It is worth noting that, on the basis of inscriptions on inscriptions by Oracle Bone Inscriptions in Shang Dynasty and Western Zhou Dynasty, people gradually learned to write on bamboo (or wood chips) and thread them into books with ropes. This is an early book. Bamboo pieces with words written on them are called Jane, or bamboo slips. A large number of mathematical achievements in the Spring and Autumn Period and the Warring States Period were handed down through bamboo slips.
Geometry and Logic The concept of geometry discussed by Mo Jing can be regarded as the first attempt to study China's mathematical theory. Mo Jing is a Mohist work headed by Mo Zhai (about 490 BC-405 BC), including optics, mechanics, logic, geometry and so on. It tries to apply formal logic to geometry research, which is the remarkable feature of this book. In this respect, it is the same as Euclid.
(1) "Flat, same height"-the height between two lines is equal, which is called flat. This is actually the definition of parallel lines.
(2) "Same length and same phase"-if two line segments overlap, it is called the same length.
(3) "middle, equal length"-the point with equal distance to both ends of the line segment is called the midpoint.
(4) "A circle is equal in length"-a figure with equal distance from the center of the circle is called a circle.
The definition of basic geometric figures such as point, line and surface is given in Mojing. The names of these figures are end, ruler and surface respectively. In the process of studying the line, Mohism clearly gave the definitions of "poverty" and "infinity": "Either being a ruler or being poor; That is, if you measure an area with a line segment, if you can reach the level less than a line from the edge, it is called poverty; If you never reach this state, it is called infinity.
There is also an important record in the Mohist Classic: "Small causes are not inevitable and do everything. One of the main reasons is inevitable. " In modern languages, major causes are "sufficient conditions" and minor causes are "necessary conditions". The distinction between primary cause and secondary cause is a very important event in the history of philosophy and mathematics.
Unfortunately, with the decline of Mohism, Mohism's mathematical theory died before it formed a system.
arithmetic
By the 4th and 5th centuries BC, fractions had been widely used in China, and some fractions had special names, such as "half", "less than half" and "more than half". Numerical system and four integer operations have been mastered skillfully, and there are simple fractional operations in the exam notes, such as (the original book is expressed in Chinese characters).
During the Spring and Autumn Period and the Warring States Period, "Jiujiuge" was a well-known common sense. Books such as Guanzi recorded the songs of 1999. The order is different from today's, from "9981" to "11". As for the order from "one for one" to "9981", it was in the Song and Yuan Dynasties.
Understanding of "infinity" in mathematics
The contradiction between finiteness and infinity is a pair of basic contradictions in mathematics. The deepening understanding of this problem has promoted the development of ancient and modern mathematics.
According to the Zhuangzi of the Warring States Period, Hui Shi once put forward that "nothing more than being a freshman; From the smallest to the smallest, it is called the little one. Among them "the big one" and "the small one" can be understood as infinity and infinitesimal. Small enough to have no interior is called infinitesimal. The famous proposition in the book "one foot pestle, half a day, inexhaustible" can be regarded as a "small" display. A foot-long stick, take half the first day, take the rest the next day, and you will never pick it up. That is, the first day, the second day and the nth day, no matter how big N is, it is always not 0, which embodies the idea that matter is infinitely divisible.
Like Zhuangzi, the issue of dividing things is also discussed in Mohist classics. But Mohism opposes the infinite division of matter. They think that if a line segment is divided into two halves (for example, the left is the front and the right is the back), the first half is kept, the second half (OB in Figure 4.4) is discarded, and then the second half (CO) of the first half is discarded, so that the remaining part is too small to be selected. That is, the first half of the line segment, the second half of the first half, the second half of the second half ... at the end, there will be an inseparable end, which is in the middle of the line segment, not at the edge (between CO's). The Mohist classic says: "The former is half-hearted, and the Jewish side is also; Obviously, this idea is consistent with modern limit theory. In mathematical analysis, the method of defining a real point on the number axis with interval sets is similar. Therefore, we can regard this division idea as the prototype of interval set principle, which contains the idea that points are the limit of infinite division of line segments.
The germination of combinatorial mathematics
Combinatorial mathematics is a branch of modern mathematics, but its idea can be traced back to ancient times. The Book of Changes in the Spring and Autumn Period contains the seeds of combinatorial mathematics.
The Book of Changes is one of the oldest books in China. It predicts good or bad luck through yin and yang divination. "-"is the yang, and "-"is the yin, which are collectively called "two instruments". Take two books at a time and arrange them in different order to generate "four images". Take three at a time to generate gossip (Figure 4.5); Take six at a time and sixty-four hexagrams will be produced. The arrangement of four images, human hexagrams and sixty-four hexagrams is equivalent to multiple arrangements in combinatorial mathematics: r is taken from n elements at a time, and * * * has nr arrangements. For example, take three of the two hexagrams at a time, and * * * has 23 = 8 permutations, which is gossip.
Shortly after German mathematician G.W. Leibniz (1646- 17 16) invented the binary system, he met the missionary J. Bouvet (1656-1730).
000 (Kun) 00 1 (earthquake) 0 10 (Kan) 0 1 1 (exchange)
100 (Chuang) 10 1 (Li)10 (Xun)11(dry)
Leibniz said that gossip is "the oldest monument in science circulating in the universe", and this invention "is really something that the German people should be grateful for", which led to his reverence for China's ancient civilization and ardent hope of coming to China. For various reasons, he didn't get his wish, so he asked someone to send his hand-cranked computer to China, which became a much-told story in the history of Sino-German relations.
Early Mathematical Tools-Calculation, Planning, Moment
Calculation is a small bamboo stick (also made of wood, bone or metal) used for calculation, and it is a calculation tool created by China people. In the Spring and Autumn Period and the Warring States Period, the use of calculation has been quite common, and there are many records in the book, such as "Mencius calculated by holding funds" (ten rounds), "Those who are good at calculating, don't count" ("Laozi"), and so on. 59660.6886868866 1
(The first row is vertical and the second row is horizontal. From right to left: one line is vertical, ten lines are horizontal, one hundred lines are vertical, one thousand lines are horizontal ... when it meets zero, it will be empty. For example, set 256 1 and set 308. The addition and subtraction in the calculation are similar to today's abacus, and can be added and subtracted from left to right. One number is lower than the other, and the last bit of the low order is aligned with the first bit of the high order (Figure 4.6 (1)). Multiply each digit in the lower order from left to right by the first digit in the upper order, place the product in the middle of the upper and lower digits (Figure 4.6 (2)), then remove the first digit in the upper order and move the lower order to the right (Figure 4.6 (3)). The middle layer is a dividend, called reality; The next level is divisor, which is called divisor.
Calculation occupies a very important position in the history of Chinese mathematics. For two thousand years, calculation has been the main calculation tool in China, and it was not until the Yuan and Ming Dynasties that it was gradually replaced by abacus.
The advantage of calculation is that it is simple and flexible, and complex calculation can be carried out with some small bamboo sticks. Its disadvantage is that the intermediate steps cannot be retained and it is not convenient to view. In addition, relying too much on calculation tools is not conducive to the symbolization and abstraction of mathematics. (Source: College)
Gauges and moments are two surveying and mapping tools. Gauges are compasses and moments are right-angle crutches, which are used to draw straight lines. There were pictographs of measuring instruments and moments in Oracle Bone Inscriptions in Shang Dynasty, so they appeared in Shang Dynasty at the latest. During the Spring and Autumn Period and the Warring States Period, these two tools were widely used in surveying and geometric drawing. (Refer to other websites)