Milü
| Milü | |||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Chinese | 密率 | ||||||||||||||
| Literal meaning | close ratio | ||||||||||||||
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Milü (Chinese: 密率; pinyin: mìlǜ; lit. 'close ratio'), also known as Zulü (Zu's ratio), is the name given to an approximation of π (pi) found by the Chinese mathematician and astronomer Zu Chongzhi during the 5th century. Using Liu Hui's algorithm, which is based on the areas of regular polygons approximating a circle, Zu computed π as being between 3.1415926 and 3.1415927[a] and gave two rational approximations of π, 22/7 and 355/113, which were named yuelü (约率; yuēlǜ; 'approximate ratio') and milü respectively.[1]
355/113 is the best rational approximation of π with a denominator of four digits or fewer, being accurate to six 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, though it is still only correct to six decimal places. To be accurate to seven decimal places, one needs to go as far as 86953/27678. For eight, 102928/32763 is needed, and for nine, 103993/33102 is required.[2]
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, ...] (sequence A001203 in the OEIS). A property of continued fractions is that truncating the expansion of a given number at any point will give the best rational approximation of 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, 1/292, to the overall fraction), this convergent will be especially close to the true value of π:[3]
Zu's contemporary calendarist and mathematician He Chengtian invented a fraction interpolation method called 'harmonization of the divisor of the day' (调日法; diaorifa) to increase the accuracy of approximations of π by iteratively adding the numerators and denominators of fractions. Zu's approximation of π ≈ 355/113 can be obtained with He Chengtian's method.[1]
See also
[edit]Notes
[edit]- ↑ Specifically, Zu found that if the diameter of a circle has a length of , then the length of the circle's circumference falls within the range . It is not known what method Zu used to calculate this result.
References
[edit]- 1 2 Martzloff, Jean-Claude (2006). A History of Chinese Mathematics. Springer. p. 281. ISBN 9783540337829.
- ↑ "Fractional Approximations of Pi".
- ↑ Weisstein, Eric W. "Pi Continued Fraction". mathworld.wolfram.com. Retrieved 2017-09-03.