{{Short description|Chinese mathematician-astronomer (429–500)}}
{{family name hatnote|[[Zu (surname)|Zu]]|lang=Chinese}}
[[File:祖冲之铜像.jpg|thumb|upright=1.2|Statue of Zu Chongzhi]]
{{Infobox Chinese
| title = Zu Chongzhi
| t = 祖沖之
| s = 祖冲之
| p = Zǔ Chōngzhī
| w = Tsu Ch'ung-chih
| altname = Wenyuan {{small|([[courtesy name]])}}
| t2 = 文遠
| s2 = 文远
| p2 = Wényuǎn
}}

'''Zu Chongzhi''' ({{zh|c=祖沖之}}; 429–500<ref>Zu's biography in ''Book of the Southern Qi'' indicate that he was 72 (by East Asian reckoning) when he died in the 2nd year of the ''Yong'yuan'' era of [[Xiao Baojuan]]'s reign. (永元二年，冲之卒。年七十二。) ''Nan Qi Shu'', vol.52</ref>), [[courtesy name]] '''Wenyuan''' ({{zh|c=文遠}}), was a Chinese astronomer, inventor, mathematician, politician, and writer during the [[Liu Song]] and [[Southern Qi]] dynasties. He was most notable for calculating [[pi]] as between 3.1415926 and 3.1415927, a record in precision which would not be surpassed for nearly 900 years.

==Life and works==
{{more citations needed section |date=October 2018}}
Chongzhi's ancestry was from modern [[Baoding]], Hebei.<ref>(祖冲之字文远，范阳蓟人也。) ''Nan Qi Shu'', vol.52</ref> To flee from the ravages of war, Zu's grandfather Zu Chang ({{lang|zh-hant|祖昌}}) moved to the [[Yangtze]], as part of the massive population movement during the [[Eastern Jin]]. Zu Chang at one point held the position of Chief Minister for the Palace Buildings ({{lang|zh-hant|大匠卿}}) within the Liu Song<ref>(祖昌，宋大匠卿。) ''Nan Qi Shu'', vol.52 and '' Nan Shi'', vol.72</ref> and was in charge of government construction projects. Zu's father, Zu Shuozhi ({{lang|zh-hant|祖朔之}}), also served the court and was greatly respected for his erudition.

Zu was born in [[Jiankang]]. His family had historically been involved in astronomical research, and from childhood Zu was exposed to both astronomy and mathematics. When he was only a youth, his talent earned him much repute.<ref>(沖之稽古，有机思，...) ''Nan Shi'', vol.72</ref>  When [[Emperor Xiaowu of Song]] heard of him, he was sent to the Hualin Xuesheng ({{lang|zh-hant|華林學省}}) academy, and later the Imperial Nanjing University (Zongmingguan) to perform research. In 461 in Nanxu (today [[Zhenjiang|Zhenjiang, Jiangsu]]), he was engaged in work at the office of the local governor. In 464, Zu moved to Louxian (today Songjiang district, Shanghai), there, he compiled the Daming calendar and calculated π.

Zu Chongzhi, along with his son [[Zu Gengzhi]], wrote a mathematical text entitled ''Zhui Shu'' ({{lang|zh-hant|綴述}}; "''Methods for Interpolation''").<ref name="Ho1987"/> Though originally one of the ''[[Ten Computational Canons]]'', this book has been lost since the [[Song dynasty]].

== Astronomy ==
Zu was an accomplished astronomer who calculated the time values with unprecedented precision. His methods of interpolation and the use of integration were far ahead of his time. Even the results of the astronomer [[Yi Xing]] (who was beginning to utilize foreign knowledge) were not comparable. The Song dynasty calendar was backwards to the "Northern barbarians" because they were implementing their daily lives with the {{tlit|zh|Daming}} calendar.{{clarify|date=May 2019}} It is said{{by whom?|date=June 2026}} that his methods of calculation were so advanced that the scholars of the Song dynasty and Indian-influenced astronomers of the Tang dynasty found them confusing.

