Comparison of rational numbers with the same signs. Comparison of rational numbers

We continue to study rational numbers. IN this lesson we will learn to compare them.

From previous lessons we learned that the further to the right a number is located on the coordinate line, the larger it is. And accordingly, the further to the left the number is located on the coordinate line, the smaller it is.

For example, if you compare the numbers 4 and 1, you can immediately answer that 4 is more than 1. This is a completely logical statement and everyone will agree with it.

As proof, we can cite the coordinate line. It shows that the four lies to the right of the one

For this case, there is also a rule that can be used if desired. It looks like this:

Of two positive numbers, the number whose modulus is greater is greater.

To answer the question which number is greater and which is less, you first need to find the modules of these numbers, compare these modules, and then answer the question.

For example, compare the same numbers 4 and 1, applying the above rule

Finding the modules of numbers:

|4| = 4

|1| = 1

Let's compare the found modules:

4 > 1

We answer the question:

4 > 1

For negative numbers There is another rule, it looks like this:

Of two negative numbers, the number whose modulus is smaller is greater.

For example, compare the numbers −3 and −1

Finding the modules of numbers

|−3| = 3

|−1| = 1

Let's compare the found modules:

3 > 1

We answer the question:

−3 < −1

The modulus of a number should not be confused with the number itself. A common mistake many newbies make. For example, if the modulus of −3 is greater than the modulus of −1, this does not mean that −3 is greater than −1.

The number −3 is less than the number −1. This can be understood if we use the coordinate line

It can be seen that the number −3 lies further to the left than −1. And we know that the further to the left, the less.

If you compare a negative number with a positive one, the answer will suggest itself. Any negative number will be less than any positive number. For example, −4 is less than 2

It can be seen that −4 lies further to the left than 2. And we know that “the further to the left, the less.”

Here, first of all, you need to look at the signs of the numbers. A minus sign in front of a number indicates that the number is negative. If the number sign is missing, then the number is positive, but you can write it down for clarity. Recall that this is a plus sign

As an example, we looked at integers of the form −4, −3 −1, 2. Comparing such numbers, as well as depicting them on a coordinate line, is not difficult.

It is much more difficult to compare other types of numbers, such as fractions, mixed numbers And decimals, some of which are negative. Here you will basically have to apply the rules, because it is not always possible to accurately depict such numbers on a coordinate line. In some cases, a number will be needed to make it easier to compare and understand.

Example 1. Compare rational numbers

So, you need to compare a negative number with a positive one. Any negative number is less than any positive number. Therefore, without wasting time, we answer that it is less than

Example 2.

You need to compare two negative numbers. Of two negative numbers, the one whose magnitude is smaller is greater.

Finding the modules of numbers:

Let's compare the found modules:

Example 3. Compare the numbers 2.34 and

You need to compare a positive number with a negative one. Any positive number is greater than any negative number. Therefore, without wasting time, we answer that 2.34 is more than

Example 4. Compare rational numbers and

Finding the modules of numbers:

We compare the found modules. But first let's bring them to in a clear way, to make it easier to compare, let’s translate it into improper fractions and lead to common denominator

According to the rule, of two negative numbers, the number whose modulus is smaller is greater. This means rational is greater than , because the modulus of the number is less than the modulus of the number

Example 5.

You need to compare zero with a negative number. Zero is greater than any negative number, so without wasting time we answer that 0 is greater than

Example 6. Compare rational numbers 0 and

You need to compare zero with a positive number. Zero is less than any positive number, so without wasting time we answer that 0 is less than

Example 7. Compare rational numbers 4.53 and 4.403

You need to compare two positive numbers. Of two positive numbers, the number whose modulus is greater is greater.

Let's make the number of digits after the decimal point the same in both fractions. To do this, in the fraction 4.53 we add one zero at the end

Finding the modules of numbers

Let's compare the found modules:

According to the rule, of two positive numbers, the number whose absolute value is greater is greater. This means that the rational number 4.53 is greater than 4.403 because the modulus of 4.53 is greater than the modulus of 4.403

Example 8. Compare rational numbers and

You need to compare two negative numbers. Of two negative numbers, the number whose modulus is smaller is greater.

Finding the modules of numbers:

We compare the found modules. But first, let’s bring them to a clear form to make it easier to compare, namely, let’s convert the mixed number into improper fraction, then we bring both fractions to a common denominator:

According to the rule, of two negative numbers, the number whose modulus is smaller is greater. This means rational is greater than , because the modulus of the number is less than the modulus of the number

Comparing decimals is much easier than comparing fractions and mixed numbers. In some cases, by looking at the whole part of such a fraction, you can immediately answer the question of which fraction is larger and which is smaller.

