1, classification of real numbers

Positive rational number

Rational Numbers Zero Finite Decimals and Infinite Cyclic Decimals

Real negative rational number

Positive irrational number

Irrational number infinite acyclic decimal

Negative irrational number

2. Irrational number

When understanding irrational numbers, we should grasp the moment of "infinite non-circulation", which can be summarized into four categories:

(1) An inexhaustible number, such as;

(2) Numbers with specific meanings, such as pi, or simplified numbers with pi, such as+8;

(3) Numbers with specific structures, such as 0.101001001… etc.

(4) some trigonometric functions, such as sin60o, etc.

Test center two, the reciprocal, reciprocal and absolute value of real numbers (3 points)

1, reciprocal

A real number and its inverse are a pair of numbers (only two numbers with different signs are called inverse numbers, and the inverse of zero is zero). Seen from the number axis, the points corresponding to two opposite numbers are symmetrical about the origin. If a and b are opposites, then a+b=0, A =-B, and vice versa.

2. Absolute value

The absolute value of a number is the distance between the point representing the number and the origin, |a|≥0. When the absolute value of zero is itself, it can also be regarded as its inverse. If |a|=a, then a ≥ 0; If |a|=-a, then a≤0. Positive numbers are greater than zero, negative numbers are less than zero, positive numbers are greater than all negative numbers, and the absolute values of the two negative numbers are smaller.

Step 3 count down the seconds

If A and B are reciprocal, there is ab= 1, and vice versa. The numbers whose reciprocal equals itself are 1 and-1. Zero has no reciprocal.

Test center 3, square root, arithmetic square root and cube root (3- 10)

1, square root

If the square of a number is equal to a, then this number is called the square root (or sum of squares) of a.

A number has two square roots, and the two square roots are in opposite directions; The square root of zero is zero; Negative numbers have no square root.

The square root of the positive number a is recorded as "".

2. Arithmetic square root

The positive square root of a positive number is called the arithmetic square root of a, and it is recorded as "".

Positive numbers and zeros have only one arithmetic square root, and the arithmetic square root of zero is zero.

( 0)

; Double nonnegativity of attention:

-(& lt； 0) 0

3. Cubic root

If the cube of a number is equal to A, then this number is called the cube root of A (or the cube root of A).

Positive numbers have positive cubic roots; Negative numbers have negative cubic roots; The cube root of zero is zero.

Note: This shows that the negative sign in the cube root symbol can be moved outside the root sign.

Test site four, scientific notation and approximate method (3-6 points)

1, valid number

A divisor is said to be accurate to the nearest round. At this time, all digits from the first non-zero digit on the left to the exact digit on the right are called the effective digits of this digit.

2. Scientific symbols

Write a number in a form where n is an integer. This notation is called scientific notation.

Test 5. Comparison of real numbers (3 points)

1, number axis

The straight line that specifies the origin, positive direction and unit length is called the number axis (pay attention to the three elements specified above when drawing the number axis).

When solving problems, we should really master the idea of combining numbers with shapes, understand the one-to-one correspondence between real numbers and points on the number axis, and use them flexibly.

2. Several common methods of comparing real numbers.

(1) axis comparison: the number on the right is always greater than the number on the left of the two numbers represented on the axis.

(2) difference comparison: let a and b be real numbers,

(3) quotient comparison method: let a and b be two positive real numbers,

(4) absolute value comparison method: let a and b be two negative real numbers, then.

(5) Flat method: Let A and B be two negative real numbers, then.

Test center 6, the operation of real numbers (the basis of the problem, the score is quite large)

1, additive commutative law

2. Additive associative law

3. Multiplicative commutative law

4. Multiplicative associative law

5. Distribution law from multiplication to addition

6. Real number operation sequence

Calculate the power first, then multiply and divide, and finally add and subtract. If there are brackets, count them first.

Knowledge Summary (2) Algebraic Formula

Test center 1. Related concepts of algebraic expressions (3 points)

1, algebraic expression

Expressions that connect numbers or letters representing numbers with operational symbols are called algebraic expressions. A single number or letter is also algebraic.

2. Single item

Algebraic expressions that contain only the product of numbers and letters are called monomials.

Note: The monomial is composed of coefficients, letters and indices of letters, in which the coefficients cannot be expressed by fractions. For example, this expression is wrong and should be written as. In a monomial, the sum of the exponents of all the letters is called the degree of the monomial. If it's a 6-degree monomial.

