Arithmetic generally refers to the four operations of natural numbers and positive fractions. At the same time, as the content of modern primary school curriculum, it mainly introduces some simple application problems through counting and measurement. Although the main content of arithmetic is not difficult, it is the oldest branch of mathematics. After thousands of years, it gradually accumulated and solidified in people's consciousness as experience. Natural number is an abstract concept to meet the needs of calculation and counting in production and life. In addition to counting requirements, various quantities including length, weight and time are also calculated, so scores appear further. The development of modern elementary arithmetic operation method originated from India in 10 century or 1 1 century. Spread to Europe through Arabs. In the15th century, it was transformed into its present form. /kloc-In the middle of the 9th century, grassmann successfully chose a basic axiomatic system for defining addition and multiplication operations for the first time. As a result of logic, other propositions of arithmetic can be derived from this system. Later, piano further improved grassmann's system. The basic concepts of arithmetic and the rules of logical reasoning, based on human practical activities, profoundly reflect the objective laws of the world and form the most solid foundation for other branches of mathematics.
Elementary algebra is the evolution, popularization and development of ancient arithmetic. In ancient times, when arithmetic accumulated a wealth of solutions to quantitative problems, in order to find a more systematic and universal method to solve various quantitative relations problems, elementary algebra centered on the solution of equations was produced. So that for a long time, mathematicians have understood algebra as the science of equations and focused on the study of equations. That is to say, the theory and method of algebraic operation of numbers and words, more precisely, the theory and method of algebraic operation of polynomials, and its research method is computational.
To discuss the equation, first of all, how to express the actual quantitative relationship as algebraic expression, and list the equation according to the equivalence relationship. Algebraic expressions include algebraic expressions, fractions and roots. Algebraic expressions can perform four operations of addition, subtraction, multiplication and division, as well as multiplication and square root operations, and abide by the basic operational rules.
In the development of solving equation problems, the number system has been expanded. The concepts of integers and fractions discussed in arithmetic are extended to the scope of rational numbers, so elementary algebra can solve more problems. However, there are still some equations that have no solutions within the rational number range. Thus, the concept of number is once again extended to real numbers and further extended to complex numbers.
Then, will there still be an equation with no solution in the range of complex numbers? Do complex numbers need to be expanded? Don't! A famous theorem in algebra-the basic theorem of algebra shows that an equation of degree n has n roots. 1742 15 February 15, Euler clearly expounded the basic theorem of algebra in a letter, and the German mathematical prince Gauss gave a strict proof in 1799.
Based on the above description, the basic contents of elementary algebra are:
With the above basic contents, we can see that the study of elementary algebra is set up in the curriculum of modern middle schools. As the continuation and extension of arithmetic, the main problems are the finite algebraic operation of algebra and the solution of generating equations.
A brief history of solving algebraic equations;
Elementary algebra further develops in two directions: linear equations with many unknowns; Higher order equations with higher unknowns. The development of these two directions makes algebra develop to the stage of advanced algebra. Advanced algebra, as a general term for the development of algebra to an advanced stage, includes many branches. Higher algebra offered by universities now generally includes two parts: linear algebra and polynomial algebra.
The research object of advanced algebra is further expanded on the basis of elementary algebra, and new concepts including set, vector, vector space, matrix and determinant are introduced. These new concepts have similar operational characteristics to numbers, but their research methods and operational means are more abstract and complex. The operation of new objects is not always the basic algorithm of symbolic numbers. So algebra was brought into the algebraic system, including group theory, ring theory and field theory. Among them, group theory is a powerful tool to study the symmetry law of mathematical and physical phenomena, and it has also become the most common and important mathematical concept in modern mathematics, and has been widely used in other departments.
Basic contents of higher algebra
Polynomial can be regarded as a simple function, and its application is very extensive. The central problem of polynomial theory is the calculation and distribution of algebraic equation roots, also known as equation theory. The study of polynomial theory mainly lies in discussing the properties of algebraic equations and finding solutions.
The research contents of polynomial algebra include divisibility theory, greatest common factor, multiple factors and so on. Divisibility is very useful for solving algebraic equations. Solve the zero problem of polynomial corresponding to algebraic equation, the zero does not exist, and the corresponding algebraic equation has no solution.
The most important concepts in linear algebra are determinant and matrix. The concept of determinant was first put forward by Japanese mathematician Guan Xiaohe in the book Method of Solving Problems published in 1683, and it was described in detail. Leibniz was the first European to put forward the concept of determinant. 184 1 year, the German mathematician Jacobi summarized and put forward the system theory of determinant.
Determinant has certain calculation rules, which can be used as a tool to solve linear equations, and the solution of a linear equations is expressed as a formula, which also means that determinant is a number or an operation.
Because the determinant has the same number of rows and columns, the arranged table is square. Through the study of determinant, the theory of matrix is discovered. A matrix is an array, and the number of rows and columns is not required to be equal. Using matrix, the coefficients in linear equations can be formed into vectors in vector space; Based on the matrix theory, the structural problem of the solution of multivariate linear equations has been completely solved. In addition, matrices are widely used in mechanics, physics, science and technology.
Abstract algebra is also called modern algebra. One of its founders is Galois, who is known as a gifted mathematician. By studying the conditions satisfied by the existence of the root solution of algebraic equations, Galois gave a comprehensive and thorough answer, solved the problem that puzzled mathematicians for hundreds of years, and put forward "Galois Field", "Galois Group" and "Galois Theory" as the most important topics in modern algebraic research. Galois Group Theory is recognized as one of the most outstanding mathematical achievements in19th century. Galois group theory also gives a general method to judge whether geometric figures can be drawn with a ruler, which satisfactorily solves the problem of bisection of arbitrary angles and multiplication of cubes. More importantly, group theory has opened up a brand-new research field, replaced calculation with structural research, changed the way of thinking from emphasizing calculation research to using structural concept research, and classified mathematical operations, which made group theory develop rapidly into a brand-new branch of mathematics and had a great influence on the formation and development of modern algebra.
1843, Hamilton invented "quaternion" which does not satisfy the multiplicative commutative law. The following year, grassmann deduced several more general algebras. 1857, Gloria designed another noncommutative matrix algebra. These studies have opened the door to abstract algebra. In fact, many kinds of algebraic systems can be obtained by weakening or deleting some assumptions of ordinary algebra or replacing some assumptions with other compatible assumptions.
Founder and Theory of Abstract Algebra
The research object of abstract algebra is all kinds of abstract and axiomatic algebraic systems. Because algebra can handle objects such as vectors, matrices and transformations other than real numbers and complex numbers, and rely on their own laws of calculus, mathematicians sublimate and abstract their common points, reaching a higher level of abstract algebra, making it the common language of most contemporary mathematics. Abstract algebra itself contains many branches such as group, ring, Galois theory and lattice theory, and intersects with other branches of mathematics to produce new mathematical disciplines such as algebraic geometry, algebraic number theory, algebraic topology and topological groups.