Chapter 16 Score 1. Definition of fraction: If A and B represent two algebraic expressions, and B contains letters, then the formula BA is called fraction. The meaningful condition of a fraction is that the denominator is not zero, the numerator of the fraction is zero and the denominator is not zero. 2. The basic nature of the fraction: the numerator of the fraction is multiplied by the denominator or divided by the algebraic expression that is not equal to 0, and the value of the fraction remains unchanged. (0≠C) 3。 General and approximate fractions of fractions: the key is to decompose the factors first. 4. Fractional operation: Fractional multiplication rule: Fractional multiplication, with the product of molecules as the numerator of the product and the product of denominator as the denominator. Law of fractional division: a fraction is divided by a fraction, and the numerator and denominator of the divisor are in turn multiplied by the divisor. Fractional power law: Fractional power should be numerator and denominator respectively. Ababacadbcdbcccbdbd = = Addition and subtraction of fractions: Addition and subtraction of fractions with the same denominator and addition and subtraction of molecules with the same denominator. Fractions with different denominators are added and subtracted, first divided by fractions with the same denominator, and then added and subtracted: the operation order is the same as before. It can be simplified by the operation speed. 5. The zeroth power of any number that is not equal to zero is equal to 1, that is,) 0 (10 ≦ = aa; When n is a positive integer, nnaa 1=? ()0≠a 6。 Positive integer exponential power operation property Positive integer exponential power operation property can also be extended to integer exponential power operation property. (m, n is an integer) (1) is multiplied by the power of the same base: nmnmnaaa+=? ; (2) Power of power: mnnmaa =) (; (3) Power of product: nnnbaab =) (; (4) Power division with the same radix: nmnmaaa? =(a≠0)； (5) Power of quotient: nnnbaba =) ((); (b≠0) 7。 Fractional equation: an equation with a fraction and an unknown number in the denominator-fractional equation. The process of solving the fractional equation is essentially to multiply both sides of the equation by an algebraic expression (the simplest common denominator) and transform the fractional equation into an integral equation. When solving a fractional equation, when both sides of the equation are multiplied by the simplest common denominator, the simplest common denominator may be 0, which increases the root, so the fractional equation must be tested. Steps to solve the fractional equation: (1) Simplification (2) Multiplying both sides of the equation by the simplest common denominator to become an integral equation; (3) solving the integral equation; (4) check the roots. There are two conditions to add a root: one is that its value should make the simplest common denominator 0, and the other is that its value should be the root of the whole equation after removing the denominator. Test method of fractional equation: bring the solution of the whole equation into the simplest common denominator. If the value of the simplest common denominator is not 0, the solution of the whole equation is the solution of the original fractional equation; Otherwise, this solution is not the solution of the original fractional equation. What are the steps of applying the equation? (1) trial; (2) setting; (3) column; (4) solutions; (5) Answer: There are several types of application questions; What is the basic formula? There are basically five kinds: (1) Travel problems: basic formula: distance = speed × time, and travel problems are divided into meeting problems and catching up problems. (2) numerical problems should master the representation of decimals in numerical problems. (3) Basic formula of engineering problem: workload = working hours × working efficiency. (4) The countercurrent problem is smooth = static. ≤a, n is an integer) is called scientific notation. When an N-bit integer whose absolute value is greater than 10 is expressed by scientific notation, the exponent of 10 is 1? N When the absolute value in scientific notation is less than 1, the exponent of 10 is the number of zeros before the first non-zero number (including a zero before the decimal point). Chapter 17 Inverse proportional function 1. Definition: A function with the shape y = xk (k where k is a constant and k≠0) is called an inverse proportional function. Other forms xy=k 1? =kxyxky 1= 2。 Image: The image of inverse proportional function belongs to hyperbola. The image of inverse proportional function is both axisymmetric and centrally symmetric. There are two symmetrical axes: straight lines y=x and y =-X, and the center of symmetry is: origin 3. Properties: when k > 0, the two branches of hyperbola are located in the first and third quadrants respectively, and the y value of each quadrant decreases with the increase of x value; When k < 0, the two branches of hyperbola are located in the second and fourth quadrants respectively, and the y value of each quadrant increases with the increase of x value ... 4.