# Zu Chongzhi - Life and Works

Life and Works

Chongzhi's ancestry was from modern Baoding, Hebei. To flee from the ravage of war, Zu's grandfather Zu Chang moved to the Yangtze, as part of the massive population movement during the Eastern Jin. Zu Chang (祖昌) at one point held the position of "Minister of Great Works" ) within the Liu Song and was in charge of government construction projects. Zu's father, Zu Shuo (祖朔) also served the court and was greatly respected for his erudition.

Zu was born in Jiankang. His family had historically been involved in astronomy research, and from childhood Zu was exposed to both astronomy and mathematics. When he was only a youth his talent earned him much repute. When Emperor Xiaowu of Liu Song heard of him, he was sent to an Academy, the Hualin Xuesheng (華林學省), and later at the Imperial Nanjing University (Zongmingguan) to perform research. In 461 in Nanxu (today Zhenjiang, Jiangsu), he was engaged in work at the office of the local governor.

Zu Chongzhi, along with his son Zu Gengzhi, wrote a mathematical text entitled Zhui Shu (Method of Interpolation). It is said that the treatise contains formulas for the volume of the sphere, cubic equations and the accurate value of pi. This book didn't survive to the present day; it has been lost since the Song Dynasty.

His mathematical achievements included:

• the Daming calendar (大明曆) introduced by him in 465 A.D.
• distinguishing the Sidereal Year and the Tropical Year, and he measured 45 years and 11 months per degree between those two, and today we know the difference is 70.7 years per degree.
• calculating one year as 365.24281481 days, which is very close to 365.24219878 days as we know today.
• calculating the number of overlaps between sun and moon as 27.21223, which is very close to 27.21222 as we know today; using this number he successfully predicted an eclipse four times during 23 years (from 436 to 459).
• calculating the Jupiter year as about 11.858 Earth years, which is very close to 11.862 as we know of today.
• deriving two approximations of pi, which held as the most accurate approximation for π for over nine hundred years. His best approximation was between 3.1415926 and 3.1415927, with 355⁄113 (密率, Milü, detailed approximation) and 22⁄7 (約率, Yuelü, rough approximation) being the other notable approximations. He obtained the result by approximating a circle with a 12,288 (= 212 × 3) sided polygon. This was an impressive feat for the time, especially considering that the device Counting rods he used for recording intermediate results were merely a pile of wooden sticks laid out in certain patterns. Japanese mathematician Yoshio Mikami pointed out, " was nothing more than the π value obtained several hundred years earlier by the Greek mathematician Archimedes, however Milu could not be found in any Greek, Indian or Arabian manuscripts, not until 1585 Dutch mathematician Adriaan Anthoniszoom obtained this fraction; the Chinese possessed this most extraordinary fraction over a whole millennium earlier than Europe". Hence Mikami strongly urged that the fraction be named after Zu Chongzhi as Zu Chongzhi fraction. In Chinese literature, this fraction is known as "Zu rate". Zu rate is a best rational approximation to π, and is the closest rational approximation to π from all fractions with denominator less than 16600.
• finding the volume of a sphere as πD3/6 where D is diameter (equivalent to 4πr3/3).

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