- See also
- References
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The name Milü ({{zh|c=密率|p=mì lǜ}}; "close ratio"), also known as Zulü (Zu's ratio), is given to an approximation to {{pi}} (pi) found by Chinese mathematician and astronomer, Zǔ Chōngzhī (祖沖之). Using Liu Hui's algorithm (which is based on the areas of regular polygons approximating a circle), Zu famously computed {{pi}} to be between 3.1415926 and 3.1415927 and gave two rational approximations of {{pi}}, {{sfrac|22|7}} and {{sfrac|355|113}}, naming them respectively Yuelü 约率 (approximate ratio) and Milü.
{{sfrac|355|113}} is the best rational approximation of {{pi}} with a denominator of four digits or fewer, being accurate to 6 decimal places. It is within 0.000009% of the value of {{pi}}, or in terms of common fractions overestimates {{pi}} by less than {{sfrac|1|{{val|3748629}}}}. The next rational number (ordered by size of denominator) that is a better rational approximation of {{pi}} is {{sfrac|{{val|52163}}|{{val|16604}}}}, still only correct to 6 decimal places and hardly closer to {{pi}} than {{sfrac|355|113}}. To be accurate to 7 decimal places, one needs to go as far as {{sfrac|{{val|86953}}|{{val|27678}}}}. For 8, we need {{sfrac|{{val|102928}}|{{val|32763}}}}.
The accuracy of Milü to the true value of {{Pi}} can be explained using the continued fraction expansion of {{Pi}}, 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 {{Pi}} immediately before the term 292; that is, {{Pi}} is approximated by the finite continued fraction 
, which is equivalent to Milü. Since 292 is an unusually large term in a continued fraction expansion, this convergent will be very close to the true value of {{Pi}}.[1]
An easy mnemonic helps memorize this useful 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. Alternatively, {{sfrac|1|{{pi}}}} ≈ {{sfrac|113|355}}.
Zu's contemporary calendarist and mathematician He Chengtian (何承天) invented a fraction interpolation method called "harmonization of the divisor of the day" to obtain a closer approximation by iteratively adding the numerators and denominators of a "weak" fraction and a "strong" fraction.[2] Zu Chongzhi's approximation {{pi}} ≈ {{sfrac|355|113}} can be obtained with He Chengtian's method.[3]
See also
- Continued fraction expansions of {{pi}}
- History of approximations of {{pi}}
- Pi Approximation Day
References
2. ^{{cite book|last=Martzloff|first=Jean-Claude|title=A History of Chinese Mathematics|date=2006|publisher=Springer|page=281}}
3. ^Wu Wenjun ed Grand Series of History of Chinese Mathematics vol 4 p125
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