(A summary of rational numbers and their operations
First, the basic knowledge of rational numbers
1, three important definitions:
(1) positive number: numbers greater than 0 like 1 and 2.5 are called positive numbers; (2) Negative number: put a "-"sign before the positive number, indicating that the number less than 0 is called negative number; (3)0 is neither positive nor negative.
2, the classification of rational numbers:
(1) Define classification:
(2) Classification by natural symbols:
3. Counting axes
The number axis has three elements: origin, positive direction and unit length. Draw a horizontal straight line, take a point on the straight line to represent 0 (called the origin), select a certain length as the unit length, and specify the right direction on the straight line as the positive direction to get the number axis. The number on the right side of the number axis is always greater than the number on the left, so all positive numbers are greater than 0, all negative numbers are less than 0, and all positive numbers are greater than negative numbers.
4. Inverse number
If two numbers differ only in sign, then one of them is called the inverse of the other. The antipodes of 0 are 0, two antipodes, two origins on the number axis, and the distance from the origin is equal.
5. Absolute value
Geometric meaning of (1) absolute value: the absolute value of a number is the distance between the point representing the number on the number axis and the origin.
(2) Algebraic significance of absolute value: the absolute value of a positive number is itself; The absolute value of 0 is 0; The absolute value of a negative number is its reciprocal, which can be expressed by the letter A as follows:
(3) Comparing two negative numbers, the larger absolute value is smaller.
Second, the operation of rational numbers
1, addition of rational numbers
(1) rational number addition rule: add two numbers with the same symbol, take the same symbol, and add the absolute values; Add two numbers with different absolute values, take the 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; When a number is added to 0, it still gets the number.
(2) Arithmetic of rational number addition:
The commutative law of addition: a+b = b+a; The associative law of addition: (a+b) +c = a+(b +c)
The basic idea of simple operation with addition algorithm is: first, add the numbers that are opposite to each other; First, add the scores of the same denominator; First, add the numbers with the same sign; First add up the numbers that add up to integers.
2. Rational number subtraction
(1) rational number subtraction rule: subtracting a number equals adding its reciprocal.
(2) Common mistakes in rational number subtraction: paying attention to one thing and losing another, and not paying attention to the sign of the result; Still using the habit of primary school calculation, without changing subtraction into addition; Only the sign of operation is changed, but the sign of subtraction is not changed, and subtraction does not become reciprocal.
(3) mixed operation step of rational number addition and subtraction: firstly, the subtraction is changed into addition, and then the operation is carried out according to the rational number addition rule;
3. Rational number multiplication
(1) rational number multiplication rule: two rational 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.
(2) Arithmetic of rational number multiplication: exchange law: AB = BA associative law: (AB) C = A (BC); Exchange law: a(b+c)=ab+ac.
(3) Definition of reciprocal: If two rational numbers whose product is 1 are reciprocal, that is, ab= 1, then A and B are reciprocal; Countdown can also be seen as reversing the position of numerator and denominator.
4. Division of rational numbers
The division law of rational numbers: dividing by a number is equal to multiplying the reciprocal of this number, and 0 cannot be divided. This law can convert division into multiplication; The law of division can also be regarded as: divide two numbers, the same sign is positive, the different sign is negative, and divide by the absolute value. Dividing 0 by any number that is not equal to 0 is equal to 0.
5. Rational number multiplication
(1) Definition of rational number multiplication: The operation of finding several identical factors A is called multiplication, which is a kind of operation, and it is a special multiplication operation of several identical factors, and it is recorded as "",where A is called base, which means identical factors, and n is called exponent, which means the number of identical factors. It means n times a, not n times a, and the result of multiplication is called power.
(2) Any power of a positive number is positive, even power of a negative number is positive, and odd power of a negative number is negative.
6. Mixed operation of rational numbers
The key of (1) rational number mixed operation is to master the operation rules, operation rules and operation sequence of addition, subtraction, multiplication, division and multiplication. For complex mixed operations, the formula can be divided into several segments according to the addition and subtraction in the stem. When calculating, start with the power of each paragraph, calculate in turn, use parentheses first, and pay attention to the flexible use of operation.
