Pi is the ratio of the circumference of a circle to its diameter, which is a constant, expressed by the Greek letter "π" and obtained by the formula 355÷ 1 13. In astronomical calendar and production practice, all problems involving circles should be calculated by pi.
How to correctly calculate the value of pi is an important topic in the history of mathematics in the world. Mathematicians in ancient China attached great importance to this problem and began to study it very early. The ratio of the ancient diameter to one week and three weeks was put forward in Parallel Calculation of Weeks and Nine Chapters Arithmetic, and the pi was set at three, that is, the circumference of a circle was three times the diameter. Since then, after successive explorations by mathematicians of past dynasties, the calculated pi value has become more and more accurate. At the end of the Western Han Dynasty, in the process of designing and manufacturing a round bronze tiger (a measuring instrument) for Wang Mang, Liu Xin found that the ancient scale of one diameter and three measurements was too rough. After further calculation, the value of pi is 3. 1547. The value of pi calculated by Zhang Heng, a famous scientist in the Eastern Han Dynasty, is 3. 162. During the Three Kingdoms period, the value of pi calculated by mathematician Wang Fan was 3. 155. Liu Hui, a famous mathematician in Wei and Jin Dynasties, created a new method to calculate pi when he annotated Nine Chapters Arithmetic. He set the radius of the circle as 1, divided the circle into six equal parts, made the inscribed regular hexagon of the circle, and calculated the circumference of the inscribed regular hexagon by pythagorean theorem. Then inscribed with dodecagon, icosahedron, etc. In turn, until the circle is inscribed with 192 polygons, its side length is 6.282048, and the more sides inscribed with regular polygons in the circle, the closer its side length is to the actual circumference of the circle, so the value of pi at this time is the side length divided by 2, and its approximate value is 3.14; It shows that this value is less than the actual value of π. Liu Hui realized the concept of limit in modern mathematics in secant. The tangent circle method he founded is a major breakthrough in the process of exploring the value of pi. In order to commemorate this achievement of Liu Hui, later generations called the value of pi he obtained "Hui rate" or "Hui technique".
After Liu Hui, scholars who have made great achievements in exploring pi have successively included He Chengtian, Pi Yanzong and others in the Southern Dynasties. The pi calculated by He Chengtian is 3.1428; Pi Yanzong calculated pi as 22/7 ≈ 3. 14. All the above scientists have made great contributions to the research and calculation of pi, but compared with Zu Chongzhi's pi, they are much inferior.
Zu Chongzhi thinks that Liu Hui is a scholar who has made the greatest achievements in the study of pi during the hundreds of years from Qin and Han Dynasties to Wei and Jin Dynasties, but it has not reached an accurate level, so he makes further in-depth research in order to find a more accurate value. The research and calculation results prove that pi should be between 3. 14 15926 and 3. 14 15927. He became the first person in the world to calculate the exact value of pi to seven decimal places. It was not until a thousand years later that this record was broken by Arabian mathematician Al Cassie and French mathematician Viette. Zu Chongzhi's "secret rate" was not called "Antuoni's rate" by German until 1000 years later, and some people with ulterior motives said that Zu Chongzhi's pi was forged after western mathematics was introduced into China in the late Ming Dynasty. This is a deliberate fabrication. The ancient book that records Zu Chongzhi's research on pi is the history book of Sui Shu in Tang Dynasty, and the current Sui Shu was published in Bingwu Year (A.D. 1306). Like other modern versions, the record of Zu Chongzhi's pi happened more than 300 years before the end of Ming Dynasty. Moreover, many mathematicians before the Ming Dynasty quoted Zu Chongzhi's pi in their works, which proved Zu Chongzhi's outstanding achievements in the study of pi.
