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

algebra Algebra () is one of the 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 mathematics. Elementary a ...

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 exa ...

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 Mongol-led imperial dynasty of China and a successor state to the Mongol Empire after its division. It was established by Kublai, the fifth ...

. 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 Zhu Shijie (, 1249–1314), courtesy name Hanqing (), pseudonym Songting (), 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 ha ...

(''

Jade Mirror of the Four Unknowns ''Jade Mirror of the Four Unknowns'', ''Siyuan yujian'' (), also referred to as ''Jade Mirror of the Four Origins'', is a 1303 mathematical monograph by Yuan dynasty mathematician Zhu Shijie. Zhu advanced Chinese algebra with this Magnum opus. ...

'') and

Li Zhi Li Zhi may refer to: *Emperor Gaozong of Tang (628–683), named Li Zhi, Emperor of China *Li Ye (mathematician) (1192–1279), Chinese mathematician and scholar, birth name Li Zhi *Li Zhi (philosopher) (1527–1602), Chinese philosopher from the M ...

(''

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 from ...

''), 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 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 ...

(''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 Variable may refer to: * Variable (computer science), a symbolic name associated with a value and whose associated value may be changed * Variable (mathematics), a symbol that represents a quantity in a mathematical expression, as used in many ...

, usually written in modern notation. It is a positional system of

rod numeral Counting rods () are small bars, typically 3–14 cm long, that were used by mathematicians for calculation in ancient East Asia. They are placed either horizontally or vertically to represent any integer or rational number. The written fo ...

s 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 equation' ...

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 var ...

of the quadratic () 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