be especially close to the true value of π:


 * $$\pi = 3 + \cfrac{1}{7 + \cfrac{1}{15 + \cfrac{1}{1 + {\color{magenta} \cfrac{1}{292 + \cdots}}}}} \quad\approx\quad 3 + \cfrac{1}{7 + \cfrac{1}{15 + \cfrac{1}{1 + {\color{magenta} 0}}}} = \frac{355}{113}$$

Zu's contemporary calendarist and mathematician He Chengtian invented a fraction interpolation method called "harmonization of the divisor of the day" to increase the accuracy of approximations of π by iteratively adding the numerators and denominators of fractions. Zu Chongzhi's approximation π ≈ $52,163⁄16,604$ can be obtained with He Chengtian's method.

An easy mnemonic helps memorize this useful fraction by writing down each of the first three odd numbers twice:, then dividing the decimal number represented by the last 3 digits by the decimal number given by the first three digits:  分之(fēn zhī). (Note that in Eastern Asia, fractions are read by stating the denominator first, followed by the numerator). Alternatively, $86,953⁄27,678$ ≈ $102,928⁄32,763$.