ZǔChōngzhī (429 AD - 500 AD) was an outstanding mathematician and scientist in my country. A native of the Southern and Northern Dynasties, Han nationality, with the courtesy name Wenyuan. He was born in the sixth year of Yuanjia, Emperor Wen of Song Dynasty, and died in Yongyuan, the second year of Qihunhou. His ancestral home is Qiuxian County, Fanyang County (now Laishui County, Hebei Province). In order to avoid the war, Zu Chongzhi's grandfather Zuchang moved from Hebei to Jiangnan. Zuchang once served as the "Great Craftsman" of Liu Song Dynasty, in charge of civil engineering; Zu Chongzhi's father was also an official in the court. Zu Chongzhi received scientific knowledge passed down from his family since childhood. When he was young, he entered Hualin Academy and engaged in academic activities. Throughout his life, he successively served as a historian in Southern Xuzhou (today's Zhenjiang City), a member of the army in the government, the magistrate of Lou County (today's northeast of Kunshan City), the Yezhe Pushe, and the captain of Changshui School. His main contributions are in mathematics, astronomy, calendar and mechanics. In terms of mathematics, he wrote the book "Zhu Shu", which was included in the famous "Ten Books on Calculation" and was used as a textbook for the Imperial Academy in the Tang Dynasty. Unfortunately, it was later lost. "Book of Sui·Lü Li Zhi" left a short record about pi (π). Zu Chongzhi calculated that the true value of π is between 3.1415926 (朒号) and 3.1415927 (surplus number), which is equivalent to the 7th decimal place. It became the most advanced achievement in the world at that time. This record was not broken until the 15th century by the Arab mathematician Cassidy. Zu Chongzhi also gave two fractional forms of π: 22/7 (approximate rate) and 355/113 (density rate). The density rate is accurate to the 7th decimal place. In the West, it was not rediscovered in the West until the 16th century by the Dutch mathematician Otto. Discover. Zu Chongzhi and his son Zu Xun also successfully solved the problem of calculating the volume of a sphere by using the "Mou He Square Cover" and obtained the correct formula for the volume of a sphere. In terms of astronomical calendar, Zu Chongzhi created the "Da Ming Calendar", which was the first to introduce precession into the calendar; adopted the new leap week of 391 years plus 144 leap months; and for the first time accurately measured the number of days in the nodal month (27.21223), the number of days in the tropical year (365.2428), etc. Data, and also invented the method of using a standard table to measure the length of the noon sun's shadow in several days before and after the winter solstice to determine the time of the winter solstice. In terms of mechanics, he designed and manufactured water mills, compasses driven by copper parts, thousand-mile ships, timers, etc. In addition, he also had attainments in music, literature, and textual research. He was proficient in music, good at playing chess, and wrote the novel "Shu Yi Ji". He is one of the few knowledgeable and talented figures in history.
To commemorate this great ancient scientist, people named a crater on the back of the moon "Zu Chong's Crater" and the asteroid 1888 "Zu Chong's Asteroid".
Through arduous efforts, Zu Chongzhi calculated the value of pi (π) to seven decimal places for the first time in the history of world mathematics, that is, between 3.1415926 and 3.1415927. He proposed an approximate rate of 22/7 and a density of 355/113. This density value was the first to be proposed in the world, more than a thousand years earlier than in Europe, so some people advocate calling it the "ancestral rate". He compiled his mathematical research results into a book called "Zhushu", which was once designated as a mathematics textbook in the Tang Dynasty Chinese Studies. The "Da Ming Calendar" compiled by him introduced "precession" into the calendar for the first time. It is proposed to set up 144 leap months in 391 years. The length of a tropical year is calculated to be 365.24281481 days, with an error of only about 50 seconds. He was not only an outstanding mathematician and astronomer, but also an outstanding mechanical expert. Recreate a variety of ingenious machines such as the long-lost compass and the thousand-mile boat. In addition, he also studied music. His works include "The Analects of Confucius", "The Classic of Filial Piety", "Yiyi", "Laoziyi", "Zhuangziyi" and the novel "Shuyiji", etc., all of which have been lost long ago.
