Chapter 1 Rational Numbers★ Classification of Rational Numbers 1. If divided by definition, rational numbers can be divided into integers (positive integers; Negative integer; 0) and score (positive score, negative score). If divided by positive and negative, rational numbers can be divided into positive rational numbers (positive integer; Positive fraction), 0, negative rational number (negative integer; Negative score). 2. All rational numbers can be expressed by fractions, and π is not rational. Number axis ★ 1. Definition of number axis: The straight line that defines the origin, positive direction and unit length is called number axis. The inverse number is 1. Numbers with only two different signs are called opposites. (The antonym of 0 is 0) Absolute value 1. The distance from point A to the origin on the number axis represents the absolute value of A. ★2. The nature of absolute value: non-negative. 3. The absolute value of a positive number is itself, the absolute value of a negative number is its reciprocal, and the absolute value of 0 is 0. The size of the rational number is 1. Positive numbers are greater than 0, negative numbers are less than 0, and positive numbers are greater than negative numbers. 2. Two negative numbers, the larger one has the smaller absolute value. Addition of rational number 1 Add two numbers with the same sign, take the same sign, and then add the absolute values. 2. Add the numbers of two different symbols with different absolute values, take the addend symbol with larger absolute value, and subtract the one with smaller absolute value from the one with larger absolute value; Two opposite numbers add up to 0. Add a number to 0 and you still get the number. 3. In the addition of rational numbers, the addition exchange rate: two numbers are added, and the position of the exchange addend remains unchanged. Law of addition and association: when three numbers are added, the first two numbers are added first, or the last two numbers are added first, and the sum is unchanged. Subtracting a number from a rational number is equal to adding the reciprocal of this number. ★ Multiply rational numbers by two numbers, the same sign is positive, the different sign is negative, and the absolute value of multiplication. Multiply any number by 0 to get 0. Reciprocal: Two numbers whose product is 1 are reciprocal. Multiplication exchange law: Multiplication exchange law multiplies two numbers, and the position of exchange factor and product remains unchanged. Multiplication and association law: when three numbers are multiplied, the first two numbers are multiplied first, or the last two numbers are multiplied first, and the product is unchanged. Multiplication and distribution law: a number multiplied by the sum of two numbers is equivalent to multiplying this number by these two numbers respectively, and then adding the products. ★ Dividing a rational number by a number that is not 0 is equal to multiplying and dividing by the reciprocal two numbers of this number. The same sign is positive, different signs are negative, and dividing the absolute value by 0 and any number that is not equal to 0 is equal to 0. Mixed operation of rational number 1. Operation sequence: calculate the power first, then multiply and divide, and finally add and subtract. If it is the same level operation, it is calculated from left to right. If there are brackets, count the brackets first, then the brackets and finally the braces. Power of rational number ★ 1. The operation of finding the product of n identical factors is called power, and the result of power is called power. In, it is called base and exponent. When 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. Scientific symbol 1. Scientific notation represents a number in the form of a× 10 to the nth power, where a is a number with only one integer bit and n is a positive integer. This notation is called scientific notation. Factor 1. A number that is close to the exact number (slightly more or less than the exact number) is called a divisor. ★2. Significant digits: In a number, all digits from the first digit on the left that is not 0 to the exact digit are called the significant digits of the number. Chapter II Addition and subtraction monomials of algebraic expressions 1. Definition of monomial: the product of numbers or letters is called monomial, and a single number or letter is also monomial. 2. Coefficient: the numerical factor in item 3. Times: Exponential sum of all letters in a single polynomial 1 The sum of several monomials is called polynomial. 2. Each monomial is called a polynomial term. 3. Items without letters are called constant items. 4. The degree of the highest term in a polynomial is called the degree of the polynomial. The highest term in a polynomial is called the highest term of a polynomial. ★5. There are no degrees in the polynomial. Algebraic expression 1 Monomial and polynomial are collectively called algebraic expressions. Addition and subtraction of algebraic expressions 1 Items with the same letters and the same letter index are called similar items, and several constant items are also similar items. 