To do this, you need to compare the modules of the entire parts. This will allow you to quickly answer the question in the task. After all, as you know, whole parts in decimal fractions have more weight than fractional parts.

Example 9. Compare rational numbers 15.4 and 2.1256

The modulus of the whole part of the fraction is 15.4 greater than the modulus of the whole part of the fraction 2.1256

therefore the fraction 15.4 is greater than the fraction 2.1256

15,4 > 2,1256

In other words, we didn’t have to waste time adding zeros to the fraction 15.4 and comparing the resulting fractions like ordinary numbers

154000 > 21256

The comparison rules remain the same. In our case we compared positive numbers.

Example 10. Compare rational numbers −15.2 and −0.152

You need to compare two negative numbers. Of two negative numbers, the number whose modulus is smaller is greater. But we will compare only the modules of integer parts

We see that the modulus of the whole part of the fraction is −15.2 greater than the modulus of the whole part of the fraction −0.152.

This means rational −0.152 is greater than −15.2 because the modulus of the integer part of the number −0.152 is less than the modulus of the integer part of the number −15.2

−0,152 > −15,2

Example 11. Compare rational numbers −3.4 and −3.7

You need to compare two negative numbers. Of two negative numbers, the number whose modulus is smaller is greater. But we will compare only the modules of integer parts. But the problem is that the moduli of integers are equal:

In this case, you will have to use the old method: find modules rational numbers and compare these modules

Let's compare the found modules:

According to the rule, of two negative numbers, the number whose modulus is smaller is greater. This means rational −3.4 is greater than −3.7 because the modulus of the number −3.4 is less than the modulus of the number −3.7

−3,4 > −3,7

Example 12. Compare rational numbers 0,(3) and

You need to compare two positive numbers. Moreover, compare a periodic fraction with a simple fraction.

Let's convert the periodic fraction 0,(3) into an ordinary fraction and compare it with the fraction . After converting the periodic fraction 0,(3) into an ordinary fraction, it becomes the fraction

Finding the modules of numbers:

We compare the found modules. But first, let’s bring them to an understandable form to make it easier to compare, namely, let’s bring them to a common denominator:

According to the rule, of two positive numbers, the number whose absolute value is greater is greater. This means that a rational number is greater than 0,(3) because the modulus of the number is greater than the modulus of the number 0,(3)

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Work progress: draw a coordinate line. Use a coordinate line to compare numbers:
Fill the table:
Example
7 and 5
5 and 0
7 and 0
4 and 6
9 and 10
8 and 3
Compare
modules
Number sign with big
module
­
­
­
|4| |6|
|9| |10|
|8| |3|
­
­
­
Answer
7 5
5 0
7 0
4 6
9 10
8 3


________________________________________________________________________________________

signs
More ______ ________ ________;

Laboratory and practical work Group 2.
Topic: “Comparison of rational numbers”
Task: Derive a rule for comparing rational numbers.
Progress: Using a thermometer scale, compare the numbers:
Fill the table:
Example
7 and 5
5 and 0
7 and 0
4 and 6
9 and 10
8 and 3
Compare
modules
Number sign with big
module
­
­
­
|4| |6|
|9| |10|
|8| |3|
­
­
­
Answer
7 5
5 0
7 0
4 6
9 10
8 3
Pay attention to the moduli of the numbers being compared.
Draw a conclusion: of two positive numbers, the greater is
________________________________________________________________________________________
Draw a conclusion: of two negative numbers, the greater is
________________________________________________________________________________________
Positive number negative

Based on your results, compare:
36 (33) 92 12 15 (18) 44 56
Try to formulate a rule for comparing numbers with different signs: made of two numbers with different signs
More ______ ________ ________;

Try to formulate a rule for comparing numbers with negative signs: of two numbers with negative
signs
More ______ ________ ________;
Laboratory and practical work Group 1.
Topic: “Comparison of rational numbers”
Task: Derive a rule for comparing rational numbers.
Progress: Using the concepts of income and debt, compare the numbers:
Fill the table:
Example
7 and 5
5 and 0
7 and 0
4 and 6
9 and 10
8 and 3
Compare
modules
Number sign with big
module
­
­
­
|4| |6|
|9| |10|
|8| |3|
­
­
­
Answer
7 5
5 0
7 0
4 6
9 10
8 3
Pay attention to the moduli of the numbers being compared.
Draw a conclusion: of two positive numbers, the greater is
________________________________________________________________________________________
Draw a conclusion: of two negative numbers, the greater is
________________________________________________________________________________________
Positive number negative