Test point 2, polynomial (1 1)

1, polynomial

The sum of several monomials is called polynomial. Where each monomial is called a term of this polynomial. Items without letters in polynomials are called constant terms. The degree of the term with the highest degree in a polynomial is called the degree of the polynomial.

Monomial and polynomial are collectively called algebraic expressions.

Replacing the letters in the algebraic expression with numerical values and calculating the results according to the operations specified in the algebraic expression are called algebraic values.

Note: (1) To find the value of the algebraic expression, it is generally to simplify the algebraic expression first and then substitute the value of the letter.

(2) To find an algebraic value, sometimes we can't find the value of its letters, so we need to use skills and "whole" substitution.

2. Similar projects

Items with the same letter and the same letter index are called similar items. Several constant terms are similar.

3. Rules for removing brackets

(1) brackets are preceded by "+". Remove the brackets together with the preceding "+"sign, and all items in the brackets remain unchanged.

(2) There is a "-"before the brackets. Remove the brackets together with the "-"in front, and all the items in the brackets have changed.

4. Algebraic expression algorithm

Addition and subtraction of algebraic expressions: (1) bracket removal; (2) Merge similar items.

Multiplication of algebraic expressions:

Division of algebraic expressions:

Note: (1) The result of single item multiplied by single item is still single item.

(2) Multiply the monomial with the polynomial to get a polynomial with the same number of terms as the polynomial in the factor.

(3) Pay attention to the symbol problem when calculating. Each term of a polynomial contains the symbol before it, and we should also pay attention to the symbol of a single term.

(4) In the expansion of polynomial multiplication, if there are similar items, they should be merged.

(5) The letters in the formula can represent numbers, monomials or polynomials.

(6)

(7) Polynomial divided by monomial, first divide each term of this polynomial by this monomial, and then add the obtained quotients. The division of a monomial by a polynomial cannot be calculated in this way.

Test site 3, factorization (1 1)

1, factorization

Turning a polynomial into the product of several algebraic expressions is called decomposing this polynomial, which is also called decomposing this polynomial.

2. Common methods of factorization

(1) common factor method:

(2) Using the formula method:

(3) Grouping decomposition method:

(4) Cross multiplication:

3, the general steps of factorization:

(1) If every term of the polynomial has a common factor, then extract the common factor first.

(2) After the common factor is put forward or not, observe the number of terms in the polynomial: two terms can be decomposed by formula; We can try to find these three terms by formula method and cross multiplication decomposition factor; For four or more items, you can try the group decomposition method to decompose the factors.

(3) The factory must be decomposed until every element can no longer be decomposed.

Test site 4, score (8~ 10)

1, the concept of fraction

Generally speaking, two algebraic expressions are represented by a and b respectively, and A \b can be represented as a form. If b contains letters, the formula is called a fraction. Where a is called the numerator of the fraction and b is called the denominator of the fraction. Fractional and algebraic expressions are usually called rational forms.

2, the nature of the score

Basic properties of (1) score:

Both the numerator and denominator of a fraction are multiplied (or divided) by the same algebraic expression that is not equal to zero, and the value of the fraction remains unchanged.

(2) The sign change rule of the score:

The numerator, denominator and sign of the fraction itself have changed, while the value of the fraction remains unchanged.

3. Fractional algorithm

Test site five, quadratic root (junior high school mathematics foundation, high score)

1, quadratic radical

The formula is called quadratic radical, which must satisfy the following requirements: it contains quadratic radical sign ""; The root sign a must be non-negative.

2. The simplest quadratic radical

If the square root satisfies: the factor of the square root is an integer and the factor is an algebraic expression; The square root number contains no factor or can be completely opened. Such a quadratic root is called the simplest quadratic root.

Methods and steps of transforming quadratic form into simplest quadratic form;

(1) If the square root of the quotient is a fraction (including decimals) or a fraction, first use the nature of the arithmetic square root of the quotient to write it in the form of a fraction, and then simplify it with the denominator.

(2) If the number of roots is integer or algebraic expression, first decompose them into factors or factors, and then extract the factors or factors that can be opened to the maximum extent.

