Book One, Chapter One, Rational Numbers 1. 1 Positive Numbers and Negative Numbers All the numbers with the minus sign "-"except 0 I have learned before are called negative numbers. Numbers other than 0 that I learned before are called positive numbers. The number 0 is neither positive nor negative, it is the dividing line between positive and negative numbers. In the same problem, positive numbers and negative numbers have opposite meanings. 1.2 rational number 1.2. 1 rational number Positive integers, 0 and negative integers are called integers, and positive and negative fractions are called fractions. Integers and fractions are collectively called rational numbers. 1.2.2 The straight line with the origin, positive direction and unit length defined by the number axis is called the number axis. Function of number axis: All rational numbers can be represented by points on the number axis. Note: The origin, positive direction and unit length of (1) axis are indispensable. ⑵ The unit length of the same shaft cannot be changed. Generally speaking, if it is a positive number, the point representing a on the number axis is on the right side of the origin, and the distance from the origin is a unit length; The point representing the number -a is on the left of the origin, and the distance from the origin is one unit length. 1.2.3 Numbers with only two different symbols are called reciprocal. Two points representing the opposite number on the number axis are symmetrical about the origin. Add a "-"sign before any number, and the new number represents the antonym of the original number. 1.2.4 absolute value Generally speaking, the distance from the point representing the number A on the number axis is called the absolute value of the number 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. Rational numbers are represented on the number axis, and the order from left to right is from small to large, that is, the number on the left is smaller than the number on the right. Compare the sizes of rational numbers: (1) Positive numbers are greater than 0, 0 is greater than negative numbers, and positive numbers are greater than negative numbers. (2) Two negative numbers, the larger one has the smaller absolute value. 1.3 addition and subtraction of rational numbers 1.3. 1 addition rule of rational numbers: (1) Add two numbers with the same sign, take the same sign and add the absolute values. ⑵ Add two numbers with different absolute values, take the sign of the addend with larger absolute value, and subtract the number with smaller absolute value from the number with larger absolute value. Two opposite numbers add up to 0. (3) When a number is added to 0, the number is still obtained. When two numbers are added, the positions of the addend are exchanged and the sum is unchanged. Additive commutative law: A+B = B+A three numbers add, first add the first two numbers, or add the last two numbers, and the sum remains the same. Law of addition and association: (a+b)+c = a+(b+c) 1.3.2 The subtraction of rational numbers can be converted into addition. Rule of rational number subtraction: subtracting a number is equal to adding the reciprocal of this number. A-B = A+(-B) 1.4 rational number multiplication 1.4. 1 rational number multiplication: when two numbers are multiplied, the same sign is positive, the different sign is negative, and the absolute value is multiplied. Any number multiplied by 0 is 0. Two numbers whose product is 1 are reciprocal. Multiply several numbers that are not 0. When the number of negative factors is even, the product is positive. When the number of negative factors is odd, the product is negative. When two numbers are multiplied, the exchange factor and the product are in the same position. AB = BA Multiply three numbers, first multiply the first two numbers, or multiply the last two numbers, and the products are equal. (ab) c = a (BC) Multiplying a number by the sum of two numbers is equivalent to multiplying this number by these two numbers respectively, and then adding the products. A (B+C) = AB+AC number multiplied by letters: (1) number multiplied by letters, omitting the multiplication sign, or ""(2) number multiplied by letters, and omitting 1 when the coefficient is 1 or-1. (3) The band score is multiplied by letters, and the band score becomes a false score. If any rational number is represented by the letter X, the product of 2 and x is 2x, and the product of 3 and x is 3x, then the formula 2x+3x is the sum of 2x and 3x, 2x and 3x are the terms of this formula, and 2 and 3 are the coefficients of these two terms respectively. Generally speaking, when formulas with the same letter factor are combined, it is only necessary to combine their coefficients, and the result is taken as the coefficient, and then multiplied by the letter factor, that is, ax+bx = (a+b) x, where X is the letter factor, and A and B are the coefficients of ax and bx respectively. Rule of removing brackets: put "+"before brackets. Remove brackets and the "+"before brackets, and the contents in brackets will not change the symbol. There is a "-"before the brackets. Remove brackets and the "-"sign in front of brackets, and change all the symbols in brackets. The factors outside brackets are positive numbers, and the symbols of the items in the formula after removing brackets are the same as those of the corresponding items in the original brackets; The factor outside the bracket is negative, and the sign of each item in the formula after the bracket is opposite to that of the corresponding item in the original bracket. 