Milü

⁠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, and for nine, ⁠103993/33102⁠ is required.

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:

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

Zu's contemporary calendarist and mathematician He Chengtian invented a fraction interpolation method called 'harmonization of the divisor of the day' (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.