Based on your results, compare:
36 (33) 92 12 15 (18) 44 56

Try to formulate a rule for comparing numbers with different signs: from two numbers with different signs
More ______ ________ ________;
Try to formulate a rule for comparing numbers with negative signs: of two numbers with negative
signs
More ______ ________ ________;
Laboratory and practical work Group 1.
Topic: “Comparison of rational numbers”
Task: Derive a rule for comparing rational numbers.
Progress: Using the concept of winning and losing, compare numbers:
Fill the table:
Example
7 and 5
5 and 0
7 and 0
4 and 6
9 and 10
8 and 3
Compare
modules
Number sign with big
module
­
­
­
|4| |6|
|9| |10|
|8| |3|
­
­
­
Answer
7 5
5 0
7 0
4 6
9 10
8 3
Pay attention to the moduli of the numbers being compared.
Draw a conclusion: of two positive numbers, the greater is
________________________________________________________________________________________
Draw a conclusion: of two negative numbers, the greater is
________________________________________________________________________________________
Positive number negative

Based on your results, compare:
36 (33) 92 12 15 (18) 44 56
Try to formulate a rule for comparing numbers with different signs: from two numbers with different signs
More ______ ________ ________;
Try to formulate a rule for comparing numbers with negative signs: of two numbers with negative
signs
More ______ ________ ________;
1. Org. moment.
2. Lesson motivation.
During the classes.
You have heard the phrase “Everything is known by comparison” more than once. And indeed, you can only evaluate something, whether it is good or bad, by comparing it with
any other. For example, Natasha received an “5” for working at the board. Is it good or bad?
Is it a big pencil or a small one? You can compare objects only on a certain basis.
For example: sweet ice cream and negative numbers?
And it is necessary to compare mathematical objects, because only in comparison do we understand them most important properties, we study them.
Today we will continue to study rational numbers.
3. Updating of basic knowledge.
What topic are we covering?
Even without knowing about negative numbers, we have already encountered them in life, in what situations?
How are positive and negative numbers located on the coordinate line?

How to draw a coordinate line?
What number is called negative?
What is the modulus of a number?
The modulus of which number is greater: 3 or 2; 6 or -4. Which number is greater?
The modulus of what number is –20?
For the numbers 8, 4, 2/3, 0, select opposites and inverses.
What numbers do we call rational?
What numbers did people first become familiar with and why did other numbers arise?
(11), +(7), (+3)
What is greater and why: 0 or 7; 3 or 29?
Mathematical dictation:
Write using rational numbers:
1. Kolya lost his wallet containing 150 rubles. (150)
2. This morning it was 150 below zero (15)
3. Chicken body temperature 400 (400)
4. In winter in Khandyga there is 580 frost (580)
5. And in the summer it reaches 350 (+350)
6. The height of Mount Kozbek is 5033 m (5033)
7. Height itself deep place Pacific Ocean 11022m (11022)

8. Mom received a bonus of 300 rubles. (+300)
9. Sasha grew by 3 cm (+3)
10. The ice on the river has become 8 cm thinner (8)
11. The tourists stopped at the post with the 40 km mark, and then continued their journey at a speed of 3 km/h. What mark will the pillar have?
Are there tourists in 2 hours?
Decide:
a) |x| = 3; b) |z| = 2; c) |a| = 8; d) |c| = 6; e) |m| = 0; e) |n| = 0;

In the article we will consider the main points on the topic of comparing rational numbers. Let's study the scheme for comparing numbers with various signs, comparisons of zero with any rational number, and we will also examine in more detail the comparison of positive rational numbers and the comparison of negative rational numbers. We will reinforce the entire theory with practical examples.

Comparison of rational numbers with different signs

Comparing given numbers with different signs is simple and obvious.

Definition 1

Any positive number is greater than any negative number, and any negative number is less than any positive number.

Let's give simple examples for illustration: from two rational numbers 4 7 and - 0, 13 larger number 4 7 , because it is positive. When comparing the numbers - 6, 53 and 0, 00 (1), it is obvious that the number - 6, 53 is smaller, because it is negative.

Comparing a rational number with zero

Definition 2

Any positive number Above zero; any negative number is less than zero.

Simple examples for clarity: the number 1 4 is greater than 0. In turn, 0 is less than

number 1 4. The number - 6, 57 is less than zero, on the other hand, zero is greater than the number - 6, 57.

