Algebra started from the general problems of higher algebra and developed into a large-scale mathematical discipline including many independent branches, such as polynomial algebra and linear algebra. The research object of algebra is not only numbers, but also the transformations of matrix, vector and vector space, which can be operated. Although it is also called addition or multiplication, the basic arithmetic of numbers is sometimes no longer effective. Therefore, the content of algebra can be summarized as studying some sets with operations, which are mathematically called algebraic systems. Such as groups, rings, domains, etc.

Polynomial is the most common and simplest function, and it is widely used. Polynomial theory is based on the calculation and distribution of roots of algebraic equations, which is also called equation theory. The study of polynomial theory mainly lies in discussing the properties of algebraic equations, so as to find a simple method to solve the equations.

The research contents of polynomial algebra include divisibility theory, greatest common factor, multiple factors and so on. These are basically the same as middle school algebra. The divisibility of polynomials is very useful for solving algebraic equations. Solving an algebraic equation is nothing more than finding the zero of the corresponding polynomial. When the zero point does not exist, the corresponding algebraic equation has no solution.

We know that a linear equation is called a linear equation, and the algebra that discusses linear equations is called a linear algebra. Determinants and matrices are the most important contents in linear algebra.

The concept of determinant was first put forward by Japanese mathematician Guan Xiaohe in17th century. 1683, he wrote a book called Method of Solving Problems, the title of which means "Determinant Method of Solving Problems". The concept of determinant and its development have been clearly stated in the book. The concept of determinant was first put forward by German mathematician Leibniz in Europe. Jacoby, a German mathematician, summarized and put forward the system theory of determinant in 184 1.

Determinant has certain calculation rules, and the solution of a linear system of equations can be expressed as a formula by determinant, so determinant is a tool for solving linear systems of equations. Determinant can represent the solution of linear equations as a formula, that is, determinant represents a number.

Because the determinant requires the same number of rows and columns, the arranged table is always square. Through the study of it, the theory of matrix is discovered. A matrix is also a table of numbers in which the numbers are arranged in rows and columns. The number of rows and columns can be equal or different.

Matrix and determinant are two completely different concepts. A determinant represents a number, while a matrix is just an ordered arrangement of some numbers. Using the tool of matrix, the coefficients in linear equations can be formed into vectors in vector space; In this way, a series of theoretical problems such as the solution of a multivariate linear equation system and the relationship between different solutions can be completely solved. Matrix is widely used in many aspects, not only in the field of mathematics, but also in the fields of mechanics, physics, science and technology.

Combined with the above analysis, the basic content of elementary algebra is:

Three Numbers-Rational Number, Irrational Number and Complex Number

Three forms-algebra, fraction and root.

The central content is equation-integral equation, fractional equation, radical equation and equation.

The content of elementary algebra is roughly equivalent to the content of algebra courses offered in modern middle schools, but it is not exactly the same. For example, strictly speaking, the concept, arrangement and combination of numbers should be classified as the content of arithmetic; Function is the content of analytical mathematics; The solution of inequality is a bit like the method of solving equations, but inequality, as a method of estimating values, essentially belongs to the category of analytical mathematics; Coordinate method is the study of analytic geometry. These are just a sorting method formed in history.

Elementary algebra is the continuation and expansion of arithmetic, and the research object of elementary algebra is algebraic operation and equation solving. Algebraic operations are characterized by a limited number of operations. All elementary algebra has ten rules. This is the key point to understand and master when learning elementary algebra.