Answer: mykyl - Level 3 Newcomer in the Workplace 2009-9-27 16:46
Zǔ Chōngzhī (429 AD - 500 AD) is an outstanding mathematician in my country. the scientist. A native of the Southern and Northern Dynasties, Han nationality, with the courtesy name Wenyuan. He was born in the sixth year of Yuanjia, Emperor Wen of Song Dynasty, and died in Yongyuan, the second year of Qihunhou. His ancestral home is Qiuxian County, Fanyang County (now Laishui County, Hebei Province). In order to avoid the war, Zu Chongzhi's grandfather Zuchang moved from Hebei to Jiangnan. Zuchang once served as the "Great Craftsman" of Liu Song Dynasty, in charge of civil engineering; Zu Chongzhi's father was also an official in the court.
Zu Chongzhi received scientific knowledge passed down from his family since childhood. When he was young, he entered Hualin Academy and engaged in academic activities. Throughout his life, he successively served as a historian in Southern Xuzhou (today's Zhenjiang City), a member of the army in the government, the magistrate of Lou County (today's northeast of Kunshan City), the Yezhe Pushe, and the captain of Changshui School. His main contributions are in mathematics, astronomy, calendar and mechanics. In terms of mathematics, he wrote the book "Zhu Shu", which was included in the famous "Ten Books on Calculation" and was used as a textbook for the Imperial Academy in the Tang Dynasty. Unfortunately, it was later lost. "Book of Sui·Lü Li Zhi" left a short record about pi (π). Zu Chongzhi calculated that the true value of π is between 3.1415926 (朒号) and 3.1415927 (surplus number), which is equivalent to the 7th decimal place. It became the most advanced achievement in the world at that time. This record was not broken until the 15th century by the Arab mathematician Cassidy. Zu Chongzhi also gave two fractional forms of π: 22/7 (approximate rate) and 355/113 (density rate). The density rate is accurate to the 7th decimal place. In the West, it was not rediscovered in the West until the 16th century by the Dutch mathematician Otto. Discover. Zu Chongzhi and his son Zu Xun also successfully solved the problem of calculating the volume of a sphere by using the "Mou He Square Cover" and obtained the correct formula for the volume of a sphere. In terms of astronomical calendar, Zu Chongzhi created the "Da Ming Calendar", which was the first to introduce precession into the calendar; adopted the new leap week of 391 years plus 144 leap months; and for the first time accurately measured the number of days in the nodal month (27.21223), the number of days in the tropical year (365.2428), etc. Data, and also invented the method of using a standard table to measure the length of the noon sun's shadow in several days before and after the winter solstice to determine the time of the winter solstice. In terms of mechanics, he designed and manufactured water mills, compasses driven by copper parts, thousand-mile ships, timers, etc. In addition, he also had attainments in music, literature, and textual research. He was proficient in music, good at playing chess, and wrote the novel "Shu Yi Ji". He is one of the few knowledgeable and talented figures in history.
To commemorate this great ancient scientist, people named a crater on the back of the moon "Zu Chong's Crater" and the asteroid 1888 "Zu Chong's Asteroid".
Through arduous efforts, Zu Chongzhi calculated the value of pi (π) to seven decimal places for the first time in the history of world mathematics, that is, between 3.1415926 and 3.1415927. He proposed an approximate rate of 22/7 and a density of 355/113. This density value was the first to be proposed in the world, more than a thousand years earlier than in Europe, so some people advocate calling it the "ancestral rate". He compiled his mathematical research results into a book called "Zhushu", which was once designated as a mathematics textbook in the Tang Dynasty Chinese Studies. The "Da Ming Calendar" compiled by him introduced "precession" into the calendar for the first time. It is proposed to set up 144 leap months in 391 years. The length of a tropical year is calculated to be 365.24281481 days, with an error of only about 50 seconds. He was not only an outstanding mathematician and astronomer, but also an outstanding mechanical expert. Recreate a variety of ingenious machines such as the long-lost compass and the thousand-mile boat. In addition, he also studied music. His works include "The Analects of Confucius", "The Classic of Filial Piety", "Yiyi", "Laoziyi", "Zhuangziyi" and the novel "Shuyiji", etc., all of which have been lost long ago.
