I related concepts of positive numbers and negative numbers

(1) positive number: a number greater than 0 is called a positive number; Negative number: a number less than 0 is called a negative number; 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 is negative, -a is positive; When a represents 0, -a is still 0. (If it is judged that the number with a positive sign is positive and the number with a negative sign is negative, this statement is wrong. For example, +a and -a cannot make simple judgments. )

② Sometimes "+"can be added before positive numbers, and sometimes "+"can be omitted. Therefore, the positive sign omitting "+"is a positive sign.

(2) Positive numbers and negative numbers represent quantities with opposite meanings. For example:

8℃ above zero means:+8℃; 8℃ below zero means: -8℃ (don't forget to add units).

(3) 0 is the dividing line between positive and negative numbers, and 0 is neither positive nor negative.

0 doesn't mean nothing, but also a real object, such as 0 degrees Celsius and 0 meters above sea level.

Second, the concept and classification of rational numbers

Rational number is a general term for integer and fraction. There are usually two types:

Note: 1. After the introduction of negative numbers, the range of odd and even numbers is also expanded. For example, -2, -4, -6, -8… are even numbers, and-1, -3, -5… are also odd numbers. 2. Both finite decimals and infinite cyclic decimals are fractions.

Summary: ① Positive integers and 0 are collectively called non-negative integers (also called natural numbers).

② Negative integers and 0 are collectively referred to as non-positive integers.

③ Positive rational numbers and 0 are collectively called non-negative rational numbers.

④ Negative rational numbers and 0 are collectively called non-positive rational numbers.

Third, the number axis

The concept of number axis: the straight line defining the origin, positive direction and unit length is called number axis.

Note: the (1) axis is a straight line extending infinitely to both ends; (2) Origin, positive direction and unit length are the three elements of the number axis, which are indispensable; (3) The unit length on the same axis should be unified; (4) The three elements of the number axis are specified according to actual needs.

2. The relationship between points on the number axis and rational numbers.

(1) All rational numbers can be represented by points on the number axis, positive rational numbers can be represented by points on the right of the origin, negative rational numbers can be represented by points on the left of the origin, and 0 can be represented by the origin.

⑵ All rational numbers can be represented by points on the number axis, but not all points on the number axis represent rational numbers, that is, there is no one-to-one correspondence between rational numbers and points on the number axis. (For example, point π on the number axis is not a rational number)

3. Use the number axis to represent the size of two numbers.

(1) Comparing the numbers on the number axis, the number on the right is always greater than the number on the left;

2 positive numbers are greater than 0, negative numbers are less than 0, and positive numbers are greater than negative numbers;

(3) Comparing two negative numbers, the number far from the origin is less than the number close to the origin.

4. Special maximum (minimum) number on the number axis

(1) The minimum natural number is 0, and there is no maximum natural number; (2) The minimum positive integer is 1, and there is no maximum positive integer;

(3) The largest negative integer is-1, and there is no smallest negative integer.

5. Motion law of points on the number axis

According to the movement of the point, if you move a few unit lengths to the left, you will subtract a few, and if you move a few unit lengths to the right, you will add a few, so that you can get the required point position. (Pay attention to the moving direction)

The number axis often works with absolute values, especially to judge the symbols in absolute values. In this regard, we generally use the assignment method, that is, the letters on the number axis, and assign him a specific number according to the actual situation, so that students will feel much more relaxed when solving problems.

Fourth, absolute value and reciprocal and reciprocal.

(1) reciprocal

Only two numbers with different signs are called reciprocal, one of which is the reciprocal of the other, and the reciprocal of 0 is 0.

Note: (1) Antiquities appear in pairs; (2) The opposite numbers only have different symbols. If one is positive and the other is negative;

(3) The inverse of 0 is itself; The inverse itself is a number of 0.

2. The nature and judgment of reciprocal

(1) Any number has a reciprocal and only one; The reciprocal of 0 is 0;

(3) If the sum of the two opposites is 0 and the two opposites are 0, that is, A and B are opposites, then a+b=0.

