The concepts of ⒈, positive number and negative number
Negative number: a number less than 0. Positive number: a number greater than 0. 0 is neither positive nor negative.
Note: ① The letter A can represent any number. When a represents a positive number, -A is a negative number; When a stands for negative number, -a is positive number; When a represents 0, -a is still 0. (If the judgment topic is: the number with a positive sign is positive and the number with a negative sign is negative, this statement is wrong. For example, +a, -A cannot make a simple judgment. )
② Sometimes "+"can be added before positive numbers, and sometimes "+"can be omitted. Therefore, the positive sign omitting "+"is a positive sign.
2. Quantities with opposite meanings
If a positive number means a quantity with a certain meaning, a negative number can mean a quantity opposite to a positive number, such as:
8℃ above zero means:+8℃; 8 degrees below zero means 8 degrees below zero.
The meaning of 3 and 0
(1)0 means "none", for example, there are 0 people in the classroom, which means there is no one in the classroom;
(2)0 is the dividing line between positive and negative numbers, and 0 is neither positive nor negative. For example:
(3)0 represents the exact quantity. For example, 0℃ and the benchmark in some topics, such as taking sea level as the benchmark, 0 meters is sea level.
rational number
1, the concept of rational number
(1) Positive integers, 0 and negative integers are collectively called integers (0 and positive integers are collectively called natural numbers).
(2) Positive and negative scores are collectively referred to as scores.
(3) Positive integers, 0, negative integers, positive fractions and negative fractions can all be written in the form of fractions, and such numbers are called rational numbers.
Understanding: Only numbers that can change the number of components are rational numbers. ① π is an infinite acyclic decimal, which cannot be written in fractional form and is not a rational number. (2) Finite decimal and infinite cyclic decimal can be converted into component numbers, both of which are rational numbers. (3) Integers can also be converted into component numbers, and component numbers are also rational numbers.
Note: After the introduction of negative numbers, the range of odd and even numbers is also expanded. For example, -2, -4, -6 and -8 are even numbers, and-1, -3 and -5 are also odd numbers.
Summary of compulsory knowledge points of rational number in mathematics 2 of grade one in junior high school
1. 1 positive and negative numbers
A number with a negative sign "-"in front of a number that is not 0 is called a negative number.
It has the opposite meaning to negative number, that is, I learned that numbers other than 0 are called positive numbers (sometimes "+"is added before positive numbers as needed).
1.2 rational number
Positive integers, 0 and negative integers are collectively called integers, and positive and negative fractions are collectively called fractions.
Integers and fractions are collectively called rational numbers.
Numbers are usually represented by points on a straight line, which is called the number axis.
Three elements of number axis: origin, positive direction and unit length.
Take any point on a straight line to represent the number 0, and this point is called the origin.
Numbers with only two different signs are called opposites. (Example: the reciprocal of 2 is-2; The reciprocal of 0 is 0)
The distance between the point representing the number A on the number axis and the origin is called the absolute value of the number A, and it is recorded as |a|.
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. Two negative numbers, the larger one has the smaller absolute value.
Plane rectangular coordinate system:
Draw two mutually perpendicular number axes on the plane, and their origins coincide to form a plane rectangular coordinate system.
The horizontal axis is called X axis or horizontal axis, the vertical axis is called Y axis or vertical axis, and the intersection of the two coordinate axes is the origin of the plane rectangular coordinate system.
Elements of a plane rectangular coordinate system: ① On the same plane; ② Two axes of numbers are perpendicular to each other; ④ The origin coincides.
Three rules:
① The specified positive direction: the horizontal axis is right, and the vertical axis is oriented in the positive direction.
(2) the provisions of the unit length; Generally speaking, the unit length of the horizontal axis and the vertical axis is the same; In fact, sometimes it can be different, but it must be on the same axis.
③ Quadrant definition: the upper right is the first quadrant, the upper left is the second quadrant, the lower left is the third quadrant, and the lower right is the fourth quadrant.
I believe that the students have mastered the knowledge of plane rectangular coordinate system, and I hope they can all be admitted.
