By the time of the Three Kingdoms and the Northern and Southern Dynasties, China's mathematical sciences had shone with dazzling light, and outstanding mathematicians Liu Hui and Zu Chongzhi in history appeared. These two immortal figures laid a solid foundation for mathematics in our country.
Let's talk about Liu Hui first, he is a native of Wei during the Three Kingdoms era. We know very little about his life and life due to limited information. His area of activity is roughly in the Shandong Peninsula and the northern part of Jiangsu. Liu Hui has been familiar with the "Nine Chapters of Arithmetic" since he was a child, and around the fourth year of Wei Chenliu's reign (263), he annotated the classic work of ancient mathematics in China, "Nine Chapters of Arithmetic", and did a lot of creative mathematical theoretical work, which had a great impact on the formation and development of China's ancient mathematical system, and occupied a prominent position in the history of mathematics.
The Nine Chapters of Arithmetic embodies the mathematical achievements of ancient China from the pre-Qin period to the Eastern Han Dynasty. But at that time, there was no way to print books, and such good books could only be copied by a pen. In the process of copying, it is inevitable that there will be many mistakes, and the original book is compiled in the form of a collection of problems, the text is too simple, and there is no scientific explanation of the theory of the solution. This state of affairs is a clear obstacle to the further development of the mathematical sciences.
Liu Hui's annotation of the Nine Chapters of Arithmetic largely made up for this major shortcoming. In the Nine Chapters of Arithmetic Notes, he brilliantly expounded the principles of various methods of solving problems, presented brief proofs, and pointed out the errors of individual solutions. What is particularly valuable is that he also did a lot of creative work and put forward many new theories that far exceeded the original work. It can be said that Liu Hui's mathematical theory work has laid a solid foundation for the establishment of a theoretical system of ancient mathematical science in China with a unique style.
Liu Hui's most important contribution in the "Nine Chapters of Arithmetic Notes" is the creation of the "circumcision", which established a rigorous theory and perfect algorithm for calculating pi, and created a new stage of pi research. Pi is the ratio of the circumference and diameter of a circle, it is an important data in mathematics, therefore, the calculation of its accurate value has important significance and contribution in theory and practice.
In the history of mathematics in the world, mathematicians in many countries have taken pi as an important research topic and have made great efforts to find its exact value. In a sense, the accuracy of the exact value of pi in a country's history can measure the development of mathematics in that country.
In the original work of "Nine Chapters of Arithmetic", the data from ancient times is used, that is, the so-called "diameter of three times a week" is taken as =3, which is very inaccurate. Later, Wang Fan (230 266) of the Three Kingdoms period adopted 31566, which is an improvement over the "Path One Week Three", is still not precise enough, and has no theoretical basis.
How can you calculate pi with a more precise level? Liu Hui pondered bitterly.
One day, Liu Hui walked out of the door to breathe fresh air in nature. In front of his eyes, the mountains stretched out continuously, as if the mystery of mathematical philosophy. Liu Hui's thoughts seem to enter the majesty of the crowd mountain, attesting to the incredible creation of nature. Liu Hui looked up, and there was a small temple on a towering peak in the distance, and he wondered if the temple of mathematics was the same as this temple, with beautiful and tortuous scenery.
A clanging sound caught Liu Hui's attention, and he walked towards the sound, which turned out to be a stone processing plant. Master stonemasons here are hewn square stones into cylindrical pillars.
Liu Hui was quite interesting, squatting beside the master stonemason and ** seriously. I saw a square stone, after the stonemason master cut off the four corners, it became an octagonal stone, and then removed the octagonal and became a hexagonal shape, so that a chisel and an axe were dried, and a square stone was processed into a smooth cylinder.
Liu Hui suddenly realized, immediately ran home, seriously on the ground analogy, the original square and circle can be transformed into each other, he divided a circle into equal 6 segments, connected these points to form a regular hexagonal in the circle, and then divided each arc in two, and then obtained a circle with a regular 12 sides, so that the endless division, you can get a regular "polygon" that is completely consistent with the circle.
