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1. Number line
(1) The concept of number axis: the straight line that stipulates the origin, positive direction, and unit length is called the number axis.
The three elements of the number axis: the origin, the unit length, and the positive direction.
(2) Points on the number line: All rational numbers can be represented by points on the number line, but the points on the number line do not all represent rational numbers.
(3) Use the number axis to compare the size: Generally speaking, when the number axis is facing the right, the number on the right is always larger than the number on the left.
2. Reciprocal number
(1) The concept of opposite numbers: only two numbers with different signs are called opposite numbers.
(2) The meaning of opposite numbers: grasp that opposite numbers appear in pairs and cannot exist alone, from the number axis, except for 0, two numbers that are opposite to each other, they are on both sides of the origin and the distance to the origin is equal.
(3) Simplification of multiple symbols: It has nothing to do with the number of "+", there are odd "-" signs and the result is negative, and there are even numbers of "-" signs, and the result is positive.
(4) Summary of the law method: The method of finding the opposite number of a number is to add "-" in front of this number, such as the opposite number of a is -a, and the opposite number of m+n is -(m+n), then m+n is a whole, and when adding a negative sign in front of the whole, small parentheses should be used.
3. Absolute
1. Concept: The distance between a number and the origin on the number line is called the absolute value of the number.
(1) The absolute value of two numbers that are opposite to each other is equal;
(2) There are two numbers whose absolute value is equal to a positive number, there is one number whose absolute value is equal to 0, and there is no number whose absolute value is equal to a negative number.
(3) The absolute values of rational numbers are all non-negative numbers.
2. If the letter a is used to represent a rational number, the absolute value of the number a is determined by the value of the letter a itself:
(1) When a is a positive rational number, the absolute value of a is its own a;
(2) When a is a negative rational number, the absolute value of a is its opposite - a;
(3) When a is zero, the absolute value of a is zero.
即|a|={a(a>0)0(a=0)-a(a<0)
4. Comparison of the size of rational numbers
1. Comparison of the size of rational numbers
To compare the size of rational numbers, you can use the number line, their order from left to right, that is, from large to small (two rational numbers represented on the number line, the number on the right is always larger than the number on the left), or you can use the properties of numbers to compare the size of two different numbers and 0, and use absolute values to compare the size of two negative numbers.
2. The law of comparison of the size of rational numbers:
(1) All positive numbers are greater than 0;
(2) Negative numbers are less than 0;
(3) Positive numbers are greater than all negative numbers;
(4) Two negative numbers, the absolute value is larger, the value is smaller.
There are three ways to compare the size of rational numbers by regular methods:
(1) Law comparison: positive numbers are greater than 0, negative numbers are less than 0, positive numbers are greater than all negative numbers.
(2) Number line comparison: The number represented by the dot on the right on the number line is greater than the number represented by the dot on the left.
(3) Comparison of differences:
If a-b>0, then a>b;
If a-b<0, then a<b;
If a-b=0, then a=b.
5. Subtraction of rational numbers
The law of subtraction of rational numbers
Subtracting a number is equal to adding the opposite of the number. i.e.: a-b=a+(-b)
Methodological guidelines:
(1) When performing subtraction operations, first figure out the symbol of the subtraction;
(2) When converting a rational number into addition, two symbols should be changed at the same time: one is the operation symbol (minus sign becomes plus), and the other is the property sign of the subtraction (the subusal number becomes the opposite number);
Note: In the subtraction of rational numbers, the positions of the subtracted and subtracted numbers cannot be swapped at will, because there is no commutative law in subtraction.
The law of subtraction cannot be compared with the law of addition, 0 plus any number does not change, and 0 minus any number should be calculated according to the law.
6. Multiplication of rational numbers
(1) The rule of multiplication of rational numbers: multiply two numbers, the same sign is positive, the different sign is negative, and the absolute value is multiplied.
(2) Any number multiplied by zero gives 0.
(3) The law of multiplication of multiple rational numbers:
(1) Multiply several numbers that are not equal to 0, the sign of the product is determined by the number of negative factors, when there are odd numbers of negative factors, the product is negative, and when there are even numbers of negative factors, the product is positive.
(2) Multiply several numbers, and if there is a factor of 0, the product is 0.
(4) Methodological guidelines
(1) Using the multiplication rule, first determine the symbols, and then multiply the absolute values.
(2) Multiply multiple factors, and look at the sign of 0 factor and product first, so that the operation is accurate and simple.
7. Mixed operations of rational numbers
1. The mixed operation order of rational numbers: first calculate the multiplication, then calculate the multiplication and division, and finally calculate the addition and subtraction; the same level operation should be calculated in the order from left to right; if there are brackets, the operation in the brackets should be done first.
2. When performing mixed operations on rational numbers, pay attention to the application of each operation law to simplify the operation process.
Four Techniques for Mixing Rational Numbers:
(1) Transformation method: one is to convert division into multiplication, the other is to convert the power into multiplication, and the third is to convert decimal into fraction for reduction calculation in the multiplication and division mixed operation.
