China's ancient idea of infinitesimal division is not only one of the achievements of China's classical mathematics, but also contains profound philosophical truth, which plays an active role in the process of people discovering, analyzing and solving practical problems.
Liu Hui introduced infinitesimal division into mathematical proof for the first time in human history, which is the most wonderful embodiment of ancient infinitesimal division thought in mathematics.
According to legend, a long time ago, there was a river god in the Yellow River. People call him Hebo. Hebo stood on the bank of the Yellow River. Watching the rolling waves coming from the west and rushing eastward, he said excitedly, "The Yellow River is so big that no river in the world can compare with it." I am the biggest water god! "
Someone told him, "What you said is wrong. There is a place east of the Yellow River called Bohai Sea. That's really big! " "Herb said," I don't believe this. No matter how big the Bohai Sea is, can it be bigger than the Yellow River? "The man said," Don't mention a Yellow River. Even if several Yellow Rivers flow into the Bohai Sea, they can't be satisfied with it. " Herb stubbornly said, "I have never seen the Bohai Sea. I don't believe it. "The man was helpless and told him," If you have a chance to see the Bohai Sea, you will understand what I said. "Autumn has arrived, and the torrential rains for several days have caused large and small rivers to flood into the Yellow River. The surface of the Yellow River is wider. Looking across the river, you can't tell the cattle and horses on the other side. As a result, Hebo is even more proud that the most spectacular scenery in the world is in his own place. While he was smug, he remembered the Bohai Sea that someone told him, so he decided to go there and have a look.
Hebo follows the flowing water eastward and reaches the Bohai Sea. His face faces east and he can't see the water. I saw the vast sea, soaring into the sky, and I was greatly shocked.
Hebo had put away his happy face, looked at the ocean and sighed at the Bohai Sea: "Now I see your vastness. If I don't come to your door, it will be dangerous, because I will always be laughed at by people who know the truth. "
When Hebo said this, Bohai God knew that his understanding had improved, so he planned to answer some of his questions.
There is a paragraph like this.
Hebo asked: "People in the world always say,' The smallest thing has no form to find, and the largest thing has no definite scope'. Is this true and credible? "
Bohai God replied: "From a small point of view, a huge thing can't be comprehensive. From a big point of view, a tiny thing can't be true. Fine is small in the middle; Huge, it's the biggest. Although different in size, each has its own appropriateness. This is the inherent situation of things. "
"The so-called fine and thick, limited to tangible things, as for intangible things, can't be divided by number; Things that cannot be limited in scope cannot be accurately calculated by quantity. "
The above story is selected from "Zhuangzi Qiushui", which is called "the first wonderful book in the world". This article is recognized as the first paragraph in Zhuangzi. Because the most wonderful thing about this article is Zhuang Yang's arbitrariness and elegance.
In fact, the essence, invisibility and inseparability mentioned in this dialogue can be regarded as the author's interpretation of the infinitesimal division thought at that time through the dialogue between the river god and Poseidon.
As early as the pre-Qin and Western Zhou Dynasties in China, the mathematician Shang Gao also discussed the relationship between circle and square with Duke Zhou. In "Shu Jing Zhou", Shang clearly answered the question.
Since the circle comes from the square, why can't the circle return to the square? This is because the world did not understand the principle that "the circle is square" and wrongly determined the pi.
Shang Gao's "Fiona Fang method", that is, the method of finding a circle in a square, is permeated with dialectical thinking. "Everything is round and round" means that you can know the world in a round way; "Masters make systems and rules set" means that producers must use rules to make goods.
It can be seen that "Fang" contains the understanding of the spatial form and quantitative relationship of the real world, while "number method comes from Fang" means that the object of mathematical research is Fang, that is, the world, and mathematical methods come from Fang. In other words, mathematical methods come from the understanding of nature.
"Destroying a square is a circle and breaking a circle is a square" means that the contradiction between a circle and a square can be transformed into each other through "destroying" and "breaking". It is considered that "there is a circle in a square" or "there is a square in a circle", that is to say, "circle" and "square" are the unity of opposites.
This is the "circle theory" of Shang Gao. It emphasizes the flexible application of mathematical thinking, thus revealing the role of human intelligence and mathematical thinking in learning mathematics. Knowing the circle, people began to calculate the circle, especially the area of the circle.
The "contention of a hundred schools of thought" in the Warring States period also promoted the development of mathematics, especially the dispute of rectifying the name and some propositions directly related to mathematics.
Famous scholars believe that abstract noun concepts are different from their original entities, and they put forward that "it is impossible to be square if the moment is not right;" If the rules are not correct, it will not be a circle, and the circle can be infinitely divided.
Mohism believes that names come from things, and names can reflect things from different sides and depths. Mohist school gave some mathematical definitions, such as circle, square, flat, straight, secondary and end.
Mohism disagrees with the proposition that the circle can be infinitely divided, and puts forward the proposition of "non-half" to refute: if a line segment is infinitely divided into two halves, there will be a "non-half" that can no longer be divided, and this "non-half" is a point.
The famous scholar's proposition discusses that a finite length can be divided into an infinite sequence, while the Mohist proposition points out the changes and results of this infinite division. The discussion on the definition and proposition of mathematics by famous scholars and Mohists is of great significance to the development of China's ancient mathematical theory.
Han Sima Qian's Historical Records? Biography of Corrupt Officials is a metaphor for the abolition of Qin criminal law in Han Dynasty. Breaking a circle contains a simple idea of infinitesimal division, which is roughly summarized by Sima Qian from the practice of turning a square into a circle and a straight line into a curved shape when a craftsman processes a circular object.
These propositions about "division" have a far-reaching influence on the idea of infinitesimal division in mathematics later.
China's ancient mathematical classic "Nine Chapters Arithmetic" wrote in the first chapter "Square Field" that "product step of semi-circle radius" is the formula we are familiar with now.
In order to prove this formula, Liu Hui, a mathematician in Wei and Jin Dynasties, wrote nine chapters of arithmetic notes, and wrote more than 1800 words after this formula. This note is the famous "cyclotomy" in the history of mathematics.
Liu Hui reprocessed the data cut into 192 polygons with "difference power". Through simple operation, he can get a high-precision result of 3072 polygons, and the extra calculation is almost negligible. This is the most wonderful embodiment of ancient infinitesimal division thought in mathematics.
Liu Hui introduced infinitesimal division into mathematical proof for the first time in human history and became an immortal chapter in the history of human civilization.