祖冲之是如何得到圆周率π=355/113的?

How Did Zu Chongzhi Find His Value π=355/113 ?

中国古代历法家在公元 5世纪初发明了一种推求新的闰周的简单算法。通过构造的一个定理对这个算法进行的分析讨论证实 ,闰周算法是一种很好的实数有理逼近算法 ,事实上 ,从某种意义上讲 ,它与连分数展开算法是等价的。通过反复地使用这个算法 ,可以非常容易地求得被逼近实数的一系列渐近分数。这个结果导致的结论是 :在中国古代数学与历法史上出现大量的渐近分数并非偶然事件。闰周算法的产生时期 ,正好是祖冲之生活的年代 ,通过具体的验算 ,推断祖冲之著名的圆周率π =35 5 /

In early 5th century AD, for deriving the cycle of the intercalary month in a calendar-making system, a numerical method was invented by Chinese Mathematician. With a constructed theorem, the author comes to a conclusion that this method, in a sense, equals to the algorithm of the continued fraction. A series convergent to any given number could be derived with this method. The result explains a fact, which has been verified being served as convergent to some constants, appeared in the texts of mathematics or astronomy in ancient China, but on evidence to show there was an algorithm of continued fraction in ancient China. By the use of this method, a possible way for deriving Zu Chongzhi's value π=355/113 is demonstrated.