[Exploring the origin of scientific and technological civilization]
The seemingly simple numbers in today's eyes are actually the wisdom crystallization of human mathematical development for tens of thousands of years. How to express numbers, record numbers, and which base system to use directly determine the way of mathematical operation, application, and even record inheritance. Arabic numerals have the advantages of clear carry and easy writing, and have become the universal counting symbols in the modern world.
In fact, Arabic numerals were invented by Indians, and got their name because they were introduced into Europe by Arabs. Academics call it Indian Arabic numerals. Prior to the Indian invention, China had developed a practical and perfect numeration system - Arithmetic. Zou Dahai, a researcher at the Institute of Natural Science History of the Chinese Academy of Sciences, said that arithmetic was a commonly used computing tool in ancient China, also called arithmetic, strategy, strategy, operator, etc. Counting, calculating and the corresponding method by using the calculation plan are called preparation. Calculation is the characteristic of ancient Chinese mathematics.
In ancient times, Chinese mathematics was the representative of East Asian mathematics and had distinctive characteristics in the history of world mathematics. Zou Dahai believes that, compared with ancient mathematics in other regions, ancient Chinese mathematics is good at calculation and algorithm. One of the main reasons is that preparation is widely used, which not only produces advanced numeration methods, but also has a set of methods to express unknown quantities and algebraic expressions. It realizes the function of algebraic symbols in a unique way.
Decimal notation first appeared in China
Base system is one of the basic concepts of mathematics, which means "accumulating several basic units will form the next unit". People are familiar with the decimal system, including the most common decimal system, the hexadecimal system for time and angle measurement, and the binary system and hexadecimal system commonly used in the computer field.
The concept of place value system was developed on the basis of the decimal system, that is, the position of the number can represent the corresponding multiple of the number unit. For example, in the Arabic numeral 5234, the position of 5 is the "thousand" digit, representing five thousand, while 2, 3, and 4 are in the "hundred", "ten", and "several" digits, representing two hundreds, three tens, and four ones, respectively. Many numeration systems do not have the concept of place value system. For example, ancient Egyptian numerals are very cumbersome because they do not have the concept of place value system.
Zou Dahai introduced that the decimal system was first seen in ancient Egypt; The position value system was first seen in the ancient Lianghe River Basin, where people adopted the position value system of 60 degrees; The modern decimal numeration system originated in China.
The Chinese people have long adopted the decimal system, and the number notation of Chinese characters is the decimal system. The first is not only the basic number, but also the number unit. In addition, the common number units are 10, 100, 1000, 10000, 100 million, trillion, Beijing, etc.
For example, the number "one trillion eight hundred five hundred three hundred twenty-one million eight thousand eight hundred fifty one", as long as the empty digit is replaced with zero, and the number units of trillion, hundred million, ten thousand, hundred, ten, etc. are removed, it will become "one hundred eighty five three hundred eighty eight five one", which corresponds to the modern "1853020188851". It can be seen that the Chinese character notation is quite advanced and can be easily transformed into the popular Arabic numeral notation. However, the Chinese character representation is only a quasi decimal value system and is not yet perfect. The real decimal value system is the arithmetic notation. Before the 14th century AD, the abacus was a widely used computing tool, and then it was gradually replaced by the abacus.
The image of Suanzhou is similar to chopsticks, which originates from the ancient people's habit of counting with long sticks. One to nine can be expressed in vertical and horizontal forms by means of arithmetic chips.
The Xiahouyang Arithmetic Sutra says: "When the number is more than six, five is above, six is not accumulated, and five is not single." It means that for one to five, the number is expressed by several pieces of arithmetic; For six to nine, put the five contained in them on the top and the rest on the bottom vertically. For example, ╥ represents seven, the upper one represents five, and the lower two represent two.
The Sun Tzu Suan Jing records the calculation method of many digit numbers: "When you calculate, you should first know the position. One vertical and ten horizontal, hundreds of feet stand in a thousand stiff, thousands of faces face each other, and thousands of feet are equal." The ancients placed vertical and horizontal forms in the individual position, and the hundred used vertical forms, and the thousand used horizontal forms, so the vertical and horizontal patterns were staggered. Zou Dahai stressed that it was enough to use only one of the two methods, but the crisscross method was clearer and not easy to confuse. The method of counting and preparing numbers probably came into being in the Western Zhou Dynasty, the Spring and Autumn Period, or even earlier. The traces of this notation can be seen on the unearthed pottery pieces and coins of the Pre Qin period.