1.4.2 division of rational numbers: dividing by a number that is not equal to 0 is equal to multiplying the reciprocal of this number. A ÷ b = a b 1 (b ≠ 0) Two numbers are divided, the same sign is positive, the different sign is negative, and the absolute value is divided. Divide 0 by any number that is not equal to 0 to get 0. Because the division of rational numbers can be converted into multiplication, the operation can be simplified by using the operational nature of multiplication. The mixed operation of multiplication and division often turns division into multiplication first, then determines the sign of the product, and finally calculates the result. The power of 65438+the power of 0.5 rational number 1.5. 1 The operation of finding the product of n identical factors is called power, and the result of power is called power. In, a is called the base and n is called the exponent. When an is regarded as the result of the n power of A, it can also be read as the n power of A. The odd power of a negative number is negative and the even power of a negative number is positive. Any power of a positive number is a positive number, and any power of a positive integer is 0. The operation sequence of rational number mixed operation: (1) multiply first, then multiply and divide, and finally add and subtract; (2) unipolar operation, from left to right; (3) If there are brackets, do the operation in brackets first, and then 1.5.2 with brackets, brackets and braces in turn. Numbers greater than 10 are expressed in the form of a× 10n (where a is a number with only one integer and n is a positive integer), and scientific notation is used. Use scientific notation to represent n-bit integers, where the exponent of 10 is n- 1. 1.5.3 The divisor and the significant number are close to the actual number, but the number different from the actual number is called the divisor. Accuracy: an approximate value is rounded to the nearest place, so it is accurate to the nearest place. From the first non-zero digit to the last digit on the left of a number, all digits are valid digits of this number. For the number a× 10n expressed by scientific notation, its effective number is defined as the effective number in A. Chapter II One-dimensional linear equation 2. 1 From formula to equation 2. 1. 1 An equation with unknown numbers is called an equation. There is only one unknown (yuan), and the exponent of the unknown is 1 (degree). Such an equation is called a one-dimensional linear equation. It is a method to solve practical problems by analyzing the quantitative relations in practical problems and listing equations by using their equal relations. Solving the equation is to find the value of the unknown quantity that makes the left and right sides of the equation equal, and this value is the solution of the equation. 2. 1.2 Properties of the equation 1 Add (or subtract) the same number (or formula) on both sides of the equation, and the results are still equal. Properties of Equation 2 Multiply both sides of the equation by the same number, or divide by the same number that is not 0, and the results are still equal. 2.2 Starting from the ancient number books-discussion on the linear equation of one yuan (1) Moving the sign of an item on one side of the equation to the other side is called shifting the item. 2.3 Starting from the "problem of buying cloth"-discussion on the linear equation of one yuan (2) When there are brackets in the equation, the method of removing brackets is similar to that in rational number operation. Solving an equation is to find an unknown number (such as x). By removing the denominator, brackets, shifting the term, merging and converting the coefficient into 1, the linear equation can be gradually transformed into the form of X = A, which mainly depends on the properties and operation rules of the equation. Denominator removal: (1) Specific practice: multiply both sides of the equation by the least common multiple of each denominator; (2) Basis: the properties of the equation; (3) Note: (1) brackets are placed on the molecule; (2) Items without denominator should also be multiplied by 2.4; (3) Exploring practical problems and linear equations; (3) Preliminary graphic understanding 3. 1 color graphics The graphics we get in real life are called geometric graphics. 3. 1. 1 Three-dimensional graphics and plane graphics Cubes, cubes, spheres, cylinders and cones are all three-dimensional graphics. In addition, prisms and pyramids are also common three-dimensional figures. Rectangular, square, triangle and circle are all plane figures. Many three-dimensional graphics are surrounded by some plane graphics, which can be expanded into plane graphics by proper cutting. 3. 1.2 Point, line, surface and volume geometry are also called volume. Cuboid, cube, cylinder, cone, sphere, prism and pyramid are all geometric bodies. What surrounds the body is the surface. There are two kinds of face shapes: flat and curved. Lines are formed at the intersection of faces. The intersection of lines is a point. Geometric figures are all composed of points, lines, surfaces and bodies, and points are the basic elements of figures. 3.2 Lines, rays and line segments pass through two points and there is only one straight line. Two points define a straight line. The line segment AB at point C is divided into two equal line segments AM and MB, and point M is called the midpoint of line segment AB. Similarly, line segments have bisectors and quartiles. The point of a straight line and the part next to it are called rays. Among the connecting lines between two points, the line segment is the shortest. To put it simply: between two points, the line segment is the shortest. 