Key outline of mathematics in the first volume of the second day of junior high school
Real number knowledge point
1, classification of real numbers: rational numbers and irrational numbers.
2. Number axis: A straight line with origin, positive direction and unit length is called number axis. Real numbers correspond to points on the number axis one by one.
3. Ancient numbers: Two numbers with different symbols are called reciprocal numbers. The reciprocal of a is -a, and the reciprocal of 0 is 0. (If A and B are opposites, then a+b=0)
4. Absolute value: the distance from the point representing the number A to the origin on the number axis is called the absolute value of the number A, which is marked as ∣a∣, and the absolute value of a positive number is itself; The absolute value of a negative number is its reciprocal; The absolute value of 0 is 0.
5. Reciprocal: Two numbers whose product is 1.
6. Power: the operation of seeking common ground factor product is called power, and the result of power operation is called power (square and cube).
7. Square root: Generally speaking, if the square of a number X is equal to A, that is, x2=a, then this number X is called the square root of A (also called quadratic square root). Positive numbers have two square roots, and the two square roots are in opposite directions; 0 has only one square root, which is 0 itself; Negative numbers have no square root. (Arithmetic square root: Generally speaking, if the square of a positive number X is equal to A, that is, x2=a, then this positive number X is called the arithmetic square root of A, and the arithmetic square root of 0 is 0. )
Real number is a general term for rational number and irrational number. Mathematically, real numbers are defined as the number of corresponding points on the number axis. Real numbers can be intuitively regarded as finite decimals and infinite decimals, and can "fill" the number axis. But the whole of real numbers can't be described only by enumeration. Real and imaginary numbers * * * together form a complex number.
Real numbers can be used to measure continuous quantities. Theoretically, any real number can be expressed as an infinite decimal, and to the right of the decimal point is an infinite series (cyclic or acyclic). In practice, real numbers are often approximate to a finite decimal (n digits are reserved after the decimal point, and n is a positive integer, including integers). In the computer field, because computers can only store a limited number of decimal places, real numbers are often represented by floating-point numbers.
1) inverse number (there are only two numbers with different signs, and their sum is zero. We say that one of them is the inverse number of the other, which is called reciprocal inverse number). The reciprocal of the real number a is -a, and the distance between a and -a on the number axis is equal to the origin 0.
2) Absolute value (the distance between a number A and the origin 0 on the number axis) The absolute value of the real number A is |a|
(1) When a is a positive number, |a|=a (constant), and a is itself;
② When a is 0, |a|=0, and A is itself;
③ When A is negative, | a | =-a(a (the absolute value of A), and -a is the inverse of A. ..
The absolute value of any number is greater than or equal to 0, because there is no negative number in the distance. )
3) Reciprocal (the product of two real numbers is 1, so these two numbers are reciprocal) The reciprocal of real number A is: 1/a(a≠0).
4) Counting axes
Definition: The straight line defining the origin, positive direction and unit length is called the number axis.
Three elements of (1) axis: origin, positive direction and unit length.
(2) There is a one-to-one correspondence between points on the number axis and real numbers.
Square root and cube root knowledge points
Square root:
Generalization 1: Generally speaking, if the square of a number is equal to A, this number is called the square root (or quadratic root) of A, that is, if x=a, then X is called the square root of A, and 23 and -23 are the square roots of 529.
Because (23) = 529, 23 is the square root of 529. Q: (1)16,49, 100,100 are all positive numbers. How many square roots do they have? What does the square root matter? (2) What is the square root of 0?
Summary 2: a positive number has two square roots, and the two square roots are in opposite directions; 0 has a square root, and the square root itself is 0; Negative numbers have no square root.
Summary 3: The operation of finding the square root of a number a(a≥0) is called square root.
Square root operation is the basis of known exponents and powers. The sum of squares and squares are reciprocal operations. A number can be positive, negative or 0, and its square is only one. The square of both positive and negative numbers is positive, and the square of 0 is 0. But positive numbers have two square roots. These two numbers are opposite, and the square root of 0 is 0. Negative numbers have no square root. Because the sum of squares and the square root are inverse operations, we can find the square root of a number through the square operation, and we can also check whether a number is the square root of another number through the square operation.
First, the concept of arithmetic square root
The positive number a has two square roots (expressed as?
Root, expressed as.
