Liu Hui

Hui's Life
There is not much information recorded about Liu Hui’s life. We know he lived in the kingdom of Wei which is now the Shansi province in north-central China. The kingdom came about after the Han Empire collapsed around 220 AD and lasted about 280 years. Liu Hui lived and did his work sometime within the 280 years the kingdom of Wei existed. He was one of the two great mathematicians of the ancient world.


His Work
Liu Hui was the first mathematician know to leave roots unevaluated, giving more accurate results rater than approximations. He expressed all his mathematical results as decimal fractions which are also known as metrological units. He provided a commentary to the mathematical proof that is identical to the Pythagorean theorem discovered more than 700 years before his life time. The figure drawn diagram was explained by Hui as “a diagram giving the relation between the hypotenuse and the sum and difference of the other two sides where one can find the unknown from the known.” Also, Hui remains one of the greatest contributors to empirical solid geometry nearly 2000 years after his death. For example, he found that a wedge with a rectangular base and both sides sloping could be broken down into a pyramid and a tetrahedral wedge along with thousands of other geometric discoveries.
Liu Hui studied much work during his lifetime. One of the most significant was the 9 chapters of the Jiuzhang suanshu. The oldest surviving and most important in the 10 ancient Chinese mathematical books. These books have become the basis for all modern mathematics. Hui revised, edited and added to these chapters which have been studied by many famous later mathematicians.

The most significant work of Liu Hui was his work with pi in chapter 1. Many mathematicians before him tried to find the exact value of pi but none were as accurate as Hui. Guesses before him were numbers such as the square root of 10 and simply the number 3. Hui came up with many formulas and used them on multiple sided polygons. The successful formula was much to long and complicated to show but there are steps he took. He found recurrence relation to express the length of the side of a regular polygon with $3(2^n)$ sides in terms of the length of the side of a regular polygon with $3(2^n^-^1)$ sides. This was achieved with the Pythagorean theorem. He came up with n = x, where x(element of the reals), $N=3(2^n)$, and a formula for p(n) which is much to complicated to explain (look at if you are interested). When numbers for x were used Hui could determine the value or pi for a polygon with the $N=3(2^n^-^1)$ number of sides. Here are a few examples:
n = 1, N = 6, pn=1, N pn/2 =3
n = 5, N = 96, pn=0.06543816562, N pn/2 =3.141031950
n = 10, N = 3072, pn=0.002045307359, N pn/2 =3.141592104
When he used n=3072 he had an output of 3.14159. This result is 1 number of the computer calculation.

Then What?
Much work was done with Liu Hui’s results after his death. His answers that we left in decimal fractions were later accurately calculated into full decimal expressions by Yang Hui more than 1000 years later. His other mathematical works included several problems related to surveying which are found in a separate appendix called Haidao suanjing. The book contained many practical problems related to geometry including the measurement of the Chinese pagoda towers. Liu Hui’s work came centuries before his time and he is still commonly referred to as one of the greatest mathematicians of all time.

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1) (2003, 12). Liu Hui. Retrieved March 15, 2008, from University of Saint Andrews, Scotland Web site: 2) Wagner, D, B, (1985). Historia math: A proof of the pythagorean theorem by liu hui. Fort Smith, Arkansas: 3)
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