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How is Jia Xian’s triangle related to Pascal’s triangle?

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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.

Jia Xian’s triangle and Pascal’s triangle are essentially the same mathematical construct, representing a triangular arrangement of binomial coefficients used to expand binomial expressions like $(x + y)^{n}$ .12

Jia Xian, a Chinese mathematician active around 1050, devised this triangular representation in the 11th century to assist in extracting roots of polynomials. His work predates the 17th-century French mathematician Blaise Pascal, who is the namesake for the triangle in the West. Because the triangle was independently discovered in China centuries earlier, it is often referred to in China as the Yang Hui triangle, named after the 13th-century mathematician Yang Hui who further popularized and studied it.12

Both versions of the triangle are constructed using the same method: placing 1s along the outer edges and filling the interior by adding the two numbers directly above each position. This shared structure demonstrates that the pattern of coefficients was recognized and utilized by mathematicians across different cultures and eras, including Omar Khayyam in the 11th century.2