== Mathematics ==
{{further|Milü}}
{{Pi box}}
The majority of Zu's great mathematical works are recorded in his lost text the ''Zhui Shu''. It is said that the treatise contained formulas for the volume of a sphere, cubic equations and an accurate value of [[pi]].<ref name="Ho1987">{{ cite book | last= Ho | first=Peng Yoke| author-mask=[[Ho Peng Yoke]] | title= Li, Qi & Shu: An Introduction to Science & Civilization in China |publisher=Hong Kong University Press | orig-date=1985|edition=University of Washington Press |date=1987| isbn= 9780295963624 |page= 76 | oclc=17656687 }}</ref> Most schools argue about his complexity since traditionally the Chinese had developed mathematics as algebraic and equational. Logically, scholars assume that the ''Zhui Shu'' yields methods of cubic equations. His works on the accurate value of pi describe the lengthy calculations involved. Zu used [[Liu Hui's π algorithm|Liu Hui's {{pi}} algorithm]] described earlier by [[Liu Hui]] to inscribe a 12,288-gon. Zu's value of pi is precise to six decimal places and for almost nine hundred years thereafter no subsequent mathematician computed a value this precise.{{sfnp|Du|He|2000}} Zu also worked on deducing the formula for the volume of a sphere with his son Zu Gengzhi. In their calculation, Zu used the concept that two solids with equal cross-sectional areas at equal heights must also have equal volumes to find the volume of a Steinmetz solid. And further multiplied the volume of the Steinmetz solid with π/4, therefore found the volume of a sphere as πd^3/6 (d is the diameter of the sphere).

Zu Chongzhi's mathematical achievements included:
*the {{tlit|zh|Daming}} calendar ({{lang|zh-hant|大明曆}}) introduced by him in 465.
*distinguishing the [[sidereal year]] and the [[tropical year]]. He measured 45 years and 11 months per degree between those two; 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 [[Lunar_month#Draconic_month|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 π|approximations of pi]], (3.1415926535897932...) which held as the most accurate approximation for {{pi}} for over nine hundred years. His best approximation was between 3.1415926 and 3.1415927, with [[Milü|{{sfrac|355|113}}]] ({{lang|zh-hant|密率}}, [[milü]], close ratio) and [[Proof that 22/7 exceeds π|{{sfrac|22|7}}]] ({{lang|zh-hant|約率}}, yuelü, approximate ratio) being the other notable approximations. He obtained the result by approximating a circle with a 24,576 (= 2<sup>13</sup> × 3) sided polygon.<ref>{{Cite news |last=Strogatz |first=Steven |date=2024-03-07 |title=Pi Day: How One Irrational Number Made Us Modern |url=https://www.nytimes.com/article/pi-day-math-geometry-infinity.html |access-date=2024-03-15 |work=The New York Times |language=en-US |issn=0362-4331}}</ref> This was an impressive feat for the time, especially considering that the [[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, "{{sfrac|22|7}} was nothing more than the {{pi}} value obtained several hundred years earlier by the Greek mathematician [[Archimedes]], however milü {{pi}} = {{sfrac|355|113}} could not be found in any Greek, Indian or Arabian manuscripts, not until 1585 [[Netherlands|Dutch]] mathematician [[Adriaan Anthonisz|Adriaan Anthoniszoon]] obtained this fraction; the Chinese possessed this most extraordinary fraction over a whole millennium earlier than Europe". Hence Mikami strongly urged that the fraction {{sfrac|355|113}} be named after Zu Chongzhi as ''Zu's fraction''.<ref>
{{cite book
 | author = Yoshio Mikami
 | title = Development of Mathematics in China and Japan
 | publisher = B. G. Teubner
 | page = 50
 | year = 1913
 | url = https://books.google.com/books?id=4e9LAAAAMAAJ&q=intitle:Development+intitle:%22China+and+Japan%22+355 }}</ref> In Chinese literature, this fraction is known as "Zu's ratio". Zu's ratio is a [[best rational approximation]] to {{pi}}, and is the closest rational approximation to {{pi}} from all fractions with denominator less than 16600.<ref>The next "best rational approximation" to {{pi}} is {{sfrac|52163|16604}} = 3.1415923874.</ref>
*finding the volume of a [[sphere]] as {{pi}}D<sup>3</sup>/6 where D is diameter (equivalent to 4{{pi}}r<sup>3</sup>/3).