2. Merging similar terms in polynomials into one term is called merging similar terms. 3. After merging similar items, the coefficient of the obtained item is the sum of the coefficients of similar items before merging, and the letter part remains unchanged. Merge similar items-remove brackets★1. If the factor outside the brackets is positive, the symbols of the items in the original brackets are the same as the original symbols after the brackets are removed; If the factor outside the brackets is negative, the symbols of the items in the original brackets are opposite to those after the brackets are removed. Chapter 3 One-dimensional linear equation One-dimensional linear equation 1. This equation is an equation with unknowns. 2. An equation is an equation, and an equation is not necessarily an equation. 3. There is only one unknown (element), and the number of unknowns is 1. Such an equation is called a one-dimensional linear equation. Column equation 1. It is a method to solve practical problems by analyzing the quantitative relations in practical problems and listing equations by using their equal relations. 2. Column equation is an important method to solve problems, and unknowns can be solved by using equations. Solve the equation 1. Solving an equation is to find the value of an unknown quantity with equal signs on both sides of the equation, and this value is the solution of the equation. Properties of equation ★ 1 The nature of the equation 1 Add (subtract) the same number (or formula) on both sides of the equation at the same time, and the results are still equal. ★2. Properties of Equation 2 When both sides of the equation are multiplied by the same number or divided by the same number that is not 0, the results are still equal. Merge similar terms 1. Merging similar terms in polynomials into one term is called merging similar terms. Adding (or subtracting) the same number or the same algebraic expression on both sides of the equation is equivalent to changing the sign of some items in the equation and moving them from one side of the equation to the other. This deformation is called a shift term. ★ Remove the bracket 1. There is a "+"sign in front of the bracket. Remove the bracket and the "+"sign in front, and the symbols of the items in the bracket will remain unchanged. 2. There is a "-"sign in front of the bracket. Remove the bracket and the "-"sign in front, and change the symbols of the items in the bracket to the opposite symbols. The fourth chapter is the preliminary geometry of graphic understanding 1. Points, lines, faces and bodies are all called geometric figures. 2. Geometric graphics are generally divided into three-dimensional graphics and plane graphics. 3. Some parts of geometric figures are not on the same plane. They are three-dimensional figures. 4. Some geometric figures are all in a plane, which is a plane figure. Development map 1. Some three-dimensional graphics are composed of some plane graphics, and they can be developed into plane graphics by cutting their surfaces properly. Such a plane figure is called the expanded figure of the corresponding three-dimensional figure. Point, line, surface and body 1. Cuboid, cube, cylinder, cone, sphere, prism and pyramid are all geometric bodies. Geometry is also called volume for short. What surrounds the body is the surface. There are two kinds of face shapes: flat and curved. 3. Point to line, opposite to line, facing the body. 4. Geometry consists of points, lines, faces and bodies. 5. Points are the basic elements of graphics. Line, ray, line segment 1 There is a straight line passing through two points and there is only one straight line. Simply put: two points determine a straight line. 2. When two different straight lines have a common point, we say that the two straight lines intersect, and this common point is called their intersection. 3. The point where a line segment is divided into two equal line segments is called the midpoint of this line segment. 4. In the connection between two points, the line segment is the shortest. Simply put: between two points, the line segment is the shortest. 5. The length of the line segment connecting two points is called the distance between these two points. Angle 1. A figure composed of two rays with a common endpoint is called an angle. This common endpoint is the vertex of the angle, and these two rays are the two sides of the angle. 2. The degrees, minutes and seconds of an angle are all in hexadecimal. (for example:1= 60 ′,1′ = 60 ″) 3. Starting from the vertex of an angle, the ray that divides the angle into two equal angles is called the bisector of the angle. If the sum of two angles is equal to 90 degrees (right angle), it is said that these two angles are complementary angles, that is, each angle is the complementary angle of the other angle. 5. If the sum of two angles is equal to 180 (flat angle), it is said that the two angles are complementary, that is, one of them is complementary to the other. 6.★ The complementary angles of equal angles are equal. ★ The complementary angles of equal angles are equal.