Chapter 1 Real Numbers ★ Key Points ★ Related concepts and properties of real numbers, operation of real numbers ☆ Abstract ☆ 1. Key concepts 1. Classification of Numbers and Table of Conceptual Number Series: Description: Principle of "Classification": 1) Proportionality (no weight, no leakage) 2) Standard 2. Non-negative number: the collective name of positive real number and zero. (Table: x≥0) Common non-negative numbers are: Nature: If the sum of several non-negative numbers is 0, then each non-burden number is 0. 3. Reciprocal: ① Definition and representation ② Property: a.a ≠1/a (a ≠1); B. 1/a，a≠0； c . 0 1； When a > 1,1/a <1; D. the product is 1. 4. Inverse number: ① Definition and representation ② Properties: A ≠≠ 0, a ≠-a; The position of a and -a on the number axis; The sum of c is 0 and the quotient is-1. 5. Number axis: ① Definition ("three elements") ② Function: a. Visually compare the size of real numbers; B. clearly reflect the absolute value; C. establish a one-to-one correspondence between points and real numbers. 6. Definition and expression of odd number, even number, prime number and composite number (positive integer-natural number): odd number: 2n- 1 even number: 2n(n is a natural number) 7. Absolute value: ① Definition (two kinds): Algebraic definition: Geometric definition: The geometric meaning of the absolute value top of number A is the distance of the point corresponding to the origin of real number A on the number axis. ② A ≥ 0, and the symbol "│ │" is a sign of "non-negative number"; ③ There is only one absolute value of number A; ④ When dealing with any type of topic, as long as "│ │" appears, the key step is to remove the "│ │" symbol. Second, the operation of real numbers is 1. Operation rules (addition, subtraction, multiplication, division, power, root) 2. Algorithm (five plus [multiplication] commutative law, associative law; [Distribution Law of Multiplication to Addition] 3. Operation sequence: a. Advanced operation to low-level operation; B. (operation at the same level) from "left" to "right" (for example, 5 ÷ 5); C (when there are brackets) from "small" to "medium" to "large". Three. Application example (omitted) Attachment: typical example 1. Known: the positions of A, B and X on the number axis are as follows, and verified: │ X-A │+│ X-A │+│ X-B │ = B-A.2 Known: a-b=-2 and AB 0, > 0; ② When a < 0, > 0 (n is an even number), < 0 (n is an odd number) (2) Zero index: = 1(a≠0) Negative integer index: =1(a ≠ 0, p is a positive integer) II. Operating rules. 2 Simplified methods (two kinds) 3. Algebraic expression algorithm (bracket removal and bracket addition) 4. The essence of power operation: ① =; ② ÷ = ; ③ = ; ④ = ; ⑤ Skills: 5. Multiplication rule: (1) single× single; (2) single × many; 3 more x more. 6. Multiplication formula: (plus or minus) (A+B) (A-B) = (A B) = 7. Division rule: (1) single-single; (2) Too many orders. 8. Factorization: (1) definition; ⑵ Methods: A. Common factor method; B. formula method; C. cross multiplication; D. group decomposition method; E. find the root formula method. 9. The nature of arithmetic roots: =; ; (a≥0，b≥0)； (a ≥ 0, b > 0) (positive and negative) 10. Root algorithm: (1) addition rule (merging similar quadratic roots); (2) multiplication and division; (3) The denominator is reasonable: a; b； C.11.Scientific notation: (1 ≤ A < 10, n is an integer = 3. Application examples (omitted) IV. Comprehensive operation of numbers (omitted) Chapter III Preliminary statistics ★ Main points ★☆ Summary ☆ I. Important concepts 65438. 2. Individuals: each respondent in the group. 3. Sample: a part of individuals extracted from the population. 4. Sample size: the number of individuals in the sample. 5. Mode: the data with the highest frequency in a set of data. 6. Median: arrange a group of data in order of size, with a number in the middle (or the average of two data in the middle) 2. Calculation method 1. Sample average:1; (2) If,,,, then (A- constant,,,, is close to a more integer constant A); (3) Weighted average: (4) Average is a characteristic number that describes the trend (concentration position) in data concentration. Sample average is usually used to estimate the overall average. The larger the sample size, the more accurate the estimation. 2. Sample variance: (1); (2) If,,,,, then (a) is a relative "integer" constant, which is close to the average value of,,,; If … is less than "whole", then; ⑶ Sample variance is a characteristic number that describes the degree of data dispersion (fluctuation). When the sample size is large, the sample variance is very close to population variance, which is usually used to estimate population variance. 