| k |: represents the area of a rectangle surrounded by a point on the inverse proportional function image and two coordinate axes and vertical line segments made by the two coordinate axes. Chapter 18 Pythagorean Theorem 1. Pythagorean Theorem: If the lengths of two right angles of a right triangle are A and B and the length of the hypotenuse is C, then A2+B2 = C2. 2. The inverse theorem of Pythagorean theorem: If the lengths of triangle A, B and C satisfy A2+B2 = C2. Then this triangle is a right triangle. A proposition that is proved to be correct is called a theorem. We call two propositions with opposite topics and conclusions reciprocal propositions. If one of them is called the original proposition, then the other is called its inverse proposition. (Example: Pythagorean Theorem and Inverse Theorem of Pythagorean Theorem) Chapter 19 Definition of Quadrilateral Parallelogram: A quadrilateral with two groups of opposite sides parallel to each other is called a parallelogram. The nature of parallelogram: the opposite sides of parallelogram are equal; Diagonal angles of parallelogram are equal. Diagonal bisection of parallelogram. Determination of parallelogram 1. Two sets of quadrilaterals with equal opposite sides are parallelograms. 2. The quadrilateral with bisector is a parallelogram. 3. Two groups of quadrangles with equal diagonal are parallelograms; 4. A set of quadrilaterals with parallel and equal opposite sides is a parallelogram. The center line of the triangle is parallel to the third side of the triangle. The center line of the triangle is parallel to the third side of the triangle. The midline of the triangle is parallel to the third side of the triangle, which is equal to half of the third side and half of the third side. . . . The center line of the hypotenuse of a right triangle is equal to half of the hypotenuse. . . . Definition of rectangle: a parallelogram with a right angle. The nature of the rectangle: all four corners of the rectangle are right angles; The diagonals of a rectangle are equally divided. AC=BD rectangle judgment theorem: 1. A parallelogram with a right angle is called a rectangle. 2. Parallelograms with equal diagonals are rectangles. A quadrilateral with three right angles is a rectangle. Definition of diamond: parallelogram with equal adjacent sides. The nature of the diamond: all four sides of the diamond are equal; The two diagonals of the diamond are perpendicular to each other, and each diagonal bisects a set of diagonals. Decision theorem of diamond: 1. A set of parallelograms with equal adjacent sides is a diamond. 2. Parallelograms with diagonal lines perpendicular to each other are diamonds. A quadrilateral with four equilateral sides is a diamond. S diamond = 1/2×ab(a and B are two diagonal lines) Square definition: a diamond with right angles or a rectangle with equal adjacent sides. The essence of a square: all four sides are equal and all four corners are right angles. A square is both a rectangle and a diamond. Square judgment theorem: 1. A rectangle with equal adjacent sides is a square. Diamonds with right angles are squares. Definition of trapezoid: A set of quadrangles with parallel opposite sides and another set of quadrangles with non-parallel opposite sides are called trapezoid. Definition of right-angled trapezoid: Definition of isosceles trapezoid with a right angle: isosceles trapezoid. The nature of isosceles trapezoid: the two angles on the same base of isosceles trapezoid are equal; The two diagonals of an isosceles trapezoid are equal. Judgment theorem of isosceles trapezoid: two trapezoid with equal angles on the same base are isosceles trapezoid. Auxiliary lines commonly used to solve trapezoidal problems: the center of gravity of the line segment is the midpoint of the line segment. The center of gravity of a parallelogram is the intersection of its two diagonals. The point of doubt when three center lines of a triangle meet is the center of gravity of the triangle. A rectangle with an aspect ratio of 2 1-5 (about 0.6 18) is called a golden rectangle. Chapter 20 Data Analysis 1. Weighted average: the calculation formula of weighted average. Understanding of weight: It reflects the importance of a certain data in the whole data. The right to learn does not directly give numbers, but appears in the form of ratio or percentage, and the weighted average is obtained by using the frequency distribution table. 2. Arrange a set of data in order from small to large (or from large to small). If the number of data is odd, the middle number is median); This set of data. If the number of data is even, the average of the middle two data is the median of this set of data. 3. The data with the highest frequency in a set of data is the pattern of this set of data. 4. The difference between the maximum data and the minimum data in a set of data is called the range of this set of data. 5. The greater the variance, the greater the data fluctuation; The smaller the variance, the smaller the data fluctuation and the more stable it is. Data collection and collation steps: 1. Collect data. Organize data 3. Description data 4. Analyze data 5. Write an investigation report 6. Communication 6. The average value is affected by the extreme value, and the mode number is not affected by the extreme value, which is an advantage. The calculation of median is rarely affected by extreme value.