Real number:
Rational numbers and irrational numbers are collectively called real numbers.
Rational number:
Integers and fractions are collectively called rational numbers.
Irrational number:
Irrational numbers refer to infinite cyclic decimals.
Natural number:
The numbers 0, 1, 2, 3, 4 ~ (including 0) representing objects are all called natural numbers.
Number axis:
The straight line that defines the point, the positive direction and the unit length is called the number axis.
Countdown:
Two numbers with different symbols are opposite.
Countdown:
Two numbers whose product is 1 are reciprocal.
Absolute value:
The distance between the point representing the number A and the point on the number axis is called the absolute value of A, the absolute value of a positive number is itself, the absolute value of a negative number is its inverse, and the absolute value of 0 is 0.
Mathematical theorem formula
Rational number arithmetic
(1) addition rule: add two numbers with the same symbol, take the same symbol, and add the absolute values; Add two numbers with different signs, take the sign of the addend with larger absolute value, subtract the smaller absolute value from the larger absolute value, and add two numbers with opposite numbers to get 0.
(2) Law of subtraction: subtracting a number is equal to adding the reciprocal of this number.
(3) Multiplication rule: two numbers are multiplied, the same sign is positive and the different sign is negative, and the multiplication takes the absolute value; Multiply any number by 0 to get 0.
(4) Division rule: dividing by a number is equal to multiplying the reciprocal of this number; Divide two numbers, the same sign is positive and the different sign is negative, and divide by the absolute value; Divide 0 by any number that is not equal to 0 to get 0.
Infinitely cyclic decimals and numbers with infinite roots are called irrational numbers.
Integers and fractions are collectively called rational numbers.
Mathematically, a rational number is the ratio of two integers, usually written as a/b, where b is not zero. Fraction is a common expression of rational number, while integer is a fraction with denominator of 1, which is also a rational number.
Mathematically, a rational number is the ratio of an integer a to a nonzero integer b, which is usually written as a/b, so it is also called a fraction. The Greek name is λ ο γ ο? The original meaning is "rational number", but the Chinese translation is not appropriate, and it has gradually become "rational number". Real numbers that are not rational numbers are called irrational numbers.
The set of all rational numbers is expressed as q, and the fractional part of rational numbers is finite or cyclic.
Rational number is a number in real number that cannot be accurately expressed as the ratio of two integers, that is, infinite acyclic decimal. Such as pi, the square root of 2, etc.
Real numbers can be divided into rational numbers and irrational numbers.
The difference between irrational numbers and rational numbers:
1. When both rational and irrational numbers are written as decimals, rational numbers can be written as finite decimals and infinite cyclic decimals.
For example, 4 = 4.0, 4/5 = 0.8, 1/3 = 0.33333 ... and irrational numbers can only be written as infinite acyclic decimals.
For example, √ 2 =1.414213562. ..............................................................................................................................
2. All rational numbers can be written as the ratio of two integers; And irrational numbers can't. Accordingly, it is suggested that irrational numbers should be labeled as "unreasonable", and rational numbers should be renamed as "comparative numbers" and irrational numbers as "non-comparative numbers". After all, irrational numbers are not unreasonable, but people didn't know much about them at first.
Using the main differences between rational numbers and irrational numbers, it can be proved that √2 is irrational.
Proof: Suppose √2 is not an irrational number, but a rational number.
Since √2 is a rational number, it must be written as the ratio of two integers:
Real numbers include rational numbers and irrational numbers. Among them, irrational numbers are infinite cyclic decimals and mantissas, and rational numbers include infinite cyclic decimals, finite decimals and integers.
Natural number (natural number)
A number used to measure the number of things or to indicate the order of things. That is, the numbers represented by the numbers 0, 1, 2, 3, 4, ... natural numbers start from 0, one by one, forming an infinite set. There are addition and multiplication operations in the set of natural numbers. The result of addition or multiplication of two natural numbers is still a natural number, and subtraction or division can also be done, but the result of subtraction and division is not necessarily a natural number, so subtraction and division operations are not always effective in the set of natural numbers. Natural numbers are the most basic of all numbers that people know. In order to make the number system have a strict logical basis, mathematicians in the19th century established two equivalent theories of natural numbers, namely ordinal number theory and cardinal number theory, which made the concept, operation and related properties of natural numbers strictly discussed.
Ordinal theory was put forward by Italian mathematician G. piano. He summed up the nature of natural numbers and gave the following definition of natural numbers by axiomatic method.
The set n of natural numbers refers to a set that meets the following conditions: ① There is an element in n, which is recorded as 1. ② Every element in n can find an element in n as its successor. ③ 1 is the successor of 0. ④0 is not the successor of any element. ⑤ Different elements have different successors. ⑥ (inductive axiom) Any subset m of n, if 1∈M, and as long as X is in M, it can be deduced that the successor of X is also in M, then M = N. ..
Cardinality theory defines natural numbers as the cardinality of finite sets. This theory puts forward that two finite sets that can establish one-to-one correspondence between elements have the same quantitative characteristics, which are called cardinality. In this way, all the single element sets {x}, {y}, {a}, {b} and so on have the same cardinality, which is recorded as 1. Similarly, whenever two fingers can establish a one-to-one set, their cardinality is the same, recorded as 2, and so on. The addition and multiplication of natural numbers can be defined by ordinal number or cardinal number theory, and the operations under the two theories are consistent.
