**Jia Xian** (simplified Chinese :贾宪; traditional Chinese :賈憲; pinyin :*Jiǎ Xiàn*; Wade–Giles :*Chia Hsien*; ca. 1010–1070) was a Chinese mathematician from Kaifeng of the Song dynasty.

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 Pascal. Jia used it as a tool for extracting square and cubic roots. 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*."^{ [1] } A page from the * Yongle Encyclopedia * preserved this historic fact.

Jia Xian's additive-multiplicative method implemented the "Horner" rule.^{ [2] }

**Arithmetic** is a branch of mathematics that consists of the study of numbers, especially the properties of the traditional operations on them—addition, subtraction, multiplication, division, exponentiation and extraction of roots. Arithmetic is an elementary part of number theory, and number theory is considered to be one of the top-level divisions of modern mathematics, along with algebra, geometry, and analysis. The terms *arithmetic* and *higher arithmetic* were used until the beginning of the 20th century as synonyms for *number theory*, and are sometimes still used to refer to a wider part of number theory.

The area of study known as the **history of mathematics** is primarily an investigation into the origin of discoveries in mathematics and, to a lesser extent, an investigation into the mathematical methods and notation of the past. Before the modern age and the worldwide spread of knowledge, written examples of new mathematical developments have come to light only in a few locales. From 3000 BC the Mesopotamian states of Sumer, Akkad and Assyria, followed closely by Ancient Egypt and the Levantine state of Ebla began using arithmetic, algebra and geometry for purposes of taxation, commerce, trade and also in the patterns in nature, the field of astronomy and to record time and formulate calendars.

In mathematics and computer science, **Horner's method** is an algorithm for polynomial evaluation. Although named after William George Horner, this method is much older, as it has been attributed to Joseph-Louis Lagrange by Horner himself, and can be traced back many hundreds of years to Chinese and Persian mathematicians. After the introduction of computers, this algorithm became fundamental for computing efficiently with polynomials.

**Zhu Shijie**, courtesy name **Hanqing** (漢卿), pseudonym **Songting** (松庭), was a Chinese mathematician and writer. He was a Chinese mathematician during the Yuan Dynasty. Zhu was born close to today's Beijing. Two of his mathematical works have survived. *Introduction to Computational Studies*, and *Jade Mirror of the Four Unknowns*.

* The Nine Chapters on the Mathematical Art* is a Chinese mathematics book, composed by several generations of scholars from the 10th–2nd century BCE, its latest stage being from the 2nd century CE. This book is one of the earliest surviving mathematical texts from China, the first being

The **suanpan**, also spelled **suan pan** or **souanpan**) is an abacus of Chinese origin first described in a 190 CE book of the Eastern Han Dynasty, namely *Supplementary Notes on the Art of Figures* written by Xu Yue. However, the exact design of this suanpan is not known. Usually, a suanpan is about 20 cm tall and it comes in various widths depending on the application. It usually has more than seven rods. There are two beads on each rod in the upper deck and five beads on each rod in the bottom deck. The beads are usually rounded and made of a hardwood. The beads are counted by moving them up or down towards the beam. The suanpan can be reset to the starting position instantly by a quick jerk around the horizontal axis to spin all the beads away from the horizontal beam at the center.

**Mathematics in China** emerged independently by the 11th century BC. The Chinese independently developed a real number system that includes significantly large and negative numbers, more than one numeral system, algebra, geometry, number theory and trigonometry.

**Yang Hui**, courtesy name **Qianguang** (謙光), was a Chinese mathematician and writer during the Song dynasty. Originally, from Qiantang, Yang worked on magic squares, magic circles and the binomial theorem, and is best known for his contribution of presenting Yang Hui's Triangle. This triangle was the same as Pascal's Triangle, discovered by Yang's predecessor Jia Xian. Yang was also a contemporary to the other famous mathematician Qin Jiushao.

The * Book on Numbers and Computation*, or the

Algebra can essentially be considered as doing computations similar to those of arithmetic but with non-numerical mathematical objects. However, until the 19th century, algebra consisted essentially of the theory of equations. For example, the fundamental theorem of algebra belongs to the theory of equations and is not, nowadays, considered as belonging to algebra.

**Rod calculus** or rod calculation was the mechanical method of algorithmic computation with counting rods in China from the Warring States to Ming dynasty before the counting rods were replaced by the more convenient and faster abacus. Rod calculus played a key role in the development of Chinese mathematics to its height in Song Dynasty and Yuan Dynasty, culminating in the invention of polynomial equations of up to four unknowns in the work of Zhu Shijie.

**Wang Xiaotong** (王孝通), also known as Wang Hs'iao-t'ung, was a Chinese mathematician, calendarist, politician, and writer of the early Tang dynasty. He is famous as the author of the *Jigu Suanjing* one of the *Ten Computational Canons*.

A timeline of **numerals** and **arithmetic**

This is a timeline of pure and applied mathematics history. It is divided here into three stages, corresponding to stages in the development of mathematical notation: a "rhetorical" stage in which calculations are described purely by words, a "syncopated" stage in which quantities and common algebraic operations are beginning to be represented by symbolic abbreviations, and finally a "symbolic" stage, in which comprehensive notational systems for formulas are the norm.

The mathematical field of combinatorics was studied to varying degrees in numerous ancient societies. Its study in Europe dates to the work of Leonardo Fibonacci in the 13th century AD, which introduced Arabian and Indian ideas to the continent. It has continued to be studied in the modern era.

* Tian yuan shu* is a Chinese system of algebra for polynomial equations. Some of the earliest existing writings were created in the 13th century during the Yuan dynasty. However, the tianyuanshu method was known much earlier, in the Song dynasty and possibly before.

* Jigu suanjing* was the work of early Tang dynasty calendarist and mathematician Wang Xiaotong, written some time before the year 626, when he presented his work to the Emperor.

The * Ten Computational Canons* was a collection of ten Chinese mathematical works, compiled by early Tang dynasty mathematician Li Chunfeng (602–670), as the official mathematical texts for imperial examinations in mathematics.

During the Mongol-ruled Yuan dynasty (1271–1368), many scientific and technological advancements were made in areas such as mathematics, medicine, printing technology, and gunpowder warfare.

- 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).
- 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.

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