Separately, it is necessary to say about the comparison of zero with zero: zero is equal to zero, i.e. 0 = 0 .

It is also worth clarifying that the number zero can be represented in a form other than 0. Zero will correspond to any entry of the form 0 n (n is any natural number) or 0, 0, 0, 00, ..., up to 0, (0). Thus, comparing two rational numbers that have entries, for example, 0, 00 and 0 3, we conclude that they are equal, because These records correspond to the same number – zero.

Comparison of positive rational numbers

When performing the operation of comparing positive rational numbers, you must first compare their integer parts.

Definition 3

The largest number is the one with whole part more. Accordingly, the smaller number is the number whose integer part is smaller.

Example 1

It is necessary to determine which of the rational numbers is less: 0, 57 or 3 2 3 ?

Solution

The rational numbers given for comparison are positive. Moreover, it is obvious that the integer part of the number 0, 57 (equal to 0) is less than the integer part of the number 3 2 3 (equal to three). So 0.57< 3 2 3 , т.е. из двух заданных чисел меньшим является число 0 , 57 .

Answer: 0 , 57

Let's look at a practical example of one nuance of the rule used: a situation where one of the numbers being compared is a periodic decimal fraction with a period of 9.

Example 2

It is necessary to compare the rational numbers 17 and 16, (9).

Solution

16 , (9) – this is periodic fraction with period 9, which is one of the forms of writing the number 17. Thus, 17 = 16, (9).

Answer: the given rational numbers are equal.

We have reviewed practical examples, when the integer parts of rational numbers are not equal and must be compared. If the integer parts of the given numbers are equal, comparing the fractional parts of the given numbers will help you get the result. The fractional part can always be written in the form common fraction type m\n, final fraction or periodic decimal fraction. Those. Essentially, comparing the fractional parts of positive numbers is comparing ordinary or decimal fractions. It is logical that the larger of two numbers with equal integer parts is the one whose fractional part is larger.

Example 3

It is necessary to compare positive rational numbers: 4, 8 and 4 3 5

Solution

It is obvious that the integer parts of the numbers to be compared are equal. Then the next step is to compare the fractional parts: 0, 8 and 3 5. There are two ways to use this:

1. Let's convert the decimal fraction to an ordinary fraction, then 0, 8 = 8 10. Let's compare ordinary fractions 8 10 and 3 5. Bringing them to a common denominator, we get: 8 10 > 6 10, i.e. 8 10 > 3 5, respectively 0, 8 > 3 5. Thus, 4, 8 > 4 3 5.
2. Let's convert an ordinary fraction to a decimal, we get: 3 5 = 0.6. Let's compare the resulting decimal fractions 0, 8 and 0, 6: 0, 8 > 0, 6. Therefore: 0, 8 > 3 5, and 4, 8 > 4 3 5.

We see that as a result of applying both methods, the same result was obtained when comparing the given initial rational numbers.

Answer: 4 , 8 > 4 3 5 .

If the integer and fractional parts of the positive rational numbers that we are comparing are equal, then these numbers are equal to each other. In this case, the numbers can be written differently (for example, 6, 5 = 6 1 2), or completely coincide (for example, 7, 113 = 7, 113 or 51 3 4 = 51 3 4).

Comparison of negative rational numbers

Definition 4

When comparing two negative numbers, the larger number will be the one whose modulus is smaller and, accordingly, the smaller number will be the one whose modulus is larger.

In essence, this rule leads the comparison of two negative rational numbers to the comparison of positive ones, the principle of which we discussed above.

Example 4

It is necessary to compare the numbers - 14, 3 and - 3 9 11.

Solution

The given numbers are negative. For comparison, let's define their modules: | - 14, 3 | = 14, 3 and - 3 9 11 = 3 9 11 _formula_. We begin the comparison by evaluating the integer parts of the given numbers: it is obvious that 14 > 3, thus 14, 3 > 3 9 11. Let's apply the rule for comparing negative numbers, which states that the larger number is the one whose modulus is smaller, and then we get: - 14, 3 > - 3 9 11.

Answer: - 14 , 3 > - 3 9 11 .

Example 5

It is necessary to compare the negative rational numbers - 2, 12 and - 2 4 25.