(2) Integer method: In the addition and subtraction mixed operation, two numbers with a sum of zero, two numbers with the same denominator, two numbers with an integer, and two numbers with a product of integers are combined into a group to solve.
(3) Splitting method: first split the fraction into the form of an integer and the sum of a true fraction, and then calculate.
(4) Clever use of arithmetic law: The clever use of addition arithmetic or multiplicative arithmetic law in calculation often makes the calculation easier.
8. Scientific notation – to represent larger numbers
1. Scientific notation: Write a number greater than 10 in the form of a×10n, where a is a number with only one digit of the integer digit, and n is a positive integer, this notation is called scientific notation. (Scientific notation form: a×10n, where 1≤a<10, n is a positive integer)
2. Summary of regular methods
(1) The requirement of a in scientific notation and the representation law of the exponent n of 10 are the key, because the exponent of 10 is less than the original integer digit by 1; according to this law, the integer digits of the original number can be counted first, and the exponent n of 10 can be obtained.
(2) The notation requires that the number greater than 10 can be represented by the scientific notation, and the negative number with an absolute value greater than 10 can also be represented by this method, but there is an extra negative sign in front.
9. Algebraic evaluation
(1) The value of the algebraic formula: the numerical algebra is used to replace the letters in the numerical formula, and the result obtained after calculation is called the value of the algebraic formula.
(2) Algebraic evaluation: the value of the algebraic formula can be directly substituted and calculated.
The following three types of questions can be summarized simply:
(1) The known conditions are not simplified, and the algebraic formula given is simplified;
(2) the known conditions are simplified, and the given algebraic formula is not simplified;
(3) Both the known conditions and the given algebraic formulas should be simplified.
10. Regularity: Variation of figures
First of all, we should find out which parts of the graph have changed and what laws have changed, and then directly use the laws to solve the changes of each part through analysis. To explore the law, we should carefully observe, think carefully, and make good use of associations to solve such problems.
11. Nature of the Equation
1. The nature of the equation
Property 1: Add the same number (or sub-equation) to both sides of the equation, and the result is still the equation;
Property 2 Multiply the same number on both sides of the equation or divide by a number that is not zero, and the result is still the equation.
2. Use the properties of the equation to solve equations
The equation is deformed by the properties of the equation, so that the form of the equation is transformed into the form of x=a.
When applying, we should pay attention to two levels:
(1) how to deform;
(2) According to which article, only when the deformation is done step by step, can it be guaranteed to be correct.
12. The solution of a one-dimensional equation
Definition: The value of an unknown that equalizes the left and right sides of a unary equation is called the solution of a unary equation.
Substituting the solution of the equation into the original equation, the left and right sides of the equation are equal.
13. Solve unary equations
1. General steps for solving unary equations
Removing the denominator, removing the parentheses, shifting the terms, merging the same terms, and turning the coefficients into 1 are only the general steps of solving the unary equation, and the various steps are applied flexibly according to the characteristics of the equation, and all kinds of steps are to gradually transform the equation into the form of x=a.
2. When solving a one-dimensional equation, observe the form and characteristics of the equation first, if there is a denominator, the denominator is generally removed first, if there is both a denominator and a parentheses, and the terms outside the parentheses can remove the denominator after multiplying the items in the parentheses, the parentheses should be removed first.
3. When solving an equation similar to "ax+bx=c", combine the left side of the equation into one term according to the method of merging similar terms, that is, (a+b)x=c.
The equation is gradually transformed into the simplest form of ax=b, which embodies the idea of naturalization.
When the ax=b coefficient is reduced to 1, it is necessary to calculate accurately, first, to find out whether the two sides of the equation are divided by a or b when finding x, especially when a is a fraction, and second, to accurately judge the symbol, a and b with the same sign x are positive, and a and b with different signs x are negative.
14. Application of unary equations
1. Types of problems for solving unary equations
(1) Exploring regular problems;
(2) numerical issues;
(3) Sales problems (profit = selling price - purchase price, profit margin = profit purchase price ×100%);
(4) engineering problems ((1) workload = per capita efficiency ×number of people × time; (2) if a job is completed in several stages, then the sum of the workload of each stage = total amount of work);
(5) Itinerary problem (distance = speed × time);
(6) Equivalent transformation problems;
(7) Sum, difference, multiple, and point problems;
(8) distribution issues;
(9) Match points;
(10) Current navigation problem (downstream velocity = hydrostatic velocity + current velocity; reverse velocity = hydrostatic velocity - current velocity).
2. The basic idea of using equations to solve practical problems
First of all, find out the unknown quantity and all the known quantities in the question, directly set the required unknown quantity or indirectly set a key unknown quantity as x, and then use the formula containing x to represent the relevant quantities, find out the equality relationship between the series of equations, solve, and answer, that is, set, column, solution, and answer.
