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c. 200 BCE (later commentary 263 CE)Reputable source · 2 sourcesWell documented

China's Nine Chapters Solves Linear Systems and Names Negative Numbers

A problem-based textbook reaches Gaussian elimination and negative-number arithmetic centuries before either has a name in the West

On the timeline · around c. 200 BCE (later commentary 263 CE) · Greek and Hellenistic MathematicsGreek and Hellenistic MathematicsChina's Nine Chapters Solves Linear Systems and Names Negative Numbers200 BCE100 BCE1 CE100 CE

Quick facts

Estimated original composition
c. 200 BCE
Key commentary
Liu Hui, 263 CE
Problems included
246, covering engineering, trade, taxation
Notable methods
Matrix elimination (chapter 8); negative number rules

What happened

The Nine Chapters on the Mathematical Art, the most famous Chinese mathematics book of all time, most historians believe originated around 200 BCE, after the burning of books ordered by Shih Huang Ti destroyed much earlier material, though the mathematician Liu Hui, who wrote an influential commentary on it in 263 CE, believed the core text dated back a thousand years earlier and had simply absorbed material from later eras. The book poses 246 practical problems covering engineering, surveying, trade, and taxation, unlike Greek mathematics, it develops no axiomatic system of proof, working instead from problems toward methods. Its eighth chapter solves systems of simultaneous linear equations by arranging coefficients in an augmented matrix and reducing it to triangular form using column operations, the same procedure now called Gaussian elimination, seventeen centuries before Gauss. Solving these systems required working with negative quantities partway through the calculation, and the chapter includes explicit rules for computing with negative numbers, one of the earliest formal treatments of negative numbers anywhere in the historical record.

Why it matters

The Nine Chapters dominated Chinese mathematical education and style for roughly 1,500 years and shows a mathematical tradition built around solving concrete problems rather than proving general theorems from axioms, a genuinely different approach from the Greek model that nonetheless reached some of the same destinations, including matrix elimination and negative-number arithmetic, well before either appeared in the West.

How we know

The Nine Chapters survives through Liu Hui's 263 CE commentary and later Chinese manuscript and print editions, and the matrix-based elimination method described in chapter 8 has been directly compared by modern historians of mathematics to Gaussian elimination, confirming the technique's substance independent of when it was first named.

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Part of a timelineHistory of Mathematics26 events · A number system built for taxes, a theorem older than the man it's named for, a proof too long for a margin, and an infinity too big to countView all →
China's Nine Chapters Solves Linear Systems and Names Negative Numbers · History of Mathematics · SourcedStory