3. Similar quadratic roots

After several quadratic roots are transformed into the simplest quadratic roots, if the number of roots is the same, these quadratic roots are called similar quadratic roots.

4. Properties of quadratic roots

( 1)

(2)

(3)

(4)

5. Quadratic radical mixed operation

The mixed operation of quadratic root is in the same order as that in real number. Multiply first, then divide, and finally add and subtract. If there are brackets, count them first (or remove them first).

Knowledge Summary (3) Equation (Group)

Test center 1, the concept of linear equation with one variable (6 points)

1, equation

Equations with unknowns are called equations.

2, the solution of the equation

The value of the unknown quantity that can make both sides of the equation equal is called the solution of the equation.

3. Properties of the equation

Adding (or subtracting) the same number or the same algebraic expression on both sides of the (1) equation, the result is still an equation.

(2) Both sides of the equation are multiplied (or divided) by the same number (the divisor cannot be zero), and the result is still an equation.

4. One-dimensional linear equation

An integral equation with only one unknown number and the highest order of the unknown number is 1 is called a unitary linear equation, where the equation is called the standard form of a unitary linear equation, a is the coefficient of the unknown number x, and b is a constant term.

Test site two, a quadratic equation (6 points)

1, unary quadratic equation

An integral equation with an unknown number and the highest degree of the unknown number is 2 is called a quadratic equation.

2. The general form of quadratic equation with one variable

It is characterized in that there are eleven quadratic polynomials on the left side of the equation about the unknown x, and the right side of the equation is zero, which is called quadratic term, and a is called quadratic term coefficient; Bx is called a linear term, and b is called a linear term coefficient; C is called a constant term.

Test center 3. Solution of quadratic equation in one variable (10)

1, direct Kaiping method

Using the definition of square root to find the solution of quadratic equation in one variable is called direct Kaiping method. The direct Kaiping method is suitable for solving quadratic equations with one variable. According to the definition of square root, it is the square root of B, when,,, when B.

2. Matching method

Matching method is an important mathematical method, which is not only suitable for solving quadratic equations with one variable, but also widely used in other fields of mathematics. The theoretical basis of the matching method is the complete square formula. If a in the formula is regarded as an unknown x and replaced by x, there is.

3. Formula method

Formula method is a method to solve the quadratic equation of one variable by finding the root formula, and it is a general method to solve the quadratic equation of one variable.

The formula for finding the root of quadratic equation with one variable;

4, factorization method

Factorization is to find the solution of the equation by factorization. This method is simple and easy to use, and it is the most commonly used method to solve the quadratic equation of one variable.

Test center 4. Discriminating formula for roots of quadratic equation with one variable (3 points)

discriminant

In the unary quadratic equation, the discriminant called the root of the unary quadratic equation is usually represented by "",that is,

Test the relationship between roots and coefficients of quintic quadratic equation (3 points)

If the two real roots of the equation are, then. That is to say, for the real root of any unary quadratic equation, the sum of the two roots is equal to the reciprocal of the quotient obtained by dividing the coefficient of the first term of the equation by the coefficient of the second term; The product of two roots is equal to the quotient obtained by dividing the constant term by the coefficient of quadratic term.

Test site six, score equation (8 points)

1, fractional equation

Equations with unknowns in the denominator are called fractional equations.

2. The general method of fractional equation

The idea of solving fractional equation is to transform "fractional equation" into "integral equation". Its general solution is:

(1) denominator, and both sides of the equation are multiplied by the simplest common denominator.

(2) Solve the whole equation.

(3) Root test: substitute the obtained root into the simplest common denominator. If it is equal to zero, it is the increase of roots and should be discarded; If it is not equal to zero, it is the root of the original equation.

3. The Special Solution of Fractional Equation

Alternative method:

Method of substitution is an important mathematical thought in middle school mathematics, and its application is very extensive. When the fractional equation has a special form and it is difficult to remove the denominator, method of substitution can be considered.

Test 7. Binary linear equation (8~ 10)

1, binary linear equation

An integral equation containing two unknowns with the highest degree of 1 is called a binary linear equation, and its general form is (

2, the solution of binary linear equation

The values of a pair of unknowns that make the values of the left and right sides of the binary linear equation equal are called the solutions of the binary linear equation.

3. Binary linear equation

Two (or more) binary linear equations are combined into one binary linear equation group.