Composition of plane rectangular coordinate system
Two number axes perpendicular to each other on the same plane and having a common origin form a plane rectangular coordinate system, which is called rectangular coordinate system for short. Usually, the two number axes are placed in the horizontal position and the vertical position respectively, and the right and upward directions are the positive directions of the two number axes respectively. The horizontal axis is called X axis or horizontal axis, the vertical axis is called Y axis or vertical axis, and the X axis or Y axis is collectively called coordinate axis, and their common origin O is called the origin of rectangular coordinate system.
Through the explanation and study of the composition knowledge of plane rectangular coordinate system, I hope students can master the above contents well and study hard.
Properties of point coordinates
After the plane rectangular coordinate system is established, the coordinates of any point on the coordinate system plane can be determined. Conversely, for any coordinate, we can determine a point it represents on the coordinate plane.
For any point C on the plane, the intersection point C is perpendicular to the X-axis and Y-axis respectively, and the corresponding points A and B perpendicular to the X-axis and Y-axis are respectively called the abscissa and ordinate of the point C, and the ordered real number pairs (A, B) are called the coordinates of the point C. ..
A point is in different quadrants or coordinate axes, and its coordinates are different.
I hope that the students can master the knowledge of the above coordinate nature, and I believe that the students will achieve excellent results in the exam.
General steps of factorization
If the polynomial has a common factor, first mention the common factor, and then consider the formula method if there is no common factor. If it is a polynomial with four or more terms,
Usually, the group decomposition method is used, and finally the cross multiplication factor is used to decompose the factors. So it can be summarized as "one mention", "two sets", "three groups" and "forty words".
Note: Factorization must be decomposed until each factor can no longer be decomposed, otherwise it is incomplete factorization. If the topic does not clearly indicate the scope of factorization, it should refer to factorization within rational numbers, so the result of factorization must be the product of several algebraic expressions.
I believe the students have mastered the general steps of factorization, and I hope they will do well in the exam.
factoring
Definition of factorization: transforming a polynomial into the product of several algebraic expressions is called factorization of this polynomial.
Factorizing elements: ① The result must be an algebraic expression ② The result must be a product ③ The result is an equation ④.
The relationship between factorization and algebraic expression multiplication: m(a+b+c)
Common factor: The common factor of each term of a polynomial is called the common factor of each term of this polynomial.
Determination of common factor: ① When the coefficient is an integer, take the greatest common factor of each term. The product of the greatest common divisor of the same letter and the lowest power of the same letter is the common factor of this polynomial.
To select a common factor:
① Determine the common factor. ② Determine the quotient formula ③ The common factor formula and the quotient formula are written in the form of product.
Factorizing attention;
(1) Lost letters are not allowed.
(2) It is not allowed to lose the same items. Please check the quantity of items.
③ Change the double brackets into single brackets.
(4) The results are arranged in the order of number, single letter and single polynomial.
⑤ The same factor is written as a power.
⑥ The first minus sign is placed outside the brackets.
⑦ Similar items in brackets are merged.
Summary of compulsory knowledge points in the first grade of mathematics 3 Chapter 1 Rational Numbers
1, numbers greater than 0 are positive numbers.
2. Rational number classification: positive rational number, 0, negative rational number.
3. Classification of rational numbers: integer (positive integer, 0, negative integer) and fraction (positive fraction, negative fraction).
4. Specify the origin and unit length, and the straight line in the positive direction is called the number axis.
5. Comparison of figures:
① Positive number is greater than 0, 0 is greater than negative number, and positive number is greater than negative number.
② When comparing two negative numbers, the larger absolute value is smaller.
6. Numbers with only two different symbols are called antonyms.
7. If a+b=0, A and B are reciprocal.
8. The distance from the point representing the number A to the origin is called the absolute value of the number A..
9. Three sentences of absolute value: 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.
10, calculation of rational number: calculate the symbol first, then calculate the value.
