Multiplication and division of positive numbers is a rule. Division of numbers with different signs: rule and examples

In this article, we will give a definition of dividing a negative number by a negative one, formulate and justify a rule, give examples of dividing negative numbers and analyze the course of their solution.

Division of negative numbers. The rule

Let us recall the essence of the division operation. This action is to find an unknown multiplier by a known product and a known other multiplier. The number c is called the quotient of the division of the numbers a and b, if the product c · b = a is true. Moreover, a ÷ b = c.

The rule for dividing negative numbers

The quotient of dividing one negative number by another negative number is equal to the quotient of dividing the absolute values ​​of these numbers.

Let a and b be negative numbers. Then

a ÷ b = a ÷ b.

This rule reduces division of two negative numbers to division of positive numbers. It is true not only for integers, but also for rational and real numbers. The result of dividing a negative number by a negative number is always a positive number.

Here is another formulation of this rule, suitable for rational and real numbers. It is given using reciprocal numbers and reads: to divide a negative number a by the number undefined, multiply by the number b - 1, the reciprocal of b.

a ÷ b = a b - 1.

The same rule, which reduces division to multiplication, can also be used to divide numbers with different signs.

The equality a ÷ b = a b - 1 can be proved using the property of multiplication of real numbers and the definition of mutually inverse numbers. Let's write the equalities:

a b - 1 b = a b - 1 b = a 1 = a.

By virtue of the definition of the division operation, this equality proves that there is a quotient from dividing a number by a number b.
Let's move on to examining examples.

Let's start with simple cases and move on to more complex ones.

Example 1. How to divide negative numbers

Divide - 18 by - 3.
The moduli of the divisor and the dividend are 3 and 18, respectively. Let's write down:

18 h - 3 = - 18 h - 3 = 18 h 3 = 6.

Example 2. How to divide negative numbers

Divide - 5 by - 2.
Similarly, we write according to the rule:

5 ÷ - 2 = - 5 ÷ - 2 = 5 ÷ 2 = 5 2 = 2 1 2.

The same result will be obtained if we use the second formulation of the rule with a reciprocal.

5 ÷ - 2 = - 5 · - 1 2 = 5 · 1 2 = 5 2 = 2 1 2.

Dividing rational fractional numbers is most convenient to represent them in the form of ordinary fractions. However, final decimal fractions can also be divided.

Example 3. How to divide negative numbers

Divide - 0.004 by - 0.25.

First, we write the modules of these numbers: 0, 004 and 0, 25.

Now you can choose one of two ways:

1. Divide decimal fractions longitudinally.
2. Go to fractions and divide.

Let's look at both methods.

1. Performing column division of decimal fractions, move the comma two digits to the right.

Answer: - 0.004 ÷ 0.25 = 0.016

2. Now we give the solution with the conversion of decimal fractions to ordinary ones.

0.004 = 4 1000; 0.25 = 25 100 0.004 ÷ 0.25 = 4 1000 ÷ 25 100 = 4 1000 100 25 = 4 250 = 0.016

The results are the same.

In conclusion, we note that if the dividend and the divisor are irrational numbers and are specified as roots, degrees, logarithms, etc., the result of the division is written as a numerical expression, the approximate value of which is calculated if necessary.

Example 4. How to divide negative numbers

Let's calculate the quotient of dividing the numbers - 0, 5 and - 5.

0.5 ÷ - 5 = - 0.5 ÷ - 5 = 0.5 ÷ 5 = 1 2 1 5 = 1 2 5 = 5 10.

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In this article, we will formulate and explain the rule for multiplying negative numbers. The process of multiplying negative numbers will be discussed in detail. The examples show all possible cases.

Multiplying negative numbers

Definition 1

The rule for multiplying negative numbers is that in order to multiply two negative numbers, you must multiply their modules. This rule is written as follows: for any negative numbers - a, - b, this equality is considered true.

(- a) (- b) = a b.

Above is the rule for multiplying two negative numbers. Based on it, we prove the expression: (- a) (- b) = a b. The article multiplication of numbers with different signs tells that the equalities a (- b) = - a b are fair, as well as (- a) b = - a b. This follows from the property of opposite numbers, due to which the equalities will be written as follows:

(- a) (- b) = (- a (- b)) = - (- (a b)) = a b.

