At the same time, in order to solve the problem of pi, Liu Hui's preliminary concept of limit and the idea of straight curve transformation were also very valuable in ancient times.
After Liu Hui, Zu Chongzhi, an outstanding mathematician in the Southern and Northern Dynasties in China, calculated pi to a more accurate degree, more than 800 years earlier than Europeans, and made extremely brilliant achievements.
Liu Hui was a great mathematician in Wei and Jin Dynasties and one of the founders of China's classical mathematical theory. He has made many achievements in mathematics, among which his contribution to pi also stems from his painstaking research.
Once, Liu Hui saw a stonemason working on a stone, which made her feel very interesting, so she observed it carefully. Masons cut down one by one, and a square stone was processed into a smooth cylinder.
Who would have thought that the original square stone was turned into an octagonal stone after four corners were chiseled off by the masons? Eight more corners and it's a hexagon again. This is a very common thing in the eyes of ordinary people, but it triggered a spark of Liu Hui's wisdom.
He thought, "Why can't the stonemason's method of processing stones be used in the study of pi?"
So, Liu Hui used this method to divide circles gradually and gave it a try, and the effect was good. Liu Hui had a unique eye and finally invented the secant, which improved the calculation accuracy of pi to a new level in the world.
Modern mathematical research has proved that pi is a concept of "transcendental number", and it is data that cannot be generated by a limited algebraic operation of addition, subtraction, multiplication and division. Before the Han Dynasty in China, the commonly used pi was "three circumference and one diameter". Obviously, this value is very rough, and using it to calculate will cause great errors.
With the development of production and science, the estimation of "three circumference diameter one" can no longer meet the requirements of accurate calculation, so people begin to explore more accurate pi.
Although the accuracy was improved later, it was mostly the result of experience and lacked a solid theoretical basis. Therefore, it is still very important to study the scientific method of calculating pi.
Liu Hui was an outstanding mathematician in Wei and Jin Dynasties, and made a very outstanding contribution to the calculation of pi.
When he annotated the ancient mathematical masterpiece Nine Chapters Arithmetic, he pointed out that "three circumference diameter one" is not the value of pi, but the ratio of the circumference to the diameter of a regular hexagon inscribed with a circle. The result of calculating the circular area by the ancient method is not the circular area, but the area inscribed by a regular dodecagon.
After in-depth study, Liu Hui found that when the number of sides inscribed with a regular polygon in a circle is infinitely increased, the perimeter of the polygon is infinitely close to the perimeter of the circle, thus establishing secant technology, and establishing quite strict theory and perfect algorithm for calculating pi and the area of the circle.
The basic idea of Liu Hui's cyclotomy is:
If you cut carefully, you will lose a little. You can't cut it again, and there's nothing to lose.
That is to say, the finer the subdivision, the smaller the error, and the infinite subdivision can gradually approach the actual value of pi. He knows very well that the more sides of a circle inscribed in a regular polygon, the more accurate the value of pi will be.
Liu Hui used secant circle to increase the number of sides by multiples of 12, 24, 48, 96, so as to calculate the side lengths of regular hexagon, regular dodecagon and regular quadrangle. One by one, the ratio of "circumferential diameter" gradually approaches pi.
When he carved a 96-sided circle, the pi obtained was 3. 14, which was much more accurate than the ancient pi. Liu Hui used "power" and "difference power" instead of the circumscribed approximation of the circle, which cleverly avoided the calculation of the circumscribed polygon and got twice the result with half the effort in the process of calculating the area of the circle. Liu Hui initiated the secant method, which can be said to be an outstanding representative of China's ancient extreme thoughts and occupies a very important position in the history of mathematics. The results he got were also very advanced in the world at that time.
Liu Hui lived in an era of warlord separatism, especially in Wei, Shu and Wu dynasties. At this time, great changes have taken place in China's society, politics and economy, especially in the ideological circle, where scholars are arguing endlessly.
So at that time, it became the wind of debate, and a group of scholars came together, just like our university debate, one for and one against, and put forward a proposition for everyone to debate with each other. In the process of debate, people should study and discuss the technology of debate and the law of thinking, so people's ideological liberation in this period should be said to have never happened since the Spring and Autumn Period and the Warring States Period. At this time, people's research on the law of thinking is particularly developed. Some people think that people's abstract thinking ability far exceeds that of the Spring and Autumn Period and the Warring States Period.
Liu Hui pointed out in the preface of Nine Chapters Arithmetic Notes that exploring the roots of mathematics is his highest task in mathematics research. The purpose of his annotation of Nine Chapters of Arithmetic is to "analyze words and use pictures to disintegrate". At that time, "anatomy" was synonymous with scholars' difficulties in arguing with each other. Through the analysis of mathematical theory, Liu Hui established the theoretical system of China's traditional mathematics.
After Liu Hui, the value of pi obtained by Zu Chongzhi can be said to be a leap in the calculation of pi. According to Sui Shu? Zu Chongzhi determined the approximate value of pi as 3. 14 15926 and the approximate value of pi as 3. 14 15927. The true value is between these two approximations, which is the most advanced achievement in the world at that time.