Liu Hui pointed out that the area of the circle connected to the regular polygon is smaller than the area of the circle, but "the cut is fine, and the loss is small." Cut and cut, so that it cannot be cut, then it becomes one with the circumference, and nothing is lost. ”
This passage contains a preliminary idea of the limit, and the idea is very clear, which establishes a theoretical basis for the calculation of pi in ancient China.
Summarizing the above discussion, Liu Hui actually established the following inequality:
s2n＜s＜s2n+（s2n-sn）
Here s s is the area of the circle, s2n, sn is the area of the circle bordered by regular polygons, and n is the number of sides. Liu Hui used this method, counting from the circle with a regular 6-sided shape, the number of sides doubled in turn, until the area of the regular 192-sided polygon, and the approximate value of pi obtained is 157 50, which is equivalent to =&314。
He continued to calculate until he found the area of the regular 3072 polygon, and further obtained an approximation of 3927 1250, which is equivalent to =&31416。
3.14 and 31416 These two data were relatively accurate, and they were the most advanced data in the world at that time. Liu Hui also clearly summarized the law of addition and subtraction of positive and negative numbers, proposed the calculation procedure of the system of multivariate equations, demonstrated the principle of finding the greatest common divisor, and also studied the algorithm of the least common multiple. These are all creative achievements, so it can be said that Liu Hui enriched and perfected the ancient Chinese mathematical science system through the annotation of the Nine Chapters of Arithmetic, and laid the foundation for the development of mathematics in later generations.
Liu Hui's "Heavy Difference" was originally the tenth volume of the "Nine Chapters of Arithmetic Notes", and was later published separately and was called "Island Arithmetic". It is a work that illustrates the methods of measurement and calculation of various heights or distances. It is a work on geometric measurements.
Once, Liu Hui and his friends went for a walk on the beach, and Liu Hui looked up and saw that it was a magnificent and tranquil, boundless blue sea. It was in the distance, connected to the pale blue clouds. The breeze caressed the sea's satin-like chest lovingly, and the sun warmed it with its own warm rays. And the sea, wheezing sleepily under the gentle power of these caresses, fills the boiling air with the smell of evaporating salt.
The pale green waves ran onto the yellow sand, throwing snow-white foam, kissing the feet of Liu Hui and his friends, Liu Hui was relaxed and happy, and simply sat on the beach, letting the brackish sea water wet his trouser legs.
At this time, a friend pointed to an isolated island in the middle of the sea and asked, "Who knows how tall the island is?" How far? Another friend thought for a moment: "As long as I have a small boat and enough rope, I can get out of the distance and height of the island." ”
Everyone laughed out loud, how much rope is needed, and even if you are given a rope, you can't measure the distance and height of the island. Because the rope is elastic, while the islet has a slope. Besides, it's too stupid.
At this time, Liu Hui was silent on the side, and someone asked him to express his opinion. Liu Hui said: "I don't need to go to the island at all, I only need two bamboo poles to measure its height and distance." ”
The friends looked at Liu Hui with wide eyes and stunned, and Liu Hui saw that his friends didn't believe him, so he drew a picture on the beach. Then he explained: "It is sufficient to erect two poles of the same length, Gh and EF, vertically on the shore so that they are in the same direction as the island AB, and then mark points C and D on the ground in line with the top of the two poles E, G and the tip of the island A."
In this way, we can measure the length of CF, DH, HF, EF, and now calculate the distance of the island bf and the height of the island ab, and the result calculated by Liu Hui is:
ab=( ef* hf)/(dh-hf)+ef
bf =(cf* hf)/(dh -cf)
We will not repeat the specific calculation, if the reader is interested, you may wish to give it a try to prove Liu Hui's formula.
Liu Hui said in the preface to the "Notes on Nine Chapters of Arithmetic": "Things are similar, and each has its own benefit. Therefore, although the branches are divided, and those who are the same as the stem, know only one end. "Liu Hui's research methods and research results have had a very profound impact on the development of ancient mathematics in China, and have added a glorious page to the history of mathematical science in China.
In recent years, many kinds of monographs and monographs on research have been published at home and abroad, and his "Notes on Nine Chapters of Arithmetic" and "Island Arithmetic" have been translated into the languages of many countries, showing the world the splendid ancient civilization of the Chinese nation.
