What was the status of arithmetic in ancient China in human development?

Ancient Chinese mathematics is actually an algorithm system based on calculation and operation. In the history of human civilization, the Chinese nation.

For more than 2,000 years, it has relied on an intuitive, symbolic, and operable calculator for a long time, which shows a representative tendency of ancient human mathematics.

It and ancient Greek mathematics represent two trends in the development of algorithms and deductions in ancient human mathematics.

There are two inevitable development directions for the algorithm system of calculation, one of which is to continue to create and develop the law of calculation and operation to solve problems on the basis of calculation and operation (which requires the promotion of practical problems and the accumulation of operation experience).

The second is that the tools of calculation and performance have been improved or innovated in the operation of the operation (this is similar to the change in the form of Western logical performance, that is, the change of strictness, formalization, and symbolization). Throughout human history, humans have been characterized by continuous improvement and innovation in any application tool.

The inconvenient and easy-to-change characteristics of the calculation arrangement and its operation will inevitably be continuously improved with the development of the calculation and performance. The abacus, which was widely used in the Song and Yuan dynasties and in the Ming Dynasty, is a milestone achievement in the improvement and innovation of the calculator itself in ancient Chinese mathematics.

Ancient Chinese mathematics is a mathematical model composed of algorithms centered on algorithms, and when algorithms form certain structural laws (such as the achievements of Song and Yuan Dynasty mathematics), people give high praise to this, but the results of the fundamental change of calculators (from calculation and calculation to abacus performance) are so plain, which is very difficult to correctly understand the algorithm system of ancient Chinese mathematics with calculators as operational tools.

From the analysis of the law of the development of mathematics in ancient China, it is an inevitable trend in the development of Chinese calculators, and it is a major progress in the algorithm system in the form of calculators. It is believed that the development of traditional Chinese mathematics was interrupted after the Song and Yuan dynasties, and that the abacus of the Ming Dynasty was only a kind of civil and commercial mathematics when the development of ancient Chinese mathematics was interrupted, then this at least shows that the important characteristics of ancient Chinese mathematics and its development laws have not been paid attention to by theoretical judgment.

Calculate the chips. According to the records of historical books and the discovery of archaeological materials, the ancient arithmetic chips are actually small sticks of the same length and thickness, generally 13-14cm in length, 0 2 0 3cm in diameter, mostly made of bamboo, but also made of wood, animal bones, ivory, metal and other materials, about 270 pieces for a bunch, put in a cloth bag, tied around the waist to carry around. When you need to count and calculate, take them out and put them on the table, on the kang, or on the floor.

Don't look at these are inconspicuous little sticks, but they have made great contributions in the history of Chinese mathematics. And their invention has also gone through a long historical development process.

In the calculation chip counting method, the number of units is expressed in two ways, vertical and horizontal, in which 1-5 are represented by the corresponding number of chips arranged in the vertical and horizontal way, and 6-9 is represented by the above calculation chips and the corresponding calculation chips below. When representing multiple digits, the single digit is vertical, the ten digits are horizontal, the hundreds are vertical, the thousand digits are horizontal, and so on. This notation follows the decimal system.

It is impossible to determine the date of the appearance of the abacus, but according to historical data, the arithmetic chip appeared in the late Spring and Autumn Period and the early Warring States period (722 BC and 221 BC) at the latest, and it was the most important calculation tool in China until the invention and popularization of the abacus.

The invention of arithmetic was gradually produced in the historical development of these counting methods. It is no longer possible to find out when it first appeared, but at the latest it was the Spring and Autumn Period and the Warring States Period; The use of arithmetic chips has become very common. As mentioned earlier, the arithmetic chips are small sticks of the same length and thickness, so how can these small sticks be used to represent various numbers?

So why are there two different pendulum methods, vertical and horizontal? This is because of the need for the decimal system. The so-called decimal system, also known as the decimal value system, has two meanings.

