First, we can correctly understand the related concepts of real numbers.

We already know that integers and collectively referred to as. It is also stipulated that infinite acyclic numbers are irrational numbers, so we call rational numbers and irrational numbers real numbers, that is, there are two members in the real number family: rational numbers and irrational numbers. When learning, we should pay attention to distinguish between rational numbers and irrational numbers, that is to say, if a number is rational, it must not be irrational. On the contrary, if a number is irrational, it must not be rational.

Second, correctly understand the classification of real numbers

The classification of real numbers can be considered from two angles: (1) classification by definition; (2) according to the positive and negative classification. However, it should be noted that 0 also plays an important role in real numbers. We usually call positive real numbers and 0 as non-negative numbers, and negative real numbers and 0 as non-positive numbers.

Third, correctly understand the relationship between real number and number axis.

There is a one-to-one correspondence between real numbers and points on the number axis, which means that all real numbers can be represented by points on the number axis; On the contrary, every point on the number axis represents a real number. The number represented by any point on the number axis is rational and irrational.

On the number axis, two points representing the opposite number are on both sides of the origin, and the distance from the two points to the origin is equal. The absolute value of real number A is the distance between the point corresponding to this number and the origin on the number axis.

Using the number axis, we can compare the sizes of any two real numbers, that is, the two real numbers represented on the number axis. The larger the absolute value, the smaller it is.

Fourth, master the related properties of real numbers.

Like rational numbers, real numbers have many important properties. Specifically, we can think about them from the following aspects:

1, the inverse of real number a is -a, and the inverse of 0 is 0. Specifically, if A and B are reciprocal, A+B = 0; On the other hand, if a+b=0, then a and b are reciprocal.

2. Absolute value The absolute value of a positive real number is itself, the absolute value of a negative real number is its reciprocal, and the absolute value of 0 is 0. The absolute value of real number A can be expressed as, that is, the absolute value of real number A must be non-negative.

3. Two real numbers whose reciprocal product is 1 are reciprocal, that is, if A and B are reciprocal, AB =1; On the other hand, if ab= 1, A and B are reciprocal. It should be noted that 0 is not a reciprocal.

4. Comparison of the size of real numbers Any two real numbers can compare the sizes. Positive real numbers are all greater than 0, negative real numbers are all less than 0, positive real numbers are greater than all negative real numbers, and the absolute values of the two negative real numbers are larger but smaller.

5. The operation of real numbers is the same as that of rational numbers. It is worth mentioning that real numbers can be added, subtracted, multiplied, divided and squared, and positive real numbers can be squared. Real numbers, like rational numbers, operate from high to low, that is, power and square, then multiplication and division, and finally addition and subtraction, using parentheses.

2. Knowledge points in the first volume of ninth grade mathematics

The concept of 1 square

A group of parallelograms with equal adjacent sides and a right angle is called a square.

2, the nature of the square

(1) has all the properties of parallelogram, rectangle and diamond;

(2) All four corners of a square are right angles and all four sides are equal;

(3) The two diagonals of a square are equal and vertically bisected, and each diagonal bisects a set of diagonals;

(4) A square is an axisymmetric figure with four axes of symmetry;

(5) A diagonal line of a square divides the square into two isosceles right-angled triangles, and two diagonal lines divide the square into four isosceles right-angled triangles;

(6) The distance from one point on one diagonal of a square to both ends of the other diagonal is equal.

3. Determination of the square

(1) The main basis for judging whether a quadrilateral is a square is definition, and there are two ways:

First prove that it is a rectangle, and then prove that a group of adjacent sides are equal.

First prove that it is a diamond, and then prove that an angle is a right angle.

(2) The general order of judging a quadrilateral as a square is as follows:

First prove that it is a parallelogram;

Then prove to be a diamond (or rectangle);

It turned out to be a rectangle (or a diamond).

3. Knowledge points in the first volume of ninth grade mathematics

First, the theorem of circle angle

In the same circle or equal circle, the circumferential angle of the same arc or equal arc is equal, which is equal to half the central angle of the arc.

(1) the theorem has three meanings:

A. The central angle and the circumferential angle are in the same or equal circle; (How to prove four * * * circles for related knowledge points)

B. They are facing the same arc or two arcs are equal.

C. Conditions A and B have the same circumferential angle, which is equal to half the central angle.

Because the degree of the central angle is equal to the degree of the arc it faces, the degree of the central angle is equal to half the degree of the arc it faces.

Second, the inference of the theorem of circle angle

Inference 1: The circular angles of the same or equal circular arc are equal, and so are the circular arcs of the same or equal circular arc.

Inference 2: the circumferential angle of a semicircle (or diameter) is equal to 90; A chord with a circumferential angle of 90 is a diameter.

Inference 3: If the median line of one side of a triangle is equal to half of this side, then this triangle is a right triangle.

Third, inference and explanation.

The theorem of circle angle is an important part of geometry in ninth grade mathematics knowledge points.

① Inference 1 is the most commonly used method to prove the equality of angles in a circle. If the "same arc or equal arc" in the inference 1 is changed to "same chord or equal chord", the conclusion will not hold, because a chord has two circumferential angles.

(2) The premise of "arcs with equal circumferential angles are equal" in Inference 2 is "in the same circle or in the same circle".

(3) Inference 2 of the theorem of circle angle is widely used. It is necessary to relate the diameter to the 90 fillet. Generally speaking, when there is a diameter in the condition, a fillet opposite to the diameter is usually made, so that a right triangle can be obtained, which creates conditions for further solving problems.

Inference 3 is essentially an inverse theorem that the median line on the hypotenuse of a right triangle is equal to half of the hypotenuse.

4. Knowledge points of the first volume of ninth grade mathematics

The concept of inequality

1. Inequality: A formula that represents inequality relations with inequality symbols is called inequality.

2. Solution set of inequality: For an unknown inequality, any unknown value suitable for this inequality is called the solution of this inequality.

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

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

5. The method of expressing inequality with number axis.

Basic properties of inequality

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

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

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

4. Explanation: ① In the unary linear inequality, unlike the equality, the equal sign is unchanged, but changes with the operation of addition or multiplication. (2) If the inequality is multiplied by 0, the inequality becomes an equal sign. Therefore, in the problem, if the number of multiplication is required, it depends on whether there is a one-dimensional inequality in the problem. If there is, then the number multiplied by the inequality is not equal to 0, otherwise the inequality is not established.

linear inequality

1, the concept of one-dimensional linear inequality: Generally speaking, an inequality contains only one unknown, the degree of the unknown is 1, and both sides of the inequality are algebraic expressions. This inequality is called one-dimensional linear inequality.

2. The general steps to solve the linear inequality of one variable are: 1, denominator 2, bracket 3, shift term 4, merge similar terms 5, and convert the coefficient of x term into 1.

Unary linear inequality system

1, the concept of one-dimensional linear inequality group: combine several one-dimensional linear inequalities into one one-dimensional linear inequality group.

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

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

4. 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.

5. 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.

6. Inequality and unequal groups

Inequalities: ① Use symbols > =,