Solutions of four binary linear equations

The values of two unknowns that make the left and right sides of two equations of binary linear equations equal are called the solutions of binary linear equations.

5. Solving the Binary Linear Founder Group

(1) substitution method (2) addition and subtraction

6, ternary linear equation

Let the whole equation have three unknowns, and the terms of the unknowns are all 1.

7, ternary linear equations

An equation group consisting of three (or more) linear equations with three unknowns is called a ternary linear equation group.

Knowledge Summary (IV) Inequality (Group)

Test the concept of 1 and inequality (3 points)

1, inequality

A formula that uses an inequality symbol to express inequality relations is called inequality.

2. Solution set of inequality

For an inequality with an unknown number, any value of the unknown number suitable for this inequality is called the solution of this inequality.

For an unknown inequality, the set of all its solutions is called the solution set of this inequality.

The process of finding the solution set of inequality is called solving inequality.

3. The method of expressing inequality with number axis.

Test site 2, the basic properties of inequality (3~5 points)

Add (or subtract) the same number or the same algebraic expression on both sides of 1. inequality, and the direction of inequality remains unchanged.

2. Both sides of the inequality are multiplied by (or divided by) the same positive number, and the direction of the inequality remains unchanged.

3. When both sides of the inequality are multiplied by (or divided by) the same negative number, the direction of the inequality will change.

Examination questions:

Test site 3, unary linear inequality (6~8 points)

1, the concept of one-dimensional linear inequality

Generally, an inequality has only one unknown number, the degree of the unknown number is 1, and both sides of the inequality are algebraic expressions. This inequality is called unary linear inequality.

2. Solving one-dimensional linear inequality

General steps to solve one-dimensional linear inequality;

(1) Denominator removal (2) Parentheses removal (3) Moving items (4) Merging similar items (5) The coefficient of the x item is changed to 1.

Test site 4, unary linear inequality group (8 points)

1, the concept of one-dimensional linear inequality system

Several unary linear inequalities are combined into a unary linear inequality group.

The common part of the solution set of several linear inequalities is called the solution set of linear inequalities.

The process of finding the solution set of inequality group is called solving inequality group.

When any number x can't make the inequality hold at the same time, we say that the inequality group has no solution or its solution is an empty set.

2. Solving one-dimensional linear inequalities.

(1) Find the solution set of each inequality in the inequality group.

(2) Use the number axis to find the common part of the solution set of these inequalities, that is, the solution set of this inequality group.

Knowledge Summary (5) Preliminary Statistics and Probability

Test center, average (3 points)

1, the concept of average value

(1) average: Generally speaking, if there are n numbers, it is called the average of these n numbers, which is pronounced as "x-pull".

(2) Weighted average: If there are times, times, …, times in n numbers (here), then according to the definition of average, the average value of n numbers can be expressed as, and the average value thus obtained is called weighted average, which is called weight.

2, the average calculation method

(1) definition method

When the given data is scattered, generally choose to define the formula:

(2) Weighted average method:

When the given data appears repeatedly, the weighted average formula: is generally selected, where.

(3) New data method:

When all the given data fluctuate up and down the constant a, the simplified formula is generally selected.

Where the constant a usually takes an "integer" closer to the average value of this set of data,,,,. Is the average of new data (usually called original data, called new data).

Test center 2. Several basic concepts in statistics (4 points)

1, in general

The sum of all the research objects is called the population.

2. Individuals

Every object in a group is called an individual.

3. Samples

Some individuals extracted from the population are called population samples.

4. Sample size

The number of individuals in a sample is called sample size.

5. Average number of samples

The average of all individuals in a sample is called the sample average.

6. Overall average

The average of all individuals in a population is called the population average. In statistics, the population average is usually estimated by the sample average.

Test center 3, mode and median (3~5 points)

1, mode

In a set of data, the data with the highest frequency is called the pattern of this set of data.

2. Central value

Arrange a set of data in order of size, and one data in the middle (or the average of the two data in the middle) is called the median of this set of data.

Test site four, variance (3 points)

1, the concept of variance

In a set of data, the average value of the square of the difference between each data and its average value is called the variance of this set of data. Usually represented by "",that is

2. Calculation of variance

(1) Basic formula:

(2) Simplified calculation formula (Ⅰ):

It can also be written as

The memory method of this formula is: the variance is equal to the average of the square of the original data MINUS the square of the average.