==Inventions and innovations==
===Hammer mills===
In 488, Zu Chongzhi was responsible for erecting water powered trip [[Hammermill|hammer mills]] which was inspected by [[Emperor Wu of Southern Qi]] during the early 490s.<ref>{{cite journal | title="The Water Mill" and Northern Song Imperial Patronage of Art, Commerce, and Science |journal = The Art Bulletin|volume = 84|issue = 4| publisher=CAA | author=Liu, Heping | pages=574|jstor = 3177285|year = 2002|doi = 10.2307/3177285}}</ref><ref name="Needham">[[Joseph Needham|Needham, Joseph]] (1965). ''Science and Civilization in China, Vol. IV: Physics and Physical Technology'', p.400. {{ISBN|978-0-521-05802-5}}.</ref><ref name=lu>{{cite book |url=https://books.google.com/books?id=d__HBAAAQBAJ&pg=PA289 |title=A History of Chinese Science and Technology, Volume 3 |editor=Yongxiang Lu |page=280 |publisher=Springer |year=2014|isbn=9783662441664 }}</ref>

===Paddle boats===
Zu is also credited with inventing Chinese paddle boats or [[Qianli chuan]] in the late 5th century AD during the Southern Qi dynasty.<ref name="needham 416">Needham, 416</ref><ref>{{Cite book |title=Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures  |url=https://archive.org/details/encyclopaediahis00seli  |url-access=limited  |last=Selin |first=Helaine |author-link=Helaine Selin |publisher=Springer |year=2008 |isbn=978-1402045592 |edition=2nd |publication-date=April 16, 2008 |page=[https://archive.org/details/encyclopaediahis00seli/page/n1076 1061]|bibcode=2008ehst.book.....S }}</ref><ref>{{Cite journal |last=Wang  |first=Hsien-Chun |date=January 1, 2019 |title=Discovering Steam Power in China, 1840s–1860s |url=https://muse.jhu.edu/article/375784 |journal=Technology and Culture |publisher=Johns Hopkins University Press |volume=51 |pages=38}}</ref><ref name=lu/> The boats made sailing a more reliable form of transportation and based on the shipbuilding technology of its day, numerous paddle wheel ships were constructed during the Tang era as the boats were able to cruise at faster speeds than the existing vessels at the time as well as being able to cover hundreds of kilometers of distance without the aid of wind.<ref name="needham 416">Needham, 416</ref>

===South pointing chariot===
The [[south-pointing chariot]] device was first invented by the Chinese mechanical engineer [[Ma Jun (mechanical engineer)|Ma Jun]] (c. 200–265 AD). It was a wheeled vehicle that incorporated an early use of [[differential (mechanical device)|differential gears]] to operate a fixed figurine that would constantly point south, hence enabling one to accurately measure their directional bearings. This effect was achieved not by [[magnetic]]s (like in a [[compass]]), but through intricate mechanics, the same design that allows equal amounts of torque applied to wheels rotating at different speeds for the modern [[automobile]]. After the [[Three Kingdoms]] period, the device fell out of use temporarily. However, it was Zu Chongzhi who successfully re-invented it in 478, as described in the texts of the ''[[Book of Song]]'' and the ''[[Book of Qi]]'', with a passage from the latter below:

<blockquote>
When [[Emperor Wu of Liu Song]] subdued [[Guanzhong]] he obtained the south-pointing carriage of Yao Xing, but it was only the shell with no machinery inside. Whenever it moved it had to have a man inside to turn (the figure). In the Sheng-Ming reign period, Gao Di commissioned Zi Zu Chongzhi to reconstruct it according to the ancient rules. He accordingly made new machinery of bronze, which would turn round about without a hitch and indicate the direction with uniformity. Since Ma Jun's time such a thing had not been.<ref name="needham volume 4 part 2 289">Needham, Volume 4, Part 2, 289.</ref><ref>''Book of Qi'', '''52'''.905</ref>
</blockquote>

