Jia Xian

Jia Xian was an 11th-century Chinese mathematician known for methods of extracting square and cube roots and a triangular arrangement of binomial coefficients.

Chinese remainder theorem: A theorem giving conditions for solving several congruences with different moduli. It represents the broader tradition of sophisticated algorithmic arithmetic in which Jia Xian worked.
Horner's method: An algorithm for evaluating polynomials or approximating their roots through successive arithmetic steps. Jia Xian's root-extraction procedure is an early method of this general algorithmic kind.
Yang Hui: A 13th-century Chinese mathematician who wrote influential works on arithmetic and mathematical methods. Yang Hui preserved and discussed methods attributed to Jia Xian.
Binomial theorem: A formula expanding an integer power of a binomial using binomial coefficients. The triangular coefficients associated with Jia Xian encode these expansions.
Binomial coefficient: A number counting selections of a fixed size from a larger set, appearing as a coefficient in a binomial expansion. These numbers form the triangular arrangement associated with Jia Xian.
Synthetic division: A compact procedure for dividing a polynomial by a linear factor using its coefficients. Its coefficient-by-coefficient arithmetic helps explain the structure of Jia Xian's extraction method.
Yang Hui suanfa: A 13th-century Chinese mathematical collection by Yang Hui, containing earlier methods and mathematical diagrams. Its surviving account is a key source for Jia Xian's triangular arrangement and root methods.
Chinese mathematics: The mathematical traditions developed in China across arithmetic, algebra, geometry, and computation. Jia Xian's algorithms belong to a long and distinctive history of Chinese mathematics.
Pascal's triangle: A triangular array whose entries are binomial coefficients and whose rows follow an addition rule. Jia Xian's arrangement gave an early Chinese presentation of this pattern.
Positional notation: A numeral system in which a digit's value depends on its position. Place-value notation makes the staged calculations in his algorithms practical.

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