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How did Seki Takakazu solve equations with more than one unknown?
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East Asian mathematics
Counting boards and markers, or counting rods, were used in China to solve systems of linear equations. This is an example from the 1st century ce.
Seki Takakazu, a foundational figure in the Japanese wasan tradition, revolutionized the handling of equations with multiple unknowns by moving away from the cumbersome Chinese method of counting rods.12
To solve these problems, Seki implemented the following innovations:
- Tabular Notation: He replaced counting rods with a tabular notational system, which simplified the manipulation of complex equations.1
- Literal Algebra: He developed a "literal" written algebra using characters to represent coefficients as numbers, parameters, or other unknowns.2
- Elimination Procedures: When establishing equations involving several unknowns, Seki introduced procedures equivalent to the computation of determinants. These methods allowed him to systematically eliminate unknowns between simultaneous equations, a technique for which he is historically recognized as a pioneer.23
These advancements were detailed in his works, such as Kaifukudai no hō (1683; "Method for Solving Concealed Problems"), and were further refined by his disciples, including Takebe Katahiro.12