1 1, plus or minus: ① positive+② big-small ③ small-big =-(big-small) ④-☆-о =-(☆+о)
12. Multiplication and division: the same sign is positive and the different sign is negative.
13, power: represents the product of n identical factors.
14, the odd power of negative number is negative, and the even power of negative number is positive.
15, mixed operation: multiply first, then multiply and divide, and then add and subtract. The operation at the same level is from left to right, with parentheses first.
16, scientific counting method: use ax 10n to represent a number. (where a is an integer with only one digit)
17, all numbers are valid from the first non-zero number on the left.
Knowledge carding
1. number axis: three elements of number axis: origin, positive direction and unit length; There is a one-to-one correspondence between points on the number axis and real numbers.
2. the reciprocal of real number a is-a; If A and B are opposites, then a+b=0, and vice versa; Geometric meaning: on the number axis, two points representing the opposite number are located on both sides of the origin, and the distance to the origin is equal.
3. Reciprocal: If the product of two numbers is equal to 1, then these two numbers are reciprocal.
4. Absolute value: Algebraic meaning: the absolute value of a positive number is itself, the absolute value of a negative number is its opposite number, and the absolute value of 0 is 0;
Geometric meaning: the absolute value of a number is the distance from the point representing this number to the origin on the number axis.
5. Scientific symbol:, in which.
6. Real number comparison: compare the size according to the law; Use the number axis to compare sizes.
7. In the range of real numbers, addition, subtraction, multiplication, division and power operations can be performed, but the root operation may not work, for example, negative numbers can't even be opened. The operation basis of real numbers is rational number operation, and all the operation properties and laws of rational numbers are applicable to real number operation. It is the key to master the real number operation to correctly determine the symbol of the operation result and flexibly use the algorithm.
Knowledge points of unary linear equation
Knowledge point 1: the concept of equality: an equation with an equal sign is called an equality.
Knowledge point 2: the concept of equation: an equation with unknown number is called an equation, and the equation must contain unknown number, and it must be an equation, and both are indispensable.
Note: algebraic expressions do not contain equal signs, equations are connected by equal signs, and must contain unknowns.
Knowledge point 3: The concept of linear equation of one variable: An equation with only one unknown number and the degree of the unknown number is 1 is called linear equation of one variable. Any form of linear equation can always be transformed into the form of ax=b(a≠0, A and B are known numbers). This form of equation is called the general formula of linear equation. Note a.
Example 2: If (a+ 1) +45=0 is a linear equation, then a _ _ _ _ _ _ _ _ _ _.
Analysis: The condition that a linear equation with one variable needs to meet: the unknown coefficient is not equal to 0, and the degree is 1. ∴ A+ 1 ≠ 0，2b- 1 = 1。 ∴a≦- 1。
Knowledge point 4: Basic properties of the equation (1) Add (or subtract) the same number or the same algebraic expression on both sides of the equation, and the result is still the equation. That is, if a=b, then a m = b m
(2) Multiply (or divide) both sides of the equation by the same number or algebraic expression that is not 0, and the result is still an equation.
That is, if a=b and am=bm. Or ... In addition, the equation has other properties: if a=b, b = a. If a=b, b=c, a = c. 。
Note: the nature of the equation is an important basis for solving the equation.
Example 3: The following deformation is correct ()
A. if ax=bx, then a=b B. if (a+ 1)x=a+ 1, then x= 1.
C. if x=y, then x-5 = 5-y D. if.
Analysis: Solving problems by using the properties of equations. D.
Note: It is impossible for both sides of an equation to be divisible into zero numbers or formulas at the same time, which must be highly valued by students.
Knowledge point 5: Solution of the equation and solving the equation: The value of the unknown quantity that makes the two sides of the equation equal is called the solution of the equation, and the process of solving the equation is called solving the equation.
Knowledge point 6: About the shift term: (1) The essence of the shift term is the application of the basic properties of the equation 1
⑵ When moving items, be sure to change the symbol of the moved items.