[Edit this paragraph] Biographies of characters
During the 170 years from the fall of the Eastern Jin Dynasty in 420 AD to the unification of the country by the Sui Dynasty in 589 AD, the opposition between the north and the south formed in the history of our country. This period is called the Northern and Southern Dynasties. The Southern Dynasties began in 420 AD when General Liu Yu of the Eastern Jin Dynasty seized the throne and established the Song Dynasty. It went through four dynasties: Song, Qi, Liang, and Chen. Confronting the Southern Dynasties was the Northern Dynasties, which experienced the Northern Wei, Eastern Wei, Western Wei, Northern Qi, Northern Zhou and other dynasties. Zu Chongzhi was a native of the Southern Dynasties. He was born in the Song Dynasty and died in the Southern Qi Dynasty.
At that time, due to the relatively stable society in the Southern Dynasties, significant progress was made in agriculture and handicrafts, and the economy and culture developed rapidly, which also promoted the advancement of science. Therefore, during this period, some very accomplished scientists emerged in the Southern Dynasties, and Zu Chongzhi was one of the most outstanding figures.
Zu Chongzhi’s native place is Qiuxian County, Fanyang County (now Laishui County, Hebei Province). At the end of the Western Jin Dynasty, the ancestral family moved to the south of the Yangtze River because their hometown was destroyed by war.
Zu Chongzhi's grandfather, Zuchang, served as a master craftsman in the Song Dynasty government and was responsible for presiding over construction projects. He had some scientific and technical knowledge. At the same time, the ancestors of the family had studied astronomy and calendar throughout the generations. Therefore, Zu Chongzhi had the opportunity to be exposed to science and technology since he was a child.
Zu Chongzhi had extensive interests in natural sciences, literature, and philosophy, especially astronomy, mathematics, and mechanical manufacturing. He had a strong interest and in-depth research. As early as his youth, he had a reputation for being knowledgeable and talented, and was sent by the government to the Hualin Academy, an academic research institution at the time, to do research. Later he held a local official position. In 461 AD, he was appointed as the governor of Southern Xuzhou (now Zhenjiang, Jiangsu Province). In 464, the Song Dynasty government transferred him to Lou County (now northeast of Kunshan County, Jiangsu Province) as a county magistrate.
During this period, although Zu Chongzhi's life was very unstable, he still continued to pursue academic research and made great achievements. His attitude towards academic research is very rigorous. He attaches great importance to the results of ancient research, but he is never superstitious about the ancients. In his own words, he would never "pursue (blindly worship) the ancients", but "search for the ancient and modern (draw the essence from a large number of ancient and modern works)". On the one hand, he conducted in-depth research on the writings of ancient scientists Liu Xin, Zhang Heng, Kan Ze, Liu Hui, Liu Hong and others, and fully absorbed all useful things from them. On the other hand, he dared to doubt the conclusions of his predecessors in scientific research and corrected and supplemented them through actual observation and research, thus achieving many extremely valuable scientific results. In terms of astronomical calendars, the "Da Ming Calendar" compiled by him was the most precise calendar at that time. In mathematics, he calculated the circumference of pi accurate to six decimal places and achieved the best results in the world at that time.
At the end of the Song Dynasty, Zu Chongzhi returned to Jiankang (today's Nanjing) and took up the official position of Visitor Pushe. From this time until the early years of the Qi Dynasty, he spent more energy on the study of machinery manufacturing, rebuilt the compass, invented the thousand-mile ship, the water mill, etc., and made outstanding contributions.
When Zu Chongzhi was in his later years, civil strife broke out in the ruling group of the Qi Dynasty. Political corruption was dark and people's lives were very painful. Wei of the Northern Dynasty took the opportunity to send troops to attack south.