Here you can clearly see the proof of the rule for multiplying negative numbers. Based on the examples, it is clear that the product of two negative numbers is a positive number. When multiplying the absolute values ​​of numbers, the result is always a positive number.

This rule applies to the multiplication of real numbers, rational numbers, and whole numbers.

Now let's take a closer look at examples of multiplying two negative numbers. When calculating, you must use the rule written above.

Example 1

Multiply the numbers - 3 and - 5.

Solution.

Modulo the data being multiplied, two numbers are equal to positive numbers 3 and 5. Their product results in 15. It follows that the product of the given numbers is 15

Let us briefly write down the multiplication of negative numbers itself:

(- 3) (- 5) = 3 5 = 15

Answer: (- 3) (- 5) = 15.

When multiplying negative rational numbers, applying the analyzed rule, you can mobilize yourself to multiply fractions, multiply mixed numbers, multiply decimal fractions.

Example 2

Calculate the product (- 0, 125) · (- 6).

Solution.

Using the rule for multiplying negative numbers, we get that (- 0, 125) (- 6) = 0, 125 6. To get the result, you need to multiply the decimal fraction by the natural number of columns. It looks like this:

We got that the expression will take the form (- 0, 125) · (- 6) = 0, 125 · 6 = 0.75.

Answer: (- 0, 125) (- 6) = 0, 75.

In the case when the factors are irrational numbers, then their product can be written as a numerical expression. The value is calculated only when needed.

Example 3

It is necessary to multiply negative - 2 by non-negative log 5 1 3.

Solution

We find modules of given numbers:

2 = 2 and log 5 1 3 = - log 5 3 = log 5 3.

Following the rules for multiplying negative numbers, we get the result - 2 · log 5 1 3 = - 2 · log 5 3 = 2 · log 5 3. This expression is the answer.

Answer: - 2 log 5 1 3 = - 2 log 5 3 = 2 log 5 3.

To continue studying the topic, you must repeat the section on multiplying real numbers.

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§ 1 Multiplication of positive and negative numbers

In this lesson, we will get acquainted with the rules for multiplying and dividing positive and negative numbers.

It is known that any product can be represented as a sum of identical terms.

The term -1 needs to be added 6 times:

(-1)+(-1)+(-1) +(-1) +(-1) + (-1) =-6

So the product of -1 and 6 is equal to -6.

The numbers 6 and -6 are opposite numbers.

Thus, we can conclude:

When you multiply -1 by a natural number, you get the opposite number.

For negative numbers, as well as for positive numbers, the displacement law of multiplication is fulfilled:

If a natural number is multiplied by -1, then the opposite number will also be obtained.

When you multiply any non-negative number by 1, you get the same number.

For instance:

For negative numbers, this statement is also true: -5 ∙ 1 = -5; -2 ∙ 1 = -2.

When you multiply any number by 1, you get the same number.

We have already seen that when multiplying minus 1 by a natural number, we get the opposite number. When multiplying a negative number, this statement is also true.

For example: (-1) ∙ (-4) = 4.

Also -1 ∙ 0 = 0, the number 0 is the opposite of itself.

When you multiply any number by minus 1, you get the opposite number.

Let's move on to other cases of multiplication. Find the product of the numbers -3 and 7.

A negative factor of -3 can be replaced by the product of -1 and 3. Then the combination law of multiplication can be applied:

1 ∙ 21 = -21, i.e. the product of minus 3 and 7 is equal to minus 21.

When multiplying two numbers with different signs, a negative number is obtained, the modulus of which is equal to the product of the moduli of the factors.

And what is the product of numbers with the same signs equal to?

We know that when you multiply two positive numbers, you get a positive number. Find the product of two negative numbers.

Replace one of the factors with a product with a factor of minus 1.

We apply our rule, when multiplying two numbers with different signs, a negative number is obtained, the modulus of which is equal to the product of the moduli of the factors,

you get -80.

Let's formulate a rule:

When two numbers with the same signs are multiplied, a positive number is obtained, the modulus of which is equal to the product of the moduli of the factors.