3.3 The measuring angle of an angle is also a basic geometric figure. Degrees, minutes and seconds are commonly used units of angle measurement. Divide a fillet into 360 equal parts, each equal part is an angle of one degree, and record it as1; Divide the angle of 1 degree into 60 equal parts, each part is called the angle of 1 minute, and it is recorded as1; Divide the angle of 1 into 60 equal parts, each part is called 1 sec, and it is recorded as 1. 3.4 Comparison and operation of angles 3.4. 1 Comparison of angles starts from the vertex of an angle, and the ray that divides this angle into two equal angles is called the bisector of this angle. Similarly, there is the so-called bisector. 3.4.2 Complementary Angle and Complementary Angle If the sum of two angles is equal to 90 (right angle), they are said to be complementary angles. If the sum of two angles is equal to 180 (flat angle), the two angles are said to be complementary. The complementary angles of equal angles are equal. The complementary angles of equal angles are equal. This chapter knowledge structure diagram geometry figure three-dimensional figure plane figure unfold three-dimensional figure plane figure three-dimensional figure plane figure angle measurement comparison complementary angle is equal to the bisector of complementary angle. Chapter IV Data collection and collation Collecting, collating, describing and analyzing data is the basic process of data processing. 4. 1 What kind of animals do students like best? Take a comprehensive survey as an example, and record the data by crossing lines. Each stroke of the word "positive" represents a data. The survey of all subjects is a comprehensive survey. 4.2 Survey of primary and secondary school students' eyesight-sampling survey, such as sampling survey, is a survey in which samples are taken from the population and the population is estimated according to the samples. Statistical survey is a common method to collect data, which generally includes comprehensive survey and sampling survey, and sampling survey is often used in practice. In the process of investigation, data can be obtained in different ways. Besides questionnaires and interviews, consulting literature and experiments is also an effective way to obtain data. Organizing data with tables can help us find the distribution law of data. Using statistical charts to represent the sorted data can reflect the data law more intuitively. 4.3 Project research survey "How do you dispose of waste batteries?" The survey activities mainly include the following five steps: 1. Design the questionnaire ① Design the questionnaire ① Determine the purpose of the survey; (2) Select the survey object; (3) Designing survey questions (2) When designing questionnaires, it should be noted that (1) questions should not involve the personal opinions of the questioner; Don't ask questions that others don't want to answer; ③ The choice answers provided should be as comprehensive as possible; ④ Ask questions concisely; ⑤ The questionnaire should be short. Second, the implementation of the survey will copy enough questionnaires and send them to the respondents. Attention should be paid to: (1) explain to the respondents who are the respondents and why they are the respondents; (2) Tell the interviewee the purpose of your data collection. Three. Data processing According to the collected questionnaires, the collected data are sorted, described and analyzed. Four. According to the survey results, discuss what findings and suggestions your group has. V. Write a simple investigation report, Volume II, Chapter V Intersection and parallel lines 5. 1 intersection line 5. 1. 1 intersection line has a common vertex, and a common side and the other side are opposite extension lines. Such two angles are called adjacent complementary angles. There are four pairs of adjacent complementary angles when two straight lines intersect. There is a vertex with a common * * *, and both sides of the corner are opposite extension lines. These two angles are called antipodal angles. Two straight lines intersect and have two opposite angles. The vertex angles are equal. 5. 1.2 Two straight lines intersect, and one of the four angles formed is a right angle, so the two straight lines are perpendicular to each other. One of the straight lines is called the perpendicular of the other straight line, and their intersection point is called the vertical foot. Note: (1) The vertical line is a straight line. The four angles formed by two vertical lines are all 90. (3) Verticality is a special case of intersection. (4) Vertical symbols: a⊥b, AB⊥CD. There are countless vertical lines to draw known straight lines. One and only one straight line is perpendicular to the known straight line. Of all the line segments connecting points outside the straight line and points on the straight line, the vertical line segment is the shortest. Simply put: the vertical line segment is the shortest. The length from a point outside a straight line to the vertical section of the straight line is called the distance from the point to the straight line. 5.2 parallel lines 5.2. 