From AD 494 to AD 500, the Jiangnan area fell into war again. Zu Chongzhi was very concerned about this political situation of heavy internal and external troubles. From about AD 494 to 498, he held the official position of captain of Changshui. At that time, he wrote an article "On Anbian", suggesting that the government reclaim wasteland, develop agriculture, enhance national strength, stabilize people's livelihood, and consolidate national defense. Emperor Qi Ming saw this article and planned to send Zu Chongzhi to patrol around the world to start some undertakings that were beneficial to the national economy and people's livelihood. However, due to years of war, his suggestions have never been realized. Not long after, this outstanding scientist lived to the age of seventy-two and passed away in AD 500.
Reform the calendar and introduce precession
Due to the needs of animal husbandry and agricultural production, the working people of ancient my country discovered the basic laws of the movement of the sun and the moon after long-term observation. They define the period from the first full moon or waning moon to the second full moon or waning moon as one month. Each month is a little more than twenty-nine days, and twelve months are called a year. This method of counting years is called the lunar calendar. They also observed that it takes 365 days and a quarter from the first winter solstice to the next (actually the time it takes for the earth to orbit the sun), so they also called this period Work for one year. The calendar calculated according to this method is usually called the solar calendar. However, the number of days in a lunar year and in a solar year are not exactly equal. According to the lunar calendar, a year is 354 days long; according to the solar calendar, a year is 365 days, five hours, 48 ??minutes and 46 seconds. A year in the lunar calendar is more than eleven days shorter than a year in the solar calendar. In order to make the number of days in the two calendars consistent, it is necessary to find a way to adjust the number of days in the lunar calendar year. Our ancestors found a solution to this problem very early, which is to use the "leap month" method. Arrange a leap year within a number of years and add a leap month to each leap year. Every leap year, there are thirteen months in a year.
Due to the adoption of this leap year method, the lunar year and the solar year are more consistent.
In ancient times, Chinese calendarists always set nineteen years as the unit for calculating leap years, called "one chapter", and there were seven leap years in each chapter. In other words, among the nineteen years, seven of them will be thirteen months. This leap method has been used for more than a thousand years, but it is not thorough and accurate enough. In 412 AD, Zhao Nuan of Northern Liang created the "Yuanshi Calendar", which broke the restrictions of the calendar and stipulated that 221 leap months should be inserted in the middle of six hundred years. It is a pity that Zhao Nuan's reforms did not attract the attention of people at the time. For example, when the famous calendar calculator He Chengtian made the "Yuan Jiali" in 443 AD, he still used the ancient method of seven leaps in nineteen years.
Zu Chongzhi absorbed Zhao Nuan's advanced theory and added his own observations. He believed that there are too many leaps in the seventeenth year, and there is a difference of one day every two hundred years, while Zhao Guan's six hundred years The leap numbers of 221 are a bit rare and not very precise. Therefore, he proposed a new leap law of 144 leaps in 391 years. This leap method was considered the most sophisticated at the time.
In addition to reforming the leap method, another major achievement of Zu Chongzhi in calendar research was the unprecedented application of "precession."
According to the principles of physics, a rigid body rotates When moving, if it is not affected by external force at all, the direction and speed of rotation should be consistent; if it is affected by external force, its rotation speed will change periodically. The earth is a rigid body with an uneven surface and irregular shape. It is often affected by the attraction of other planets during its operation. Therefore, the rotation speed always undergoes some periodic changes and cannot be absolutely uniform. Therefore, every year the sun orbits once (actually the earth orbits the sun once), it is impossible to completely return to the winter solstice point of the previous year. There is always a slight difference. According to current astronomers' precise calculations, the difference is approximately 50.2 seconds per year, and it moves backward by one degree every seventy-one years and eight months. This phenomenon is called precession.
With the gradual development of astronomy, ancient Chinese scientists gradually discovered the phenomenon of precession. Deng Ping in the Western Han Dynasty, Liu Xin, Jia Kui and others in the Eastern Han Dynasty all observed the phenomenon of the winter solstice moving backward, but they have not yet clearly pointed out the existence of precession. It was not until the early years of the Eastern Jin Dynasty that the astronomer Yu Xi began to affirm the existence of the precession phenomenon, and first advocated the introduction of precession into the calendar. He provided the first data for precession, calculating that the winter solstice moved back one degree every fifty years. Later, in the early years of the Southern Song Dynasty, He Chengtian believed that the precession varied by one degree every hundred years, but he did not apply the precession in the "Yuan Jiali" he formulated.