§ 2 Division of positive and negative numbers

Let's move on to division.

By selection, we find the roots of the following equations:

y ∙ (-2) = 10. 5 ∙ 2 = 10, so x = 5; 5 ∙ (-2) = -10, so a = 5; -5 ∙ (-2) = 10, so y = -5.

Let's write down the solutions of the equations. The factor is unknown in each equation. We find the unknown factor by dividing the product by a known factor; we have already selected the values ​​of the unknown factors.

Let's analyze.

When dividing numbers with the same signs (and these are the first and second equations), a positive number is obtained, the modulus of which is equal to the quotient of the moduli of the dividend and the divisor.

When dividing numbers with different signs (this is the third equation), a negative number is obtained, the modulus of which is equal to the quotient of the moduli of the dividend and the divisor. Those. when dividing positive and negative numbers, the quotient sign is determined according to the same rules as the product sign. And the modulus of the quotient is equal to the quotient of the moduli of the dividend and the divisor.

Thus, we have formulated the rules for multiplying and dividing positive and negative numbers.

List of used literature:

1. Mathematics. Grade 6: lesson plans for the textbook I.I. Zubareva, A.G. Mordkovich // compiled by L.A. Topilin. - Mnemosyne, 2009.
2. Mathematics. Grade 6: a textbook for students of educational institutions. I.I. Zubareva, A.G. Mordkovich. - M .: Mnemosina, 2013.
3. Mathematics. Grade 6: a textbook for students of educational institutions. / N. Ya. Vilenkin, V.I. Zhokhov, A.S. Chesnokov, S.I. Schwarzburd. - M .: Mnemosina, 2013.
4. Mathematics reference - http://lyudmilanik.com.ua
5. Handbook for high school students http://shkolo.ru

This article provides a detailed overview of division of numbers with different signs... First, there is a rule for dividing numbers with different signs. Below are examples of dividing positive numbers into negative and negative numbers by positive.

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The rule for dividing numbers with different signs

In the article dividing integers, a rule was obtained for dividing integers with different signs. It can be extended to both rational numbers and real numbers by repeating all the arguments from the above article.

So, rule for dividing numbers with different signs has the following formulation: in order to divide a positive number by a negative or negative number by a positive one, you must divide the dividend by the modulus of the divisor, and put a minus sign in front of the resulting number.

Let's write this division rule using letters. If the numbers a and b have different signs, then the following formula is valid a: b = - | a |: | b | .

From the stated rule, it is clear that the result of dividing numbers with different signs is a negative number. Indeed, since the modulus of the dividend and the modulus of the divisor are more positive than the number, then their quotient is a positive number, and the minus sign makes this number negative.

Note that the considered rule reduces division of numbers with different signs to division of positive numbers.

You can give another formulation of the rule for dividing numbers with different signs: to divide the number a by the number b, you need to multiply the number a by the number b −1, the reciprocal of the number b. That is, a: b = a b −1 .

This rule can be used when it is possible to go beyond the set of integers (since not every integer has an inverse). In other words, it is applicable on the set of rational numbers, as well as on the set of real numbers.

It is clear that this rule for dividing numbers with different signs allows you to go from division to multiplication.

The same rule applies when dividing negative numbers.

It remains to consider how this rule for dividing numbers with different signs is applied when solving examples.

Examples of dividing numbers with different signs

Consider solutions to several typical examples of division of numbers with different signs to learn the principle of applying the rules from the previous paragraph.

Example.

Divide the negative number −35 by the positive number 7.

Solution.

The rule for dividing numbers with different signs dictates that you first find the modules of the dividend and the divisor. The modulus of -35 is 35, and the modulus of 7 is 7. Now we need to divide the modulus of the dividend by the modulus of the divisor, that is, we need to divide 35 by 7. Remembering how the division of natural numbers is performed, we get 35: 7 = 5. The last step of the rule for dividing numbers with different signs remains - put a minus in front of the resulting number, we have −5.

Here's the whole solution:.

