Before Zu Chongzhi, China had reached the world advanced level in mathematics, and many outstanding mathematicians and excellent mathematical works emerged. As early as the end of primitive society, various geometric patterns appeared on the pottery of "Longshan Culture". The decimal system was used in mathematical operations in Shang Dynasty, which was the earliest decimal system in the world, and its adoption greatly facilitated mathematical calculations. Zhouyi in the Spring and Autumn Period is the first book in the world to study permutation and combination. During the Warring States period, a hundred schools of thought contended, mathematics further developed, and the formula of "99" multiplication appeared. In geometry, rulers have been widely used in drawing, thus promoting the development of geometry. At the same time, many valuable mathematical theories have been put forward in the works of a hundred schools of thought contend. For example, in the Mohist classic Mozi, there are many places that involve some basic problems in geometry, and all of them have accurate definitions, and their accuracy is comparable to Euclid's Elements of Geometry, which is popular in ancient western countries. Zhuangzi, written by Taoism, put forward the limit theory, and often quoted a famous example when explaining the limit of sequence: "There is a foot-long stick, and if you cut it in half every day, it will never be cut off".
During the Qin, Han, Wei and Jin dynasties, with the great development of feudal economy, mathematics closely related to it also developed greatly, and a large number of mathematical works and famous mathematicians emerged. Among them, the most important works are Zhoupian suan Jing, Nine Chapters of Arithmetic and Island Suan Jing. The book "Zhou Bi suan Jing" was written not later than the first century BC. The author doesn't know. Zhao, a famous mathematician in the Eastern Han Dynasty, commented on this. His main achievement is that he put forward the famous Pythagorean Theorem and adopted more complicated fractional operation. Nine Chapters of Arithmetic was written at about the same time as Zhou Kuai suan Jing, whose original author is unknown. Many mathematicians in the Western Han Dynasty, such as Zhang Cang, Geng Shouchang, Xu Shang and Du Zhong, added or deleted this book, and Liu Hui, a famous mathematician in the Three Kingdoms period, annotated this book. This book brings together the outstanding achievements of mathematics in the pre-Qin, Qin and Han dynasties, and has had a very far-reaching influence on ancient mathematics in China in the future. The book is divided into square fields (mainly the method of calculating fields), small plots (mainly the method of calculating square roots and square roots), commercial workers (mainly calculating various volumes to solve practical problems in construction projects such as building cities and building water conservancy projects), millet (mainly the method of calculating the conversion between various cereals), difference (mainly the method of calculating the storage and transportation of cereals) and even loss (mainly the method of calculating the storage and transportation of cereals). From the point of view of mathematical achievements, the first thing that this book should mention is that it recorded the most advanced fractional arithmetic and proportional algorithm in the world at that time. In addition, the method of square root and square root recorded in the book is actually to solve the quadratic equation of one variable; The solution of simultaneous equations for solving equations is earlier than similar algorithms in Europe 1500 years. The book also put forward the concept of negative number and the addition and subtraction algorithm of positive and negative numbers for the first time in the history of mathematics in the world. Nine Chapters Arithmetic not only occupies an important position in the history of Chinese mathematics, but also has far-reaching influence abroad. North Korea and Japan have taken Nine Chapters of Arithmetic as textbooks, and some of its calculation methods have also spread to India, Arabia and Europe.
The author of Archipelago Calculation is Liu Hui in the Three Kingdoms Period. In this book, he mainly talked about how to use benchmarks to solve various problems in measurement mathematics for the second, third or even fourth time. His accomplishments in this respect far exceeded those of western mathematicians at that time. And this measurement mathematics is the mathematical basis of cartography.
In addition to these three works, the more important mathematical works include Sun Tzu's Art of War, Cao Wubing Law and Xiahou Yangbing Law.
After studying hard, Zu Chongzhi inherited and developed the outstanding achievements of previous scientists. His research on pi is an outstanding contribution to China and even the world. Zu Chongzhi's accurate calculation of the value of pi was named "Zu Chongzhi Pi" after him, or "Ancestral Rate" for short.
What is pi? A circle has a circumference and a center. The distance from any point on the circumference to the center of the circle is called radius, and doubling the radius is the diameter. The diameter is a line segment passing through the center of the circle, and the circumference is an arc. How many times is an arc a straight line? It's mathematically called pi. Simply put, pi is the ratio of the circumference to the diameter of a circle. It is a constant, represented by the Greek letter "π". 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 making a round bronze tiger (a measuring instrument) for Wang Mang, Liu Xin found that the ratio of one diameter to three circumference in ancient times 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 pi 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 and Pi Yan of the Southern Dynasties. 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 results of research and calculation prove that pi should be between 3. 14 15926 and 3. 14 15927; To show. 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 put forward the "secret rate" until the Germans one thousand years later? It is called "Antuoni rate", and some people with ulterior motives say that Zu Chongzhi's pi was forged after western mathematics was introduced to 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.
So, how did Zu Chongzhi achieve such great scientific achievements? It is true that his achievements are based on previous studies. Judging from the mathematical level at that time, Zu Chongzhi probably inherited and developed the secant technology that Liu Hui founded and first used, so he made great achievements beyond his predecessors. When the secant technique was mentioned earlier, we already know a conclusion that the more sides of a circle inscribed by a regular N polygon, the closer the sum of the sides is to the actual length of the circle. But because it is inscribed, it is impossible to increase the number of sides to infinity, so the sum of the sides is always less than the perimeter.
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 units of length are no longer universal. In other words, if the diameter of a circle is 1, then the circumference is less than 3. 14 15927, and the size is less than 10 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 to make 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 is admired not only by the people of China, but also by the scientific community all over the world. 1960, after studying the photos on the back of the moon, Soviet scientists named the valley above it with the names of some of the most contributing scientists in the world, and one of them was named "Zu Chongzhi Crater".
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 measuring volume with the latest pi results.
In ancient times, there was a measuring instrument called "(pot)", 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.