Li Zhi (mathematician)
Encyclopedia
Li Zhi (1192–1279), later Li Ye , was a Chinese
mathematician
and scholar, who published and improved the tian yuan shu
method for solving polynomial equations of one variable.
with the third T'ang emperor
who was also named Li Zhi, removing one stroke from his original name to change the character. His name is also sometimes written as Li Chih or Li Yeh. The literary name was (Renqing, Jen-ch’ing) and appellation
was (Jingzhai, Ching-chai).
in 1230 at the age of 38, and was administrative prefect of Jun prefecture in Henan
province until the Mongol invasion in 1233. He then lived in poverty in the mountainous Shanxi province. In 1248 he finished his most known work Ceyuan haijing . Li then returned to Hebei
.
In 1257 Kublai Khan
, grandson of Genghis Khan
, sent for Li to get advice on government and science. In 1259 Li completed Yigu yanduan , also an important mathematics text. After becoming Khan
, Kublai twice offered Li government positions. He politely declined, referring to his age and ill health. In 1264 Li finally accepted a position at the Hanlin Academy
, writing official histories. He was critical of the politics and resigned after a few months, again citing ill health. He spent his final years teaching at his home near Feng Lung mountain in Yuan, Hebei
. Li told his son to burn all of his books except for Sea mirror of circle measurements. Fortunately other texts, both mathematical and literary, still survived.
Ceyuan haijing (Sea mirror of circle measurements) is a collection of 170 problems, all related to the same example of a circular city wall 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 in a certain spot. It is not a geometry
book, the purpose of the geometrical problems are to create algebraic problems.
Many of the problems are solved by polynomial equations, which are represented using a method called tian yuan shu
, "coefficient array method" or literally "method of the celestial unknown". Li Zhi is the earliest known source of this method, though it was known before him in some form. It is a positional system of rod numerals to represent polynomial equations.
For example 2x2 + 18x − 316 = 0 is represented as
which in Arabic numerals is
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. Later, the line order was reversed so that the first line is the lowest exponent.
Li does not explain how to solve equations in general, but shows it with the example problems. Most of the equations can be reduced to the second or sometimes third order. It is often assumed that he used methods similar to Ruffini's rule
and Horner scheme
.
Yigu yanduan
(New steps in computation) is a work of more basic mathematics written soon after Li Zhi completed Ceyuan haijing, and was probably written to help students who could not understand Sea mirror of circle measurements. Yigu yanduan consists of three volumes
dedicated to solving geometrical problems on two tracks, through Tian yuan shu
and geometry.
China
Chinese civilization may refer to:* China for more general discussion of the country.* Chinese culture* Greater China, the transnational community of ethnic Chinese.* History of China* Sinosphere, the area historically affected by Chinese culture...
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
and scholar, who published and improved the tian yuan shu
Tian yuan shu
' is a Chinese system of algebra for polynomial equations created in the 13th century. It is first known from the writing of Li Zhi , though it was created earlier....
method for solving polynomial equations of one variable.
Name
He was born Li Zhi, but later changed his name to Li Ye to avoid confusionNaming taboo
Naming taboo is a cultural taboo against speaking or writing the given names of exalted persons in China and neighboring nations in the ancient Chinese cultural sphere.-Kinds of naming taboo:...
with the third T'ang emperor
Emperor Gaozong of Tang
Emperor Gaozong of Tang , personal name Li Zhi , was the third emperor of the Tang Dynasty in China, ruling from 649 to 683...
who was also named Li Zhi, removing one stroke from his original name to change the character. His name is also sometimes written as Li Chih or Li Yeh. The literary name was (Renqing, Jen-ch’ing) and appellation
Art-name
An art-name is a pseudonym, or penname, used by an East Asian artist, which they sometimes change. The word and the idea to use a pseudonym originated from China, then became popular in other East Asian countries ....
was (Jingzhai, Ching-chai).
