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remove Erik9bot category and add appropriately dated unref tag using AWB

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[[File:Diaorifa.GIF|thumb|right|Fractional approximations to {{pi}}.]]

The name '''Milü''' ({{zh|c=密率|p=mì lǜ|l=detailed (approximation) ratio}}), also known as '''Zulü''' ([[Zu Chongzhi|Zu]]'s ratio), is given to an approximation to [[pi|{{pi}}]] ([[pi]]) found by Chinese mathematician and [[Chinese astronomy|astronomer]] [[Zu Chongzhi|Zǔ Chōngzhī (祖沖之)]]. He 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 (literally "approximate ratio") and '''Milü'''.

{{sfrac|355|113}} is the best [[rational number|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|3 748 629}}. The next rational number (ordered by size of denominator) that is a better [[Best rational approximation|rational approximation]] of {{pi}} is {{sfrac|52 163|16 604}}, 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|86 953|27 678}}. For 8, we need {{sfrac|102 928|32 763}}.

: <math>\begin{align}\pi & \approx 3.141\ 592\ 653\ 5\dots \\
\\
\frac{355}{113} & \approx 3.141\ 592\ 920\ 3\dots \\
\\
\frac{52163}{16604} & \approx 3.141\ 592\ 387\ 4\dots \\
\\
\frac{86953}{27678} & \approx 3.141\ 592\ 600\ 6\dots\end{align}</math>

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.

Zu's contemporary calendarist and mathematician [[He Chengtian]] 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.<ref>Jean claude Martzloff, A History of Chinese Mathematics p281</ref> [[Zu Chongzhi]]'s approximation {{pi}} ≈ {{sfrac|355|113}} can be obtained with He's method<ref>Wu Wenjun ed Grand Series of History of Chinese Mathematics vol 4 p125</ref>

== See also ==
*[[Continued fraction#Continued fraction expansions of π|Continued fraction expansions of {{pi}}]]
*[[Numerical approximations of π|History of numerical approximations of {{pi}}]]
*[[Pi Approximation Day]]

==External links==
*[http://qin.laya.com/tech_projects_approxpi.html Fractional Approximations of Pi]

== References ==
{{reflist}}

{{DEFAULTSORT:Milu}}
[[Category:Pi]]

{{num-stub}}

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en>Rich Farmbrough: remove Erik9bot category and add appropriately dated unref tag using AWB