1 parallel lines are in the same plane, and there is no intersection point between the two straight lines, so these two straight lines are parallel to each other, which is recorded as: a ∑ B. There are only two relationships between two straight lines in the same plane: intersection or parallelism. Parallelism axiom: after passing a point outside a straight line, there is one and only one straight line parallel to this straight line. If both lines are parallel to the third line, then the two lines are also parallel to each other. 5.2.2 Conditions of parallel lines Two straight lines are cut by a third straight line, and such two angles are called congruent angles on the same side of the two cut lines and on the same side of the cut lines. Two straight lines are cut by a third straight line, and between the two cutting lines, on both sides of the cutting line, such two angles are called inscribed angles. Two straight lines are cut by a third line, and between the two cut lines, on the same side of the cut line, such two angles are called ipsilateral internal angles. Method for judging that two straight lines are parallel: Method 1 Two straight lines are cut by a third straight line. If congruent angles are equal, two straight lines are parallel. To put it simply: the same angle is equal and two straight lines are parallel. Method 2 Two straight lines are cut by a third straight line. If the internal dislocation angles are equal, two straight lines are parallel. To put it simply: the internal dislocation angles are equal and the two straight lines are parallel. Method 3 Two straight lines are cut by a third straight line. If they are complementary, then these two straight lines are parallel. To put it simply: the internal angles on the same side are complementary and the two straight lines are parallel. 5.3 Properties of parallel lines Parallel lines have properties: properties 1 Two parallel lines are cut by a third line, and their congruence angles are equal. To put it simply: two straight lines are parallel and have the same angle. Property 2 Two parallel lines are cut by a third straight line, and their internal angles are equal. To put it simply: two straight lines are parallel and their internal angles are equal. Property 3 Two parallel lines are cut by a third straight line and complement each other. Simply put, two straight lines are parallel and complementary. The length of a line segment perpendicular to and sandwiched between two parallel lines is called the distance between two parallel lines. A statement that judges a thing is called a proposition. 5.4 Translation (1) Move a whole figure in a certain direction, and you will get a new figure, which is exactly the same in shape and size as the original figure. ⑵ Every point in the new graph is obtained by moving a point in the original graph. These two points are corresponding points, and the line segments connecting each group of corresponding points are parallel and equal. This movement of graphics is called translation transformation, or translation for short. Chapter VI Planar Cartesian Coordinate System 6. 1 Planar Cartesian Coordinate System 6. 1. 1 Ordered number pairs A number pair consisting of two numbers A and B in turn is called an ordered number pair. 6. 1.2 Plane Cartesian Coordinate System Draw two mutually perpendicular number axes with coincident origins on the plane to form a plane Cartesian coordinate system. The horizontal axis is called the X axis or the horizontal axis, and it is customary to take the right as the positive direction; The vertical axis is called the Y axis or the vertical axis takes 2 as the positive direction; The intersection of the two coordinate axes is the origin of the plane rectangular coordinate system. Any point on the plane can be represented by an ordered number pair. After the rectangular coordinate system is established, the coordinate plane is divided into four parts, I, II, III and IV, which are called the first quadrant, the second quadrant, the third quadrant and the fourth quadrant respectively. The points on the coordinate axis do not belong to any quadrant. 6.2 Simple application of coordinate method 6.2. 1 Representing geographical position with coordinates The process of drawing a plan of some places in the area with a plane rectangular coordinate system is as follows: (1) Establish a coordinate system, select a suitable reference point as the origin, and determine the positive directions of the X axis and the Y axis; ⑵ Determine the appropriate scale according to specific problems and mark the unit length on the coordinate axis; (3) Draw these points on the coordinate plane and write down the coordinates of each point and the name of each place. 6.2.2 Coordinate translation. In the plane rectangular coordinate system, the corresponding point (x+a, y) (or (x-a, y)) can be obtained by translating the point (x, y) to the right (or to the left) by a unit length. The corresponding point (x, y+b) (or (x, y-b)) can be obtained by translating the point (x, y) up (or down) by b unit lengths. In the plane rectangular coordinate system, if a positive number A is added (or subtracted) to the abscissa of each point of the graph, the corresponding new graph is to translate the original graph to the right (or left) by a unit length; If a positive number A is added (or subtracted) to the ordinate of each point, the corresponding new figure is to translate the original figure up (or down) by a unit length. Chapter VII Triangle 7. 1 Line segments related to the triangle 7. 1. 