Integers, fractions, finite decimals and cyclic decimals are called rational numbers. Numbers like-1, -2.5, -4/3 are called negative numbers, and negative numbers >; 0; Numbers like 12. +5.4 and +2/5 are called positive numbers, and positive numbers are less than 0. 0 is neither positive nor negative, it is the boundary between positive and negative numbers. Positive and negative numbers are widely used in life, for example, the height of Mount Everest is 8850 meters, which is recorded as +8850 meters; Take out 400 yuan from the bank and record it as -400 yuan.

People often draw pictures to visualize numbers, and use points on straight lines to represent numbers. This straight line is the number axis, and the place indicated by 0 is called the origin.

Like -2 2, -4/5 4/5, only two numbers with different symbols are called antonyms. The reciprocal of 0 is itself. The distance from the origin to a number is the absolute value of this number, the absolute value of negative number is its reciprocal, and the absolute values of positive number and zero are itself.

Positive number > 0> negative number. Compared with negative numbers, the larger the absolute value, the smaller it is.

A number plus a equals negative a: a number minus a equals positive a.

Multiplication and division of rational numbers, with odd negative sign, the result is negative, with even sign, the result is positive, with 0, the result is 0.

A number whose product is 1 is reciprocal.

The algorithm is applicable to all rational number operations. Example 1 If 8 km east is marked as +8 km, and 5 km west is marked as -5 km, what do the following figures mean respectively?

( 1)+4km； (2) kilometers; (3)0 km

Solution: (1)+4km means 4km to the east.

(2) Kilometers means kilometers to the west.

(3)0 km means no movement.

Note: (1) Positive and negative numbers can be used to represent quantities with opposite meanings. (2) Positive numbers can be preceded by a+sign. Generally speaking, the+sign in front of a positive number can be omitted, but sometimes it is customary to add a+sign in front of a positive number to show emphasis. (3) In addition to indicating nothing, 0 is the boundary between positive and negative numbers. This has certain significance in practical problems.

Example 2 uses rational numbers to represent the following quantities.

(1) If the income in 200 yuan is recorded as +200 yuan, how can the expenditure be expressed as 100 yuan?

(2) If the altitude below 100 m is marked as-100 m, how can it be expressed as altitude 1000 m?

(3) If southbound 100m is marked as+100m, what about northbound 200m?

(4) What should I say if it is+10/0kg heavier than the standard weight and 5g less than the standard weight?

After analysis, every two quantities in this question are two quantities with opposite meanings. In order to distinguish quantities with opposite meanings, we use numbers with different symbols to express them.

Solution (1) expenditure 100 yuan is expressed as-100 yuan; (2) The altitude 1000m should be expressed as+1000 m; (3) 200 meters northbound, that is, -200 meters; (4) 5 grams less than the standard weight means -5 grams.

Note (1) whether a quantity is expressed as a positive number or a negative number is stipulated by people, but in terms of expression, people should also respect the habits formed in years of life. For example, the temperature above zero is generally specified as positive; Above sea level is generally defined as positive; (2) The+sign before a positive number can be omitted.

Example 3 Judge right and wrong (tick right and wrong, cross wrong).

(1)-a must be a negative number. ()

(2) Zero is a natural number.

(3) There is no minimum positive rational number. ()

Solution: (1)×(2)√(3)√.

Note: we should closely follow the concepts of reciprocal, negative number, zero sum and positive rational number to solve this kind of problem, mainly because we have studied the field of algebra. We should always pay attention to the fact that the letter A may be negative or zero and positive.

Example 4 (1) In the knowledge contest, if+10 means plus 10, how to deduct 20 points? (2) When someone turns the turntable, if +5 means that he has turned it counterclockwise for 5 times, what about turning it clockwise 12 times? (3) In a ping-pong quality test, a ping-pong ball exceeds the standard quality by 0. 02g and marked as +0. 02.-What does 02.-0.03g mean?

Solution: (1) Deduct 20 points and record it as -20 points; (2) Turn clockwise 12 turn, and record it as-12 turn; (3)-0.03g means that the quality of table tennis is 0.03g lower than the standard quality.

Explanation: Give three examples to illustrate how to express this opposite quantity with positive and negative numbers.

Example 5 Fill in the following figures in the corresponding brackets:-16,26,-12,-0.92,0.0.1008, -4.95 (Thinking: Is decimal a fraction? ).

