Elementary functions and mathematical talk

1. Elementary functions include: algebraic functions and transcendental functions.

A function that can be represented by an analytic expression.

2. Algebraic functions:

3. Transcendence function:

Transcendental function (transcendental Functions) refers to a function in which the relationship between variables cannot be represented by finite times addition, subtraction, multiplication, addition, exponentiation, and root operation.

Transcendental functions are those that do not satisfy any polynomial equation with polynomial coefficients.

4. Single-and polynomial-type

The concept of single-item (monomial): The algebraic formula consisting of a number and a letter product is called a single type, and a single number or one letter is also called a single type (example: 0 can be regarded as 0 times a,1 can be regarded as 1 times exponent 0 letters, B can be regarded as B by 1).

Polynomial: An algebraic formula consisting of a plurality of individual elements called a polynomial (subtraction: minus one number equals the inverse of it). Each item in a polynomial is called a polynomial, and the highest number of these items is the number of times in the polynomial.

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1.
A letter is a variable
Plural:
Real numbers: Rational number, irrational number (algebraic number and transcendental number)
Algebraic formula: A formula derived from the algebraic operation of a number and a number of letters by a finite addition, subtraction, multiplication, addition, exponentiation, and Root, or a mathematical expression containing letters called algebraic
Rational formula: Integral, fractional, this algebraic formula for the letter is only limited to add, subtract, multiply, divide, and the whole number of the operation of the powers.
Irrational: The algebraic formula of a non-integral number of radicals or letters containing letters is called an irrational type.
Integral: Single-entry, polynomial
A single formula: an algebraic formula consisting of the product of numbers and letters or the product of letters and letters is called a single-item (monomial).
Coefficients and times, 2abc, number of times is 1+1+1=3
Polynomial: An algebraic formula consisting of a plurality of individual elements called a polynomial (subtraction: minus one number equals the inverse of it). Each item in a polynomial is called a polynomial, and the highest number of these items is the number of times in the polynomial.
Constant term: A constant is any number other than a letter, including positive and negative integers, plus and minus decimals, fractions, 0, and irrational numbers. π and E. Not a letter, but a constant term.
Fractional: Generally, if A and b represent two integers, and B contains letters, then A/b is called a fraction, where a is called a molecule and B is called the denominator. Fractions are a class of algebraic expressions that are different from the whole equation.
Integral and fractional are collectively referred to as rational expressions.
A formula with a square root and a letter under the root of the root is called an irrational formula.
Irrational and rational are collectively referred to as algebraic expressions. [2]
2. Algebra Operations:
In the set M with an operation, any two elements still get a definite element in M through this operation, which is called an "algebraic operation" of M.
For real-number sets, the arithmetic and the whole number of powers are algebraic operations.
Mentioning common divisor, common divisor, factoring
-Pass and the peace of
Least common multiple and greatest common divisor

3. Mathematical symbols:

4. Two-item and Yang Hui triangles

5. Elementary Functions

6. Mathematical Methods
Mathematical method is to use mathematical language to express the state, relationship and process of things, and to deduce, calculate and analyze to form the method of explaining, judging and predicting the problem.

The so-called method refers to the operational rules or patterns contained in the means, ways, and modes of action taken by people in order to achieve a certain purpose. Through long-term practice, we find many means, avenues or procedures for using mathematical ideas. The same means, avenues or procedures have been reused many times, and have achieved the intended purpose, become a mathematical method. Mathematical method is a method of scientific research by means of mathematics, that is, to express the State, relation and process of things in mathematical language, through derivation, calculation and analysis, in order to form the method of interpretation, judgment and prophecy.

The mathematical method has the following three basic characteristics:
One is a high degree of abstraction and generality;
The second is accuracy, that is, the rigor of logic and the certainty of conclusion;
The third is the universality and operability of the application.

The basic mathematical methods commonly used in high school mathematics can be broadly divided into the following three categories:
(1) Methods in logic. For example, analytical method (including inverse proof method), comprehensive method, anti-absurdity method, inductive method, exhaustive method (requires classification discussion) and so on. These methods not only obey the basic laws and laws of logic, but also have mathematical features in mathematics.
(2) General methods in mathematics. For example, modeling method, elimination method, descending method, substituting method, image method (also called coordinate method, often called the image method in algebra, in our future to learn the analytic geometry is often called coordinate method), comparative method (mathematics is mainly refers to the comparative size, which is different from the logic of the multi-faceted), the shrinkage method, And in the future to learn the vector method, the mathematical induction method (which is different from the incomplete induction in logic) and so on. These methods are extremely important and widely used.
(3) Special methods in mathematics. For example, the formulation method, the undetermined coefficient method, the elimination element method, the formula method, the Exchange element method (also called the Intermediate variable method), the Split-Item method (contains the mathematics idea which adds the auxiliary element realizes the normalization), the factorization various methods, as well as the parallel movement method, the folding method and so on. These methods also play an important role in solving some mathematical problems.