(2) When mixing rational numbers, we should pay attention to the following points: first, we should pay attention to the operation order, first calculate the operation of the superior, and then calculate the operation of the subordinate; Second, we should pay attention to observation, flexibly use the algorithm to perform simple operations, and improve the operation speed and ability.
(2) Modification of algebraic expression.
(3) Review of linear equations with one variable
First, the related concepts of the equation
1, the concept of equation:
(1) An equation with an unknown number is called an equation.
(2) In an equation, there is only one unknown, whose exponent is 1 and coefficient is not 0. Such an equation is called a linear equation.
2, the basic properties of the equation:
Adding (or subtracting) the same algebraic expression on both sides of the (1) equation will still get an equation. If a=b, then a+c=b+c or A–C = B–C. 。
(2) Both sides of the equation are multiplied (or divided) by the same number at the same time (the divisor cannot be 0), and the result is still an equation. If a=b, then ac=bc or.
(3) Symmetry: the left and right sides of the equation exchange positions, and the result is still the equation. If a=b, then b = a.
(4) transitivity: if a=b and b=c, then a=c, which is called equivalent substitution.
Second, solve the equation.
1, related concepts of transferring materials:
After changing the sign of a term in an equation, it moves from one side of the equation to the other, which is called a shift term. This law comes from the property of equation 1 and is the basis of solving the equation. It should be understood that the shift term is to move an item from left to right or from right to left according to the need of solving equation deformation, and the moved item must change its sign.
2, the steps to solve the linear equation:
(1) denominator (attribute 2 of equation)
Pay attention to multiply each term of the equation by this least common multiple. Remember not to omit the multiplication term. The denominator is decimal. First, the denominator is changed into an integer by using the properties of fractions. If the molecule is algebraic, it must be enclosed in parentheses.
(2) bracket removal (bracket removal rule, multiplication and distribution rule)
Strictly enforce the rule of removing brackets. If you multiply by parentheses, remember not to omit the items in parentheses, remove the parentheses after the minus sign, and the symbols of the items in parentheses must be changed.
(3) Shift term (the property of equation 1)
Items that cross "=" are called shifted items, and items that belong to shifted items will change sign; Items that have not been moved have the same logo, so be careful not to miss them. When moving items, move unknown items to the left and known items to the right. When writing, write the untouched items first, and then write the change marks of the moved items later.
(4) Merging similar items (rules for merging similar items)
Note that when merging, only the coefficients are added together, but the letters and their indexes remain unchanged.
(5) The coefficient is 1 (property 2 of the equation)
Divide both sides by the unknown coefficient, remember that the unknown coefficient is always the denominator (divisor), and the numerator and denominator must not be reversed.
(6)
Second, the column equation solves the application problem
1, the general steps to solve application problems with column equations:
(1) Abstracts practical problems into mathematical problems;
(2) Analyze the known quantity and unknown quantity in the problem, and find out the equivalence relation;
(3) setting unknown numbers and listing equations;
(4) solving the equation;
(5) Test and answer.
2. Some practical problems in law and equivalence relation:
(1) The number arrangement rule on the calendar is: 7 consecutive numbers are arranged horizontally in each row, and the lower number in the vertical column is greater than the upper number. 7. The number range on the calendar is between 1 and 3 1, and cannot exceed this range.
(2) Several commonly used area formulas:
Rectangular area formula: S=ab, a is long, b is wide, and s is the area; Square area formula: S = a2, A is the side length, and S is the area;
Trapezoidal area formula: S =, a, b are the lengths of the upper and lower bottoms, h is the height of the trapezoid, and s is the trapezoidal area;
Formula for the area of a circle: R is the radius of the circle, and S is the area of the circle;
Triangle area formula: A is the length of one side of the triangle, H is the height of this side, and S is the area of the triangle.
