Multiply positive. Multiplication of numbers with different signs, rule, examples

In this article, we will deal with multiplying numbers with different signs. Here we will first formulate the rule for multiplying a positive and negative number, justify it, and then consider the application of this rule when solving examples.

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Rule for multiplying numbers with different signs

Multiplying a positive number by a negative one, as well as a negative number by a positive one, is carried out according to the following multiplication rule with different signs : to multiply numbers with different signs, you need to multiply, and put a minus sign in front of the resulting product.

Let's write down this rule in literal form. For any positive real number a and any negative real number −b, the equality a(−b)=−(|a|·|b|) , and for the negative number −a and the positive number b, the equality (−a)b=−(|a|·|b|) .

The rule for multiplying numbers with different signs is fully consistent with properties of actions with real numbers. Indeed, on their basis it is easy to show that for real and positive numbers a and b a chain of equalities of the form a (−b)+a b=a ((−b)+b)=a 0=0, which proves that a (−b) and a b are opposite numbers, which implies the equality a (−b)=−(a b) . And from it follows the validity of the multiplication rule under consideration.

It should be noted that the announced rule for multiplying numbers with different signs is valid both for real numbers and for rational numbers and for integers. This follows from the fact that operations on rationals and integers have the same properties that were used in the proof above.

It is clear that the multiplication of numbers with different signs according to the obtained rule is reduced to the multiplication of positive numbers.

It remains only to consider examples of applying the analyzed multiplication rule when multiplying numbers with different signs.

Examples of multiplying numbers with different signs

Let's take a look at several solutions examples of multiplying numbers with different signs. Let's start with a simple case to focus on rule steps rather than computational complexity.

Multiply the negative number −4 by the positive number 5 .

According to the multiplication rule for numbers with different signs, we first need to multiply the modules of the original factors. The modulus of −4 is 4, and the modulus of 5 is 5, and the multiplication of natural numbers 4 and 5 gives 20. Finally, it remains to put a minus sign in front of the resulting number, we have -20. This completes the multiplication.

Briefly, the solution can be written as follows: (−4) 5=−(4 5)=−20 .

(−4) 5=−20 .

When multiplying fractional numbers with different signs, you need to be able to perform multiplication ordinary fractions, multiplication of decimal fractions and their combinations with natural and mixed numbers.

Carry out the multiplication of numbers with different signs 0, (2) and.

Having completed the translation of the periodical decimal fraction into an ordinary fraction, as well as by making the transition from a mixed number to improper fraction, from the original product we will come to the product of ordinary fractions with different signs of the form. This product is equal to the multiplication rule for numbers with different signs. It remains only to multiply the ordinary fractions in brackets, we have .

.

Separately, it is worth mentioning the multiplication of numbers with different signs, when one or both factors are

Now let's deal with multiplication and division.

Suppose we need to multiply +3 by -4. How to do it?

Let's consider such a case. Three people got into debt, and each has $4 in debt. What is the total debt? In order to find it, you need to add up all three debts: $4 + $4 + $4 = $12. We have decided that the addition of three numbers 4 is denoted as 3 × 4. Since in this case we are talking about debt, there is a “-” sign in front of 4. We know the total debt is $12, so now our problem is 3x(-4)=-12.

We will get the same result if, according to the condition of the problem, each of the four people has a debt of 3 dollars. In other words, (+4)x(-3)=-12. And since the order of the factors does not matter, we get (-4)x(+3)=-12 and (+4)x(-3)=-12.

Let's summarize the results. When multiplying one positive and one negative number, the result will always be a negative number. Numerical value the answer will be the same as in the case of positive numbers. Product (+4)x(+3)=+12. The presence of the "-" sign only affects the sign, but does not affect the numerical value.

How do you multiply two negative numbers?

Unfortunately, this topic is very difficult to come up with. good example from life. It's easy to imagine $3 or $4 in debt, but it's completely impossible to imagine -4 or -3 people getting into debt.

Perhaps we will go the other way. In multiplication, changing the sign of one of the factors changes the sign of the product. If we change the signs of both factors, we must change the signs twice product sign, first from positive to negative, and then vice versa, from negative to positive, that is, the product will have its original sign.

Therefore, it is quite logical, although a bit strange, that (-3)x(-4)=+12.

Sign position when multiplied it changes like this:

• positive number x positive number = positive number;
• negative number x positive number = negative number;
• positive number x negative number = negative number;
• negative number x negative number = positive number.

In other words, multiplying two numbers with the same sign, we get a positive number. Multiplying two numbers with different signs, we get a negative number.

The same rule is true for the opposite of multiplication - for.