The square root of 0 is also called the arithmetic square root of 0, so the arithmetic square root of 0 is 0, which is 0? 0。 "
"is the symbol of the arithmetic square root, and a means the arithmetic square root of a ... one has two meanings:
A), whose square root we call positive is the arithmetic square of a.
(1) The radical A means nonnegative, that is, A ≥ 0;
(2)a also means non-negative, that is, a≥0. In other words, the non-negative "arithmetic" square root is non-negative. Negative numbers have no arithmetic square root, that is, a.
=3, 8 is the arithmetic square root of 64. 6 meaningless.
9 represents both the square root operation of 9 and the positive square root of 9.
Second, the difference between square root and arithmetic square root is that
① Different definitions;
② Outliers: A positive number has two square roots, while a positive number has only one arithmetic square root; ③ The representation method is different: the square root of a positive number is expressed as? A, the arithmetic square root of a positive number is expressed as a; ④ Different value ranges: the arithmetic square root of a positive number must be a positive number, and the square root of a positive number is a plus sign and a minus sign. ⑤ The square root and arithmetic square root of 0 are both 0. Three. Example description:
Example 1, find the arithmetic square root of the following numbers:
( 1) 100;
(2)49;
(3)0.8 164
Note: Since the arithmetic square root of positive numbers is positive and the arithmetic square root of zero is zero, they can be summarized as: non-negative number calculation.
The square root of the operation is nonnegative, that is, when a≥0, a≥0 (when a
The meaning of arithmetic square root can be expressed intuitively by geometric figures. For example, if there is an area of a(a should be non-negative), the side length is.
The arithmetic square root represented by the square of.
What needs to be explained here is that the symbol ""of arithmetic square root is not only an operation symbol, for example, when a≥0, A represents the square root operation of non-negative number A, it is also a property symbol, that is, it represents the positive square root of non-negative number A. ..
3. Cubic root
Definition of (1) cube root: If the cube of a number X equals A, this number is called the cube root of A (also called cube root), that is, if X? A, then x is called the cube root of a.
(2) The cube root of a number is pronounced as "cube root number A", where A is called the root number and 3 is called the root index, which cannot be omitted. If omitted, it means square.
(3) Positive numbers have positive cubic roots; 0 has a cube root, which is itself; Negative numbers have negative cubic roots; Any number has a cubic root.
(4) Using the reciprocal operation relationship between the issuer and the cube to find the cube root of a number, we can use this reciprocal relationship to test its correctness and find the cube root of a negative number. We can first find the cubic root of the absolute value of this negative number, and then take its reciprocal.
Right triangle knowledge point
First, solve the right triangle.
1. Definition: known edges and angles (two of which must have one side) → all unknown edges and angles.
2. Basis: ① Relationship between the two sides: review outline of junior high school mathematics
② Angle relation: A+B = 90.
③ Angular relation: the definition of trigonometric function.
Note: Try to avoid using intermediate data and division.
Second, the handling of practical problems.
1. Review outline of prone and elevation of junior high school mathematics
2. Azimuth and quadrant angle
3. Slope:
4. When both right triangles lack the conditions for solving right triangles, they can be solved by column equations.
Axisymmetric knowledge points of graphics
Perpendicular bisector of I-shaped line segment
① Definition: A straight line that is vertical and bisects a known line segment is called the median vertical line or median vertical line of the line segment.
② Nature:
A, the point on the midline of the line segment is on the midline of the line segment with the same distance from the two end points of the line segment;
B, the points with the same distance to the two end points of the line segment are on the middle vertical line of the line segment;
C, the line segment is an axisymmetric figure, the middle vertical line of the line segment is an axis of symmetry of the line segment, and the other is the straight line where the line segment is located.
Properties of bisector of angle II
① The point on the bisector of the angle is equal to the distance on both sides of the known angle.
② The points with the same distance to both sides of the known angle are on the bisector of the known angle.
③ The angle is an axisymmetric figure, and the straight line where the bisector of the angle is located is the axis of symmetry of the angle.
Quadratic radical knowledge point
1. Quadratic root: the formula (≥0) is called quadratic root.
2. The simplest quadratic root:
The definition of (1) simplest quadratic root: ① the root number is an integer and the factor is an algebraic expression; (2) The number of square roots does not include numbers or factors that can be completely opened; ③ The denominator has no radicals.
(2) The simplest quadratic radical must meet the following conditions at the same time:
(1) The root sign does not contain root factors or factors;
② Denominators are not included in the number of square roots;
③ The denominator has no radicals.