Zu Chongzhi inherited the scientific research results of his predecessors. Not only did he confirm the existence of the precession phenomenon, he calculated that the precession moved backward by one degree every forty-five years and eleven months, and in his "Da Ming Calendar" he produced Precession is applied. Because the astronomical historical data he relied on were still not accurate enough, the data he proposed naturally could not be very accurate. Despite this, Zu Chongzhi's application of precession to the calendar was a pioneering work in the history of astronomical calendars and opened a new page for the improvement of our country's calendar. After the Sui Dynasty, precession has been paid attention to by many calendarists. For example, the Sui Dynasty's "Daye Calendar" and "Huangji Calendar" all used precession.
Zu Chongzhi’s third great contribution in calendar research is his ability to find the number of days in the calendar that are often called "node months".
The so-called nodal month is the time between the moon passing through the intersection of the "ecliptic" and the "white path" twice in a row. The ecliptic refers to the orbit of the sun as seen by us on earth, and the ecliptic is the orbit of the moon as seen by us on earth. The number of days in the nodal month can be calculated. The number of days of the nodal month measured by Zu Chongzhi is 27.21223 days, which is much more precise than that measured by astronomers in the past. It is very similar to the number of days of the nodal month measured by modern astronomers, 27.21222 days. Given the level of astronomy at that time, Zu Chongzhi was able to obtain such precise figures, and his achievements were truly astonishing.
Since both solar and lunar eclipses occur near the intersection of the ecliptic and the ecliptic, after calculating the number of days in the node month, the time when the solar or lunar eclipse occurs can be more accurately calculated.
In the "Da Ming Calendar" he formulated, Zu Chongzhi used the nodal months to calculate the times of solar and lunar eclipses that were more accurate than in the past, and were very close to the actual times of solar and lunar eclipses.
Based on the above-mentioned research results, Zu Chongzhi finally succeeded in producing the most scientific and progressive calendar at that time - the "Da Ming Calendar". This is the fruit of Zu Chongzhi's scientific research genius and his most outstanding contribution to the astronomical calendar.
In addition, Zu Chongzhi also observed and calculated the orbits of the five planets in the sky, including wood, water, fire, metal, and earth, and the time required for one cycle. Ancient Chinese scientists calculated that Jupiter (called the Year Star in ancient times) rotates once every twelve years. When Liu Xin of the Western Han Dynasty was writing the "Santong Calendar", he discovered that Jupiter's rotation lasted less than twelve years. Zu Chongzhi went a step further and calculated that the time it takes for Jupiter to rotate once is 11.858 years. Modern scientists estimate that Jupiter's orbital period is approximately 11.862 years. The difference between Zu Chongzhi's calculation and this figure is only 0.04 years. In addition, Zu Chongzhi calculated that Mercury's rotation period is 115.88 days, which is completely consistent with the number determined by modern astronomers to within two decimal places. He calculated that the time it takes for Venus to move around once is 583.93 days, which is only 0.01 day different from the number determined by modern scientists.
In 462 AD (the sixth year of the Song Dynasty and the Ming Dynasty), Zu Chongzhi sent the carefully compiled "Da Ming Calendar" to the government, requesting its publication and implementation. Emperor Xiaowu of the Song Dynasty ordered officials who knew the calendar to discuss the pros and cons of this calendar. During the discussion, Zu Chongzhi encountered opposition from conservative forces represented by Dai Faxing. Dai Faxing was a trusted minister of Emperor Xiaowu of Song Dynasty and was very powerful. Since he took the lead in opposing the new calendar, officials of all sizes in the imperial court also echoed his opinion, and everyone was not in favor of changing the calendar.
In order to insist on his correct views, Zu Chongzhi confidently started a fierce debate with Dai Faxing.