Knowledge point 7: The general steps of solving a linear equation with one variable are: removing denominator, brackets, shifting terms, merging similar terms, and converting unknown coefficients into 1. When solving a specific problem, some steps may be unnecessary, some steps may be reversed, and some steps may be combined to simplify the operation, which should be applied flexibly according to the characteristics of the equation.
Example 4: Solve the equation.
Analysis: Use the steps of linear equation flexibly to solve this problem.
Solution: If the denominator is removed, 9x-6 = 2x；; If the item is shifted, 9x-2x = 6；; If similar items are merged, 7x = 6；; If the coefficient is 1, x=.
Note: after removing the denominator, some items in the left and right algebraic expressions of the equation can be easily multiplied. For example, if the denominator is removed, it is easy to get the wrong solution: 9x- 1=2x, and the constant term is easy to multiply.
Knowledge point 8: the test of equation
To test whether a number is the solution of the original equation, we should substitute it into the left and right sides of the original equation to see if the values on both sides are equal.
Note: it should be substituted into the left and right sides of the original equation for calculation, not into the left and right sides of the deformation equation.
There is only one straight line after the summary of knowledge points required for Math Test 4 in Grade One 1 2.
The line segment between two points is the shortest.
The complementary angles of the same angle or equal angle are equal.
The complementary angles of the same angle or the same angle are equal.
One and only one straight line is perpendicular to the known straight line.
Of all the line segments connecting a point outside the straight line with points on the straight line, the vertical line segment is the shortest.
7 Parallel axiom passes through a point outside a straight line, and there is only one straight line parallel to this straight line.
If both lines are parallel to the third line, the two lines are also parallel to each other.
The same angle is equal and two straight lines are parallel.
The internal dislocation angles of 10 are equal, and the two straight lines are parallel.
1 1 are complementary and two straight lines are parallel.
12 Two straight lines are parallel and have the same angle.
13 two straight lines are parallel, and the internal dislocation angles are equal.
14 Two straight lines are parallel and complementary.
Theorem 15 The sum of two sides of a triangle is greater than the third side.
16 infers that the difference between two sides of a triangle is smaller than the third side.
The sum of the internal angles of 17 triangle is equal to 180.
18 infers that the two acute angles of 1 right triangle are complementary.
19 Inference 2 An outer angle of a triangle is equal to the sum of two non-adjacent inner angles.
Inference 3 The outer angle of a triangle is greater than any inner angle that is not adjacent to it.
2 1 congruent triangles has equal sides and angles.
Axiom of Angular (SAS) has two triangles with equal angles.
The Axiom of 23 Angles (ASA) has the congruence of two triangles, which have two angles and their sides correspond to each other.
The inference (AAS) has two angles, and the opposite side of one angle corresponds to the congruence of two triangles.
The axiom of 25 sides (SSS) has two triangles with equal sides.
Axiom of hypotenuse and right angle (HL) Two right angle triangles with hypotenuse and right angle are congruent.
Theorem 1 The distance between a point on the bisector of an angle and both sides of the angle is equal.
Theorem 2 is a point with equal distance on both sides of an angle, which is on the bisector of this angle.
The bisector of an angle 29 is the set of all points with equal distance to both sides of the angle.
The nature theorem of isosceles triangle 30 The two base angles of isosceles triangle are equal (that is, equilateral and equiangular).
3 1 Inference 1 The bisector of the vertices of an isosceles triangle bisects the base and is perpendicular to the base.
The bisector of the top angle, the median line on the bottom edge and the height on the bottom edge of the isosceles triangle coincide with each other.
Inference 3 All angles of an equilateral triangle are equal, and each angle is equal to 60.
34 Judgment Theorem of an isosceles triangle If a triangle has two equal angles, then the opposite sides of the two angles are also equal (equal angles and equal sides).
Inference 1 A triangle with three equal angles is an equilateral triangle.
Inference 2 An isosceles triangle with an angle equal to 60 is an equilateral triangle.
In a right triangle, if an acute angle is equal to 30, the right side it faces is equal to half of the hypotenuse.
The center line of the hypotenuse of a right triangle is equal to half of the hypotenuse.
