Milü

Template:Short description [[File:Diaorifa.GIF|thumb|right|Fractional approximations to [[pi|Template:Pi]].]] Template:Infobox Chinese

Milü (Template:Zh; "close ratio"), also known as Zulü (Zu's ratio), is the name given to an approximation to [[pi|Template:Pi]] (pi) found by Chinese mathematician and astronomer Zu Chongzhi in the 5th century. Using Liu Hui's algorithm (which is based on the areas of regular polygons approximating a circle), Zu famously computed Template:Pi to be between 3.1415926 and 3.1415927Template:Efn and gave two rational approximations of Template:Pi, Template:Sfrac and Template:Sfrac, naming them respectively Yuelü (Template:Zh; "approximate ratio") and Milü.

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 Template: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.

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 Template:Nowrap. 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 Template:Pi immediately before the term 292; that is, Template:Pi is approximated by the finite continued fraction Template:Nowrap, 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:

$\pi =3+\frac{1}{7+\frac{1}{15+\frac{1}{1+\frac{1}{292+\cdots }}}}\approx 3+\frac{1}{7+\frac{1}{15+\frac{1}{1+0}}}=\frac{355}{113}$

Zu's contemporary calendarist and mathematician He Chengtian invented a fraction interpolation method called "harmonization of the divisor of the day" (Template:Zh) to increase the accuracy of approximations of Template:Pi by iteratively adding the numerators and denominators of fractions. Zu Chongzhi's approximation Template:Pi ≈ Template:Sfrac can be obtained with He Chengtian's method.

An easy mnemonic helps memorize this fraction by writing down each of the first three odd numbers twice: Template:Nowrap, then dividing the decimal number represented by the last 3 digits by the decimal number given by the first three digits: Template:Nowrap. (In Eastern Asia, fractions are read by stating the denominator first, followed by the numerator). Alternatively, Template:Nowrap.Template:Original research inline

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

External links

• Fractional Approximations of Pi