== Literature ==
Zu's [[Paradoxography|paradoxographical work]] ''Accounts of Strange Things'' [{{zh|labels=no|c=述異記}}] survives.<ref name=":0">{{Cite book|script-title=zh:中国大百科全书（第二版）|publisher=Encyclopedia of China Publishing House|year=2009|isbn=978-7-500-07958-3|volume=30|pages=205|language=zh|trans-title=Encyclopedia of China (2nd Edition)}}</ref><ref>{{Cite book|title=The Cambridge History of Chinese Literature|last=Owen|first=Stephen|publisher=Cambridge University Press|year=2010|isbn=978-0-521-11677-0|volume=1|pages=242}}</ref>

== Named after him ==
* {{pi}} ≈ {{sfrac|355|113}} as [[Milü|Zu Chongzhi's {{pi}} ratio]]
*The lunar crater [[Tsu Chung-Chi (crater)|Tsu Chung-Chi]]
*[[1888 Zu Chong-Zhi]] is the name of [[asteroid]] 1964 VO1
*[[ZUC stream cipher]], an encryption algorithm
*Zhuchongzi, a series of [[Quantum computing|quantum computer]]s developed by researchers at the [[University of Science and Technology of China]] in Hefei

== References ==
{{Reflist}}

=== General references ===
* {{Cite book |last=Needham |first=Joseph |date=1986 |title=Science and Civilization in China: Volume 4, Part 2 |publisher=Cambridge University Press}}
* {{ Cite book | last1= Du | first1=Shiran |last2= He |first2=Shaogeng | chapter-url=http://203.72.198.245/web/Content.asp?ID=43669&Query=1 |chapter=Zu Chongzhi| title= Encyclopedia of China (Mathematics Edition) | edition =1st |title-link= Encyclopedia of China | author1-mask= Du Shiran (杜石然); | date=2000 | archive-url=https://web.archive.org/web/20070929095635/http://203.72.198.245/web/Content.asp?ID=43669&Query=1 | archive-date=29 September 2007 | author2-mask= He Shaogeng (何紹庚)}}

==Further reading==
* {{cite book |last=Needham |first=Joseph |date=1986 |title=Science and Civilization in China: Volume 3, Mathematics and the Sciences of the Heavens and the Earth |publisher=Cambridge University Press}}
* {{cite book|script-title=zh:南齊書 |trans-title=[[Book of Qi]] |editor=Xiao Zixian |editor-link=Xiao Zixian |date=1974|location=Beijing |publisher=Zhonghua Publishing |orig-year=537 |volume=52 |pages=903–906}}
* {{cite book|script-title=zh:南史 |trans-title=[[History of the Southern Dynasties]]  |editor1=Li Dashi |editor-link1=Li Dashi  |editor2=Li Yanshou (李延壽) |editor-link2=:zh:李延壽|location=Beijing |publisher=Zhonghua Publishing |orig-year=659 |year=1975 |volume=72 |pages=1773–1774}}

==External links==
*[http://www.britannica.com/eb/article-9384156/Zu-Chongzhi Encyclopædia Britannica's description of Zu Chongzhi]
*[https://web.archive.org/web/20070705032140/http://www.chinaculture.org/gb/en_aboutchina/2003-09/24/content_26294.htm Zu Chongzhi at Chinaculture.org]
*[http://hua.umf.maine.edu/China/astronomy/tianpage/0014ZuChongzhi9296bw.html Zu Chongzhi at the University of Maine] {{Webarchive|url=https://web.archive.org/web/20100614000009/http://hua.umf.maine.edu/China/astronomy/tianpage/0014ZuChongzhi9296bw.html |date=2010-06-14 }}
*{{MacTutor Biography|id=Zu_Chongzhi}}

{{Authority control}}

{{DEFAULTSORT:Zu, Chongzhi}}
[[Category:Zu Chongzhi| ]]
[[Category:429 births]]
[[Category:500 deaths]]
[[Category:5th-century Chinese mathematicians]]
[[Category:5th-century Chinese astronomers]]
[[Category:Chinese inventors]]
[[Category:Liu Song government officials]]
[[Category:Liu Song writers]]
[[Category:Pi-related people]]
[[Category:Politicians from Nanjing]]
[[Category:Scientists from Nanjing]]
[[Category:Southern Qi government officials]]
[[Category:Writers from Nanjing]]
[[Category:Chinese geometers]]