3. The geometric meaning of reciprocal

Two points on the number axis with the same distance from the origin represent two opposite numbers; Two mutually opposite numbers, the corresponding points (except 0) on the number axis are on both sides of the origin, and the distance from the origin is equal. The reciprocal of 0 corresponds to the origin; The origin represents the reciprocal of 0. Note: On the number axis, two points representing opposite numbers are symmetrical about the origin.

4. The solution of reciprocal

(1) To find the reciprocal of a number, just add and subtract the sign "-"before it (for example, the reciprocal of 5 is-5);

(2) Find the reciprocal of the sum or difference of multiple numbers, add "-"in brackets, and then simplify (for example; The reciprocal of 5a+b is -(5a+b). Simplified to-5a-b);

(3) To find a single number with "-"in front, you should also enclose it in parentheses before adding "-",and then simplify it (for example, the inverse of -5)

-(-5), simplified to 5)

5. Representation method of reciprocal

(1) Generally speaking, the inverse of the number A is -a, where A is any rational number, which can be positive, negative or 0.

When a>0, -a

When a<0,-a > 0 (the reciprocal of a negative number is a positive number)

When a=0, -a=0, (the reciprocal of 0 is 0)

6. Simplification of multiple symbols (the same symbol is positive and different symbols are negative)

Simplification law of multiple symbols: the number of "+"signs does not affect the simplification result, and can be omitted directly; The number of "-"determines the final simplification result; That is, when the number of "-"is odd, the result is negative, and when the number of "-"is even, the result is positive.

(2) Absolute value

Geometric definition of absolute value

Generally speaking, the distance between the point representing the number A on the number axis and the origin is called the absolute value of A, which is denoted as |a|.

2. Algebraic definition of absolute value

(1) The absolute value of a positive number is itself; (2) The absolute value of a negative number is its inverse; The absolute value of 0 is 0.

Can be expressed in letters as follows:

It can be summarized as ①: a ≥ 0,

②a≤0，& lt═>； |a|=-a (the absolute value of a non-positive number is equal to its inverse; A number whose absolute value is equal to its opposite number is not positive. )

3. The essence of absolute value

The absolute value of any rational number is non-negative, that is, the absolute value is non-negative. Therefore, if a takes any rational number, there is |a|≥0. That is, the absolute value of (1)0 is 0; A number with an absolute value of 0 is 0. That is, a = 0.

The absolute value of a number is non-negative, and the number with the smallest absolute value is 0. That is | a | ≥ 0;

(3) The absolute value of any number is not less than the original number. Namely: | a | ≥ a;

(4) The absolute values of two numbers are the same positive number, and they are opposite. That is, if | x | = a(a >；; 0), then x = a；;

5] The absolute values of two opposite numbers are equal. That is: |-a|=|a| or | a | = | b | If a+b = 0;

[6] Two numbers with equal absolute values are equal or opposite. That is: |a|=|b|, then a=b or a =-b;

Once, if the sum of the absolute values of several numbers is equal to 0, then these numbers are simultaneously 0. That is |a|+|b|=0, then a=0 and b=0.

(Common properties of non-negative numbers: if the sum of several non-negative numbers is 0, then only these non-negative numbers are 0 at the same time)

4. Comparison of rational numbers

⑴ Compare the size of two numbers by using the number axis: when two numbers on the number axis are compared, the left one is always smaller than the right one;

⑵ Compare the size of two negative numbers with absolute values: two negative numbers compare the size, and the absolute value is larger than the small one; Compare the sizes of two numbers with different signs, and the positive number is greater than the negative number.

5. Simplification of absolute value (first judge whether the absolute value sign is just negative,)

(1) when a≥0, | a | =-a② when a≤0, | a | =-a.

6. Know the absolute value of a number and find it.

The absolute value of the number A is the distance from the point representing the number A on the number axis to the origin. Generally speaking, there are two rational numbers with the same positive absolute value, which are opposite to each other. A number with an absolute value of 0 is 0, and there is no number with a negative absolute value.