It was possible to proceed from a different formulation of the rule for dividing numbers with different signs. In this case, we first find the reciprocal of the divisor 7. This number is the common fraction 1/7. In this way, . It remains to perform the multiplication of numbers with different signs:. Obviously, we arrived at the same result.

Answer:

(−35):7=−5 .

Example.

Calculate the quotient 8: (- 60).

Solution.

By the rule for dividing numbers with different signs, we have 8:(−60)=−(|8|:|−60|)=−(8:60) ... The resulting expression corresponds to a negative ordinary fraction (see the division sign as a line of a fraction), you can reduce the fraction by 4, we get .

Let us write down the entire solution briefly:.

Answer:

.

When dividing fractional rational numbers with different signs, their dividend and divisor are usually represented as ordinary fractions. This is due to the fact that it is not always convenient to perform division with numbers in another notation (for example, in decimal).

Example.

Solution.

The modulus of the dividend is equal, and the modulus of the divisor is 0, (23). To divide the modulus of the divisible by the modulus of the divisor, we turn to ordinary fractions.

Let's translate the mixed number into an ordinary fraction: , as well as

Open lesson topic: "Multiplication of negative and positive numbers"

Date: 17.03.2017

Teacher: V.V. Kuts

Class: 6 g

The purpose and objectives of the lesson:

introduce the rules for multiplying two negative numbers and numbers with different signs;

promote the development of mathematical speech, working memory, voluntary attention, visual-active thinking;

the formation of internal processes of intellectual, personal, emotional development.

foster a culture of behavior in frontal work, individual and group work.

Lesson type: a lesson in the primary presentation of new knowledge

Forms of training: frontal, work in pairs, work in groups, individual work.

Teaching methods: verbal (conversation, dialogue); visual (work with didactic material); deductive (analysis, application of knowledge, generalization, project activities).

Concepts and terms : modulus numbers, positive and negative numbers, multiplication.

Planned results learning

-be able to multiply numbers with different signs, multiply negative numbers;

Apply the rule for multiplying positive and negative numbers when solving exercises, consolidate the rules for multiplying decimal and ordinary fractions.

Regulatory - be able to define and formulate a goal in the lesson with the help of a teacher; to pronounce the sequence of actions in the lesson; work according to a collectively drawn up plan; evaluate the correctness of the action. Plan your action in accordance with the task at hand; make the necessary adjustments to the action after its completion based on its assessment and taking into account the mistakes made; make your guess.Communicative - be able to formulate their thoughts orally; listen to and understand the speech of others; jointly agree on and follow the rules of conduct and communication at school.

Cognitive - be able to navigate in their system of knowledge, to distinguish new knowledge from already known with the help of a teacher; gain new knowledge; find answers to questions using the textbook, your life experience and the information received in the lesson.

Formation of a responsible attitude to learning based on motivation to learn new things;

Formation of communicative competence in the process of communication and cooperation with peers in educational activities;

Be able to carry out self-assessment based on the criterion of the success of educational activities; focus on success in educational activities.

During the classes

Structural elements of the lesson

Didactic tasks

Projected teacher activity

Projected student activities

Result

1.Organizational moment

Motivation for successful activity

Checking readiness for the lesson.

- Good afternoon guys! Have a seat! Check if everything is ready for the lesson: notebook and textbook, diary and writing materials.

I am glad to see you at the lesson in a good mood today.

Look into each other's eyes, smile, with your eyes wish your friend a good working mood.

I wish you a good job today too.

Guys, the motto for today's lesson will be a quote from the French writer Anatole France:

“Learning can only be fun. To digest knowledge, one must absorb it with appetite. "

Guys, who can tell me what it means to absorb knowledge with appetite?

So today in the lesson we will absorb knowledge with great pleasure, because they will be useful to us in the future.

Therefore, rather, we open notebooks and write down the number, great work.

Emotional attitude

-With interest, with pleasure.

Willingness to start a lesson

Positive motivation to learn a new topic

2. Activation of cognitive activity

Prepare them for the assimilation of new knowledge and methods of action.

Organize a frontal survey based on the material covered.

Guys, who can tell me what is the most important skill in mathematics? ( Check). Right.

Now I’ll check you how well you can count.

We will now perform a mathematical warm-up with you.

