Milü

Milü (Chinese: 密率; pinyin: mìlǜ; lit. 'close ratio'), also known as Zulü (Zu's ratio), is the name given to an approximation of π (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 π as being between 3.1415926 and 3.1415927^[a] and gave two rational approximations of π, ⁠22/7⁠ and ⁠355/113⁠, which were named yuelü (约率; yuēlǜ; 'approximate ratio') and milü respectively.^[1]

Fractional approximations of π

Milü
Chinese	密率
Literal meaning	close ratio
Transcriptions Standard Mandarin Hanyu Pinyin mìlǜ Wade–Giles mi4 lü4 Yue: Cantonese Yale Romanization maht léut Jyutping mat6 leot2

⁠355/113⁠ is the best rational approximation of π with a denominator of four digits or fewer, being accurate to six decimal places. It is within 0.000009% of the value of π, or in terms of common fractions overestimates π by less than ⁠1/3748629⁠. The next rational number (ordered by size of denominator) that is a better rational approximation of π is ⁠52163/16604⁠, though it is still only correct to six decimal places. To be accurate to seven decimal places, one needs to go as far as ⁠86953/27678⁠. For eight, ⁠102928/32763⁠ is needed.^[2]

The accuracy of milü to the true value of π can be explained using the continued fraction expansion of π, the first few terms of which are [3; 7, 15, 1, 292, 1, 1, ...] (sequence A001203 in the OEIS). 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 π immediately before the term 292; that is, π 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, ⁠1/292⁠, to the overall fraction), this convergent will be especially close to the true value of π:^[3]

${\displaystyle \pi =3+{\cfrac {1}{7+{\cfrac {1}{15+{\cfrac {1}{1+{\color {magenta}{\cfrac {1}{292+\cdots }}}}}}}}}\quad \approx \quad 3+{\cfrac {1}{7+{\cfrac {1}{15+{\cfrac {1}{1+{\color {magenta}0}}}}}}}={\frac {355}{113}}}$

Zu's contemporary calendarist and mathematician He Chengtian invented a fraction interpolation method called 'harmonization of the divisor of the day' (调日法; diaorifa) to increase the accuracy of approximations of π by iteratively adding the numerators and denominators of fractions. Zu's approximation of π ≈ ⁠355/113⁠ can be obtained with He Chengtian's method.^[1]

See also

• Continued fraction expansion of π and its convergents
• Approximations of π
• Pi Approximation Day

Notes

1. Specifically, Zu found that if the diameter ${\displaystyle d}$ of a circle has a length of ${\displaystyle 100,000,000}$, then the length of the circle's circumference ${\displaystyle C}$ falls within the range ${\displaystyle 314,159,260<C<314,159,270}$. It is not known what method Zu used to calculate this result.