Find the original text of the relevant chapter of "rational number" in junior high school mathematics textbook. thank you

Chapter II Rational Numbers and Their Operations 1∽4

Main contents: 1. Why is the quantity insufficient?

2. Counting axis

3. Absolute value

4. Addition of rational numbers

Second, the study guidance:

1. Why is the quantity insufficient?

In real life, we often encounter such problems:

(1) The temperature is above 10℃ and zero or minus 5℃;

(2) Income from 500 yuan or expenditure from 300 yuan;

(3) The water level rises 1.2m or falls by 0.9m

(4) 5 meters forward or 6 meters backward;

(5) Buy 20 bottles of mineral water or sell 15 bottles of mineral water.

Although the specific contents of each pair of quantities here are different, they all have a common feature: they all have opposite meanings. Zero and negative, income and expenditure, rising and falling, forward and backward trading all have opposite meanings.

It is not easy to distinguish this opposite quantity with the figures we learned in primary school. For example, if 5℃ above zero and 5℃ below zero are both represented by the number 5, there will be misunderstanding. In other words, the numbers we have learned are not enough. As we all know, in the weather forecast, minus 5℃ is expressed as minus 5℃, and "minus 5℃ is pronounced as minus 5℃. So we introduce negative numbers.

Numbers like 5 1.2500, ... are called positive numbers greater than 0.

Numbers beginning with "-"are called negative numbers, such as-10, -3, -0.3 145, ... they are less than 0.

0 is neither positive nor negative.

In order to highlight the symbols of numbers, you can also add a "+"sign before positive numbers, such as +5,+1.2,+,+500, ...

With positive and negative numbers, we can express quantities with opposite meanings:

It is usually stipulated that above zero is positive and below zero is negative. Similarly, if the income is recorded as positive, the expenditure is recorded as negative; The rise is positive and the decline is negative; Forward is positive, backward is negative, buying is positive and selling is negative. Specifically, 5℃ above zero is +5℃, and 5℃ below zero is-5℃; When the water level rises 1.2m, it is marked as+1.2m, and when it falls by 0.9m, it is marked as -0.9m, and so on.

The introduction of negative numbers, the numbers we have learned can be divided into integer, positive integer and positive fraction.

zero

Negative integer negative fraction

Integers and fractions are collectively called rational numbers.

Rational numbers can be classified in two ways:

(1) rational number positive integer (2) rational number positive rational number

Zero positive score

negative integer

The score is positive zero.

Negative rational number negative integer

Negative score

Note that 0 is a special number, it is neither a positive number nor a negative number, it is an integer, and it is also a number that we easily miss when we classify, so we should pay special attention when studying this section.

2. Counting axis

Thermometer, ruler, balance, etc. Give us the image of the number axis.

The straight line that defines the origin, positive direction and unit length is called the number axis.

Origin, positive direction and unit length are the three elements of straight line as the number axis.

Usually the number axis is placed horizontally, and the direction to the right is positive.

Every rational number can be represented by a point on the number axis.

The number axis can be used to compare the sizes of two numbers. Because the direction to the right is positive, the number on the right side of the number axis is larger than the number on the left.

Two numbers like 3 and -3, and-are only different in sign, and two numbers like this are opposite.

Generally speaking, if two numbers are only different in sign, then we say that one of them is opposite to the other, that is, the two numbers are opposite. We also stipulate that the antonym of 0 is 0.

We also see that the positions of two numbers with opposite numbers on the number axis are on both sides of the origin, and the distance to the origin is equal. We also say that points representing two numbers with opposite numbers on the number axis are symmetrical about the origin.

Note that the antipodal is a pair. You can't say that a number is an enantiomer, only that a number is an enantiomer of another number.

3. Absolute value

On the number axis, the distance between the point corresponding to a number and the origin is called the absolute value of this number. For example, the absolute value of +3 is equal to 3, and it is recorded as |+3 | = 3; The absolute value of -2 is equal to 2, and it is recorded as |-2 | = 2.

The absolute value of a number has the following relationship with this number:

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.

It is easy to see that the absolute values of two mutually opposite numbers are equal, such as |-6 | = |+6 | = 6.

Since the absolute value is the distance from the point representing the number to the origin, the point farther away from the origin represents the greater the absolute value of the number. The greater the absolute value of a negative number, the farther to the left the point representing the number. So when two negative numbers are compared, the bigger the absolute value, the smaller it is.

With the concepts of positive and negative numbers and absolute values, we know that determining a number can be considered from two aspects: sign and absolute value.