List the five steps of solving a problem with a one-dimensional equation
(1) Review: Carefully examine the problem, determine the known quantity and the unknown quantity, and find out the equivalent relationship between them.
(2) Set: Set the unknown (x), according to the actual situation, you can set the direct unknown (ask what to set what), or set the indirect unknown.
(3) Columns: List equations according to the equiquantity relationship.
(4) Solution: Solve the equation and find the value of the unknown.
(5) Answer: Check whether the value of the unknown is correct and whether it is in line with the question, and write the answer completely.
15. The text on the cube opposite the two faces
(1) For such problems, the general method is to fold the paper according to the shape of the diagram and then solve it, or directly imagine it on the basis of the understanding of the unfolded diagram.
(2) The key to solving such problems is to start from the physical object, combine with specific problems, distinguish the expansion diagram of geometric objects, and establish the spatial concept by combining the transformation of three-dimensional graphics and plane graphics.
(3) There are 11 situations in the expansion diagram of the cube, and after analyzing the various situations of the plane expansion diagram, carefully determine which two faces are opposite.
16. Straight lines, rays, line segments
(1) Representation of straight lines, rays, and line segments
(1) Straight line: represented by a lowercase letter, such as: straight line l, or represented by two uppercase letters (on a straight line), such as straight line AB.
(2) Ray: is a part of the straight line, represented by a lowercase letter, such as: ray l, represented by two uppercase letters, the endpoint is in front, such as: ray OA. Note: When represented by two letters, the letter of the endpoint is placed in front.
(3) Line segment: A line segment is a part of a straight line, represented by a lowercase letter, such as line segment a, and represented by two letters representing the endpoint, such as: line segment AB (or line segment BA).
(2) The position relationship between the point and the line:
(1) The point passes through a straight line, indicating that the point is on a straight line;
(2) If the point does not pass through a straight line, it means that the point is outside the straight line.
17. The distance between two points
(1) The distance between two points: The length of the line segment connecting two points is called the distance between two points.
(2) There is a certain distance between any two points on the plane, it refers to the length of the line segment connecting these two points, when learning this concept, pay attention to emphasizing the last two words "length", that is, it is a quantity, there is a size, different from the line segment, the line segment is a figure, the length of the line segment is the distance between the two points.
18. The concept of horns
(1) Definition of angle: A graph with a common endpoint composed of two rays is called an angle, where this common endpoint is the vertex of the angle, and these two rays are the two sides of the angle.
(2) the representation of the angle: the angle can be represented by a capital letter, can also be represented by three capital letters, in which the vertex letter should be written in the middle, only in the case of only one corner at the vertex, you can use a letter at the vertex to remember the angle, otherwise you can't tell which angle the letter represents. The angle can also use a Greek letter (such as ∠α, ∠β, ∠γ、... ) or in Arabic numerals (∠1, ∠2...) ) representation.
(3) flat angle, circumferential angle: the angle can also be regarded as a figure formed by the rotation of a ray around its endpoint, when the beginning edge and the end edge form a straight line, when the beginning edge and the end edge rotation coincide, the formation of a perimeter angle.
(4) Measurement of angles: degrees, minutes, and seconds are commonly used units of measurement of angles.1 degree = 60 minutes, that is, 1 ° = 60 ', 1 minute = 60 seconds, that is, 1 ' = 60".
19. Definition of angular bisector
Starting from the apex of an angle, the ray that divides the angle into two equal angles is called the bisector of the angle.
(1) AOB, AOC, AOC+BOC+BOC.BOC.BOC.BOC.AOB, AOC, AOB-BOC
2)若射线OC是∠AB的三等分线,则∠AOB=3∠BOC或∠BOC=13∠AB。
20. Degrees, minutes, and seconds
(1) Addition and subtraction of degrees, minutes, and seconds.
When adding and subtracting degrees, minutes and seconds, it is necessary to add and subtract degrees and degrees, minutes and minutes, seconds and seconds, add minutes and seconds, carry every 60, and borrow 1 to 60 when subtracting.
(2) Multiplication and division of degrees, minutes, and seconds
(1) Multiplication: degrees, minutes, and seconds are multiplied separately, and the result is to be carried every 60.
(2) Division: Degrees, minutes, and seconds are removed separately, and the remainder of each time is further removed as the next level unit.
21. Judge the geometry from the three views
(1) To imagine the shape of the geometry from three views, first, the shapes of the front, top, and left sides of the geometry should be imagined according to the front, top, and left views, and then the overall shape should be considered together.
(2) It is difficult to imagine the shape of a geometry from the three views of an object, which can be analyzed in the following ways:
(1) Imagine the shape of the front, top, and left sides of the geometry, as well as the length, width, and height of the geometry according to the front, top, and left views;
(2) imagining the contours of the visible and invisible parts of the geometry from the solid and dashed lines;
(3) Memorizing the three views of some simple geometries will be helpful for the imagination of complex geometries;
(4) Use the reverse process of drawing geometry from three views to drawing three views with geometry, practice repeatedly, and constantly summarize the method.
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