Solution

Let us determine the modules of the numbers being compared. | - 2, 12 | = 2, 12 and - 2 4 25 = 2 4 25. We see that the integer parts of the given numbers are equal, which means it is necessary to compare their fractional parts: 0, 12 and 4 25. Let's use the method of converting an ordinary fraction to a decimal, then: 4 25 = 0.16 and 0.12< 0 , 16 , т.е. 2 , 12 < 2 4 25 . Применим правило сравнения отрицательных рациональных чисел и получим: - 2 , 12 > - 2 4 25 .

Answer: - 2 , 12 > - 2 4 25 .

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MATHEMATICS
Lessons for 6th grade

Lesson No.68

Subject. Comparison of rational numbers

Goal: based on students’ observations and experience, derive a rule for comparing any two rational numbers and develop the ability to use it to compare rational numbers and solve exercises that involve comparing rational numbers.

Lesson type: application of knowledge, skills and abilities.

During the classes

I. Verification homework

@ According to the author, in order to save time, you need to check only No. 3, 4, 5 (especially pay attention to the use of the properties of multiplication and addition to simplify calculations in No. 5). We check everything else by collecting the students’ notebooks.

II. Updating of reference knowledge

Oral exercises

2. Name the numbers opposite numbers: 15; -3; -38; 0; a; c + d .

3. Find the modules of numbers: 13; -8; -615; 0; a, if a is positive, b, if b is negative.

4. Solve the equation: |x| = 3; |t | = 0.4; |in| = ; |u | = 0.

5. Place a “>” or “” sign instead of * to make the entry correct: 35* 0.35; 35.1* 35.01; * ; 2.7*2.

III. Application of knowledge

1. Comparing numbers using a coordinate line

Task. Mark the numbers 2 on the coordinate line; 5; 7; 4. Compare the numbers: a) 2 and 5; b) 2 and 7; c) 2 and 4. Using the coordinate line, find out how the number 2 is located in relation to each of the other numbers.

@ We see that 2 is to the left of 5; 2 to the left of 7, 2 to the left of 4. Let us remember that in grade 5, while studying the topic of comparison natural numbers we said that on coordinate ray less number always lies on the left, and more - on the contrary - on the right. In general, on a coordinate line, more than two numbers lie on the right, and fewer on the left.

Example. Compare the numbers a, b, c, d shown in the figure (write in ascending order).

Solutions. b c a d , because from left to right the numbers are in that order.

2. Rule for comparing rational numbers
Let's turn to the coordinate line.

We see that all positive numbers lie to the right of 0, and all negative numbers lie to the left of 0, therefore:

1) a positive number greater than 0; negative number less than 0;

2) any positive number is greater than any negative number.

For example, 3 > 0; -thirty; -3 3; 3 > -3.

If both numbers (a and b) are negative (see figure), then

3) of two negative numbers, the one with the smaller modulus is greater.

For example, - 3.7 > - 7.3, since|-3.7| = 3.7; 3.7 7.3, since |-7.3| = 7.3.

3. Conclusion. Rational numbers can be compared using both the coordinate line and comparison rules. In the first case: the number on the right is greater.

In the second case:

a) positive > negative; b) positive > 0; c) negative 0; d) of two negative numbers, the one with the smaller modulus is greater.

@ The issue of symbolic recording of these rules is not resolved unambiguously and the method for solving it depends on the preparation of students.

IV. Mastering skills

@ So much time in this lesson was spent on explaining new material; there is not enough time for exercises of varying content and level. That's why the main objective- practice the application of the rules for comparing rational numbers well on standard exercises.

Oral exercises

1. Read the inequalities. Are they correct?

a) 0 3; b) 0 > -5; c) -7 0; d) -3 > 2; e) -7 1; e) -2 -5; g) -5 -3.

2. It is known that a b c. Which of the pictures meets this condition?
1) 2) 3) 4)

Writing exercises

1. Place the “>” or “” sign instead of * to form the correct inequality:

d) -5.5 * -7.2;

e) -96.9 * -90.3;

yes) -100 * 0;

With) *;

To) *.

2. Arrange the following numbers in ascending order:

1) -4; 3; -2; 1; 0; -1; 2; -3; 4;

2) -5,4; 4,3; -3,2; 2,1; -1,2; 2,3; -3,4.

3. Which number is -5; -1; 8; 0; -5.3 the most? less? In which of them largest module? smallest module?

4. Fill out the table. To do this, in each cell, enter a number that satisfies both conditions:

5. It is known that x and y are positive numbers, and m and n are negative. Compare:
a) 0 and n; b) in and 0; c) -x and 0; d) 0 and -m; e) x and t; e) n and x; g) -m and n; c) -x and y; j) |m | and m; l) -|m | and m; m) x and |x|; n) x and |-x|.