How did Seki Takakazu adapt Chinese algebraic methods?

Seki Takakazu, the founder of the Japanese mathematical tradition known as wasan, significantly adapted and advanced Chinese algebraic methods by shifting away from their traditional, cumbersome reliance on counting rods.12

His key adaptations included:

• Literal Algebra: Seki developed a "literal" written algebra using characters to represent coefficients, parameters, and unknowns. This innovation liberated mathematicians from the physical constraints of counting rods and allowed for a more abstract, positional notation.12
• Generalization of Equations: While traditional Chinese methods often viewed equations as having only one root, Seki devised a classification system for equations that accounted for both negative and multiple roots, building upon observations first noted by his predecessor Sawaguchi Kazuyuki.12
• Advanced Procedures: To solve systems of equations with multiple unknowns, Seki introduced procedures equivalent to the computation of determinants, enabling the elimination of unknowns.1
• Root Extraction: He took ancient Chinese algorithms for extracting roots of higher-degree polynomial equations and extended them to identify all real roots of an equation.2

By reformulating these classical techniques, Seki transformed them into powerful tools that opened new directions for Japanese mathematics, including early work on infinite series.1