Milü, also known as Zulü, is the name given to an approximation to pi| found by Chinese mathematician and astronomer, Zǔ Chōngzhī, born 429 AD. Using Liu Hui's algorithm, Zu famously computed to be between 3.1415926 and 3.1415927 and gave two rational approximations of, and, naming them respectively Yuelü 约率 and Milü. is the best rational approximation of with a denominator of four digits or fewer, being accurate to 6 decimal places. It is within 0.000009% of the value of, or in terms of common fractions overestimates by less than. The next rational number that is a better rational approximation of is, still only correct to 6 decimal places and hardly closer to than. To be accurate to 7 decimal places, one needs to go as far as. For 8, is needed. The accuracy of Milü to the true value of can be explained using the continued fraction expansion of, the first few terms of which are. A property ofcontinued 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 immediately before the term 292; that is, isapproximated by the finite continued fraction, which is equivalent to Milü. Since 292 is an unusually large term in a continued fraction expansion, this convergent will be very close to the true value of. 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. Alternatively, ≈. Zu's contemporary calendarist and mathematicianHe 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. Zu Chongzhi's approximation ≈ can be obtained with He Chengtian's method.