We work as usual, count orally, and write down the answer in writing. I give you 1 min.

5,2-6,7=-1,5

2,9+0,3=-2,6

9+0,3=9,3

6+7,21=13,21

15,22-3,34=-18,56

Let's check the answers.

We will check the answers, if you agree with the answer, then clap your hands, if you do not agree, then stamp your feet.

Well done boys.

Tell me, what actions did we perform with the numbers?

What rule did we use when invoicing?

Formulate these rules.

Answer questions by solving small examples.

Addition and subtraction.

Add numbers with different signs, add numbers with negative signs, and subtract positive and negative numbers.

The readiness of students to pose a problematic question, to find ways to solve the problem.

3. Motivation for setting the topic and purpose of the lesson

Stimulate students to formulate the topic and purpose of the lesson.

Arrange work in pairs.

Well, it's time to move on to learning new material, but first, let's review the material from the previous lessons. A math crossword puzzle will help us with this.

But this crossword puzzle is not ordinary, it contains an encrypted keyword that will tell us the topic of today's lesson.

Guys, the crossword puzzle is on your tables, we will work with it in pairs. And once in pairs, then remind me how it is in pairs?

We remembered the rule of working in pairs, but now we are starting to solve the crossword puzzle, I give you 1.5 minutes. Who will do everything, put down the pens for me to see.

(Annex 1)

1. What numbers are used for counting?

2. The distance from the origin to any point is called?

3. Are the numbers represented by a fraction called?

4. Two numbers that differ from each other only in signs are called?

5. What numbers lie to the right of zero on the coordinate line?

6.Natural numbers, opposite numbers and zero are called?

7. What number is called neutral?

8. A number showing the position of a point on a straight line?

9. What numbers lie to the left of zero on the coordinate line?

So the time is up. Let's check.

We have solved the entire crossword puzzle and thus repeated the material of the previous lessons. Raise your hand, who made only one mistake and who made two? (So ​​you guys are great).

Well, now let's get back to our crossword puzzle. At the very beginning, I said that it contains an encrypted word that will tell us the topic of the lesson.

So what is the topic of our lesson?

And what are we going to multiply with you today?

Let's think, for this we recall the types of numbers that we already know.

Let's think, what numbers can we already multiply?

What numbers will we learn to multiply today?

Write the topic of the lesson in a notebook: "Multiplying positive and negative numbers."

So, guys, we figured out what we are going to talk about today in the lesson.

Please tell me the purpose of our lesson, what should each of you learn and what should you try to learn by the end of the lesson?

Guys, well, in order to achieve this goal, what tasks will we have to solve with you?

Quite right. These are the two tasks that we will have to solve with you today.

They work in pairs, formulate the topic and purpose of the lesson.

1.Natural

2.Module

3.Rational

4.Opposite

5.Positive

6.Integer

7.Zero

8.Coordinate

9.Negative

-"Multiplication"

Positive and negative numbers

"Multiplication of positive and negative numbers"

The purpose of the lesson:

Learn to multiply positive and negative numbers

First, to learn how to multiply positive and negative numbers, you need to get a rule.

Second, when we get the rule, what should we do next? (learn to apply it when solving examples).

4. Learning new knowledge and ways of acting

Master new knowledge on the topic.

-Organize group work (learning new material)

- Now, in order to achieve our goal, we will proceed to the first task, derive the rule for multiplying positive and negative numbers.

And research work will help us in this. And who will tell me why it is called research? - In this work we will investigate to discover the rules "Multiplication of positive and negative numbers."

Your research work will take place in groups, in total we will have 5 research groups.

They repeated in my head how we should work in a group. If someone has forgotten, then the rules are in front of you on the screen.

The purpose of your research work: While exploring the tasks, gradually deduce the rule "Multiplication of negative and positive numbers" in task number 2, in task number 1 you have 4 tasks in total. And in order to solve these problems, our thermometer will help you for this, each group has one.

Make all your notes on a piece of paper.

As soon as the group has a solution to the first problem, you show it on the board.

You are given 5-7 minutes to work.

(Appendix 2 )

Work in groups (fill in the table, conduct research)

Rules for working in groups.