(3) Reciprocity

Two numbers whose product is 1 are reciprocal, and one of them is called reciprocal of the other, which means a = 1 (a ≠ 0), that is to say, A is reciprocal, that is, A is reciprocal.

Note: ①0 has no reciprocal; If a and b are reciprocal, then a× b =1;

To find the reciprocal of a false score or a true score, just reverse the numerator and denominator of this score; When calculating the reciprocal of a fraction, first turn the fraction into a false fraction, and then reverse the positions of the numerator and denominator;

③ The reciprocal of a positive number is a positive number, and the reciprocal of a negative number is a negative number. Find the reciprocal of a number without changing its symbolic nature;

④ The number whose reciprocal equals itself is 1 or-1, excluding 0.

Absolute value, reciprocal and reciprocal often have some comprehensive problems of the distribution rate of multiplication, which should have special overall significance here. (The sum of two opposite numbers is 0, and the product of two reciprocal numbers is 1. We should have the concept of overall substitution. )

The mystery of self

① The reciprocal is one's own number, yes 1 ② The absolute value is one's own number, not negative (positive plus 0).

③ The number whose square equals itself is 0, and the number whose cube equals the warp itself is 1, 0.

⑤ The number whose even power equals itself is 0, and the number whose odd power equals itself is 1, 0.

⑦ The inverse number is itself, and the number is 0.

Highest number

① Minimum positive integer is 1 ② Maximum negative integer is-1 ③ Minimum absolute value is 0.

④ The minimum square number is 0 ⑤ The minimum non-negative number is 0 ⑤ The maximum non-positive number is 0.

⑦ There is no maximum and minimum rational number ⑧ There is no maximum positive number and minimum negative number.

Five, rational number addition (sign first, then size)

1. 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 symbols are added, if the absolute values are not equal, take the sign of the addend with the larger absolute value and subtract the one with the smaller absolute value; When the absolute values of two addends are equal, the two addends are in opposite directions and the sum is zero.

(3) The sum of two mutually opposite numbers is zero;

(4) Add a number to zero and you still get this number.

2. Arithmetic of rational number addition

(1) additive commutative law: a+b = b+a.

⑵ law of additive combination: (a+b)+c=a+(b+c)

When using the algorithm, we must use it flexibly according to the needs to achieve the purpose of simplification. Usually, there are the following laws:

(1) first add two opposites-"combination of opposites";

2. First add two numbers with the same symbol-"the combination of the same symbol";

③ The numbers with the same denominator are added first-"the combination method with the same denominator";

(4) Add several numbers to get an integer, and add them first-"rounding method";

⑤ Addition of integers to integers and decimals to decimals-"isomorphic combination method".

3. Additional nature

The sum of a number plus a positive number is greater than the original number; The sum after adding negative number is less than the original number; The sum after adding 0 is equal to the original number. Namely:

(1) when b>0, a+b > When b, A 2

6. Rational number subtraction rules

Subtracting a number is equal to adding the reciprocal of this number. Expressed in letters: a-b=a+(-b).

Seven. Significance of unifying addition and subtraction of rational numbers into addition

In the mixed operation of rational number addition and subtraction, according to the law of rational number subtraction, subtraction can be converted into addition, and then calculation can be made according to the law of addition.

In the summation formula, the parentheses of each addend and the preceding plus sign are usually omitted and written as the sum of the omitted plus signs. For example:

(-8)+(-7)+(-6)+(+5)=-8-7-6+5.

How to read the sum formula: ① According to the meaning expressed by this formula, it is read as "the sum of negative 8, negative 7, negative 6 and positive 5"

② Read it as "minus 8 minus 7 minus 6 plus 5" according to the operation meaning.

