The Nine Chapters on the Mathematical Procedures

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Figure 9: The first example of a system of linear equations in the Nine Chapters. Chinese mathematics, coefficients, algorithms, calculations.

The Nine Chapters on the Mathematical Procedures

Chinese mathematics

Also known as: “Jiuzhang suanshu”, “Nine Chapters”, “The Nine Chapters on the Mathematical Art”(Show More)

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July 1, 2026

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The Nine Chapters on the Mathematical Art is a foundational Chinese mathematical text, likely completed between the 1st century BCE and 1st century CE. It presents algorithms for solving problems related to civil administration, such as surveying, taxation, and resource management.1

Key mathematical achievements within The Nine Chapters include:
* Arithmetic and Algebra: It presupposes knowledge of basic arithmetic operations and utilizes a decimal place-value system with counting rods for computations.234
* Geometry: Formulas are provided for calculating areas of plane figures and volumes of solid figures. It also includes the Pythagorean theorem and methods for solving problems involving right triangles.32
* Systems of Equations: The text features a chapter dedicated to solving simultaneous linear equations, employing a method similar to modern Gaussian elimination. This section also introduces the concept and computation with negative numbers.4

The Nine Chapters served as a crucial reference for mathematicians in East Asia for centuries, influencing subsequent mathematical developments.1

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East Asian mathematics
The textual sources
…Chinese is Jiuzhang suanshu (The Nine Chapters on the Mathematical Art), which contains arithmetic, algebraic, and geometric algorithms, presented in relation to problems, some of which evoke the duties of the civil administration: surveying fields (areas), levying taxes according to various types of grains (ratios), determining wages for civil…
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…Chinese is Jiuzhang suanshu (The Nine Chapters on the Mathematical Art), which contains arithmetic, algebraic, and geometric algorithms, presented in relation to problems, some of which evoke the duties of the civil administration: surveying fields (areas), levying taxes according to various types of grains (ratios), determining wages for civil…
Read More

Liu Hui
…263 on Jiuzhang suanshu (The Nine Chapters on the Mathematical Art)—a mathematical canon of the 1st century bce or ce that played a similar role in the East to Euclid’s Elements in the West. Liu’s commentary on The Nine Chapters proved the correctness of its algorithms. These proofs are…
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…263 on Jiuzhang suanshu (The Nine Chapters on the Mathematical Art)—a mathematical canon of the 1st century bce or ce that played a similar role in the East to Euclid’s Elements in the West. Liu’s commentary on The Nine Chapters proved the correctness of its algorithms. These proofs are…
Read More

Pythagorean theorem
In the Nine Chapters on the Mathematical Procedures (or Nine Chapters), compiled in the 1st century ce in China, several problems are given, along with their solutions, that involve finding the length of one of the sides of a right triangle when given the other two sides.…
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In the Nine Chapters on the Mathematical Procedures (or Nine Chapters), compiled in the 1st century ce in China, several problems are given, along with their solutions, that involve finding the length of one of the sides of a right triangle when given the other two sides.…
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Zu Geng
263) on Jiuzhang suanshu (Nine Chapters on the Mathematical Procedures), a Chinese classic probably compiled in the 1st century ce (see mathematics, East Asian: Mathematics in China). Fragments of Zu Geng’s writings are also found in the astronomical chapter of Suishu (“History of the Sui Dynasty”).
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263) on Jiuzhang suanshu (Nine Chapters on the Mathematical Procedures), a Chinese classic probably compiled in the 1st century ce (see mathematics, East Asian: Mathematics in China). Fragments of Zu Geng’s writings are also found in the astronomical chapter of Suishu (“History of the Sui Dynasty”).
Read More