Natural numbers play a great role in daily life, and people use them widely.
Whether "0" is included in natural numbers is controversial. Some people think that natural numbers are positive integers, that is, counting from 1; Others think that natural numbers are non-negative integers, that is, counting from 0. There is no consensus on this issue at present. However, in number theory, the former is often used; In set theory, the latter is often used. At present, our primary and secondary school textbooks classify 0 as a natural number!
Natural numbers are integers, but integers are not all natural numbers.
For example,-1 -2 -3 ... is an integer rather than a natural number.
The set of all nonnegative integers is called the nonnegative integer set (i.e. natural number set).
The so-called prime number or prime number is a positive integer, and there is no other factor except itself and 1. For example, 2, 3, 5 and 7 are prime numbers, but 4, 6, 8 and 9 are not. The latter is called a composite number or a composite number. From this point of view, integers can be divided into two types, one is called prime number and the other is called composite number. Some people think that the number 1 should not be called prime number. ) the famous gauss "unique decomposition theorem" says that any integer. It can be written as the product of a series of prime numbers.
Chapter 5:
This chapter focuses on the solution of one-dimensional linear inequality,
Difficulties in this chapter: understand the solution set of inequality and the solution set of inequality group, and use it correctly.
Basic properties of inequality 3.
The focus of this chapter: thoroughly understand the difference between inequality and equality.
The concept of (1) inequality: use unequal symbols ("≦";
(2) The basic nature of inequality, which is the theoretical basis for solving inequality.
(3) The solution set of discriminant inequality and solution inequality are two completely different concepts.
(4) The solution of general inequality has infinite values, which are expressed by the number axis. (5) The concept and solution of one-dimensional linear inequality is the focus and core of this chapter.
(6) The solution set of one-dimensional linear inequality is the solution set of one-dimensional linear inequality on the exponential axis.
(7) A group of one-dimensional linear inequalities consisting of two one-dimensional linear inequalities. A group of one-dimensional linear inequalities can be composed of several (unknown) one-dimensional linear inequalities.
(8) Determine the solution set of the unary linear inequality group with the number axis.
Chapter VI:
1. Binary linear equations, binary linear equations and their solutions, making clear that the solution of binary linear equations is a pair of unknowns will test whether a pair of values is the solution of a binary linear equations.
2. Two basic solutions of linear equations can flexibly use method of substitution, addition and subtraction to solve binary linear equations and simple ternary linear equations.
3. According to the given application problem, the corresponding binary linear equations or ternary linear equations are listed, thus the solution of the problem is obtained, and the rationality of the result is tested according to the practical significance of the problem.
This chapter focuses on the solution of binary linear equations-substitution, addition and subtraction and simple application of solving linear equations.
The difficulties in this chapter are:
1. will solve binary linear equations and simple ternary linear equations with appropriate elimination methods;
2. Correctly find out the equation relationship in the application problem and list the linear equations.
Chapter VII
The emphasis of this chapter is: the multiplication and division of algebraic expressions, especially the operation of powers and the application of multiplication formulas should be mastered.
The difficulties in this chapter are: the structural characteristics of multiplication formula, the understanding of the meaning of letters in the formula and the flexible use of multiplication formula.
Operational properties of 1. Power, correctly express these properties, and skillfully use them for related calculations.
2. The laws of monomial multiplication (or division), polynomial multiplication (or division) and polynomial multiplication, and skillfully use them for calculation.
3. The derivation process of multiplication formula can be calculated flexibly.
4. Skillfully use algorithms and algorithms to perform operations.
5. Understand the meaning of numbers and formulas expressed by letters. Through the deformation of the formula, we can deeply understand the thinking method of transformation.
Chapter 8:
1. Several ways to understand things: observation and experimental induction and analogy, conjecture and proof of reasoning in life.
2. Definitions, propositions, axioms and theorems
3. Reasoning in simple geometric figures
4. Complementary angle, supplementary angle and turning angle
5. Determination of parallel lines
Judgment: one axiom, two theorems.
Axiom: Two straight lines are cut by a third straight line. If the same angle is equal (quantitative relationship), two straight lines are parallel (positional relationship).
Theorem: Internal dislocation angles are equal (quantitative relationship) and two straight lines are parallel (positional relationship).
Theorem: Two straight lines are parallel to each other (positional relationship).
Properties of parallel lines:
Two straight lines are parallel and have the same angle.
Two straight lines are parallel and have equal internal angles.
These two lines are parallel and complementary.
Determine the "quantity relationship" from the "position relationship" of the graph.
Chapter 9:
Key points: factorization method,
Difficulties: Analyze the characteristics of polynomials and choose the appropriate decomposition method.
1. The concept of factorization;
2. Factorial decomposition method: common factor extraction method, formula method and grouping decomposition method (cross multiplication).
3. Solve some practical problems with factorization (including graphic exercises)
Chapter 10:
The key point is to use statistical knowledge to solve practical problems in real life.
The difficulty is: solving practical problems with statistical knowledge.
1. Basic knowledge of statistics, calculation of average, median and mode, etc.
2. Understand the data collection and collation, and draw three statistical charts.
3. Applying statistical knowledge to solve practical problems can solve comprehensive problems related to statistics.
hope this helps