Theorem 39 Is the distance between the point on the vertical line of a line segment and the two endpoints of this line segment equal?
The inverse theorem and the point where the two endpoints of a line segment are equidistant are on the middle vertical line of this line segment.
The perpendicular bisector of the 4 1 line segment can be regarded as the set of all points with equal distance from both ends of the line segment.
Theorem 42 1 Two graphs symmetric about a line are conformal.
Theorem 2: If two figures are symmetrical about a straight line, then the symmetry axis is the perpendicular to the straight line connecting the corresponding points.
Theorem 3 Two graphs are symmetrical about a straight line. If their corresponding line segments or extension lines intersect, then the intersection point is on the axis of symmetry.
Summary of compulsory knowledge points in the first grade math exam 5. Grasp scientific knowledge as soon as possible and improve learning ability quickly. The editing teacher has provided you with the knowledge points of the first grade mathematics in the new semester, hoping to bring you inspiration!
I. Objectives and requirements
1. By dealing with practical problems, it is an improvement for students to experience algebraic methods from arithmetic methods;
2. Learn how to find the equation relationship in the problem, list the equations, and understand the concept of the equations;
3. Cultivate students' ability to obtain information, analyze and deal with problems.
Second, the main points
Seeking equality from practical problems;
Establish the thinking method of solving practical problems with column equations, learn to merge similar terms, and solve ax+bx=c linear equations.
Third, difficulties.
Seeking equality from practical problems;
Analyze the known and unknown quantities in practical problems, find out the equal relationship and list the equations, so that students can gradually establish a thinking method to solve practical problems by listing the equations.
Fourth, the summary of knowledge points and concepts
1. One-dimensional linear equation: An integral equation with only one unknown number and a degree of 1 and a non-zero coefficient is a one-dimensional linear equation.
2. The standard form of one-dimensional linear equation: ax+b=0(x is unknown, A and B are known numbers, a0).
3. Conditions: A linear equation with one variable must meet four conditions at the same time:
(1) It is an equation;
(2) There are no unknowns in the denominator;
(3) The highest unknown term is1;
(4) The coefficient of the term with unknown number is not 0.
4. The nature of the equation:
Properties of the equation 1: When a number is added to both sides of the equation or the same number or the same algebraic expression is subtracted, the equation is still valid.
Property 2 of the equation: both sides of the equation expand or contract at the same time by the same multiple (except 0), and the equation still holds.
Property 3 of the equation: When both sides of the equation are multiplied (or squared) at the same time, the equation still holds.
Solving the equation is based on these three properties of the equation. One of the properties of the equation: adding a number or subtracting the same number on both sides of the equation at the same time, the equation still holds.
5. Merge similar projects
(1) Basis: Multiplication and Distribution Law
(2) Combine the unknowns with the same number of times into one item; Constants are calculated and combined into one term.
(3) When merging, the number of times is unchanged, but the coefficient is added and subtracted.
Step 6 move the project
(1) After the sign change, the term with unknown number is moved to the left of the equation, while the term without unknown number is moved to the right.
(2) Basis: the nature of the equation
(3) When you move an item from one side of the equation to the other, you must change the sign.
7. General steps to solve one-dimensional linear equation:
The value of the unknown that makes the left and right sides of the equation equal is called the solution of the equation.
Universal solution:
(1) denominator: both sides of the equation are multiplied by the least common multiple of each denominator;
(2) bracket removal: first remove the bracket, then remove the bracket, and finally remove the braces; (If there is a minus sign outside the brackets, please remember to change the sign.)
(3) Move the term: move all terms containing unknowns to one side of the equation and all other terms to the other side of the equation; Move the item to change the symbol.
(4) merging similar terms: transforming the equation into the form of ax=b(a0);
(5) Coefficient division 1: divide the unknown coefficient a on both sides of the equation to get the solution of equation x = b/a. 。
8. Homotopy equation
If the solutions of two equations are the same, then these two equations are called homosolution equations.
