* Tian yuan shu* (simplified Chinese :天元术; traditional Chinese :天元術; pinyin :

The Tianyuanshu was explained in the writings of Zhu Shijie (* Jade Mirror of the Four Unknowns *) and Li Zhi (* Ceyuan haijing *), two Chinese mathematicians during the Mongol Yuan dynasty.

However, after the Ming overthrew the Mongol Yuan, Zhu and Li's mathematical works went into disuse as the Ming literati became suspicious of knowledge imported from Mongol Yuan times.

Only recently, with the advent of modern mathematics in China has the tianyuanshu been re-deciphered.

Meanwhile, *tian yuan shu* arrived in Japan, where it is called * tengen-jutsu*. Zhu's text

*Tian yuan shu* means "method of the heavenly element" or "technique of the celestial unknown". The "heavenly element" is the unknown variable, usually written *x* in modern notation.

It is a positional system of rod numerals to represent polynomial equations. For example, 2*x*^{2} + 18*x* − 316 = 0 is represented as

The 元 (*yuan*) denotes the unknown *x*, so the numerals on that line mean 18*x*. The line below is the constant term (-316) and the line above is the coefficient of the quadratic (*x*^{2}) term. The system accommodates arbitrarily high exponents of the unknown by adding more lines on top and negative exponents by adding lines below the constant term. Decimals can also be represented.

In later writings of Li Zhi and Zhu Shijie, the line order was reversed so that the first line is the lowest exponent.

**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 **Hongwu Emperor**, personal name **Zhu Yuanzhang**, was the founding emperor of the Ming dynasty, reigning from 1368 to 1398.

**Wen Tianxiang**, Duke of Xinguo (信國公), was a Chinese poet and politician in the last years of the Southern Song Dynasty. For his resistance to Kublai Khan's invasion of the Song Dynasty, and for his refusal to yield to the Yuan Dynasty despite being captured and tortured, he is a popular symbol of patriotism and righteousness in China. He is known as one of the 'Three Loyal Princes of the Song' (大宋三忠王), alongside Lu Xiufu and Zhang Shijie. Wen Tianxiang is depicted in the Wu Shuang Pu by Jin Guliang.

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* Jinshi* was the highest and final degree in the imperial examination in Imperial China. The examination was usually taken in the imperial capital in the palace, and was also called the

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**Liu Zhi**, or **Liu Chih**, was a Chinese Sunni Hanafi-Maturidi scholar of the Qing dynasty, belonging to the Huiru (Muslim) school of Neoconfucian thought. He was the most prominent of the Han Kitab writers who attempted to explain Muslim thought in the Chinese intellectual climate for a Hui Chinese audience, by frequently borrowing terminologies from Buddhism, Taoism and most prominently Neoconfucianism and aligning them with Islamic concepts. He was from the city of Nanjing. His magnum opus, T'ien-fang hsing-li or 'Nature and Principle in the Direction of Heaven', was considered the authoritative exposition of Islamic beliefs and has been republished twenty-five times between 1760 and 1939, and is constantly referred to by Muslims writing in Chinese.

The **Red Turban Rebellions** were uprisings against the Yuan dynasty between 1351 and 1368, eventually leading to the overthrow of Yuan rule in China proper.

**Emperor Xuan of Northern Zhou** ( 周宣帝) (559–580), personal name **Yuwen Yun** (宇文贇), courtesy name **Qianbo** (乾伯), was an emperor of the Xianbei-led Northern Zhou dynasty of China. He was known in history as an erratic and wasteful ruler, whose actions greatly weakened the Northern Zhou regime. As part of that erratic behavior, he passed the throne to his son Emperor Jing in 579, less than a year after taking the throne, and subsequently entitled not only his wife Yang Lihua empress, but four additional concubines as empresses. After his death in 580, the government was taken over by his father-in-law Yang Jian, who soon deposed his son Emperor Jing, ending the Northern Zhou and establishing the Sui dynasty.

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**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 increasingly 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.

The **Yuan dynasty**, officially the **Great Yuan**, was a successor state to the Mongol Empire after its division and a ruling dynasty of China established by Kublai Khan, leader of the Mongol Borjigin clan, lasting from 1271 to 1368. In Chinese historiography, this dynasty followed the Song dynasty and preceded the Ming dynasty.

**Li Ye**, born **Li Zhi**, courtesy name **Li Jingzhai**, was a Chinese scientist and writer who published and improved the tian yuan shu method for solving polynomial equations of one variable. Along with the 4th-century Chinese astronomer Yu Xi, Li Ye proposed the idea of a spherical Earth instead of a flat one before the advances of European science in the 17th century.

* Ceyuan haijing* is a treatise on solving geometry problems with the algebra of Tian yuan shu written by the mathematician Li Zhi in 1248 in the time of the Mongol Empire. It is a collection of 692 formula and 170 problems, all derived from the same master diagram of a round town inscribed in a right triangle and a square. They often involve two people who walk on straight lines until they can see each other, meet or reach a tree or pagoda in a certain spot. It is an algebraic geometry book, the purpose of book is to study intricated geometrical relations by algebra.

* Yigu yanduan* is a 13th-century mathematical work by Yuan dynasty mathematician Li Zhi.

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.

* Jade Mirror of the Four Unknowns*,

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.

* Heavenly Sword and Dragon Slaying Sabre* is a 2019 Chinese wuxia television series adapted from the novel

- Martzloff, Jean-Claude (2006).
*A History of Chinese Mathematics*. trans. Stephen S. Wilson. Springer. pp. 258–272. ISBN 3-540-33782-2 . Retrieved 2009-12-28. - Murata, Tamotsu (2003). "Indigenous Japanese mathematics, Wasan". In Ivor Grattan-Guinness (ed.).
*Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences*.**1**. JHU Press. pp. 105–106. ISBN 0-8018-7396-7 . Retrieved 2009-12-28.

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