''Tian yuan shu'' () is a Chinese system of
algebra
Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
for
polynomial
In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...
equations. Some of the earliest existing writings were created in the 13th century during the
Yuan dynasty
The Yuan dynasty ( ; zh, c=元朝, p=Yuáncháo), officially the Great Yuan (; Mongolian language, Mongolian: , , literally 'Great Yuan State'), was a Mongol-led imperial dynasty of China and a successor state to the Mongol Empire after Div ...
. However, the tianyuanshu method was known much earlier, in the Song dynasty and possibly before.
History
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
The Yuan dynasty ( ; zh, c=元朝, p=Yuáncháo), officially the Great Yuan (; Mongolian language, Mongolian: , , literally 'Great Yuan State'), was a Mongol-led imperial dynasty of China and a successor state to the Mongol Empire after Div ...
.
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 ''
Suanxue qimeng'' was deciphered and was important in the development of
Japanese mathematics (''wasan'') in the 17th and 18th centuries.
Description
''Tian yuan shu'' means "method of the heavenly element" or "technique of the celestial unknown". The "heavenly element" is the unknown
variable, usually written in modern notation.
It is a positional system of
rod numerals to represent
polynomial equation
In mathematics, an algebraic equation or polynomial equation is an equation of the form P = 0, where ''P'' is a polynomial with coefficients in some field (mathematics), field, often the field of the rational numbers.
For example, x^5-3x+1=0 is a ...
s. For example, is represented as
, which in Arabic numerals is
The (''yuan'') denotes the unknown , so the numerals on that line mean . The line below is the constant term () and the line above is the
coefficient
In mathematics, a coefficient is a Factor (arithmetic), multiplicative factor involved in some Summand, term of a polynomial, a series (mathematics), series, or any other type of expression (mathematics), expression. It may be a Dimensionless qu ...
of the
quadratic () 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.
See also
*''
Yigu yanduan''
*''
Ceyuan haijing''
References
Bibliography
*
*
Chinese mathematics
Japanese mathematics
Polynomials
13th-century Chinese books
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