Milü
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| Chinese | Mật suất | ||||||||||||||
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Milü(Chinese:Mật suất;pinyin:mìlǜ;"close ratio" ), also known asZulü(Zu's ratio), is the name given to an approximation toπ(pi) found by Chinese mathematician andastronomerZu Chongzhiin the 5th century. UsingLiu Hui's algorithm(which is based on the areas of regular polygons approximating a circle), Zu famously computedπto be between 3.1415926 and 3.1415927[a]and gave two rational approximations ofπ,22/7and355/113,naming them respectively Yuelü (Chinese:Ước suất;pinyin:yuēlǜ;"approximate ratio" ) and Milü.[1]
355/113is the bestrationalapproximation ofπwith 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 overestimatesπby less than1/3748629.The next rational number (ordered by size of denominator) that is a better rational approximation ofπis52163/16604,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,102928/32763is needed.[2]
The accuracy of Milü to the true value ofπcan be explained using thecontinued fraction expansion ofπ,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 ofπimmediately before the term 292; that is,πis 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 ofπ:[3]
Zu's contemporary calendarist and mathematicianHe Chengtianinvented a fraction interpolation method called "harmonization of the divisor of the day" (Chinese:zh: Điều nhật pháp;pinyin:diaorifa) to increase the accuracy of approximations ofπby iteratively adding the numerators and denominators of fractions.Zu Chongzhi's approximationπ≈355/113can be obtained with He Chengtian's method.[1]
An easymnemonichelps memorize this fraction by writing down each of the first threeodd numberstwice:1 13 35 5,then dividing the decimal number represented by the last 3 digits by the decimal number given by the first three digits:1 13Phân chi (fēn zhī)35 5.(Note that in Eastern Asia, fractions are read by stating the denominator first, followed by the numerator). Alternatively,1/π≈113⁄355.[original research?]
See also
[edit]Notes
[edit]- ^Specifically, Zu found that if the diameterof a circle has a length of,then the length of the circle's circumferencefalls within the range.It is not known what method Zu used to calculate this result.
References
[edit]- ^abMartzloff, Jean-Claude (2006).A History of Chinese Mathematics.Springer. p.281.ISBN9783540337829.
- ^"Fractional Approximations of Pi".
- ^Weisstein, Eric W."Pi Continued Fraction".mathworld.wolfram.com.Retrieved2017-09-03.