Zu Chongzhi determined a circle with a diameter of ten feet according to the method of Liu Hui's cyclotomy, and cut it in the circle for calculation. When he cut the circle into a polygon with 192 sides, he got the value of "emblem rate". But he was not satisfied, so he continued to cut and made 380 quadrilaterals and 768 polygons ... until he cut into 24576 polygons and calculated the side length of each inscribed regular polygon in turn. Finally, a circle with a diameter of 10 foot is obtained, and its circumference is between three feet, one foot, four inches, one minute, nine milliseconds, seven minutes to three feet, one foot, four inches, one minute, nine milliseconds and six minutes. The above unit of length is not commonly used. In other words, if the diameter of a circle is 1, then the circumference is less than 3. 14 15927, which is far less than ten million.
Making such an accurate calculation is an extremely meticulous and arduous mental work. As we know, in the era of Zu Chongzhi, abacus has not yet appeared, and the commonly used calculation tool is called calculation. It is a square or flat stick several inches long, made of bamboo, wood, iron, jade and other materials. Different calculation and financing methods are used to represent various numbers, which is called financing algorithm. If there are more digits, the larger the area needs to be placed. It is not like using a pen to calculate with a calculation formula, it can be left on paper, and every time the calculation is completed, it must be swung again for a new calculation; You can only write down the calculation results with notes, and you can't get more intuitive graphics and formulas. So as long as there are errors, such as calculation errors or calculation errors, we can only start from scratch. To get the value of Zu Chongzhi π, we need to add, subtract, multiply, divide and square the decimals with 9 significant digits. Each step needs to be repeated for more than 10 times and 50 times, and finally the calculated number reaches 16 or 17 digits after the decimal point. Today, it is not an easy task to complete these calculations with an abacus and a pen and paper. Let's think about it. 1500 years ago, in the Southern Dynasties, a middle-aged man kept calculating and remembering under a dim oil lamp. He often had to rearrange his calculations for tens of thousands of times. This is a very hard thing that needs to be repeated day after day. Without great perseverance, one can never finish the work. This brilliant achievement also fully reflects the highly developed level of ancient mathematics in China.
Zu Chongzhi's research on pi has positive practical significance and adapted to the needs of production practice at that time. He personally studied weights and measures and revised the ancient calculation of volume with the latest pi results.
In ancient times, there was a measuring instrument called "kettle", which was generally one foot deep and cylindrical. What is the volume of this measuring device? To find this value, you need to use pi. Zu Chongzhi worked out the exact value with his research. He also recalculated the "Lujialiang" made by Liu Xin in the Han Dynasty (another measuring instrument, similar to the above-mentioned "Sheng" equivalent, but all are cylinders. ), because the calculation method and pi value used by Liu Xin are not accurate enough, the volume value obtained by him is different from the actual value. Zu Chongzhi found his mistake and corrected the value with "ancestral rate".
Later, when making measuring instruments, people used Zu Chongzhi's "ancestral rate" value. On the basis of predecessors, Zu Chongzhi calculated the pi to 7 decimal places after assiduous study and repeated calculation, and obtained the approximate value in the form of pi fraction. What method did Zu Chongzhi use to get this result? It's impossible to find out now. If you imagine that he will follow Liu Hui's "secant" method, you must work out how many polygons are inscribed in the circle 16000, and how much time and energy it will take!
According to Sui Shu's law, Zu Chongzhi calculated the circumference of a circle with a diameter of ten feet in one minute (one hundredth of a foot), and obtained the true values of abundance (3. 14 15927) and pi (3. 14 15926). Sui Shu did not specify how Zu Chongzhi calculated its surplus. It is generally believed that Zu Chongzhi adopted Liu Hui's secant technique, but there are many other speculations. These two approximations are accurate to the seventh place after the decimal point, which is the most advanced achievement in the world at that time. It was not until more than a thousand years later that Cassie, an Arab mathematician in the15th century, and F. Veda, a French mathematician in the16th century, got more accurate results. Zu Chongzhi determined two asymptotic fractions of π, the approximation rate is 22/7 and the density rate is 355/ 1 13. Among them, the secret rate of 355/113 (≈ 3.1415929) was not discovered by V. Otto of Germany until16th century. It is composed of three odd pairs of 1 13355, and then folded into two sections, which is beautiful, regular and easy to remember. In order to commemorate Zu Chongzhi's outstanding contribution, some foreign mathematical historians call the density of pi "ancestral rate".