4. Addition of rational numbers

We are familiar with the law of addition, but after the introduction of rational numbers, we should add in strict accordance with the law of addition of rational numbers.

Law of rational number addition:

Add two numbers with the same sign, take the same sign, and then add the absolute values.

When two numbers with different signs are added, the sum is 0 when the absolute values are equal (that is, two numbers with opposite numbers are added to get 0);

When the absolute values are not equal, take the sign of the number with the larger absolute value and subtract the smaller absolute value from the larger absolute value.

When a number is added to 0, it still gets the number.

The law of addition points out that the result of adding two rational numbers consists of two parts:

First determine the sign of sum, and then determine the absolute value of two numbers to add or subtract to get the absolute value of sum.

In addition, in the operation, the symbol problem is the most likely to make mistakes, so we should pay special attention to the symbol problem.

Three. Typical example

Example 1 An object moves in two opposite directions. If the east direction is defined as the positive direction, what should be recorded when moving 5 meters east and 6.8 meters west? What do 6 meters,-15 meters and 0 meters mean?

Solution: moving 5 meters eastward is +5 meters; Moving 6.8 meters to the west is marked as -6.8 meters;

Moving 6 meters means that the object moves 6 meters eastward; Moving-15m means that the object has moved westward15m; A movement of 0 meters means that the object is stationary.

Note: (1) Positive and negative numbers can represent quantities with opposite meanings;

(2)0 not only means no, but also means the origin, starting position and the boundary between positive and negative numbers.

Example 2 displays the following numbers on the number axis and uses "

-2,,0,, 1, -.

Answer: -2-0 1

& lt-2 & lt； -& lt； 0 & lt 1 & lt；

Note: Numbers are represented on the number axis, and the numbers on the left are smaller than those on the right.

Example 3 If point A and point B represent two numbers in opposite directions, and the distance between these two points is 13, write these two numbers.

Solution: Because the points representing two mutually opposite numbers are symmetrical about the origin on the number axis, their distance from the origin is equal, which is equal to half the distance between the two points, so these two numbers are 6.5 and -6.5.

Example 4 Write an integer whose absolute value is not greater than 3.

Solution: Integer whose absolute value is not greater than 3 is 3,2, 1, 0.

Note: When writing such an integer, don't leave out negative numbers and don't leave out 0.

exercises

1. Fill in the blanks:

(1) If the rising flag of 20m is +20m, then the falling flag of 15m is _ _ _ _ _ _-.

(2) 4 meters forward is marked as +4 meters, and then 6 meters backward is marked as _ _ _ _ _ _ _ _ _.

(3) If the expenditure in 500 yuan is recorded as -500 yuan, then the income in 800 yuan is recorded as _ _ _ _ _ _ _.

(4) Both Party A and Party B start from place A at the same time. If Party A walks 48 meters south, it will be marked as +48 meters, while Party B walks 32 meters north, it will be marked as _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _.

(5) On the number axis, the point to the left of the origin represents _ _ _ _ _ _ _ number, the point to the right of the origin represents _ _ _ _ _ _ _ _ number, and the _ _ _ _ point represents 0.

(6) A positive integer less than 4 is _ _ _ _ _ _ _ _; Negative integers greater than -4 are _ _ _ _ _ _ _

(7) The reciprocal of-π is _ _ _ _ _ _ _, and the reciprocal of _ _ _ _ _ _ is 0.

(8)-(-8) is _ _ _ _ _ _ _

Use ">" and "< to fill in the blanks:

( 1)-9_________- 16; (3)-3. 14_________-π.

3. A point starts from the point representing-1 on the number axis, moves 6 unit lengths to the right, and then moves 5 unit lengths to the left, indicating the number represented by this point at this time.

4. How many points are there on the number axis that are 3 unit lengths away from the origin? What are the numbers they represent?

5. given, find X.

6. Calculation: (1) |+/kloc-0 |+|-8 |; (2)|-200 1|-|+ 1999|;

(3)|-6.25|×|+4|; (4)|+ |÷| |.

7. Compare the dimensions of the following pairs:

(1)|3/5| and |-2/5 | (2) |-0.02 | and |-0.2 |;

(3) |-4| and-4; (4) |-(-3 )| and |-3 |;

8. Simplification: (1)-[+(-3)]; (2)-[-(-4)].

9. Expressed by formula: the temperature reached after the temperature rose from -5℃ to 8℃.

10. Calculation:

( 1) +(-5 ); (2)(-5)+0;

(3)(+2)+(-2.2);