Milü

# Milü

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Encyclopedia
The name Milü also known as Zulü (Zu
Zu Chongzhi
Zu Chongzhi , courtesy name Wenyuan , was a prominent Chinese mathematician and astronomer during the Liu Song and Southern Qi Dynasties.-Life and works:...

's ratio), is given to an approximation to π
Pi
' is a mathematical constant that is the ratio of any circle's circumference to its diameter. is approximately equal to 3.14. Many formulae in mathematics, science, and engineering involve , which makes it one of the most important mathematical constants...

(Pi
Pi
' is a mathematical constant that is the ratio of any circle's circumference to its diameter. is approximately equal to 3.14. Many formulae in mathematics, science, and engineering involve , which makes it one of the most important mathematical constants...

) found by Chinese mathematician and astronomer
Chinese astronomy
Astronomy in China has a very long history, with historians considering that "they [the Chinese] were the most persistent and accurate observers of celestial phenomena anywhere in the world before the Arabs."...

Zu Chongzhi
Zu Chongzhi
Zu Chongzhi , courtesy name Wenyuan , was a prominent Chinese mathematician and astronomer during the Liu Song and Southern Qi Dynasties.-Life and works:...

. He computed π to be between 3.1415926 and 3.1415927 and gave two rational approximations of π, and , naming them respectively Yuelü 约率 (literally "approximate ratio") and Milü.

is by far the best rational
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...

approximation of π
Pi
' is a mathematical constant that is the ratio of any circle's circumference to its diameter. is approximately equal to 3.14. Many formulae in mathematics, science, and engineering involve , which makes it one of the most important mathematical constants...

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 . The next rational number (ordered by size of denominator) that is a better rational approximation of π is , still only correct to 6 decimal places. To be accurate to 7 decimal places, one needs to go as far as .

An easy mnemonic
Mnemonic
A mnemonic , or mnemonic device, is any learning technique that aids memory. To improve long term memory, mnemonic systems are used to make memorization easier. Commonly encountered mnemonics are often verbal, such as a very short poem or a special word used to help a person remember something,...

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.
Zu's contemporary calendarist and mathematician He 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
Zu Chongzhi
Zu Chongzhi , courtesy name Wenyuan , was a prominent Chinese mathematician and astronomer during the Liu Song and Southern Qi Dynasties.-Life and works:...

's approximation can be obtained with He's method

• Continued fraction expansions of π
• History of numerical approximations of π
Numerical approximations of π
This page is about the history of approximations for the mathematical constant pi . There is a table summarizing the chronology of computation of π. See also the history of pi for other aspects of the evolution of our knowledge about mathematical properties of pi...

• Pi Approximation Day