Milü

Milü (Chinese: 密率; pinyin: mìlǜ; "close ratio"), also known as Zulü (Zu's ratio), is the name given to an approximation to π (pi) found by Chinese mathematician and astronomer Zu Chongzhi in the 5th century.

[1] ⁠355/113⁠ is the best rational approximation of π with a denominator of four digits or fewer, being accurate to six decimal places.

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] Zu's contemporary calendarist and mathematician He Chengtian invented a fraction interpolation method called "harmonization of the divisor of the day" (Chinese: zh:调日法; pinyin: diaorifa) to increase the accuracy of approximations of π by iteratively adding the numerators and denominators of fractions.

Zu Chongzhi's approximation π ≈ ⁠355/113⁠ can be obtained with He Chengtian's method.