tiān yuán shù
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''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 ...
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 exampl ...
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 fif ...
. 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'') and Li Zhi ('' Ceyuan haijing''), two Chinese mathematicians during the Mongol
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 fif ...
. 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 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 (mathematics), field, often the field of the rational numbers. For many authors, the term '' ...
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 () 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 r ...
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.


See also

*'' Yigu yanduan'' *'' Ceyuan haijing''


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