Positive number set {0}; Negative set {0};

Integer set {0}; Positive score set {0};

Negative score set {0};

Analysis: According to the definitions of positive number, negative number, integer and fraction, it is strictly distinguished. Note that zero is neither positive nor negative, but an integer.

Solution: positive number set {26,,, 0. 1008, ...};

Negative sets {- 16,-12, -0.92, -4.95, ...};

Positive fraction set {,0. 1008, ...};

Negative score set {-0.92, -4.95, ...}.

Note: When using braces to represent a set, pay attention to the use of ellipsis. For example, "set of positive numbers" refers to a "set" containing all positive numbers. Because it is "all", only a part can be filled in, so an ellipsis should be added after it.

Selected exercises

First, multiple choice questions

1. The correct statement is ().

A. If a number is preceded by a "-",it is negative.

B.0 is neither positive nor negative.

C. rational numbers consist of negative numbers and 0. D. positive numbers and negative numbers are collectively called rational numbers.

2. If 200m above sea level is marked as +200m, 50m above sea level should be marked as ().

A.-50m B.+50m

C. It may be+50m or-50m. This is not correct.

3. The following statement is wrong ().

A.0 is the smallest integer B. 1 is the smallest positive integer.

C.0 is the smallest natural number D. Natural numbers are nonnegative integers.

Second, fill in the blanks

1. If the backward10m is marked as-10/0m, the forward10m should be marked as _ _ _ _ _ _ _ _;

2. If the standard weight of a bag of cement is 50kg, if 2kg less than the standard weight is recorded as -2kg, then 1kg more than the standard weight should be recorded as _ _ _ _ _ _ _ _;

3. If the counterclockwise rotation of the wheel is marked as+1, then the clockwise rotation of the wheel is marked as _ _ _ _ _.

Third, the judgment question

1.0 is a rational number. ()

2. Rational numbers can be divided into positive rational numbers and negative rational numbers. ()

3. The rational number starting with "+"is a positive number. ()

4.0 is the smallest rational number. ()

Fourth, answer questions.

1. Write 5 numbers (no repetition) and meet the following three conditions at the same time.

(1) where three numbers are non-positive; (2) Three numbers are nonnegative; (3) All five numbers are rational numbers.

2. If we record the sea level as positive, use rational numbers to express the following questions.

A plane flies at an altitude of 9630 meters; (2) The submarine is 60 meters underwater.

3. If the temperature in Hainan Island can be expressed by a positive number in June 5438+February every year, what number should be used to express the temperature in Harbin at this time?

4. A listed stock fell by 0.7 1% on the first day and rose by 1.25% on the second day. What should I say?

5. If we specify that the sea level is positive, can the height of the ground be expressed as a positive number?

6. A student took part in a quiz, and he answered five questions. The scoring standard is set as follows. He gets 1 if he answers a question correctly, and deducts 1 if he answers it incorrectly or fails to answer it. Students get 3 points. He answered several questions correctly.

number axis

Selected exercises

First, multiple choice questions

1. If the reciprocal of a number is itself, then the number is ().

A. positive number B. negative number C. 0 D. there is no such number

2. There are two points E and F on the number axis, and E is to the left of F, so the inverse of the number represented by point E should be the inverse of the number represented by point F ().

A. left B. right C. left or right D. none of the above is right

3. If one number is greater than another, the reciprocal of this number ()

A. the opposite number that is smaller than another number

C. the opposite number is equal to another number d. Uncertain size

Second, fill in the blanks

1. If the point representing a number on the number axis is on the left side of the origin, then the point representing the reciprocal of the number must be on the _ _ _ _ _ side of the origin;

2. Any rational number can be represented by _ _ _ _ _ on the number axis;

3. There are _ _ _ _ _ _ _ points five units away from the origin, and the rational numbers they represent are _ _ _ _ _ and _ _ _ _ _ _ _ _;

4. The number to the left of the two numbers represented on the number axis is always _ _ _ _ _ _ _ _.

Third, the judgment question

The number on the 1. number axis that is four unit lengths from the origin is 4. ()

2. The farther away from the origin on the number axis, the larger the number. ()

3. The number axis is a straight line that defines the origin and the positive direction. ()

4. The distance between two opposite points is equal to the origin. ()

This is the rational number part.