Milü
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Milü (Template:Zh), also known as Zulü (Zu's ratio), is the name given to an approximation of Template:Pi (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 Template:Pi as being between 3.1415926 and 3.1415927Template:Efn and gave two rational approximations of Template:Pi, Template:Sfrac and Template:Sfrac, which were named yuelü (Template:Zhi) and milü respectively.[1]
Template:Sfrac is the best rational approximation of Template:Pi with a denominator of four digits or fewer, being accurate to six decimal places. It is within Script error: No such module "val".% of the value of Template:Pi, or in terms of common fractions overestimates Template:Pi by less than Template:Sfrac. The next rational number (ordered by size of denominator) that is a better rational approximation of Template:Pi is Template:Sfrac, though it is still only correct to six decimal places. To be accurate to seven decimal places, one needs to go as far as Template:Sfrac. For eight, Template:Sfrac is needed.[2]
The accuracy of milü to the true value of Template:Pi can be explained using the [[Simple continued fraction#Continued fraction expansion of π and its convergents|continued fraction expansion of Template:Pi]], 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 of the number. To obtain milü, truncate the continued fraction expansion of Template:Pi immediately before the term 292; that is, Template:Pi 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, Template:Sfrac, to the overall fraction), this convergent will be especially close to the true value of Template:Pi:[3]
Zu's contemporary calendarist and mathematician He Chengtian invented a fraction interpolation method called 'harmonization of the divisor of the day' (Template:Zhi) to increase the accuracy of approximations of Template:Pi by iteratively adding the numerators and denominators of fractions. Zu's approximation of Template:Pi ≈ Template:Sfrac can be obtained with He Chengtian's method.[1]
See also
- [[Simple continued fraction#Continued fraction expansion of π and its convergents|Continued fraction expansion of Template:Pi and its convergents]]
- Approximations of π
- Pi Approximation Day
Notes
References
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