When did Zu Chongzhi put forward pi?

attention! ! ! Zu Chongzhi's specific method of seeking pi is still being debated in academic circles, and the cyclotomy is Liu Hui's. As for whether Zu Chongzhi uses cyclotomy, it is still inconclusive! Everyone just guessed that he might have used cyclotomy. "Zu Chongzhi's research work on pi and other significant contributions are recorded in the book Composing, but this rich mathematical monograph was later lost. Therefore, Zu Chongzhi's method of calculating pi is no longer available. " Related information: Liu Hui's cyclotomy often uses pi when solving problems such as circumference, area and sphere volume. Pi л can be expressed as an infinite acyclic decimal 3.1456535. Modern mathematics has proved that pi л is a number that cannot be worked out by algebraic operations such as finite addition, subtraction, multiplication, division and exponentiation, which is the so-called "transcendental number". Before the Han Dynasty in China, the commonly used pi was "diameter one on Wednesday", that is, л=3. Obviously, this value is very rough, and using it to calculate will cause great errors. With the development of production and science, the "three-week diameter one" can not meet the requirements of accurate calculation more and more. Therefore, people began to explore more accurate pi. For example, according to the Lujia Gauge (a cylindrical standard gauge) made at the beginning of the first century A.D., the pi it takes is 3.1547. At the beginning of the second century, Zhang Heng, an astronomer in the Eastern Han Dynasty, used ≈3.1466 in Lingxian and ≈3.1622 in the formula of sphere volume. Wang Fan (228-266), a native of Wu in the Three Kingdoms period, took ≈3.1556 in the theory of armillary sphere. Compared with the ancient rate of "three-week radius one", the accuracy of these approximate values of pi is improved, and the value of pi is the earliest record in the world. However, most of these values are empirical results and lack a solid theoretical basis. Therefore, it is still very important to study the scientific method of calculating pi. Liu Hui, an outstanding mathematician in Wei and Jin Dynasties, made a very outstanding contribution in calculating the square quotient of pi. When he made a note for the ancient mathematical masterpiece Nine Chapters Arithmetic, he correctly pointed out that "Three Weeks Diameter One" is not the value of pi, but actually the ratio of the circumference and diameter of a regular hexagon inscribed in a circle. The result of calculating the area of a circle by the ancient method is not the area of a circle, but the area of a circle inscribed with a regular dodecagon. After in-depth study, Liu Hui found that when the number of sides inscribed with a regular polygon in a circle increases infinitely, the circumference of the polygon approaches the circumference of the circle infinitely, thus establishing the secant technique, and establishing a fairly strict theory and perfect algorithm for calculating the pi and the area of the circle. The main contents and basis of Liu Hui's secant technique are as follows: First, the length of each side of a regular hexagon inscribed in a circle is equal to the radius. Secondly, according to Pythagorean theorem, we can find the length of each side of a regular л polygon inscribed in a circle. Thirdly, from the length of each side of a regular л polygon inscribed in a circle, the area of a regular 2л polygon inscribed in a circle can be directly calculated. As shown on the right, the area of the quadrilateral OADB is equal to half the prODuct of the radius od and the side length AB of the regular л polygon. Fourthly, the circular area s satisfies the inequality S2n <; S< S2n+(S2n-Sn)。 As shown on the right, the area of quadrilateral OADB and the schematic diagram of Liu Hui's cyclotomy. Delta △OAB area difference is equal to the area of two right triangles with AD and DB as chords. And the area of OADB plus the area of these two right-angled triangles, some of them are beyond the circumference. Fifth, Liu Hui pointed out: "If you cut it carefully, you will lose little. If you cut it and cut it, so that it can't be cut, it will be combined with the circumference and nothing will be lost. " ("Nine Chapters of Arithmetic" Fang Tian Zhang Yuan Tian Shu Liu Hui Note) That is to say, when the number of sides inscribed with regular polygons in a circle increases indefinitely, the limit of its perimeter is the circumference of a circle, and the limit of its area is the area of a circle. Liu Hui calculated the length of each side of a regular dodecagon, a regular quadrangle, ... and even a regular 96-sided polygon according to the cyclotomic technique, and calculated the area of a regular 192-sided polygon. S192=。 This is equivalent to finding л=3.14124. He used л=3.14= in his actual calculation. Not only that, Liu Hui continued to find the area of the regular polygon in the circle, verified the previous results, and got a more accurate pi value л==3.1416. Liu Hui's secant technique has laid a solid and reliable theoretical foundation for the study of pi, which 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's calculation method only uses the area of the polygon inscribed in the circle, but does not need the area of the circumscribed shape. Compared with the calculation of the regular polygon inscribed in the circle and circumscribed by the ancient Greek mathematician Archimedes (B.C. 287-B.C. 212), it is much simpler in procedure and can get twice the result with half the effort. 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 1,5 years ago. After Liu Hui, Zu Chongzhi's pi was calculated to a more accurate level by Zu Chongzhi, an outstanding mathematician in the Southern and Northern Dynasties, and he made brilliant achievements. According to the records in Sui Shu's Li Zhi, Zu Chongzhi determined that the approximate value of pi is 3.1415926, the approximate value of pi is 3.1415927, and the true value is between these two approximate values, that is, 3.145926 <: л< 3.1415927。 At the same time, Zu Chongzhi also determined two approximate forms of pi: approximate ratio л=≈3.14 and density л=≈3.1415929. Zu Chongzhi's short approximation of pi is 3.1415926 and the excess approximation is 3.1415927, accurate to seven decimal places, which was very advanced in the world at that time, until a thousand years later, the 15th century Arab mathematician Al Cassie (? -1436) and the 16th century French mathematician Veda (154-163) broke Zu Chongzhi's record. In addition, before the concept of decimal fraction was fully developed, ancient mathematicians and astronomers in China often used fractions to represent approximate values of constants. The contract rate л = proposed by Zu Chongzhi has been used by predecessors, and the secret rate л = was discovered by him. Density is the best approximation of pi in fractional form with numerator and denominator within 1. Using these two approximations, the calculation can meet the requirements of certain accuracy and is very simple. The secret rate put forward by Zu Chongzhi was only regained by German Otto (about 155-165) and Dutch Antuoni (1527-167) after 1 years. However, in the history of western mathematics, л = is often called "Antuoni's rate". As we know, pi was widely used in production practice. In ancient times, when science was not very developed, it was a very complicated and difficult job to calculate pi. Therefore, the theory and calculation of pi reflect the mathematical level of a country to some extent. Zu Chongzhi's accurate calculation of seven decimal places indicates the highly developed mathematical level in ancient China, which has attracted people's attention. Since the splendid scientific culture in ancient China was gradually recognized by the world, some people suggested that л = be called "ancestral rate" to commemorate Zu Chongzhi's outstanding contribution. Zu Chongzhi's research work on pi and other important contributions are recorded in the book Composing, but this rich mathematical monograph was later lost. Therefore, Zu Chongzhi's method of calculating pi is not available now.