The mathematical theory of natural numbers or positive integers is called arithmetic. As for the names of many branches of mathematics, such as geometry and algebra, they came into being very late.
Foreign language books that systematically collate predecessors' mathematical knowledge were first written by Euclid of Greece. The book The Elements of Geometry consists of fifteen volumes, and the last two volumes are supplemented by later generations. Most of the books belong to geometry knowledge. The seventh, eighth and ninth volumes discuss the properties and operations of numbers, which belong to arithmetic.
The Latin word "arithmetic" comes from the Greek word "number and number technology (phonology)" and the word "suan" also means "number" in China, which means bamboo chips for calculation. In ancient China, all complicated numerical calculations used calculation. So "arithmetic" includes all the mathematical knowledge and calculation skills at that time. The earliest handed down Nine Chapters Arithmetic, the lost Xu Shang Arithmetic and Du Zhong Arithmetic are all discussing the solutions to various practical mathematical problems. The basis of arithmetic is that the addition and multiplication of integers obey certain laws. In order to describe these general laws, we can't use symbols like 1, 2, 3 to represent specific numbers. Two integers have the same sum, regardless of their order. but
1+2=2+ 1
This proposition is only a special case of this universal law. Therefore, when we want to show that a certain relationship between integers-regardless of some specific integer values involved-is correct, we can use the letters A, B, C, … as symbols to represent integers. Therefore, the five familiar arithmetic laws can be described as follows:
The first two are the exchange laws of addition and multiplication, which show that people can exchange the order of elements in addition or multiplication. The third is the associative law of addition, which shows that when three numbers are added, or we add the first one to the sum of the second and the third; Or we add the first and second to the third, and the result is the same. The fourth is the law of multiplicative association. The last one is the distribution law, which shows that when a sum is multiplied by an integer, we can multiply each term of the sum by the integer and then add these products. Arithmetic is a branch of mathematics, including natural numbers, properties of various operations, operation rules and practical applications. However, in the history of mathematical development, the meaning of arithmetic is much broader.
In ancient China, it was a bamboo computing device. Arithmetic refers to the technology of operating this computing device, and also refers to all the mathematical knowledge related to calculation at that time. The word arithmetic officially appeared in Nine Chapters of Arithmetic. The Nine Chapters Arithmetic is divided into nine chapters, namely Tian Fang and Su Mi, which are mostly practical names. For example, "square field" refers to the shape of land, and the calculation of land area belongs to the scope of geometry; "Millet" is a synonym for grain, which talks about the exchange of various grains, mainly involving proportion, and belongs to the arithmetic category. It can be seen that "arithmetic" at that time refers to the whole of mathematics, which is different from the modern meaning.
It was not until the Song and Yuan Dynasties that the word "mathematics" appeared. In the field of mathematicians, mathematics and arithmetic are often used together. Of course, the mathematics here only refers to the mathematics in ancient China, which is different from the mathematical system in ancient Greece, and it focuses on the study of algorithms.
From19th century, some western mathematics disciplines including algebra and trigonometry were introduced into China. Western missionaries mostly used mathematics, Japanese later used the word mathematics, and China still used "arithmetic" in ancient arithmetic. 1953, chinese mathematical society established a committee to examine mathematical terms, and defined the meaning of "arithmetic", but arithmetic and mathematics still coexist. 1937, Tsinghua University also had a "computing department". 1939, in order to unify, a special "mathematics" was determined. About the generation of arithmetic, we still have to talk about numbers. Numbers are used to express and discuss quantitative problems. There are different types of quantities and also different types of numbers. As early as the initial stage of ancient development, due to the needs of human daily life and production practice, the simplest concept of natural numbers came into being in the initial stage of cultural development.
One of the characteristics of natural numbers is that they are composed of inseparable individuals. For example, two things, a tree and a sheep, if two trees are in tandem; If there are three sheep, it is one, one after another. But you can't say that there are half trees and half sheep. Half a tree or half a sheep can only be counted as wood or mutton at best, but not as trees and sheep.
There are different relationships between numbers. In order to calculate these numbers, there are methods of addition, subtraction, multiplication and division. These four methods are four operations.
The oldest mathematics, arithmetic, is formed by accumulating and sorting out the properties of numbers and the experience of four operations between numbers in the application process.