This debate on the pros and cons of the new calendar actually reflected the sharp struggle between science and anti-science, progressive and conservative forces at that time. Dai Faxing first wrote to the emperor and used the signs of ancient sages and sages from ancient books to suppress Zu Chongzhi. He said that the sun is always at a certain position during the winter solstice, which was determined by ancient sages and sages and cannot be changed forever. He said that Zu Chongzhi thought that the winter solstice point moved slightly every year, which was a slander against the sky and violated the scriptures of the saints. It is a kind of treasonous behavior. He also said that the nineteen-year seven-leap calendar that was popular at the time was formulated by ancient sages and sages and could never be changed. He even scolded Zu Chongzhi as a simple common man who was not qualified to talk about reforming the calendar.
Zu Chongzhi showed no fear at all against the attacks by the powerful. He wrote a famous rebuttal. Based on ancient literary records and observations of the sun at that time, he proved that the winter solstice point changed. He pointed out: The facts are very clear, how can we believe in the past but doubt the present? He also cited in detail his many years of personal observation of the changes in the length of the midday sun's shadow on various days before and after the winter solstice, and accurately calculated the date and time of the winter solstice, thus proving that the seventeenth leap year in nineteen years is very imprecise. He asked: "The old calendar is not accurate. Should it be used forever and never be reformed? Whoever wants to say that the Daming Calendar is not good should provide conclusive evidence. If there is evidence, I am willing to accept it."
At that time, Dai Faxing could not point out the shortcomings of the new calendar, so he argued over issues such as the speed of the day's movement, the length of the sun's shadow, the speed of the moon's movement, etc. Zu Chong argued one by one and refuted him.
Under Zu Chongzhi's confident rebuttal, Dai Faxing had nothing to say in reply, and said unreasonably: "No matter how good the new calendar is, it cannot be used." Zu Chongzhi was not intimidated by Dai Faxing's arrogant attitude. But he firmly stated: "We should never blindly believe in the ancients. Now that we have discovered the shortcomings of the old calendar and determined that the new calendar has many advantages, we should switch to the new one."
In this great debate , many ministers were convinced by Zu Chongzhi's incisive and thorough theory, but they did not dare to speak for Zu Chongzhi because they were afraid of Dai Faxing's power. Finally, a minister named Chao Shangzhi came out to express support for Zu Chongzhi.
He said that the "Da Ming Calendar" is the result of Zu Chongzhi's many years of research. According to the "Da Ming Calendar", he can calculate the thirteenth year of Yuanjia (436), the fourteenth year (437), the twenty-eighth year (451), and the third year of the Ming Dynasty (459). The four lunar eclipses are all accurate, but the calculation results using the old calendar have a large error. Since the "Da Ming Calendar" has been proved to be better by facts, it should be adopted.
As a result, Dai Faxing was speechless. Zu Chongzhi achieved the final victory. Emperor Xiaowu of the Song Dynasty decided to change to the new calendar in the ninth year of the Ming Dynasty (465). Unexpectedly, Emperor Xiaowu died in the eighth year of the Ming Dynasty, and then there were rebellions within the ruling group, so the matter of changing the calendar was shelved. It was not until the ninth year of the Tianjian reign of Liang Dynasty (51O) that the new calendar was officially adopted, but Zu Chongzhi had been dead for ten years by then.
The art of writing and fixing the law of circles
Zu Chongzhi was not only proficient in astronomy and calendars, but his contributions to mathematics, especially his outstanding achievements in the study of "pi", surpassed those of previous generations. Radiating splendor in the history of world mathematics.
We all know that pi is the ratio of the circumference of a circle to the diameter of the same circle. This ratio is a constant, which is now generally represented by the Greek letter "π". Pi is an infinite decimal that can never be divided. It cannot be expressed completely accurately by fractions, finite decimals or recurring decimals. Thanks to advances in modern mathematics, pi has been calculated to more than two thousand digits after the decimal point.
Pi has a wide range of applications. Especially in astronomy and calendar, all problems involving circles must be calculated using pi. The earliest pi value obtained by the working people in ancient my country in production practice was "3". This was of course very imprecise, but it was still used until the Western Han Dynasty. Later, with the development of astronomy, mathematics and other sciences, more and more people studied pi. At the end of the Western Han Dynasty, Liu Xin first abandoned the inaccurate pi value of "3". The pi value he once used was 3.547. Zhang Heng of the Eastern Han Dynasty also calculated that pi = 3.1622. These values ??are certainly a big improvement over π=3, but they are still far from precise. It was not until the end of the Three Kingdoms that the mathematician Liu Hui created a method of calculating pi by using the method of cutting a circle, and the study of pi made significant progress.
