{{Short description|Chinese mathematician}}
{{family name hatnote|[[Jia (surname)|Jia]]|lang=Chinese}}
[[Image:Yanghui triangle.gif|right|thumb|250px|Jia Xian triangle (Pascal's triangle) using [[Counting rods|rod numerals]], as depicted in a publication of [[Zhu Shijie]] in 1303 AD.]]
[[File:Jiaxian.jpg|thumb|right|250px|Yang Hui referred to Jia Xian's Shi Suo Suan Shu in the Yongle Encyclopedia]]
'''Jia Xian''' ({{zh|t=賈憲|s=贾宪|p=Jiǎ Xiàn|w=Chia Hsien}}; ca. 1010–1070) was a [[Chinese mathematics|Chinese mathematician]] from [[Kaifeng]] of the [[Song dynasty]]. He described [[Pascal's triangle]] during the 11th century.

==Biography==
According to the history of the Song dynasty, Jia was a palace [[eunuch]] of the Left Duty Group. He studied under the mathematician Chu Yan, and was well versed in mathematics, writing many books on the subject. Jia Xian described the [[Pascal's triangle]] (Jia Xian triangle) around the middle of the 11th century, about six centuries before [[Blaise Pascal|Pascal]]. Jia used it as a tool for extracting [[square root|square]] and [[cubic root]]s. The original book by Jia entitled ''Shi Suo Suan Shu'' was lost; however, Jia's method was expounded in detail by [[Yang Hui]], who explicitly acknowledged his source: "My method of finding square and cubic roots was based on the Jia Xian method in ''Shi Suo Suan Shu''."<ref>Wu Wenjun chief ed, ''The Grand Series of History of Chinese Mathematics'' Vol 5 Part 2, chapter 1, Jia Xian</ref> A page from the ''[[Yongle Encyclopedia]]'' preserved this historic fact.

Jia Xian's additive-multiplicative method implemented the "Horner" rule.<ref>Jean-Claude Martzloff, ''A History of Chinese Mathematics ''Springer, p142</ref>

==Additive-multiplicative method of square-root extraction==
[[File:JIA XIAN SQRT2.GIF|frameless|none|upright=2000|Jia Xian additive-multiplicative square-root extraction]]

==Additive-multiplicative method of cubic-root extraction==
[[File:Jia Xian cubic root.GIF|frameless|none|upright=900|Jia Xian add-multiply method for cubic roots]]


==References==
<references/>

==Bibliography==
*J-C Martzloff, A history of Chinese mathematics  (Berlin-Heidelberg, 1997).
*J-C Martzloff, Histoire des mathématiques chinoises  (Paris, 1987).
*B Qian, History of Chinese mathematics  (Chinese) (Peking, 1981).
*[[Karine Chemla|K Chemla]], Similarities between Chinese and Arabic mathematical writings  I : Root extraction, Arabic Sci. Philos.  4 (2) (1994), 207-266.
*S Guo, Preliminary research into Jia Xian's Huangdi Jiuzhang Suanjing Xicao (Chinese), Studies in the History of Natural Sciences  7 (4) (1988), 328 -334.
*S Guo, Jia Xian, in Du Shiran (ed.), Zhongguo Gudai Kexuejia Zhuanji  (Biographies of Ancient Chinese Scientists)  (Beijing, 1992), 472 -479.
*R Mei, Jia Xian's additive-multiplicative method for the extraction of roots (Chinese), Studies in the History of Natural Sciences  8 (1) (1989), 1 -8.

{{Song dynasty topics}}
{{authority control}}

{{DEFAULTSORT:Jia, Xian}}
[[Category:11th-century Chinese mathematicians]]
[[Category:Song dynasty eunuchs]]
[[Category:Song dynasty science writers]]
[[Category:Year of birth unknown]]
[[Category:Year of death unknown]]