Milü

Fractional approximations to

π

.

The name Milü (Chinese: 密率; pinyin: mì lǜ; "close ratio"), also known as Zulü (Zu's ratio), is given to an approximation to $π$ (pi) found by Chinese mathematician and astronomer, Zǔ Chōngzhī (祖沖之). Using Liu Hui's algorithm (which is based on the areas of regular polygons approximating a circle), Zu famously computed $π$ to be between 3.1415926 and 3.1415927 and gave two rational approximations of $π$ , 22/7 and 355/113, naming them respectively Yuelü 约率 (approximate ratio) and Milü.

355/113 is the best rational approximation of $π$ with a denominator of four digits or fewer, being accurate to 6 decimal places. It is within 0.000009% of the value of $π$ , or in terms of common fractions overestimates $π$ by less than 1/3 748 629. The next rational number (ordered by size of denominator) that is a better rational approximation of $π$ is 52 163/16 604, still only correct to 6 decimal places and hardly closer to $π$ than 355/113. To be accurate to 7 decimal places, one needs to go as far as 86 953/27 678. For 8, we need 102 928/32 763.

{\begin{aligned}\pi &\approx 3.141\ 592\ 653\ 5\dots \\\\{\frac {355}{113}}&\approx 3.141\ 592\ 920\ 3\dots \\\\{\frac {52163}{16604}}&\approx 3.141\ 592\ 387\ 4\dots \\\\{\frac {86953}{27678}}&\approx 3.141\ 592\ 600\ 6\dots \end{aligned}}

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 $[3;7,15,1,292,1,1,...]$ . 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 $π$ 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, this convergent will be very close to the true value of $π$ .^[1]

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, 1/ $π$ ≈ 113/355.

Zu's contemporary calendarist and mathematician He Chengtian (何承天) 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.^[2] Zu Chongzhi's approximation $π$ ≈ 355/113 can be obtained with He Chengtian's method.^[3]

References

↑ W., Weisstein, Eric. "Pi Continued Fraction". mathworld.wolfram.com. Retrieved 2017-09-03.
↑ Martzloff, Jean-Claude (2006). A History of Chinese Mathematics. Springer. p. 281.
↑ Wu Wenjun ed Grand Series of History of Chinese Mathematics vol 4 p125

External links

Fractional Approximations of Pi

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[1] W., Weisstein, Eric. "Pi Continued Fraction". mathworld.wolfram.com. Retrieved 2017-09-03.

[2] Martzloff, Jean-Claude (2006). A History of Chinese Mathematics. Springer. p. 281.

[3] Wu Wenjun ed Grand Series of History of Chinese Mathematics vol 4 p125

Milü

See also

References

External links