| C: | 密率 |
| P: | mì lǜ |
| W: | mi lü |
| Y: | maht léut |
| J: | mat leot |
Milü (; "close ratio"), also known as Zulü (Zu's ratio), is the name given to an approximation to (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 to be between 3.1415926 and 3.1415927 and gave two rational approximations of, and, naming them respectively Yuelü (; "approximate ratio") and Milü.
is the best rational approximation of with a denominator of four digits or fewer, being accurate to six decimal places. It is within % of the value of, or in terms of common fractions overestimates by less than . The next rational number (ordered by size of denominator) that is a better rational approximation of is, though it is still only correct to six decimal places. To be accurate to seven decimal places, one needs to go as far as . For eight, is needed.[1]
The accuracy of Milü to the true value of can be explained using the continued fraction expansion of, the first few terms of which are . 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 immediately before the term 292; that is, is approximated by the finite continued fraction, 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,, to the overall fraction), this convergent will be especially close to the true value of :[2]
\pi=3+\cfrac{1}{7+\cfrac{1}{15+\cfrac{1}{1+{\color{magenta}\cfrac{1}{292+ … }}}}} ≈ 3+\cfrac{1}{7+\cfrac{1}{15+\cfrac{1}{1+{\color{magenta}0}}}}=
| 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 by iteratively adding the numerators and denominators of fractions. Zu Chongzhi's approximation ≈ can be obtained with He Chengtian's method.[3]
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: . (Note that in Eastern Asia, fractions are read by stating the denominator first, followed by the numerator). Alternatively, .