3. Similar secondary roots (combinable 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, that is, two roots that can be merged.
4. Properties of quadratic roots
Non-negative number: It is non-negative.
Note: this property can be written as a formula, which is often used in radical operation.
Letters are not necessarily positive numbers.
(2) When a completely openable factor moves out of the root sign, it must be replaced by its arithmetic square root.
③ The factors that can move into the root sign must be non-negative. If the value of the factor is negative, the negative sign should stay outside the root sign.
(4) The difference and connection between formula and:
① indicates the arithmetic root of finding the square of a number, and the range of a is all real numbers.
② represents the square of the arithmetic square root of a number, and the value range of a is non-negative.
The result of sum operation is nonnegative.
Estimate knowledge points
1. Rounding
Example: arithmetic square root of 2 (reserved as 0.0 1)
Solution: Root number 2 =1.414 ≈1.41.
Step 2: Step 1.
Example: How much is one pen from 2.6 yuan and four pens (to the whole number)?
Solution: 2.6x= 10.4 yuan ≈ 1 1 yuan.
If rounded, it is 10 yuan. If it is not enough, you should go up.
3. Tailing method
Example: If you buy a pen from 20 yuan and 3 yuan, how many pens can you sell?
Solution: 20/3=6.6666...6 branches.
If rounded, it is seven, so it should be removed.
According to the general method, 854 is estimated to be 840, and 840 divided by 7 equals 120. However, this makes it difficult for students to grasp the scale. We can directly calculate that 854 divided by 7 equals 122. Then look at the nearest integer of 122. We can easily see that 122 is close to 120.
For example, if two or more numbers are multiplied or added, subtracted or divided, we can't calculate them quickly and correctly, that is, we can only calculate the open numbers.
Mathematics answering skills in senior high school entrance examination
1, direct debit method
Starting directly from the conditions given by the proposition, using concepts, formulas, theorems, etc. Reasoning or operating, drawing a conclusion and choosing the correct answer are traditional methods to solve problems.
2. Verification method
Find out the appropriate verification conditions from the questions, and then find out the correct answer through verification. You can also substitute alternative answers into conditions to verify and find the correct answer. This method is called verification method (also called substitution method). This method is often used when encountering quantitative propositions.
3. Special element method
Substitute appropriate special elements (such as figures or numbers) into the conditions or conclusions of the topic to get the answer. This method is called the special element method.
4. Exclusion method
For multiple-choice questions with only one correct answer, according to mathematical knowledge or reasoning and calculus, the incorrect conclusion is excluded, and the remaining conclusions are screened, so that the solution to the correct conclusion is called exclusion screening.
5. Graphical method
According to the nature and characteristics of the graphics or images that meet the requirements of the topic, it is called graphical method to make the right choice. Graphic method is one of the common methods to solve multiple-choice questions.
Mathematics learning method
The foundation is very important.
Do you feel that students who can get full marks in mathematics don't even have to read books? In fact, math experts pay more attention to the foundation. Mathematical formulas, properties of geometric figures, properties of functions, etc. It is the foundation of mathematics learning. It can even be said that the quality of the foundation directly determines the level of mathematics scores in the senior high school entrance examination.
Li Xianliang said that one of his classmates came to talk to him about this topic. In fact, the topic is not difficult, but this classmate did not master some basic knowledge thoroughly, which led to no ideas when doing the topic. The foundation is not firm, the earth shakes, and a small knowledge loophole may lead to your whole problem without a clue, which is very dangerous.
2. The wrong book is very important.
Of all the subjects, mathematics is the most important one. Li Xianliang also advocated sorting out the wrong problems. Li Xianliang has some tips about wrong questions, that is, if he insists on sorting out wrong questions at ordinary times, he will eventually produce many thick wrong questions. We can review regularly. For those who have a thorough grasp, we can mark them so that we don't have to review them in the future, so that the wrong questions will be used more efficiently.
3, do more reflection.
Mathematics learning should be consolidated by doing a lot of problems, but not only quantity, but also quality. When you encounter classic questions and high comprehensive questions, after each question is finished, you should analyze and reflect, and ask a few more why, in order to really do the questions thoroughly.
4. Form mathematical knowledge into a system
Li Xianliang, a master of mathematics, said that the knowledge in textbooks is scattered. I suggest you draw your own mind map to connect the knowledge. The process of drawing mind map is the process of constantly understanding and transforming knowledge into structure.
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