Zu Chongzhi and Pi
Ancient China was a land of mathematics, and the first to work out the decimal system and the binary system through the yin and yang and Eight Trigrams for divination.
The two inventions have had a great impact on contemporary life as well as the progress of history. The comments by Zhao Shuang on Zhou Bi Suan Jing (The Arithmetical Classic of the Zhou Gnomon) in the Western Han Dynasty or earlier elaborated the theorem that the sum of the squares of the lengths of the sides of a right triangle is equal to the square of the length of the hypotenuse, or the Pythagorean theorem by Greece's Pythagoras, the Chinese discovery was made the same time as the Greek one, but they were independent and not related. Liu Hui, a great Chinese mathematician of the Three Kingdoms Period, put forward his own principle on the calculation of the volume of polyhedrons in his Comments on The Nine Chapters on the mathematical Art. Liu also developed a formula for calculating the area of a circle through infinite segments and the idea of the limit, and a scientific procedure for computing approximate values of the ratio of the circumference of a circle to the diameter. He worked out the area of a regular polygon of 129 sides, and an approximate value of pi, at 3.14, or 157/50 and 3927/1250 in terms of fraction.
Based on the achievements of Liu Hui and other mathematicians, another great Chinese mathematician made an important breakthrough in the calculation of the ratio of the circumference of a circle to its diameter.
Zu Chongzhi (429-500) was born in the family of a scholar- bureaucrat of the Southern Dynasties Period. Since childhood he was diligent, eager to learn, and dare to explore. At the age of 25 he entered the Hualin Academy to continue his academic career. Later he worked under a high-ranking official as a staff member, which gave him enough time for scientific research. Zu scored a great deal in the studies of astronomy, calendrical science, mathematics and mechanics. The achievements of Zu Chongzhi and his son Zu Geng were recorded in their work Zhui Shu (Method of Interpolation) and listed among the Ten Mathematical Classics to be used by Chinese students of tile Tang Dynasty as well as Korean and Japanese students. It is a great pity that this book has been lost, and Zu's achievements can now be found only in fragments kept in other mathematical works, which mainly fall into three areas: computing the ratio of the circumference of a circle to its diameter, calculating the volume of globes, and solving higher degree equations.
In calculating approximate values of the ratio of the circumference of a circle to the diameter, ancient Greeks outdid ancient Chinese for quite a long time. In the 5th century BC, when Greek mathematicians put the value of the ratio at 3.1216, the Chinese still believed the ratio of the circumference to the diameter was three to one, and used this value till the Han Dynasty. In the Western Han Dynasty, mathematician Liu Xin got two approximate values of the ratio - 3.141547 or 3.14166, with the significant figure being 3.1. In the Eastern Han Dynasty, scientist Zhang Heng worked out two expressions for approximate values of the ratio: 92/29 and the square root of 10.Liu Hui worked out the value of 3.14, but Zu Chongzhi was not
satisfied with this result. Using Liu Hui's method of inscribing the circle, Zu calculated the areas of hexagon to 6X212-gon, and further worked out the ratio to be between 3.1215926 and 3.1415927.
To obtain such a result, Zu had undertaken extremely lengthy calculations, involving hundreds of square roots, all to 9 decimal place accuracy. This was a work that took great perseverance, resolve and energy. It was a great achievement of the time, which was more than 100 times more accurate than Liu Hui's. Zu devised a precise method to provide a range for the changes of the ratio, which is a basic method for the expression of an irrational number, and Zu was only next to the great Greek mathematician Archimedes in using this method. Zu got two values for the ratio: the "approximate value" of 22/7 and the "precise value" of 355/113. The precise value is so approximate that, some mathematicians have proved, if this value is used to calculate the area of a circle with a radius of 10 km, the error will be within millimeters. In Europe this "precise value" was recalculated by the German scholar Valentin Otto for the first time in 1573, more than 1,000 years later after Zu. |