3. Standard deviation of the sample: III. Application examples (omitted) Chapter IV Related concepts, judgments and properties of linear shapes ★ key points ★ intersection lines and parallel lines, triangles and quadrangles. ☆ Abstract ☆ 1. Straight lines, intersecting lines and parallel lines 1. This paper analyzes the differences and relations among line segments, rays and straight lines from the aspects of figure, representation, boundary, number of endpoints and basic properties. 2. The midpoint of a line segment and its representation 3. Basic properties of lines and line segments (using the basic properties of line segments to prove that the sum of two sides of a triangle is greater than the third side) 4. The distance between two points (three distances: point-point; Dotted line; Line-line) 5。 Angle (flat angle, rounded corner, right angle, acute angle, obtuse angle) 6. Complementary angle, complementary angle and their representation 7. Equiangular line and its representation 8. Vertical line and its basic properties (used to prove that "the hypotenuse in a right triangle is larger than the right") 9. Diagonal vertex angle and its properties 10. Parallel lines and their judgment and properties (reciprocal) ② Parallel to two straight lines perpendicular to a straight line. The definition, proposition and composition of 12. 13 proposition. Axioms and theorems 14. Inverse proposition 2. Triangle classification: (1) divided by edge; ⑵ Divided by the angle 1. Definition (including internal angle and external angle) 2. The relationship between angles of triangle: ⑵ Angle and angle :⑵Sum and inference of internal angles; ② sum of external angles; (3) the sum of the internal angles of the N-polygon; (4) the sum of the external angles of the N-polygon. ⑵ Edge and edge: The sum of two sides of a triangle is greater than the third side, and the difference between the two sides is less than the third side. ⑶ Angle and edge: In the same triangle, 3. Discussion on the main line segments of triangle: ① Define the intersection of ②×× lines-the× center ③ Properties of triangle ① High line ② Midline ③ Angular bisector ④ Midline ⑤ Midline ⑤ General triangle ② Special triangle: right triangle, isosceles triangle and equilateral triangle 4. Judgment and properties of special triangles (right triangle, isosceles triangle, equilateral triangle and isosceles right triangle) 5. Congruent triangles ① congruence judgment of general triangles (SAS, ASA, AAS, SSS) ① congruence judgment of special triangles: ① general method ② special method 6. The area of triangle ⑴ General calculation formula ⑴ Properties: The areas of triangles with equal base and equal height are equal. 7. The midpoint of the important auxiliary line (1) cooperates with the midpoint to form the midline; (2) Double the center line; (3) Add auxiliary parallel lines 8. Proof method (1) direct proof: synthesis method, analysis method (2) indirect proof-reduction to absurdity: 1) reverse design (2) reduction to absurdity (3) proof of line segment equality and angle equality (4) proof of line segment doubling relationship: doubling method, equal division method (5) proof of line segment and difference relationship: continuation method. Inference 1: Connect the midpoints of the sides of the quadrilateral in turn with equal diagonal lines to get a diamond. Inference 2: Connect the midpoints of the sides of the quadrilateral in turn with diagonal lines perpendicular to each other to get a rectangle. ⑶ Sum of external angles: 360 2. Special quadrangles (3) General methods to study them: (3) Parallelogram, rectangle, diamond and square; Definition, properties and judgment of trapezoid and isosceles trapezoid (3) Judgment steps: quadrilateral → parallelogram → rectangle → square ┗→ rhombus —↑ (4) Diagonal bond function: 3. Symmetric graph (1) axis symmetry (definition and properties); (2) Central symmetry (definition and nature) 4. Relevant theorems: ① the bisection theorem of parallel lines and its inference 1, ② the midline theorem of triangle and trapezoid, ③ the distance between parallel lines is equal everywhere. (For example, look for triangles with equal areas in the picture below) 5. Important auxiliary lines: ① Always connect the diagonals of quadrangles; ② Trapezoids are often transformed into triangles by translating a waist, translating a diagonal, making a height, connecting the midpoint between the vertex and the waist and extending the intersection with the bottom. 6. Drawing: Divide the line segments randomly. Four. Application examples (omitted) Chapter 5 Equation (