(3) Simplified calculation formula (Ⅱ):

When the amount of data in a group of data is large, a new group of data can be obtained by subtracting a constant a close to their average from each data according to the calculation method of simplified average, and then,

The memory method of this formula is: the variance is equal to the average of the new data square minus the square of the new data average.

(4) New data method:

The variance of original data is equal to the variance of new data, that is, according to the basic formula of variance, the variance obtained is equal to the variance of original data.

3. Standard deviation

The arithmetic square root of variance is called the standard deviation of this set of data, which is represented by "S", that is,

Five, frequency distribution test sites (6 points)

1, the significance of frequency distribution

In many problems, it is not enough to know only the mean and variance, but also the proportion of data in each small range in the sample, so it is necessary to study how to sort out a group of data to get its frequency distribution.

2. Study the general steps and related concepts of frequency distribution.

(1) The general steps to study the frequency distribution of samples are:

① Calculation range (the difference between the maximum value and the minimum value)

(2) Determine the distance and the number of groups.

③ Determine the demarcation point.

④ Column frequency distribution table

⑤ Draw the histogram of frequency distribution.

(2) Related concepts of frequency distribution.

① Range: the difference between the maximum value and the minimum value.

② Frequency: the number of times each group of data drops.

③ Frequency: the ratio of the frequency of each group to the total data (sample size n) is called the frequency of this group.

Test six, determine the events and random events (3 points)

1, determine the event

Inevitable event: an event that will inevitably occur in each test when the test is repeated under certain conditions.

Impossible events: Some events will not happen in every experiment. Such events are called impossible events.

2. Random events:

Under certain conditions, events that may or may not occur are called random events.

Test seven, the possibility of random events (3 points)

Generally speaking, the probability of random events is different, and the probability of different random events may be different.

For the probability of random events, we can use some empirical data obtained from repeated experiments to predict their probability. To judge whether certain rules of the game are fair to players is to see whether they are equally likely to happen. The so-called judging whether the possibility of events is the same is to see whether the possibility of each event is the same, and to explain the problem with data.

Test center 8. The meaning and expression of probability (5~6 points)

1, the meaning of probability

Generally speaking, in a large number of repeated experiments, if the frequency of event A will be stable near a constant p, then this constant p is called the probability of event A.

2. Representation of events and probabilities

Usually, events use capital letters A, B, C, ... in English, and the probability p of event A can be written as p (a) = p

Test 9. Determine the relationship between probability events and random events (3 points)

1, determine the event probability.

(1) when a is an inevitable event, P(A)= 1.

(2) When a is an impossible event, P(A)=0.

2. Determine the relationship between event probability and random events.

The possibility of an accident is getting smaller and smaller.

Value 0 1 probability

Impossible, inevitable.

The possibility of an accident is increasing.

Ten test sites, classical probability (3 points)

1, the definition of classical probability

If a test has: ① the number of possible structures in a test is limited; ② In an experiment, the possibility of various results is equal. We call the experiment with these two characteristics classical probability.

2. Probability solution of classical probability

Generally speaking, if there are n possible results in an experiment, and their probabilities are all equal, and event A contains the results in M, then the probability of event A is P(A)= 1

Test center eleven, list method to find the probability (10)

1, list method

The method of analyzing and solving the probability of some events by list method is called list method.

2. Application of list method

When two factors are designed in an experiment and the number of possible results is large, in order to list all possible results, the list method is usually used.

Test center 12, tree diagram method of probability (10 score)

1, tree diagram method

It is to list all possible results of an event through a tree diagram, and the method of finding its probability is called tree diagram method.

2. Conditions for finding probability by tree diagram method.

When an experiment is designed with three or more factors, it is inconvenient to use the list method. In order to list all possible results, the tree diagram method is usually used to find the probability.

Thirteen, the use of frequency estimation probability test center (8 points)

1, estimate the probability by frequency.

Under the same conditions, we can estimate the probability of a random event by doing a lot of repeated experiments and using the frequency of events to gradually stabilize to a constant.

2. In statistics, simple experimental methods are often used to complete probability estimation, rather than complex experiments in actual operation. Such an experiment is called a simulation experiment.

3. Random number

In random events, a large number of repeated experiments are needed to generate a series of random data for statistical work. These randomly generated data are called random numbers.