In the 200 years after Liu Hui, another great scientist Zu Chongzhi appeared during the Northern and Southern Dynasties of China. He believes that Liu Hui's circumcision technique only stopped when he only counted the regular 3072 polygons, and the results obtained were still not accurate enough.
If we can cut and cut on the basis of Liu Hui's 3072 polygonal shape and make a polygonal shape, wouldn't we be able to find a more accurate pi?
Zu Chongzhi was not satisfied with the achievements of his predecessors and decided to climb new peaks. Through long-term hard study, with the assistance of his son Zu Xuan, he repeatedly calculated, and finally found a higher accuracy of pi.
The Book of Sui and the Chronicles of the Law record his achievements:
At the end of the Song Dynasty, Southern Xuzhou engaged in Shi Zu Chong's more open secret method, with a circular diameter of 100 million as a zhang, a circumference of 3 zhang 1 foot 4 inches 1 minute 5 centimeters 9 millimeters 2 seconds 7 suddenly (31415927 zhang), 3 zhang 1 ft 4 inch 1 minute 5 centimeters 9 millimeters 2 seconds 6 hu (31515926 zhang), the positive number is between the yingyu. Secret law: diameter 113, circumference 355. Covenant: Circle diameter 7, Week 23. ”
Judging from the above written records, Zu Chongzhi's contribution to pi has 3 points:
1.Calculate that pi is at 31415926 to 31415927, i.e. 31415926＜π＜3.1415927, for the first time in the history of mathematics in the world, pi was estimated to 7 decimal places.
It was not until 1,000 years later, when the 15th-century Arab mathematician Al Qasi calculated to 16 decimal places, breaking Zu Chongzhi's record.
2.Zu Chongzhi clearly pointed out the upper and lower limits of pi, using two fixed numbers with high accuracy as boundaries, accurately stated the size range of pi, and actually determined the error range, which is unprecedented.
3.Zu Chongzhi proposed an approximate rate of 20 7 and a dense rate of 355 113. This density value is the first time in the world that it has been proposed, so some people advocate calling it "ancestral rate". In Europe, the German Otto and the Dutchman Antoniz achieved this result in the 16th century.
How did Zu Chongzhi arrive at this result? He should calculate from the circle with regular 6-sided, 12-sided, and 24-sided sides all the way to 12288 and 24576 sides, and find their side lengths and areas in turn.
This requires addition, subtraction, multiplication, division, and squared operations on large numbers with 9 significant digits, a total of more than 100 steps, of which nearly 50 times are multiplied and squared, and the significant figures reach as many as 17 digits.
At that time, there was no paper, pen and number for number crunching, but outdated algorithms. Through the calculation of small bamboo sticks in vertical and horizontal directions, it can be seen how arduous the labor Zu Chongzhi has paid and how serious and serious the spirit he needs to have.
Zu Chongzhi and his son Zu Yu also used ingenious methods to solve the problem of calculating the volume of the ball. Before them, the calculation of the area of a circle and the volume of a cylinder had been correctly solved in the Nine Chapters of Arithmetic. But in this book, the formula for calculating the volume of a sphere is wrong. Although Liu Hui pointed out this error in the Nine Chapters of Arithmetic Notes, he also failed to find the formula for calculating the volume of the sphere.
200 years later, Zu Chongzhi and his son continued Liu Hui's work and derived the correct formula for the volume of a sphere for the first time in the history of mathematics in China. It is worth noting that in the process of estimation and verification, Zu Wei came to the conclusion that "if the cross-sectional area at the same height is equal, then the volume of the two three-dimensional dimensions must be equal". This question was only proposed 1000 years later by the Italian mathematician Cavalielli, which is known as the "Cavalielli theorem", when in fact we have every right to call it the "Zu Huang theorem".
The research results of Zu Chongzhi's father and son were collected in a work called "Fixation", which was designated as one of the "Ten Sutras of Calculation". It is a pity that after the Song Dynasty, this great work was lost. Zu Chongzhi's scientific achievements will always shine in the history of the development of science and technology in our country and even in the world. In honor of this great scientist, a valley on the far side of the moon was named "Zu" internationally.
Chongzhi", it can be seen that people respect Zu Chongzhi.