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How did Liu Hui prove mathematical algorithms in his commentary on The Nine Chapters?
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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.
In his 3rd-century commentary on The Nine Chapters on the Mathematical Procedures, Liu Hui established the correctness of mathematical algorithms through rigorous logical demonstration, marking some of the earliest proofs in the modern sense.12
Rather than focusing on abstract theorems, Liu sought to unify mathematical knowledge by showing that diverse procedures relied on a small set of fundamental operations. His primary techniques included:12
- Geometric Dissection: He proved volume and area formulas by decomposing shapes into smaller, known pieces and recomposing them, sometimes involving an infinite number of pieces.12
- Cavalieriās Principle: He utilized a simplified version of this principle, comparing the cross-sectional areas of solids to determine their volumes.13
- Algebraic Analogy: In arithmetic and algebra, he demonstrated that different algorithmsāsuch as adding fractions and solving simultaneous linear equationsāshared identical "key steps." He showed that by multiplying sets of numbers by appropriate factors, one could manipulate them to perform operations like elimination or equalization without altering the underlying mathematical meaning.12
By organizing these proofs into a cohesive system, Liu Hui successfully reduced the complexity of ancient mathematical practice into a structured, logical framework.12