Working in groups is very easy

Be able to observe five rules:

first: do not interrupt,

when tells

friend, there must be silence around;

second: do not shout loudly,

and give the arguments;

and the third rule is simple:

decide what is important to you;

fourthly: it is not enough to know verbally,

must be recorded;

and fifthly: sum up, think,

what could you do.

Mastery

the knowledge and methods of action that are determined by the objectives of the lesson

5.Fizzy

Establish the correctness of assimilation of new material at this stage, identify misconceptions and their correction

Well, I put all your answers in the table, now, let's look at each line in our table (see the Presentation)

What conclusions can we draw when examining the table.

1 line. What numbers are we multiplying? What number is the answer?

Line 2. What numbers are we multiplying? What number is the answer?

3 line. What numbers are we multiplying? What number is the answer?

4 line. What numbers are we multiplying? What number is the answer?

And so you analyzed the examples, and are ready to formulate the rules, for this you had to fill in the gaps in the second task.

How to multiply a negative number by a positive one?

- How do I multiply two negative numbers?

Let's get some rest.

Positive answer - sit down, negative - get up.

5*6

2*2

7*(-4)

2*(-3)

8*(-8)

7*(-2)

5*3

4*(-9)

5*(-5)

9*(-8)

15*(-3)

7*(-6)

By multiplying positive numbers, the answer is always a positive number.

Multiplying a negative number by a positive one always gives a negative number in the answer.

By multiplying negative numbers, the answer will always be a positive number.

Multiplying a positive number by a negative number produces a negative number.

To multiply two numbers with different signs, you needmultiply modules of these numbers and put a "-" sign in front of the resulting number.

- To multiply two negative numbers, you needmultiply their modules and put a sign in front of the resulting number «+».

Students practice physical exercises, reinforcing the rules.

Prevent fatigue

7.Initial securing of new material

To master the ability to apply the acquired knowledge in practice.

Organize frontal and independent work on the material covered.

Let's fix the rules, and tell each other as a pair of these same rules. I'll give you a minute for that.

Tell me, can we now move on to solving examples? Yes we can.

Opening page 192 # 1121

All together we will make the 1st and 2nd lines a) 5 * (- 6) = 30

b) 9 * (- 3) = - 27

g) 0.7 * (- 8) = - 5.6

h) -0.5 * 6 = -3

n) 1.2 * (- 14) = - 16.8

o) -20.5 * (- 46) = 943

three people at the blackboard

You are given 5 minutes to solve the examples.

And we check everything together.

Creative task in pairs. (Appendix 3)

Insert the numbers so that on each floor their product equals the number on the roof of the house.

Solve examples by applying the knowledge gained

Raise your hands who have not had any mistakes, well done….

Active actions of students to apply knowledge in life.

9. Reflection (lesson summary, assessment of students' performance results)

Provide reflection of students, i.e. their assessment of their performance

Organize a wrap-up of the lesson

Our lesson has come to an end, let's summarize.

Let's remember the topic of our lesson again? What goal did we set? - Have we achieved this goal?

What difficulties did this topic cause for you?

- Guys, well, in order to evaluate your work in the lesson, you must draw a smiley face in circles that are on your tables.

A smiling emoticon means that you understand everything. Green means that you understand, but you need to practice, and a sad smiley, if you don't understand anything at all. (I give half a minute)

Well guys, are you ready to show how you did your lesson today? So, we raise and, I also raise a smiley for you.

I am very pleased with you in class today! I see that everyone has understood the material. Guys, you are great!

The lesson is over, thanks for your attention!

Answer questions, evaluate their work

Yes, we did.

The openness of students to the transmission and comprehension of their actions, to the identification of positive and negative aspects of the lesson

10 .Homework information

Provide an understanding of the purpose, content and way of doing homework

Provides an understanding of the purpose of the homework.

Homework:

1. Learn the rules of multiplication
2.No. 1121 (3 columns).
3. Creative task: make a test of 5 questions with multiple answers.

They write down their homework, trying to comprehend and understand.

Realization of the need to achieve conditions for the successful completion of homework by all students, in accordance with the task and the level of development of students