Tian yuan shu

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Tian yuan shu in Zhu Shijie's text Suanxue qimeng Suan Xue Qi Meng .jpg
Tian yuan shu in Zhu Shijie's text Suanxue qimeng
The technique described in Alexander Wylie's Jottings on the Science of the Chinese Wylie on Tian Yuen.jpg
The technique described in Alexander Wylie's Jottings on the Science of the Chinese

Tian yuan shu (simplified Chinese :天元术; traditional Chinese :天元術; pinyin :tiān yuán shù) is a Chinese system of algebra for polynomial equations created in the 13th century. It is first known from the writing of Li Zhi (Li Ye), though it was created earlier.

Simplified Chinese characters standardized Chinese characters developed in mainland China

Simplified Chinese characters are standardized Chinese characters prescribed in the Table of General Standard Chinese Characters for use in mainland China. Along with traditional Chinese characters, they are one of the two standard character sets of the contemporary Chinese written language. The government of the People's Republic of China in mainland China has promoted them for use in printing since the 1950s and 1960s to encourage literacy. They are officially used in the People's Republic of China and Singapore.

Traditional Chinese characters

Traditional Chinese characters are Chinese characters in any character set that does not contain newly created characters or character substitutions performed after 1946. They are most commonly the characters in the standardized character sets of Taiwan, of Hong Kong and Macau, and in the Kangxi Dictionary. The modern shapes of traditional Chinese characters first appeared with the emergence of the clerical script during the Han Dynasty, and have been more or less stable since the 5th century.

Hanyu Pinyin, often abbreviated to pinyin, is the official romanization system for Standard Chinese in mainland China and to some extent in Taiwan. It is often used to teach Standard Mandarin Chinese, which is normally written using Chinese characters. The system includes four diacritics denoting tones. Pinyin without tone marks is used to spell Chinese names and words in languages written with the Latin alphabet, and also in certain computer input methods to enter Chinese characters.


The mathematical culture in which it was created was lost due to war and general suspiciousness during the Ming dynasty of knowledge from the (Mongolian) Yuan dynasty. The writings of Li Zhi ( Ceyuan haijing ), Zhu Shijie ( Jade Mirror of the Four Unknowns ) and others could no longer be fully understood, until the arrival of western mathematics in China.

Ming dynasty former empire in Eastern Asia, last Han Chinese-led imperial regime

The Ming dynasty was the ruling dynasty of China – then known as the Great Ming Empire – for 276 years (1368–1644) following the collapse of the Mongol-led Yuan dynasty. The Ming dynasty was the last imperial dynasty in China ruled by ethnic Han Chinese. Although the primary capital of Beijing fell in 1644 to a rebellion led by Li Zicheng, regimes loyal to the Ming throne – collectively called the Southern Ming – survived until 1683.

Yuan dynasty former Mongolian-ruled empire in Eastern and Northeastern Asia

The Yuan dynasty, officially the Great Yuan, was the empire or ruling dynasty of China established by Kublai Khan, leader of the Mongolian Borjigin clan. It followed the Song dynasty and preceded the Ming dynasty. Although the Mongols had ruled territories including modern-day North China for decades, it was not until 1271 that Kublai Khan officially proclaimed the dynasty in the traditional Chinese style, and the conquest was not complete until 1279. His realm was, by this point, isolated from the other khanates and controlled most of modern-day China and its surrounding areas, including modern Mongolia. It was the first foreign dynasty to rule all of China and lasted until 1368, after which the rebuked Genghisid rulers retreated to their Mongolian homeland and continued to rule the Northern Yuan dynasty. Some of the Mongolian Emperors of the Yuan mastered the Chinese language, while others only used their native language and the 'Phags-pa script.

<i>Ceyuan haijing</i>

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.

Meanwhile, tian yuan shu arrived in Japan, where it is called tengen-jutsu. Zhu's text Suanxue qimeng was deciphered and was important in the development of Japanese mathematics (wasan) in the 17th and 18th centuries.

Japanese mathematics denotes a distinct kind of mathematics which was developed in Japan during the Edo period (1603–1867). The term wasan, from wa ("Japanese") and san ("calculation"), was coined in the 1870s and employed to distinguish native Japanese mathematical theory from Western mathematics.


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.

In elementary mathematics, a variable is a symbol, commonly a single letter, that represents a number, called the value of the variable, which is either arbitrary, not fully specified, or unknown. Making algebraic computations with variables as if they were explicit numbers allows one to solve a range of problems in a single computation. A typical example is the quadratic formula, which allows one to solve every quadratic equation by simply substituting the numeric values of the coefficients of the given equation to the variables that represent them.

It is a positional system of rod numerals to represent polynomial equations. For example, 2x2 + 18x − 316 = 0 is represented as

Polynomial equation with rod numerals.png , which in Arabic numerals is Polynomial equation in tian yuan shu with arabic numerals.png

The (yuan) denotes the unknown x, so the numerals on that line mean 18x. The line below is the constant term (-316) and the line above is the coefficient of the quadratic (x2) 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 mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or any expression; it is usually a number, but may be any expression. In the latter case, the variables appearing in the coefficients are often called parameters, and must be clearly distinguished from the other variables.

Quadratic function polynomial function in which the highest-degree term is of the second degree

In algebra, a quadratic function, a quadratic polynomial, a polynomial of degree 2, or simply a quadratic, is a polynomial function with one or more variables in which the highest-degree term is of the second degree. For example, a quadratic function in three variables x, y, and z contains exclusively terms x2, y2, z2, xy, xz, yz, x, y, z, and a constant:

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

See also

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Tianyuan may refer to:

Li Ye, born Li Zhi, courtesy name Li Jingzhai, was a Chinese mathematician and scholar who published and improved the tian yuan shu method for solving polynomial equations of one variable. Along with the 4th-century scholar Yu Xi, Li Ye is one of the few Chinese scholars to propose the idea of a spherical Earth instead of a flat one before the arrival of European science in China during the 17th century.

<i>Yigu yanduan</i>

Yigu yanduan is a 13th-century mathematical work by Yuan dynasty mathematician Li Zhi. Yigu yanduan was based on North Song mathematician Jiang Zhou (蒋周) Yigu Ji which was extinct. However, from fragments quoted in Yang Hui's work The Complete Algorithms of Acreage(田亩比类算法大全), we know that this lost mathematical treatise Yigu Ji was about solving area problems with geometry. Li Zhi used the examples of Yigu Ji to introduce the art of Tian yuan shu to newcomers to this field. Although Li Zhi's previous monograph Ceyuan haijing also used tian yuan shu, however it is harder to understand than Yigu yanduan.

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