Eight, rational number multiplication (sign of definite product, multiplication of absolute value)

1. Multiplication Rule of Rational Numbers

Rule 1: multiply two numbers, the same sign is positive and the different sign is negative, and the multiplication takes the absolute value; ("The same sign is positive and the different sign is negative" refers to the situation of "multiplying two numbers". If there are more than two factors, Rule 3 must be applied. )

Rule 2: any number multiplied by 0 will get 0;

Rule 3: Multiply several numbers that are not 0. When the number of negative factors is even, the product is positive. When the number of negative factors is odd, the product is negative;

Rule 4: Multiply several numbers. If one of the factors is 0, the product is equal to 0.

2. Multiplication algorithm of rational numbers

(1) Multiplication commutative law: Generally speaking, in rational number multiplication, two numbers are multiplied, and the position of the commutative factor is equal to the product. Namely ab=ba

⑵ Law of Multiplication: When three numbers are multiplied, the first two numbers are multiplied or the last two numbers are multiplied, and the products are equal. That is, (ab)c=a(bc).

(3) Multiplication and distribution law: Generally speaking, the multiplication of a number with the sum of two numbers is equivalent to the multiplication of this number with these two numbers respectively, and the products are added. That is, a(b+c)=ab+ac.

9. Division rule of rational numbers

(1) divided by a number that is not equal to 0 is equal to multiplying the reciprocal of this number.

(2) Divide two numbers, the one with the same sign is positive, and the one with different signs is negative, and divide by the absolute value.

(3) Divide 0 by any number that is not equal to 0 to get 0.

X. mixed operation of rational number multiplication and division

(1) The mixed operation of multiplication and division usually decomposes division into multiplication first, then determines the sign of the product, and finally obtains the result.

(2) The mixed operation of addition, subtraction, multiplication and division of rational numbers, if there is no bracket to indicate what operation to do first, will be carried out in the order of' multiplication and division first, then addition and subtraction'.

Summary: Determination of product logo

When several rational numbers are multiplied, the sign of the product is determined by the number of negative factors: when the negative factors are odd, the product is negative;

When there are even negative factors, the product is positive. Multiply several rational numbers, one factor is zero and the product is zero.

Xi。 Power of rational number

The operation of (1) seeking the same factor product is called power. The result of power operation is called power.

Generally speaking, the multiplication of n A is recorded as: the n power of A, which means the multiplication of n A; Where a is the base and n is the exponent, which is called power.

(2) means: multiplication. It's called base, it's called exponent, and the result of calculation is called power.

When it is a positive number, it is an arbitrary number, and the calculation result is a positive number.

When it is negative, when it is odd, the result is negative; Is it even? Yes, the result is affirmative.

When the radix is negative or fractional, the radix must be placed in parentheses.

Note: radix is, index is, and result is; The cardinality of is, the exponent is, and the result is.

Calculation:

(3) Any degree of a positive number is a positive number.

The odd power of a negative number is a negative number,

Even the power of negative numbers is positive.

(4) The square of a number is itself, and this number is 0 and1;

The cube itself is a number of 0, 1 and-1.

Twelve, the mixed operation of rational numbers

When doing the mixed operation of rational numbers, pay attention to the following operation order:

1. Multiply first, then multiply and divide, and finally add and subtract;

2. Operation at the same level, from left to right; When there is multiplication and division, it should be unified as multiplication first.

3. If there are brackets, do the operation in brackets first, and then press brackets, middle brackets and braces in turn.

Thirteen, scientific counting methods

In general, when the number greater than 10 is expressed in the form of (n is a positive integer), in order to unify the standard, the range of a is specified, (1 ≤| A | < 10), and this notation is called scientific notation.

Fourteen, there are two ways to express the precision of the divisor: accurate to one tenth or accurate to 0. 1.

When using rounding method to approximate the accuracy, first find the corresponding number as required, and then round the subsequent numbers immediately.

The divisor with numeration unit can be determined in two ways: if the number in front of numeration unit is an integer, the divisor is accurate to numeration unit; If there is a decimal in front of the numeration unit, first restore the divisor to the original number, and then look at the position of the last digit in the original number. For example, the approximate number is 65.438+0.3 billion, accurate to 1 100 million; When the approximate number is 24300, it will be accurate to 100. When determining the exact number, the divisor expressed by scientific notation should also be restored to the original number, and then where the last digit in the original number is located from left to right, it is said that the divisor is accurate to which place. For example, restore to the original number 369.0, and the last digit of "0" is in the tenth place of the original number, so it is accurate.

