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By: Tao Steven Zheng (郑涛)

Born: c. 225 AD in Shandong province

Liu Hui was a Chinese mathematician who lived during the Three Kingdoms period (220 - 280 AD) and Wei-Jin dynasty (265 - 420 AD). Little is known about Liu Hui's life aside that he was a descendant of the Marquis of Zixiang, and that he published his commentary on the Jiuzhang Suanshu (九章算術) in the year 263 AD. Despite his relative obscurity, Liu Hui showed prowess in his mathematical inventiveness. Liu Hui's contribution to Chinese mathematics was immense because he rigorously explained why many of the computational methods of the Han and Pre-Han period worked, and developed practical surveying techniques that are theoretically sound.

## Discoveries

Approximations of pi

Liu Hui developed a geometric algorithm for computing pi called the ge yuan shu (割圆术 circle cutting method). Using a 96-sided polygon, Liu obtained the bound . He later devised a quick method that involved geometric series to arrive at the approximation of . Liu Hui's quick method achieved the accuracy of a 3072-sided polygon from his original algorithm with a 96-sided polygon.

Arithmetic with Negative Numbers

Liu Hui established the rules of arithmetic with negative numbers in his commentary of the Jiuzhang Suanshu (九章算術). However, it is likely that negative numbers were known in China during the Han dynasty (206 BC - 220 AD). Liu Hui also used negative numbers to solve fangcheng problems (方程). These problems are systems of simultaneous linear equations that are solved using the elimination method.

The Cavalieri Principle

Liu Hui discovered the Cavalieri Principle (the method of indivisibles) 1400 years before Bonaventura Cavalieri. He applied this principle to derive precise formulas of cylinders, wedges, and frustums. Liu Hui also applied the Cavalieri Principle to determine the volumetric ratio of the sphere to the mou he fang gai (牟合方盖), a solid formed by the intersection of two perpendicular cylinders.

Surveying Techniques

In his book Haidao Suanjing (海島算經), Liu Hui created and solved several surveying problems by exploiting the chong cha method (重差術), a nascent form of trigonometry.