''Tian yuan shu'' () is a Chinese system of

algebra Algebra () is one of the areas of mathematics, broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathem ...

for

polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An ex ...

equations. Some of the earliest existing writings were created in the 13th century during the

Yuan dynasty The Yuan dynasty (), officially the Great Yuan (; xng, , , literally "Great Yuan State"), was a Mongols, Mongol-led Dynasties in Chinese history, imperial dynasty of China and a successor state to the Mongol Empire after Division of the M ...

. 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 ''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 fro ...

''), two Chinese mathematicians during the Mongol

. 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, often the field of the rational numbers. For many authors, the term ''algebraic equati ...

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 multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves ...

of the

quadratic In mathematics, the term quadratic describes something that pertains to squares, to the operation of squaring, to terms of the second degree, or equations or formulas that involve such terms. ''Quadratus'' is Latin for ''square''. Mathematics ...

() term. The system accommodates arbitrarily high

exponent Exponentiation is a mathematical operation, written as , involving two numbers, the '' base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". When is a positive integer, exponentiation corresponds to re ...

s 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.

References

* * {{cite book, last=Murata, first=Tamotsu, title=Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences, editor=Ivor Grattan-Guinness, publisher=JHU Press, year=2003, volume=1, pages=105–106, chapter=Indigenous Japanese mathematics, Wasan, isbn=0-8018-7396-7, chapter-url=https://books.google.com/books?id=2hDvzITtfdAC&pg=PA105, accessdate=2009-12-28 Chinese mathematics Japanese mathematics Polynomials 13th-century Chinese books

History

Description

See also

References