1 A graph composed of three line segments that are not on the same line is called a triangle. The angle formed by two adjacent sides is called the inner angle of a triangle, which is called the angle of a triangle for short. A triangle with vertices A, B and C is marked as △ABC and pronounced as "triangle ABC". The sum of two sides of a triangle is greater than the third side. 7. 1.2 The bisector of the height, midline and angle of the triangle 7. 1.3 The stability triangle of the triangle has stability. 7.2 The angle related to the triangle is 7.2. 1 The sum of the internal angles of the triangle is equal to 180. 7.2.2 Exterior Angle of Triangle The angle formed by one side of the triangle and the extension line of the other side is called the exterior angle of the triangle. The outer angle of a triangle is equal to the sum of two non-adjacent inner angles. The outer angle of a triangle is greater than any inner angle that is not adjacent to it. 7.3 A polygon and its internal angles are in the same plane as 7.3. 1 polygon, and a figure consisting of several line segments connected end to end is called a polygon. The line segment connecting two nonadjacent vertices of a polygon is called the diagonal of the polygon. Diagonal formula of n polygons: 2) 3 (-nn Polygons with equilateral angles are called regular polygons. 7.3.2 Formula for the sum of inner angles of polygons and n-polygons: 180 (n-2) The sum of outer angles of polygons is equal to 360. 7.4 Thematic learning mosaic Chapter 8 Binary linear equations 8. 1 Binary linear equations contain two unknowns, and the exponents of the unknowns are both 1, which is called binary linear equations. The values of two unknowns that make the values on both sides of a binary linear equation group equal are called the solutions of binary linear equations, and the common solutions of two equations of binary linear equations are called the solutions of binary linear equations. 8.2 Elimination An unknown number is represented by an equation in a binary linear equation set, and then it is substituted into another equation to realize elimination, and the solution of this binary linear equation set is obtained. This method is called substitution elimination method, or substitution method for short. When the coefficients of the same unknown in two binary linear equations are opposite or equal, the unknown can be eliminated by adding or subtracting the two sides of the two equations respectively, thus a univariate linear equation system can be obtained. This method is called addition, subtraction and elimination, or addition and subtraction for short. 8.3 Re-exploring Practical Problems and Binary Linear Equations Chapter 9 Inequality and Inequality Group 9. 1 Inequality 9. 1 Inequality and its solution set The formula that ""indicates the size relationship is called inequality. The value of the unknown quantity that makes the inequality valid is called the solution of the inequality. The range of unknowns that can make inequality hold is called inequality solution set, which is called solution set for short. An inequality with an unknown degree of 1 is called one-dimensional linear inequality. 9. 1.2 The nature of inequality has the following properties: the nature of inequality 1 Add (or subtract) the same number (or formula) on both sides of inequality, and the direction of inequality remains unchanged. The nature of inequality 2 Both sides of inequality are multiplied by (or divided by) the same positive number, and the direction of inequality remains unchanged. The nature of inequality 3 Both sides of inequality are multiplied (or divided) by the same negative number, and the direction of inequality changes. 9.2 Practical Problems and Unary Linear Inequality To solve the unary linear equation, the equation should be gradually transformed into the form of x = a according to its properties; To solve one-dimensional linear inequality, it is necessary to gradually transform inequality into the form of x a) according to the nature of inequality. 9.3 One-dimensional linear inequality group When two inequalities are combined, a one-dimensional linear inequality group is formed. The common part of the solution set of several inequalities is called the solution set of inequalities composed of them. Solving inequality is to find its solution set. All kinds of inequality problems can be solved by inequality groups. When solving a system of linear inequalities with one variable. Generally, the solution set of each inequality is found first, and then the common part of these solution sets is found. The number axis can be used to express the solution set of inequality groups intuitively. 9.4 Learning and Applying Inequality Analysis Contest Title Chapter 10 Real number 10. 1 square root 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. The arithmetic square root of A is recorded as "root number A" and A is called the root number. If the square of a number is equal to a, then this number is called the square root or quadratic root of a, and the operation of finding the square root of a number is called the square root. 10.2 cube root If the cube of a number is equal to A, then this number is called the cube root or cube root of A ... The operation of finding the cube root of a number is called the square root. 10.3 real number infinite acyclic decimal number is also called irrational number. Rational numbers and irrational numbers are collectively called real numbers. The absolute value of a positive real number is itself; The absolute value of a negative real number is its reciprocal; The absolute value of 0 is 0.