| Milü | |||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Chinese | 密率 | ||||||||||||||
| Literal meaning | close ratio | ||||||||||||||
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Milü (Chinese:密率;pinyin:mìlǜ;lit. 'close ratio'), also known asZulü (Zu's ratio), is the name given to an approximation ofπ (pi) found by the Chinese mathematician and astronomerZu Chongzhi during the 5th century. UsingLiu Hui's algorithm, which is based on the areas of regular polygons approximating a circle, Zu computedπ as being between 3.1415926 and 3.1415927[a] and gave two rational approximations ofπ,22/7 and355/113, which were namedyuelü (约率;yuēlǜ; 'approximate ratio') andmilü respectively.[1]
355/113 is the bestrational approximation ofπ with a denominator of four digits or fewer, being accurate to six decimal places. It is within0.000009% of the value ofπ, or in terms of common fractions overestimatesπ by less than1/3748629. The next rational number (ordered by size of denominator) that is a better rational approximation ofπ is52163/16604, though it is still only correct to six decimal places. To be accurate to seven decimal places, one needs to go as far as86953/27678. For eight,102928/32763 is needed.[2]
The accuracy ofmilü to the true value ofπ can be explained using thecontinued fraction expansion ofπ, the first few terms of which are[3; 7, 15, 1, 292, 1, 1, ...] (sequenceA001203 in theOEIS). A property of continued fractions is that truncating the expansion of a given number at any point will give thebest rational approximation of the number. To obtainmilü, truncate the continued fraction expansion ofπ immediately before the term 292; that is,π is approximated by the finite continued fraction[3; 7, 15, 1], which is equivalent tomilü. Since 292 is an unusually large term in a continued fraction expansion (corresponding to the next truncation introducing only a very small term,1/292, to the overall fraction), this convergent will be especially close to the true value ofπ:[3]
Zu's contemporary calendarist and mathematicianHe Chengtian invented a fraction interpolation method called 'harmonization of the divisor of the day' (调日法;diaorifa) to increase the accuracy of approximations ofπ by iteratively adding the numerators and denominators of fractions. Zu's approximation ofπ ≈ 355/113 can be obtained with He Chengtian's method.[1]