Life
Li Zhi was born in Ta-hsing (now Beijing). His father was a secretary to an officer in the Jurchen army. Li passed the civil service examinationImperial examination
The Imperial examination was an examination system in Imperial China designed to select the best administrative officials for the state's bureaucracy. This system had a huge influence on both society and culture in Imperial China and was directly responsible for the creation of a class of...
in 1230 at the age of 38, and was administrative prefect of Jun prefecture in Henan
Henan
Henan , is a province of the People's Republic of China, located in the central part of the country. Its one-character abbreviation is "豫" , named after Yuzhou , a Han Dynasty state that included parts of Henan...
province until the Mongol invasion in 1233. He then lived in poverty in the mountainous Shanxi province. In 1248 he finished his most known work Ceyuan haijing . Li then returned to Hebei
Hebei
' is a province of the People's Republic of China in the North China region. Its one-character abbreviation is "" , named after Ji Province, a Han Dynasty province that included what is now southern Hebei...
.
In 1257 Kublai Khan
Kublai Khan
Kublai Khan , born Kublai and also known by the temple name Shizu , was the fifth Great Khan of the Mongol Empire from 1260 to 1294 and the founder of the Yuan Dynasty in China...
, grandson of Genghis Khan
Genghis Khan
Genghis Khan , born Temujin and occasionally known by his temple name Taizu , was the founder and Great Khan of the Mongol Empire, which became the largest contiguous empire in history after his death....
, sent for Li to get advice on government and science. In 1259 Li completed Yigu yanduan , also an important mathematics text. After becoming Khan
Khan (title)
Khan is an originally Altaic and subsequently Central Asian title for a sovereign or military ruler, widely used by medieval nomadic Turko-Mongol tribes living to the north of China. 'Khan' is also seen as a title in the Xianbei confederation for their chief between 283 and 289...
, Kublai twice offered Li government positions. He politely declined, referring to his age and ill health. In 1264 Li finally accepted a position at the Hanlin Academy
Hanlin Academy
The Hanlin Academy was an academic and administrative institution founded in the eighth century Tang dynasty China by Emperor Xuanzong.Membership in the academy was confined to an elite group of scholars, who performed secretarial and literary tasks for the court. One of its main duties was to...
, writing official histories. He was critical of the politics and resigned after a few months, again citing ill health. He spent his final years teaching at his home near Feng Lung mountain in Yuan, Hebei
Hebei
' is a province of the People's Republic of China in the North China region. Its one-character abbreviation is "" , named after Ji Province, a Han Dynasty province that included what is now southern Hebei...
. Li told his son to burn all of his books except for Sea mirror of circle measurements. Fortunately other texts, both mathematical and literary, still survived.
Ceyuan haijing
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
book, the purpose of the geometrical problems are to create algebraic problems.
Many of the problems are solved by polynomial equations, which are represented using a method called tian yuan shu
Tian yuan shu
' is a Chinese system of algebra for polynomial equations created in the 13th century. It is first known from the writing of Li Zhi , though it was created earlier....
, "coefficient array method" or literally "method of the celestial unknown". Li Zhi is the earliest known source of this method, though it was known before him in some form. It is a positional system of rod numerals to represent polynomial equations.
For example 2x2 + 18x − 316 = 0 is represented as
which in Arabic numerals is
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. Later, the line order was reversed so that the first line is the lowest exponent.
Li does not explain how to solve equations in general, but shows it with the example problems. Most of the equations can be reduced to the second or sometimes third order. It is often assumed that he used methods similar to Ruffini's rule
Ruffini's rule
In mathematics, Ruffini's rule allows the rapid division of any polynomial by a binomial of the form x − r. It was described by Paolo Ruffini in 1809. Ruffini's rule is a special case of synthetic division when the divisor is a linear factor. The Horner scheme is a fast algorithm for dividing...
and Horner scheme
Horner scheme
In numerical analysis, the Horner scheme , named after William George Horner, is an algorithm for the efficient evaluation of polynomials in monomial form. Horner's method describes a manual process by which one may approximate the roots of a polynomial equation...
.
Yigu yanduan
Yigu yanduan
Yigu yanduan is a 13th-century mathematical work by Yuan dynasty mathematician Li Zhi. Yigu yanduan was based on North Song mathematician Jiang Zhou Yigu Ji which was extinct...
(New steps in computation) is a work of more basic mathematics written soon after Li Zhi completed Ceyuan haijing, and was probably written to help students who could not understand Sea mirror of circle measurements. Yigu yanduan consists of three volumes
dedicated to solving geometrical problems on two tracks, through Tian yuan shu
Tian yuan shu
' is a Chinese system of algebra for polynomial equations created in the 13th century. It is first known from the writing of Li Zhi , though it was created earlier....
and geometry.