The method of calculating pi using circle slicing is roughly as follows: first draw a circle, and then draw an inscribed regular hexagon within the circle. Suppose the diameter of the circle is 2, then the radius is equal to 1. One side of the inscribed regular hexagon must be equal to the radius, so it is also equal to 1; its perimeter is equal to 6. If we take the circumference 6 of the inscribed regular hexagon as the circumference of a circle and divide it by the diameter 2, we get the ratio of circumference to diameter π=6/2=3, which is the value of π=3 in ancient times. But this value is incorrect. We can clearly see that the circumference of the inscribed regular hexagon is much smaller than the circumference of the circle.
If we double the number of sides of the inscribed regular hexagon and change it to an inscribed regular dodecagon, and then use appropriate methods to find its perimeter, then we can see that this perimeter The length is closer to the circumference of the circle than the circumference of the inscribed regular hexagon, and the area of ??this inscribed regular dodecagon is also closer to the area of ??the circle. From here we can draw the conclusion: the more sides the inscribed regular polygon has in a circle, the smaller the difference between the total length (circumference) of its sides and the circumference of the circle. Theoretically speaking, if the number of sides of an inscribed regular polygon increases to infinity, then the perimeter of the regular polygon will closely coincide with the circumference of the circle. From this calculation, the area of ??the inscribed infinite regular polygon is equal to the area of ??the circle. Equal. But in fact, it is impossible for us to increase the number of sides of an inscribed regular polygon to infinite so that the perimeter of this infinite regular polygon coincides with the circumference of a circle. The number of sides of an inscribed regular polygon can only be increased to a certain extent so that its perimeter and the circumference of the circle nearly coincide. Therefore, if you use the method of increasing the number of sides of a regular polygon inscribed in a circle to find pi, the number you get will always be slightly smaller than the true value of π. Based on this principle, Liu Hui started from the circle inscribed in a regular hexagon, gradually doubled the number of sides, and calculated until it was inscribed in a regular hexagon, and calculated the pi to be 3.141024. Convert this number into a fraction, which is 157/50. The pi obtained by Liu Hui was later called "Hui's rate".
His calculation method actually has the concept of limit in modern mathematics. This is a glorious achievement of ancient Chinese research on pi.
Zu Chongzhi achieved another major achievement beyond his predecessors in deriving pi. According to the records in Sui Shu·Lü Li Zhi, Zu Chongzhi converted one zhang into 100 million hu, and used this as the diameter to calculate pi. The result of his calculation was to obtain two numbers: one is the surplus number (that is, the approximate value of the surplus), which is 3.1415927; the other is the number (that is, the approximate value of the deficiency), which is 3.1415926. The true value of pi is exactly between the two numbers. "Sui Shu" only has such a simple record, without specifying the method he used to calculate it. However, judging from the level of mathematics at the time, there was no better method except Liu Hui's circle cutting technique. Zu Chongzhi probably adopted this method. Because Liu Hui's method is used to increase the number of sides of the regular polygon inscribed in the circle to 24576, the result obtained by Zu Chongzhi can be obtained.
The two numbers Ying 撒 can be listed as an inequality, such as: 3.1415926 (*) < π (real pi) < 3.1415927 (Ying 撒), which shows that the pi should be between the two numbers Ying 撒. In accordance with the custom of using fractions for calculation at that time, Zu Chongzhi also used two fractional values ??of pi. One is 355/113 (approximately equal to 3.1415927). This number is relatively precise, so Zu Chongzhi called it "density". The other one is (approximately equal to 3.14). This number is relatively rough, so Zu Chongzhi called it "approximate rate". In Europe, it was not until 1573 that the German mathematician Walter calculated the value 355/113. Therefore, the Japanese mathematician Yoshio Mikami once suggested calling the pi value of 355/113 the "zu rate" to commemorate this great Chinese mathematician.
