elementary algebra
As the main content of middle school mathematics curriculum, the core content of elementary algebra is equation theory. The word algebra means "regression" in Latin. In elementary algebra, the theory of algebraic equations is extended from one-dimensional linear equations to two aspects: one is to increase the number of unknowns and investigate binary or ternary equations (mainly linear equations) composed of several equations containing several unknowns; The second is to increase the unknowns and investigate the quadratic equation or quasi-quadratic equation. The main content of elementary algebra was basically developed in16th century.
Ancient Babylon (BC19th century ~ BC17th century) solved the problems of first-order and second-order equations, and Euclid's Elements of Geometry (4th century BC) solved the second-order equations by geometric methods. In Nine Chapters of Arithmetic (A.D. 1 century), there are solutions of cubic equations and simultaneous equations of one degree, using negative numbers. In the 3rd century, Diophantine used rational numbers to find the solutions of linear and quadratic indefinite equations. China's celestial sphere technique (Ye Li Circle Sea Mirror), which appeared in the13rd century, is a numerical solution to the one-dimensional higher-order equation. Italian mathematicians discovered the solutions of cubic and quartic equations in16th century.
The development history of algebraic symbols can be divided into three stages. The first stage was before the third century, when the solution to the problem was written into a paper called narrative algebra, without abbreviations and symbols. The second stage is from the 3rd century to16th century, in which some common summation operations are simplified, which is called simplified algebra. One of the outstanding contributions of Diophantine in the third century is to simplify Greek algebra and create simplified algebra. However, in the following hundreds of years, narrative algebra was very common in other parts of the world except India, especially in western Europe, until15th century. The third stage is after16th century. The solution of the problem is mostly expressed by mathematical shorthand composed of symbols, which has no obvious connection with the content, and is called symbolic algebra. /kloc-In the 6th century, Introduction to Analytical Methods, the representative work of Vedas, made many contributions to the development of symbolic algebra. At the end of 16, Viette initiated symbolic algebra, which was improved into a modern form by Descartes.
The numbers "+"and "-"first appeared in math books and were written by Weidemann in 1489. But it is officially recognized by everyone. As a symbol of addition and subtraction, it started from Hojk in 15 14. 1540, Reckord began to use the present "=". It was not until 159 1 that Vedas were widely used in his works and gradually accepted by people. In 1600, harriot created the greater than sign ">" and the less than sign.
The extension of the concept of number is not entirely caused by solving algebraic equations in history, but it is customary to put it in elementary algebra in order to be consistent with the arrangement of this course. In the 4th century BC, the ancient Greeks discovered irrational numbers. In the 2nd century BC (Western Han Dynasty), China began to use negative numbers. 1545, Italian cardano began to use imaginary numbers. 16 14 years, Naipur of England invented logarithm. At the end of 17, the general concept of real index was gradually formed.
3. Advanced Algebra
In higher algebra, linear equations (that is, linear equations) developed into linear algebra theory; And-,quadratic equation develops into polynomial theory. The former is a branch of modern algebra, including vector space, linear transformation, type theory, invariant theory and tensor algebra, while the latter is a branch of modern algebra, which studies any degree equation with only one unknown number. Advanced algebra, as a university course, only studies their foundations.
The concept of determinant was first put forward by Guan Xiaohe (Japanese) in 1683. The most systematic exposition of determinant theory is Jacoby's book The Formation and Properties of Determinants written in 184 1. Logically, the concept of matrix precedes the concept of determinant; Historically, the order is just the opposite. Carlisle introduced the concept of matrix in 1855, and published the first important article "Research Report on Matrix Theory" in 1858.
/kloc-in the 0/9th century, people paid great attention to determinant and matrix, and there were more than 1000 articles on these two topics. However, they are not a great reform in mathematics, but a shorthand expression. But they have proved to be very useful tools.
The research of polynomial algebra begins with exploring the root formulas of cubic and quartic equations. 15 15, Philo solved the problem of solving the cubic equation simplified as missing quadratic term. 1540, Ferrari successfully discovered the algebraic solution of the general quartic equation. People are constantly seeking for the root formulas of fifth-order, sixth-order or higher-order equations, but these efforts have been in vain for more than 200 years.
1746, D'Alembert first gave the proof of "Basic Theorem of Algebra" (with some imperfections). This theorem asserts that every algebraic equation of degree n with real or complex coefficients has at least one real root or complex root. So generally speaking, algebraic equations of degree n should have n roots. 1799, 22-year-old Gauss wrote his doctoral thesis and gave the first strict proof of this theorem. 1824, 22-year-old Abel proved that the roots of all the coefficients of a general equation higher than 4 can't be its roots. In 1828, Galois, who was only 17 years old, founded the "Galois Theory", which contains the necessary and sufficient conditions for the equation to be solved by the root sign.