One is:"Decimal", that is, every full ten counts into a unit, ten into ten into ten, ten into hundred, ten hundred into thousands, ......The second is"The place-value system, i.e. the value represented by each number, depends not only on the number itself, but also on its position in the count. Such as the same number"2", put it in the single place to mean 2, put it in the 100 place to mean 20, put it in the 100 place to mean 200, and put it in the thousand place to mean 2000 ......In the text counting system of the Shang Dynasty in China, there was already a bud of the decimal value system, and when it came to counting and arithmetic, it was a standard decimal value system.

According to the ancient Chinese calculation rules, the expression method of counting the number of calculations is: vertical for single digits, horizontal for ten, vertical for hundreds, horizontal for thousands, and vertical ......for 10,000 digitsIn this way, from right to left, vertical and horizontal, and so on, you can use arithmetic chips to represent arbitrarily large natural numbers. Because of the vertical and horizontal transformations between its bits and bits, and the fixed pendulum of each bit, there is no confusion or misalignment.

There is no doubt that such an arithmetic notation is completely consistent with the modern decimal notation.

There is a record about the shape and size of the calculation chip in the Hanshu Chronicles: "The algorithm is made of bamboo, one minute in diameter, six inches long, and two hundred and seventy-one to form six goblets, which is a hand." "The Western Han Dynasty is generally a round bamboo stick with a diameter of centimeters and a length of about centimeters, and 271 chips are bundled into hexagonal bundles.

And the "Sui Book and Legal Chronicles" said: "It is calculated to use bamboo, which is two minutes wide and three inches long." The positive policy is three incorruptible, accumulating 216 pieces into six goblets, and the dry strategy is also.

Negative strategy four incorruptible, a total of 144 squares, Kun's strategy also. "By the Sui Dynasty, there were two kinds of arithmetic chips, triangular and quadrangular, to distinguish positive and negative numbers. It is about centimeters wide and centimeters long.

This shows that from the Han to the Sui Dynasty, the calculation chip changed from round to square, from long to short, so that it could be used. Wei Liu Hui's annotation "Nine Chapters of Arithmetic" said: "The positive is red, and the negative is black, otherwise it is different from evil and good."

and "Dream Creek Writings" volume 8 said: "The algorithm uses red chips and black chips to distinguish between positive and negative numbers." It can be seen that as early as before the Three Kingdoms, the Chinese arithmetic had used the red and black colors of the chips or the evil and positive (triangular and quadrangular shapes) to distinguish positive and negative numbers.

The rules of counting and counting were first contained in the "Sun Tzu Sutra": "All the methods of calculation, first know its position." One vertical and ten horizontal, hundreds of stiffs.

Thousands and ten look at each other, and ten thousand and hundreds are equivalent. "There are two ways to express numbers with chips: In ancient China, calculations were not only the four operations and squares of positive and negative integers and fractions, but also included a variety of calculations with specific chips.

Chinese economists not only use the different "bits" of chips to represent different "values", invent the decimal value notation, but also use the chips to arrange various relative positions on the board into a specific mathematical pattern to describe a certain type of practical application problem. For example, the formulas listed in the "Column Decay", "Profit and Fall" and "Equation" techniques describe the common proportional and linear problems in practice. The equations of Tianyuan, Quaternion, and Kaifang depict the problems of higher-order equations; The Great Derivation Technique is a special method designed for "multiplication". Formulas represent specific quantity relations in different positional relationships.

On the different "bits" specified in these chips, arbitrary numbers can be arranged (they take different values according to the actual problem), so that the ancient Chinese chips themselves have the nature of algebraic symbols. It can be considered a unique symbolic system. In ancient China, calculation was expressed in the form of algorithms, and had the characteristics of modeling and proceduralization.

China's calculations do not need to be calculated symbolically, and there is no need to retain the intermediate process of calculation, but only require the final solution to the problem through the gradual transformation of the formula. Therefore, the "technique" in ancient Chinese arithmetic is a programmed algorithm described by a set of "programming languages", and the arithmetic on which it is based is often implicit in the steps of calculus, playing the role of "self-evident and self-evident". It can be said that "allegorical calculation" is another characteristic of ancient calculation in the form of expression.

The calculation chip was originally created in China before the invention of the abacus and was the most effective calculation tool. The early development and sustained development of mathematics in ancient China benefited from fundraising.