9. The same solution principle of the equation:
Add or subtract the same number or the same equation from both sides of the (1) equation, and the obtained equation is the same solution equation as the original equation.
(2) The equation obtained by multiplying or dividing the same number whose two sides are not zero is the same as the original equation.
Edit the mathematics knowledge points provided by the teacher in the new semester of grade one, hoping to bring you inspiration!
Summary of compulsory knowledge points of mathematics in the first day of junior high school 6. Related concepts of equations
1. Equation: An equation with an unknown number is called an equation.
2. One-dimensional linear equation: there is only one unknown (element) X, and the exponent of the unknown X is 1 (degree). This equation is called one-dimensional linear equation. For example, 1700+50x= 1800, and 2(x+ 1.5x)=5 are all linear equations.
3. Solution of the equation: The value of the unknown that makes the left and right sides of the equation equal is called the solution of the equation.
Note: The solution of the (1) equation and the solution of the equation are different concepts. The solution of the equation is essentially the result of the solution, which is a numerical value (or several numerical values). The meaning of solving the equation refers to the process of finding the solution of the equation or judging that the equation has no solution. ⑵ Test method of equation solution: First, substitute the unknown value into the left and right sides of the equation to calculate its value, and then compare the values on both sides to draw a conclusion.
Second, the nature of the equation.
Add (or subtract) the same number (or formula) on both sides of the (1) equation, and the results are still equal. Expressed in the form of a formula: if a=b, then ac=bc.
(2) If 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, which is expressed in the form of a formula: if a=b, then ac=bc If a=b(c0), then AC = BC.
Third, the transfer rules:
Moving the sign of the term on one side of the equation to the other side is called moving the term.
Fourth, the rule of removing brackets.
1. The factors outside the brackets are positive numbers, and the symbols of the items after removing the brackets are the same as those of the corresponding items in the original brackets.
2. The factor outside the bracket is negative, and the sign of each item is changed by the sign of the corresponding item in the original bracket after the bracket is removed.
Five, the general steps to solve the equation
1. denominator (least common multiple of denominator on both sides of the equation)
2. Parenthesis deletion (according to the rules of parenthesis deletion and distribution)
3. Move the term (move the term containing the unknown to one side of the equation, and all other terms will be moved to the other side of the equation. Moving the term will change the sign).
4.Merge (transform the equation into ax=b(a0))
5. The coefficient is 1 (divide the unknown coefficient a on both sides of the equation to get the solution of equation x=ba).
Sixth, the general steps to solve practical problems with equation thought.
1. Examination: Examine the questions, analyze what is known and what is sought in the questions, and clarify the relationship between quantity and quantity.
2. Assumptions: Assumptions about the unknown (which can be divided into direct and indirect ways).
3. Column: List the equations according to the meaning of the question.
4. Solution: Solve the listed equations.
5. Check: Check whether the solution meets the meaning of the problem.
6. Answer: Write the answer (some units should indicate the answer).
Seven, the relationship between common application types and quantity.
1, sum, difference, multiplication and division:
(1) multiplicity relation: it is reflected by the key words "how many times, how many times, how many times, what percentage, growth rate …".
(2) How much relationship: it is reflected by the key words "more, less, harmony, difference, lack, surplus ……".
2, equal product deformation problem:
"Equal area deformation" is based on the premise that the shape changes but the volume remains unchanged. The commonly used equivalence relation is:
(1) The shape area has changed, but the perimeter has not changed;
② Raw material volume = finished product volume.
3, labor distribution:
This kind of problem to find out the number of changes, common problems are:
(1) There are transfers in and transfers out.
(2) Only the transfer-in did not turn out, the transfer-in part changed, and the rest remained unchanged.
(3) Only the transfer-out has not been transferred in, some of the transfers have changed, and the rest remain unchanged.
4. Quantity problem
(1) Need to know the representation of numbers: a three-digit hundreds digit is A, a tens digit is B, and a unit digit is C (where A, B and C are integers,19,09,09), then this three-digit representation is:100a+/kloc-
(2) Some representations in the number problem: the relationship between two consecutive integers, the larger one is larger than the smaller one1; Even numbers are represented by 2n, and continuous even numbers are represented by 2n+2 or 2n2; Odd numbers are represented by 2n+ 1 or 2n 1.
