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{{Short description|Pi approximations by astronomer Zu Chongzhi}}
[[File:Diaorifa.GIF|thumb|right|Fractional approximations to [[pi|{{pi}}]].]]
{{Infobox Chinese
  |c=密率
  |p=mì lǜ
  |w=mi{{sup|4}} lü{{sup|4}}
  |y=maht léut
  |j=mat{{sup|6}} leot{{sup|2}}
}}

'''Milü''' ({{zh|c=密率|p=mìlǜ}}; "close ratio"), also known as '''Zulü''' ([[Zu Chongzhi|Zu]]'s ratio), is the name given to an approximation to [[pi|{{pi}}]] (pi) found by Chinese mathematician and [[Chinese astronomy|astronomer]] [[Zu Chongzhi]] in the 5th century. Using [[Liu Hui's π algorithm#Later developments|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{{efn|Specifically, Zu found that if the diameter <math>d</math> of a circle has a length of <math>100,000,000</math>, then the length of the circle's circumference <math>C</math> falls within the range <math>314,159,260 < C < 314,159,270</math>. It is not known what method Zu used to calculate this result.}} and gave two rational approximations of {{pi}}, {{sfrac|22|7}} and {{sfrac|355|113}}, naming them respectively Yuelü ({{zh|c=约率|p=yuēlǜ}}; "approximate ratio") and Milü.<ref name=Martzloff/>

{{sfrac|355|113}} is the best [[rational number|rational]] approximation of {{pi}} with a denominator of four digits or fewer, being accurate to six decimal places. It is within {{val|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}}}}, though it is still only correct to six decimal places. To be accurate to seven decimal places, one needs to go as far as {{sfrac|{{val|86953}}|{{val|27678}}}}. For eight, {{sfrac|{{val|102928}}|{{val|32763}}}} is needed.<ref>{{Cite web|url=http://qin.laya.com/tech_projects_approxpi.html|title = Fractional Approximations of Pi}}</ref>

The accuracy of Milü to the true value of {{Pi}} can be explained using the [[Simple continued fraction#Continued fraction expansion of π and its convergents|continued fraction expansion of {{Pi}}]], the first few terms of which are {{nowrap|[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 "[[Simple continued fraction#Best rational approximations|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 {{nowrap|[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, {{sfrac|1|292}}, to the overall fraction), this convergent will be especially close to the true value of {{Pi}}:<ref>{{Cite web|url=http://mathworld.wolfram.com/PiContinuedFraction.html|title=Pi Continued Fraction|last=Weisstein|first=Eric W.|website=mathworld.wolfram.com|language=en|access-date=2017-09-03}}</ref>

:<math>\pi = 3 + \cfrac{1}{7 + \cfrac{1}{15 + \cfrac{1}{1 + {\color{magenta} \cfrac{1}{292 + \cdots}}}}} \quad\approx\quad 3 + \cfrac{1}{7 + \cfrac{1}{15 + \cfrac{1}{1 + {\color{magenta} 0}}}} = \frac{355}{113}</math>

Zu's contemporary calendarist and mathematician [[He Chengtian]] invented a fraction interpolation method called "harmonization of the divisor of the day" ({{zh|c=[[:zh:调日法]]|p=diaorifa}}) to increase the accuracy of approximations of {{Pi}} by iteratively adding the numerators and denominators of fractions. [[Zu Chongzhi]]'s approximation {{pi}}&nbsp;≈&nbsp;{{sfrac|355|113}} can be obtained with He Chengtian's method.<ref name=Martzloff>{{cite book|last=Martzloff|first=Jean-Claude|title=A History of Chinese Mathematics|url=https://archive.org/details/historychinesema00mart_058|url-access=limited|date=2006|publisher=Springer|page=[https://archive.org/details/historychinesema00mart_058/page/n298 281]|isbn=9783540337829}}</ref>

An easy [[mnemonic]] helps memorize this fraction by writing down each of the first three [[odd numbers]] twice: {{nowrap|'''{{red|1 1}} {{purple|3 3}} {{blue|5 5}}'''}}, then dividing the decimal number represented by the last 3 digits by the decimal number given by the first three digits: {{nowrap|{{red|1 1}} {{purple|3}} 分之(fēn zhī) {{purple|3}} {{blue|5 5}}}}. (In Eastern Asia, fractions are read by stating the denominator first, followed by the numerator). Alternatively, {{nowrap|{{sfrac|1|{{pi}}}} ≈ {{frac|'''{{red|11}}{{purple|3}}'''|'''{{purple|3}}{{blue|55}}'''}}}}.{{Original research inline|date=August 2024}}

== See also ==
*[[Simple continued fraction#Continued fraction expansion of π and its convergents|Continued fraction expansion of {{pi}} and its convergents]]
*[[Approximations of π]]
*[[Pi Approximation Day]]

==Notes==
{{notelist}}

== References ==
{{reflist}}

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

{{DEFAULTSORT:Milu}}
[[Category:Pi]]
[[Category:History of mathematics]]
[[Category:History of science and technology in China]]
[[Category:Chinese mathematical discoveries]]
[[Category:Chinese words and phrases]]
[[Category:Approximations]]
[[Category:Rational numbers]]
[[Category:Zu Chongzhi]]