Since the mathematical monograph "Zhu Shu" written by Zu Chongzhi has been lost, and the "Sui Shu" does not specifically record his method of calculating pi, therefore, experts in our country who study the mathematical heritage of the motherland have no idea what he asked for. There are also different opinions on the method of pi.
Some people think that Zu Chongzhi’s pi is the “number”. It is obtained by using the method of inscribing a regular polygon of a circle; while the "excess number" is obtained by using the method of using a regular polygon circumscribing a circle. If Zu Chongzhi continued to use Liu Hui's method, starting from the regular hexagon inscribed in the circle and doubling the number of sides one by one, until he calculated the regular hexagon inscribed in 24576, the sum of the lengths of each side would only be close to and smaller than The circumference of a circle and the area of ??this regular polygon can only be successively close to and smaller than the area of ??the circle. The pi calculated from this is 3.14159261, which can only be smaller than the true value of pi. This is the number. Judging from Zu Chongzhi's mathematical level, it is possible to break through Liu Hui's method and start from the circumscribed regular hexagon and try to find the pi one by one. If Zu Chongzhi doubled the number of sides of the circumscribed regular hexagon, and reached 24576 regular polygons, the pi he obtained should be 3.14159270208. This number is found using the circumcision method. Since the sum of the lengths of the sides of a circumscribed regular polygon is always greater than the length of the circle, the area of ??the regular polygon is always greater than the area of ??the circle, so this number is always greater than the real pi. Use the rounding method to round off the numbers after seven decimal places to get the surplus number.
There is no precise historical data to confirm whether Zu Chongzhi used the two methods of inscription and circumcision at the same time to calculate the negative and positive numbers of pi. However, the two values ????and Ying obtained by this method are generally consistent with the results originally obtained by Zu Chongzhi. Therefore, some historians of mathematics believe that Zu Chongzhi once used the method of constructing a regular polygon circumscribing a circle to obtain pi, which is a very reasonable conjecture.
However, according to the research of other mathematical historians, the numbers Ying and Z can also be obtained by calculating the side lengths of the regular 12288-sided polygon and the regular 24576-sided polygon inscribed in the circle. However, this kind of calculation is difficult to understand, so I won’t go into it here.
Although there are discrepancies in the statements, it is certain that Zu Chongzhi once obtained the "density" and clearly used the upper and lower limits to illustrate the range of the value of pi. Fifteen hundred years ago, he had such achievements and understanding, which is really worthy of our admiration.
Zu Chongzhi put in an unknown amount of hard work when calculating pi. If you start from the regular hexagon and count to 24576 sides, you have to repeat the same operation procedure twelve times, and each operation procedure includes more than ten steps such as addition, subtraction, multiplication, division and square root. It is extremely difficult for us to perform such calculations using pen and paper abacus. At that time, Zu Chongzhi could only use chips (small bamboo sticks) to perform such complicated calculations step by step. If your mind is not very calm and precise, and you do not have perseverance, you will never succeed. Zu Chongzhi’s tenacious and hard-working research spirit is very worthy of praise.
After Zu Chong's death, his son Zu Xuan (xuanxuan) continued his father's research and further discovered a method for calculating the volume of a sphere.
In the ancient Chinese mathematics work "Nine Chapters of Arithmetic", there was a formula for calculating the volume of a sphere, but it was very inaccurate. Although Liu Hui once pointed out its error, he did not find a solution to how it should be calculated. Jing Zuxun studied hard and finally found the correct calculation method. The formula he derived to calculate the volume of a sphere is: volume of a sphere = π/c D (D represents the diameter of the sphere). This formula is still used today.
Zu Chongzhi also wrote five volumes of "Zhu Shu", which is an extremely brilliant mathematics book and is very popular among people. In the arithmetic subjects of government-run schools in the Tang Dynasty, it was stipulated that students should study "Zhu Shu" for four years; when the government held mathematics examinations, most of the questions were from "Zhu Shu". Later this book was spread to Korea and Japan. Unfortunately, by the middle of the Northern Song Dynasty, this valuable work was lost. Until now it has remained to be investigated.