Calculation has powerful counting and calculation functions
The calculation of addition and subtraction is relatively simple, and the ancients did not record it. For the complex multiplication and division method, the ancients put the arithmetic plan into three lines, the middle line is the product or divisor, the lower line is the multiplier or divisor, and the upper line is the multiplicand or quotient. The results can be obtained by combining the 99 formula with the counting method. This process is essentially close to modern vertical multiplication and division.
Preparation can also solve linear equations. In the book Nine Chapters of Arithmetic written in the first century BC, a method called "equation" was used to express problems and solve them. Its expression and operation method are very similar to modern augmented matrices. The augmented matrix method for solving linear equations only appeared at the turn of the 18th and 19th centuries in Europe. Zou Dahai introduced that different positions in the equation have the function of indicating different unknowns and constant terms, which is equivalent to the modern separation coefficient method. It is the equation that leads to the concept of positive and negative numbers. In ancient times, black and red arithmetic chips were used to represent negative and positive numbers respectively.
"Nine Chapters of Arithmetic" also uses preparation to square and cube, as well as to solve quadratic equations with one variable. This method developed continuously, and by the Song Dynasty, it had formed a set of methods to find the numerical solution of higher order equations with one variable. In the Song and Yuan dynasties, a general method for listing equations was also developed. At first, Tianyuan technique for solving one unknown quantity appeared. Later, it developed to the formulation and solution of high-order equations with up to four unknowns - quaternion technique, which uses heaven, earth, people and things to indicate the unknown quantity. Zou Dahai stressed that, except for Suanchi and individual Chinese characters "Tai" or "Yuan", the formulas listed in this method do not use symbols, but reflect the unknown quantity and its power through position, realizing the function of algebraic symbols, which is even more advantageous than modern symbolic methods in some aspects.
In ancient China, mathematical work was carried out and inherited through the cooperation of words and calculations. With its powerful functions of counting, symbol and operation, planning played an important role in the development of ancient Chinese mathematics.
Later, with the development of business, the calculation method and corresponding pithy formula developed, and then the calculation was replaced by abacus. Abacus plays a huge role in business, accounting and daily life by using and improving the formula and method of calculation.
But planning also has its limitations. Zou Dahai pointed out that one of the shortcomings of the calculation is that the calculation process cannot be retained, and it is difficult to trace back if there are errors in the intermediate operation. Second, it takes up more space. Third, it is difficult to achieve more than quaternion expressions and operations.
Planning to influence Chinese traditional culture
In the Ming Dynasty, Tianyuan and Siyuan techniques were lost, and planning was replaced by abacus and pen calculation. However, their methods and ideas still played a role in the development of mathematics, and even penetrated into all aspects of Chinese ideology and culture.
Zou Dahai stressed that the importance of planning lies in the formulation and use of a series of rules and methods. Planning mainly depends on intelligence, and is especially precise. Therefore, in Chinese traditional culture, many expressions reflecting intellectual activities are related to planning. For example, planning is called "planning" and "planning". It means that the commander plans well in the rear to make the front win. Another example is that thorough consideration is called "no last resort" and "having a plan in mind", while planning mistakes are called "miscalculation" and "blunder". Other words, such as "calculating", "plotting", "planning", "fortune telling" and "winning", are all derived from ancient planning thinking.
"The ancients fully explored the position function in the numeration method of arithmetic chips, used arithmetic chips to express algebraic expressions, and designed corresponding operation methods, which formed the unique tradition of calculation in ancient China. The ancient Chinese arithmetic and algebra were in an advanced or even leading position in the world, which had a lot to do with calculation. Planning is an important part of Chinese culture, and the thinking of planning also has a certain impact on social culture and people's lives. " Zou Dahai said. (Source: Science and Technology Daily; Reporter Sun Mingyuan)
