Elementary algebra starts with the simplest one-dimensional linear equation. On the one hand, elementary algebra further discusses binary and ternary linear equations, on the other hand, it studies equations that are larger than quadratic and can be reduced to quadratic. Along these two directions, algebra discusses the linear equations with any number of unknowns, also known as linear equations, and also studies the univariate equations with higher degrees. This stage is called advanced algebra.
Advanced algebra is a general term for the development of algebra to an advanced stage, including many branches. Higher algebra offered by universities now generally includes two parts: linear algebra and polynomial algebra.
2. Advanced Mathematics
Refers to the object and method of mathematics, which is more complicated than elementary mathematics. Broadly speaking, all mathematics except elementary mathematics is advanced mathematics, and some call algebra, geometry, simple set theory and logic in middle school as intermediate mathematics, which is regarded as the transition between elementary mathematics in primary and secondary schools and advanced mathematics in universities.
Generally speaking, advanced mathematics is a basic subject formed by calculus, algebra, geometry and their intersection. The main contents include: series, limit, calculus, spatial analytic geometry and linear algebra, series, ordinary differential equations.
3. Probability theory and mathematical statistics
Probability theory and mathematical statistics are a unique and active branch of mathematics. On the one hand, it has a unique research topic, its own unique concepts and methods, rich content and profound results; On the other hand, it is closely related to other disciplines and is an important part of modern mathematics.
4. Theory of complex variable function
Complex variable function theory is a basic branch of mathematics, and its research object is complex variable function. The theory of complex variable function has a long history, rich content and perfect theory. It is widely used in many branches of mathematics, mechanics and engineering science. Complex numbers originate from finding the roots of algebraic equations.
The concept of complex number originated from finding the roots of equations. When finding the roots of quadratic and cubic algebraic equations, the square root of negative numbers appeared. For a long time, people couldn't understand this figure. However, with the development of mathematics, the importance of such numbers is increasingly apparent. The general form of a complex number is: a+bi, where I is an imaginary unit.
5. Analytic geometry
Analytic geometry was founded and developed by mathematicians such as Descartes and Fermat with the help of Cartesian coordinate system. It is a branch of geometry, which uses analytical expressions to study the relations and properties between geometric objects, also known as coordinate geometry.
Strictly speaking, analytic geometry does not use algebraic methods, but uses analytical formulas to study geometric figures. Analytic expressions can be algebraic or transcendental, such as trigonometric functions, logarithms, etc. Usually, the default algebraic expression only consists of four operations and the roots of finite steps, and transcendental operation generally does not belong to the research field of algebra.
6, abstract algebra
Abstract algebra, also known as modern algebra, came into being in19th century. Galois [181-1832] used the concept of "group" in1832, which completely solved the possibility of solving rooted algebraic equations. He was the first mathematician who put forward the concept of "group" and is generally called the founder of modern algebra.
He changed algebra from the science of solving equations to the science of studying algebraic operation structure, that is, he pushed algebra from elementary algebra to abstract algebra.
Abstract algebra includes group theory, ring theory, Galois theory, lattice theory, linear algebra and many other branches. Combined with other branches of mathematics, new mathematical disciplines such as algebraic geometry, algebraic number theory, algebraic topology, topological groups, etc. Abstract algebra is also one of the foundations of modern computer theory.
References:
Baidu encyclopedia-advanced algebra
References:
Baidu encyclopedia-advanced mathematics
References:
Baidu Encyclopedia-Probability Theory and Mathematical Statistics
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Baidu Encyclopedia-Theory of Complex Variable Functions
References:
Baidu encyclopedia-analytic geometry
References:
Baidu encyclopedia-abstract algebra