Regardless of natural sciences, technical sciences or social sciences, in order to obtain a more profound understanding of the quality of the objects studied, it is necessary to make a number of aspects of the characterization, which requires the help of mathematical methods. The requirements and possibilities of using mathematical methods are different for things of different nature and different complexity. In general, a science is really mature only when it achieves the ability to use mathematics. In modern science, the use of mathematics has become a measure of the degree of development of a science, especially to measure the maturity of its theory is an important symbol.
The key to the successful application of mathematical methods in scientific research lies in refining a suitable mathematical model for the problems to be studied, which can not only reflect the essence of the problem, but also make the problem be simplified to facilitate the mathematical deduction. The establishment of mathematical model is the scientific abstract process of concrete analysis of the problem, so we should be good at grasping the principal contradiction, highlighting the main factors and relations, leaving aside those secondary factors and relations. The process of building a model is a process of "simplifying and simplifying" and "making it easy". Of course, simplification is not unconditional, and reasonable simplification must take into account the range of errors that the actual problem can allow and the prerequisites for the required mathematical methods. For the same problem can establish different mathematical models, at the same time in the process of continuous testing, comparison, gradually sift out the optimal model, and in the application process continue to be tested and modified, so that gradually improve. Mathematical models abstracted from a particular problem often have some degree of universality, because a particular mathematical model can evolve into a common mathematical model for describing the same kind of phenomenon. There are generally two types of mathematical models that have been widely applied and fruitful: a class of deterministic models, which describe and study various inevitability phenomena, such as algebraic equations, differential equations, integral equations, difference equations, and so on, in which the change of things in such a model is subject to the determined mechanical regularity. Another class is called stochastic model, that is to describe and study various probability phenomena by probability theory and mathematical statistic method, the development and change of things in this kind of model shows as the stochastic process, obey the statistic law, and have many possible results. The inevitability and probability phenomena of the objective world are not completely separate. Some things are mainly manifested as inevitability phenomena, but when the influence of stochastic factors can not be neglected, it is necessary to introduce stochastic factors into deterministic model to form a kind of mathematical model such as stochastic differential equation. Since the 1970s, it has been found that in some deterministic models, such as some nonlinear equations that describe conservative systems or dissipative structures, there are no random factors attached to them, but they show "intrinsic randomness" within a certain range of parameters, i.e. stochastic behavior of bifurcation and chaos. The mechanisms of such phenomena and their mathematical problems have attracted the attention of mathematicians and scientists, and are currently being studied.
Mathematics itself is constantly developing, the study of the relationship between quantity and quantity is also getting deeper, new mathematical concept and new branch of mathematics are constantly appearing, and new mathematical methods are bred and sprouted accordingly. With the increasing penetration of mathematics into various sciences and the combination of various objects and problems, a variety of new mathematical models are being extracted and new mathematical tools are created. In particular, the use of electronic computers so that the mathematical method shows a new vitality, the so-called "mathematical experimental methods." The essence of this method is not to experiment on the actual object, but in its mathematical model "experiment", the operation of this "experiment" is to achieve a large number of numerical and logical operations on the computer. This makes it possible to complete a trial project that was difficult to carry out in the past because of the heavy workload. The development prospects of mathematical methods in this respect are considerable.

7. Mathematical Models

8. Mathematical Thinking
The mathematical thought refers to the real world space form and the quantity relation reflected to people's consciousness, through the thought activity produces the result.
9. Mathematical equations

10. Mathematical formulas

11. Mathematics: Basic Mathematics and Applied Mathematics
Basic mathematics is also called Pure mathematics, which specializes in the internal laws of mathematics itself. The algebra, Geometry, calculus, and probability theory, which are introduced in the textbooks of elementary and middle schools, belong to pure mathematics. A notable feature of pure mathematics is the temporary setting aside of specific content to study the quantitative and spatial forms of things in a purely form.

Mathematics can be divided into two categories: a class called Pure mathematics, and a class called applied mathematics.
The first large class of mathematics. It studies the intrinsic laws of the mathematical structure itself according to the needs of mathematics, or the possible future application, and does not require a direct connection with the practical problems of other disciplines.
The second largest class of mathematics. It focuses on applying mathematical tools to solve practical problems in work and life. In the process of solving the problem, the mathematical tools used are basic mathematics.

12. Meta-ontology expression form
Material Consciousness Value time
Mass movement of space quantity

Quantity is an abstraction of the amount of things in real life. The amount of (Events and objects).

13. Concrete things and abstract things

Elementary functions and mathematical talk