5, engineering problems:
The three quantities in engineering problems and their relationships are: total work = working efficiency and working time.
6. Travel problems:
(1) Three basic quantities in the travel problem and their relationships: distance = speed and time.
(2) The basic types are
(1) meeting problems;
(2) follow up the problem; Common ones are: running for opponents; Navigation problems; Circular runway problem.
7, the problem of commodity sales
Correlation:
Commodity profit = commodity price = discount rate of commodity price = commodity price
Commodity profit rate = commodity profit/commodity purchase price
Commodity price = discount rate of commodity list price
8. Savings problem
(1) The money deposited by the customer in the bank is called the principal, and the reward paid by the bank to the customer is called interest. Principal and interest are collectively referred to as the sum of principal and interest, the time of deposit in the bank is called the number of periods, and the ratio of interest to principal is called the interest rate. Interest tax is paid at 20% of interest.
(2) Interest = principal interest rate cycle
Sum of principal and interest = principal+interest
Interest tax = interest tax rate (20%)
Let's call it a day.
Summary of the knowledge points of the compulsory math test in the first grade of junior high school 1: the concept of positive and negative numbers: we call positive numbers like 3, 2, +0.5 and 0.03% numbers, which are all numbers greater than 0; Numbers like -3, -2, -0.5 and -0.03% are called negative numbers. Are all numbers less than 0. 0 is neither positive nor negative. We can use positive numbers and negative numbers to represent quantities with opposite meanings.
Knowledge point 2: the concept and classification of rational numbers: integers and fractions are collectively called rational numbers. There are two main classifications of rational numbers:
Note: Both finite decimals and infinite cyclic decimals can be regarded as fractions.
Knowledge point 3: the concept of number axis: the straight line defining the origin, positive direction and unit length as follows is called number axis.
Knowledge point 4: the concept of absolute value:
(1) Geometric meaning: The distance from the point representing A on the number axis to the origin is called the absolute value of the number A, and it is recorded as | a |
(2) Algebraic significance: the absolute value of a positive number is itself; The absolute value of a negative number is its reciprocal; The absolute value of zero is zero.
Note: The absolute value of any number is greater than or equal to 0 (i.e. non-negative).
Knowledge point 5: the concept of reciprocal:
(1) Geometric meaning: The number represented by two points on both sides of the origin with the same distance is called reciprocal;
(2) Algebraic meaning: Two numbers with different signs but equal absolute values are called reciprocal. The antonym of 0 is 0.
Knowledge point 6: Comparison of rational numbers:
The basic principle of rational number size comparison: all positive numbers are greater than zero, all negative numbers are less than zero, and positive numbers are greater than negative numbers.
Comparison of rational numbers on the number axis: the number on the right is always greater than the number on the left.
Comparison between rational number and absolute value: two positive numbers, the positive number with larger absolute value is larger; Two negative numbers, the negative number with larger absolute value is smaller.
Knowledge point 7: rational number addition rule:
(1) Add two numbers with the same symbol, take the same symbol, and add the absolute values;
(2) When two numbers with different signs are added and the absolute values are equal, the sum is 0; When the absolute values are not equal, take the sign of the addend with larger absolute values and subtract the addend with smaller absolute values from the addend with larger absolute values;
(3) Adding a number to 0 still gets this number.
Knowledge point 8: rational number addition algorithm:
Additive commutative law: When two numbers are added, the position of the addend is reversed and the sum remains the same.
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.
Knowledge point 9: rational number subtraction rule: subtracting a number is equal to adding the inverse of this number.
Knowledge point 10: rational number addition and subtraction mixed operation: according to the law of rational number subtraction, all addition and subtraction operations can be unified into addition operations, and then brackets and plus signs are omitted, and calculations are made by using the law of addition and the law of addition operation.