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| Chinese | 密率 | ||||||||||||||
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Milü (Chinese: 密率; pinyin: mìlǜ; "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, 22/7 and 355/113, naming them respectively Yuelü (Chinese: 约率; pinyin: yuēlǜ; "approximate ratio") and Milü.
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.
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 . 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 , 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:
Zu's contemporary calendarist and mathematician He Chengtian invented a fraction interpolation method called "harmonization of the divisor of the day" (Chinese: zh:调日法; pinyin: diaorifa) to increase the accuracy of approximations of π by iteratively adding the numerators and denominators of fractions. Zu Chongzhi's approximation π ≈ 355/113 can be obtained with He Chengtian's method.
An easy mnemonic helps memorize this 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: 1 1 3 分之(fēn zhī) 3 5 5. (In Eastern Asia, fractions are read by stating the denominator first, followed by the numerator). Alternatively, 1/π ≈ 113⁄355.
See also
Notes
- 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.
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
- ^ 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.