Summary: The methods to compare the sizes of two rational numbers are:

(1) directly compare the positions of points corresponding to rational numbers on the number axis;

(2) Comparison according to regulations: two positive numbers; Positive numbers and zeros; Negative numbers and zeros; Positive and negative numbers; Two negative numbers reflect the mathematical idea of classified discussion;

(3) Difference method: A-B >; 0 ? a & gtb；

(4) Business Law: A/B > 1，b & gt0 ? a & gtb.

(5) Compare sizes with absolute values.

Comparison of two positive numbers: the one with the largest absolute value is the largest;

Comparison of two negative numbers: first calculate their absolute values, the larger the absolute value, the smaller.

Typical case study:

Xiaoshi, a taxi driver, was operating on the east-west renmin street one afternoon. If it is stipulated that east is due west and negative, his mileage this afternoon (unit: km) is as follows:

+ 15,-3,+ 14,- 1 1,+ 10,- 12,+4,- 15,+ 16,- 18.

(1) When the last passenger arrived at the destination, how many kilometers was Xiaoshi from the departure point in the afternoon?

(2) If the fuel consumption of a car is one liter/km, how many liters will the car consume this afternoon?

Analysis: (1) Sum the known numbers of 10 to get the distance from Xiaoshi to the starting point in the afternoon;

(2) The fuel consumption is required, and the distance the car travels is required, regardless of the driving direction, that is, the sum of the absolute values of 10 is calculated and then multiplied by one liter.

Pay attention to the difference between the two questions.

Solution: (1) (+15)+(-3)+(+14)+(-1)+(-)

=( 15+ 14+ 10+4+ 16)+(-3)+(- 1 1)+(- 12)+(- 15)+(- 18)

=59+(-59)

=0 km

(2)

=118km

118× a =118a (length)

Answer: (1) When the last passenger is sent to the destination, the distance from Xiaoshi to the departure point in the afternoon is 0 km, that is, he returns to the departure point;

(2) If the fuel consumption of the car is one liter/km, then the fuel consumption of the car this afternoon is *** 1 18a liter.

Typical case study:

In the calculation of power, the "symbol problem" is the most likely to make mistakes. The key to solving the problem is to accurately understand the concept of power, always keep a clear head, don't add, subtract or change symbols at will, let alone "skip", and strictly follow the algorithm.

Solution:

Typical case study:

1 564.2 million in scientific notation.

Analysis: The following two points should be noted: ① "10,000,000,000,000,000" will appear in some data, which should be noted; ② Scientific notation has its standard form:, where n is a positive integer.

Solution: 564200000 = 564200000 =

Typical case study:

(1) How many points have a distance equal to 4 from the origin? What is the number it represents?

(2) The number represented by point A on the number axis is -3. What is the number represented by a point two units away from point A?

Analysis: For beginners, you can draw a number axis. Seen from the number axis, there are two points from the origin 4, which are located on both sides of the origin, and the numbers they represent are +4 and -4 respectively. Never ignore the point to the left of the origin, which means -4. In this way, the second problem is solved.

Solution: (1) There are two points whose distance from the origin is equal to 4, and the numbers they represent are +4 and -4.

(2) On the number axis, the number represented by point A is -3, and the number represented by points two units away from point A is-1 and -5.

3. The reciprocal of-(-3) is _ _ _ _ _. (Analysis: Simplify -(-3) first, and then find the inverse of the calculation result)

Typical case study:

Known, find the values of x and y.

Analysis: this question examines the application of the concept of absolute value, because the absolute value of any rational number a is non-negative, that is.

Therefore, if the sum of two non-negative numbers is 0, then both numbers are 0, so the values of x and y can be found.

Solution: Again

∴, namely

∴

Typical case study:

If △ is specified as an operation, a△b=, find the value of: 3△(4△).