During the development of the algorithm, many new problems are raised due to the needs of practice and theory. In the process of solving these new problems, ancient arithmetic has been further developed from two aspects.
On the one hand, in learning the four operations of natural numbers, it is found that only division is more complicated, some can be divided, some can be divided, some cannot be decomposed, some are greater than the common divisor of 1, and some cannot. In order to seek the laws of these numbers, it has developed into an independent branch of mathematics, called integer theory or elementary number theory, which specializes in the properties of numbers and is independent from ancient arithmetic, and has made new development in the future.
On the other hand, various types of application problems and various methods to solve these problems are discussed in ancient arithmetic. In the long-term research, it will naturally inspire people to seek general methods to solve these application problems. That is to say, can we find a universal and more universal method to solve the same type of application problems, so we invented abstract mathematical symbols, which developed into another ancient branch of mathematics, namely elementary algebra.
With the development of mathematics, arithmetic is no longer a branch of mathematics. What we usually call arithmetic is only a teaching subject in primary schools. The purpose is to enable students to understand and master the most basic knowledge about quantitative relations and spatial forms, correctly and quickly perform the four operations of integers, decimals and fractions, initially understand some of the simplest ideas in modern mathematics, and have preliminary logical thinking ability and spatial concepts. The specific content of modern primary school mathematics is basically the knowledge of ancient arithmetic, which means that many contents of ancient arithmetic and modern arithmetic are the same. However, there are differences between modern arithmetic and ancient arithmetic.
First of all, the content of arithmetic is the research object of ancient adults, including mathematicians, and it has become children's mathematics. Secondly, in modern primary school mathematics, the basic operation properties summarized for a long time are summarized, namely, the exchange law and associative law of addition and multiplication, and the distribution law of multiplication to addition. These five basic algorithms are not only the important properties of number operation learned in primary school mathematics, but also the main properties of the whole mathematics, especially algebra.
Thirdly, in modern primary school mathematics, the ideas of basic mathematical concepts such as set and function in modern mathematics are also bred. For example, the change of sum, difference, product and quotient, the corresponding relationship between numbers, ratio and proportion.
In addition, elementary school mathematics also includes decimal fraction and its four operations, which only appeared in16th century. It should be pointed out that the decimal fraction is not a new number, and it can be regarded as another way to write the fraction with the denominator of 10.
Modern algebra and number theory were originally developed from arithmetic. Later, the concepts of arithmetic and mathematics appeared, which replaced the meaning of arithmetic and included all mathematics, and arithmetic became a branch. So it can also be said that arithmetic is the oldest branch. Arithmetic is a masterpiece of Diophantine, a mathematician in the late ancient Greece. Original 13 volume. For a long time, it was thought that only the first six volumes of Greek manuscripts were found in Venice in 1464, and then four volumes of Arabic translations were found in Mashhad (northeast Iran).
Arithmetic is actually an algebraic work, which includes linear equations of one or more variables, quadratic indefinite equations and number theory. There are six volumes 189 questions, almost one question and one method, each with its own differences. Although later generations classified it into more than 50 categories, there is still no universal method to find. Moreover, many abbreviations are quoted in the book, such as the unknowns and their powers with symbols such as S, △r, Kr, △r△, △Kr, KrK, etc. In content and form, this completely divorced from geometry is quite different from the prevailing fashion of Euclidean geometry in ancient Greece at that time. Therefore, although Diophantine's arithmetic represented the highest level of ancient Greek algebra, it was far beyond its contemporaries and was not accepted by them, so it was quickly forgotten and did not have much influence on the development of mathematics at that time.
It was not until the15th century that arithmetic was rediscovered, which inspired a large number of mathematicians to greatly promote algebra. First of all, the French mathematician Pompeii realized the great value of arithmetic, and his compatriot Veda made a contribution to symbolic algebra inspired by Diophantine's abbreviated algebra. In the17th century, Fermat had a book Arithmetic in his hand, drawing pictures in the blank, and put number theory on the modern track. Indefinite analysis in arithmetic also has a far-reaching influence on modern mathematics. In different number domains, any problem involving the solution of indefinite equations is called Diophantine equation or Diophantine analysis.