(3) Several commonly used perimeter formulas:
The circumference of a rectangle: L=2(a+b), A and B are the length and width of the rectangle, and L is the circumference.
Perimeter of a square: L=4a, where a is the side length of the square and l is the perimeter.
Circle: L=2πr, r is the radius and l is the circumference.
(4) The volume of a cylinder is equal to the bottom area multiplied by the height. When the volume is constant, the larger the bottom surface, the lower the height. Therefore, the equal relation of equal product change is generally: the volume before deformation = the volume after deformation.
(5) The equivalent relationship of discounted sales is: profit = selling price-cost.
(6) The equivalence relation of key points in the travel problem: distance = speed × time, and the derived relation.
(7) In some complex problems, we can use tables to analyze the quantitative relationship in complex problems, find out some more direct equivalent relationships, and list equations, which can help us analyze the relationship between quantities.
(In the trip problem, the numerical language in the topic can be expressed by "line graph". Analyze the quantitative relationship in the problem, find out the equivalence relationship, and list the equations.
(9) Some concepts about savings:
Principal: the money deposited by the customer in the bank; Interest: fees paid by banks to customers; Principal and interest: the sum of principal and interest; Number of periods: deposit time; Interest rate: the ratio of interest to principal per period; Interest = principal × interest rate × number of periods; Principal and interest = principal+interest.
(4) generally review the preliminary understanding of graphics.
(A) colorful graphics
Three-dimensional figure: prism, pyramid, cylinder, cone, sphere, etc.
1, geometry
Plane graphics: triangle, quadrilateral, circle, etc.
2. Front view-looking from the front.
Three views of geometry (left and right)-viewed from the left (right) side.
Top view-from above.
(1) will judge three views of a simple object (straight prism, cylinder, cone and ball).
(2) can describe the basic geometric or physical prototype according to the three views.
3, the three-dimensional graphic plane expansion diagram
(1) The same three-dimensional figure is unfolded in different ways, and the plane figure obtained is also different.
(2) Understand the plane development diagrams of straight prisms, cylinders and cones, and judge and make three-dimensional models according to the development diagrams.
4. Points, lines, surfaces and bodies
Synthesis of (1) Geometry
Point: the intersection of straight lines is the point, which is the most basic figure of geometric figures.
Line: The intersection line between faces is a line, which can be divided into straight lines and curves.
Face: Surrounding the body is a face, which is divided into plane and curved surface.
Volume: Geometry is also called volume for short.
(2) Points move into lines, lines move into planes, and planes move into adults.
(2) Lines, rays and line segments
1, basic concept
Graphics: lines, rays, line segments
The number of endpoints is zero, one or two.
Represents a straight line a
Line AB(BA), ray AB, line segment a.
AB line (BA)
Practice is described as a straight line ab;
Let straight line a be ray ab and line segment a;
Making line segment ab;
Connecting AB
Prolonging narration cannot prolong the extension line AB of the reverse extension line AB;
Reverse extension line segment BA
2, the nature of the line
There is a straight line after two o'clock, and there is only one straight line.
To put it simply: two points determine a straight line.
3. Draw a line segment equal to the known line segment.
(1) metric method
(2) Drawing with a ruler and a ruler
4, the size of the line comparison method
(1) metric method
(2) Overlapping method
5. The midpoint (bisector), bisector, quadrant, etc. of a line segment.
Definition: the point that divides a line segment into two equal lines.
Symbol: If point M is the midpoint of line segment AB, then AM=BM=AB, AB=2AM=2BM.
6, the nature of the line segment
Among the connecting lines between two points, the line segment is the shortest. Simply put, the line segment between two points is the shortest.
7. Distance between two points
The length of the line segment connecting two points is called the distance between two points.
8, the position of the point and the straight line.
(1) point is on a straight line (2) point is outside the straight line.
(3) Angle
1, angle: A figure composed of two rays at a common endpoint is called an angle.
2, the angle of representation (4):
3. Angle measurement unit and conversion
4. Classification of angles
∠ β acute angle right angle obtuse angle right angle fillet.