You can easily verify this by running inverse multiplication operations. If in each of the examples above you multiply the quotient by the divisor, you get the dividend, and make sure it has the same sign, like (-3)x(-4)=(+12).

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

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Rule for dividing numbers with different signs

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

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

We write this division rule using letters. If the numbers a and b have different signs, then the formula is valid a:b=−|a|:|b| .

From the voiced 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 the division of numbers with different signs to the 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. I.e, 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 is used when dividing negative numbers.

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

Examples of dividing numbers with different signs

Let us consider solutions of several characteristic examples of dividing numbers with different signs to grasp the principle of applying the rules from the previous paragraph.

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

The rule for dividing numbers with different signs prescribes first to find the modules of the dividend and 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 is the whole solution: .

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

(−35):7=−5 .

Calculate the quotient 8:(−60) .

By the rule of 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 fraction bar), you can reduce the fraction by 4, we get .

We write down the whole solution briefly: .

.

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 a different notation (for example, in decimal).

The modulus of the dividend is, and the modulus of the divisor is 0,(23) . To divide the modulus of the dividend by the modulus of the divisor, let's move on to ordinary fractions.

Back forward

Attention! The slide preview is for informational purposes only and may not represent the full extent of the presentation. If you are interested in this work, please download the full version.

Lesson goals.

Subject:

• formulate a multiplication rule negative numbers and numbers with different signs,
• teach students to apply this rule.

Metasubject:

• to form the ability to work in accordance with the proposed algorithm, draw up a plan-scheme of their actions,
• develop self-control skills.

Personal:

• develop communication skills,
• form cognitive interest students.

Equipment: computer, screen, multimedia projector, PowerPoint presentation, handout: table for writing rules, tests.

(Textbook by N.Ya. Vilenkin “Mathematics. Grade 6”, M: “Mnemosyne”, 2013.)

During the classes

I. Organizational moment.

Reporting the topic of the lesson and recording the topic in notebooks by students.

II. Motivation.

Slide number 2. (Lesson goal. Lesson plan).

Today we will continue to study an important arithmetic property - multiplication.

You already know how to multiply natural numbers - verbally and in a column,

Learn how to multiply decimal and common fractions. Today you have to formulate a multiplication rule for negative numbers and numbers with different signs. And not only to formulate, but also to learn how to apply it.

III. Knowledge update.

1) Slide number 3.

Solve the equations: a) x: 1.8 = 0.15; b) y: = . (Student at the blackboard)

Conclusion: to solve such equations, you need to be able to multiply different numbers.

2) Checking home independent work. Repetition of the rules for multiplying decimals, common fractions and mixed numbers. (Slides #4 and #5).

IV. Rule formulation.

Consider task 1 (slide number 6).

Consider task 2 (slide number 7).

In the process of solving problems, we had to perform the multiplication of numbers with different signs and negative numbers. Let's take a closer look at this multiplication and its results.

Having multiplied numbers with different signs, we got a negative number.

Let's consider another example. Find the product (-2) * 3, replacing the multiplication with the sum of the same terms. Find the product 3 * (–2) in the same way. (Check - slide number 8).

Questions:

1) What is the sign of the result when multiplying numbers with different signs?

2) How is the result module obtained? We formulate the rule for multiplying numbers with different signs and write the rule in the left column of the table. (Slide number 9 and Appendix 1).

Multiplication rule for negative numbers and numbers with different signs.

Let's return to the second problem, in which we performed the multiplication of two negative numbers. It is rather difficult to explain this multiplication in another way.

Let's use the explanation given back in the 18th century by the great Russian scientist (born in Switzerland), mathematician and mechanic Leonard Euler. (Leonhard Euler left behind not only scientific works, but also wrote a number of textbooks on mathematics intended for pupils of the academic gymnasium).

So, Euler explained the result approximately as follows. (Slide number 10).

It is clear that –2 · 3 = – 6. Therefore, the product (–2) · (–3) cannot be equal to –6. However, it must be somehow related to the number 6. One possibility remains: (–2) · (–3) = 6. .

Questions:

1) What is the sign of the product?

2) How is the product module obtained?

We formulate the rule for multiplying negative numbers, fill in the right column of the table. (Slide number 11).

To make it easier to remember the rule of signs for multiplication, you can use its formulation in verse. (Slide number 12).

Plus by minus, multiplying,
We put a minus without yawning.
Multiply minus with minus
In response, we will put a plus!

V. Formation of skills.

Let's learn how to apply this rule for calculations. Today in the lesson we will perform calculations only with integers and with decimal fractions.

1) Drawing up a scheme of actions.

A scheme for applying the rule is drawn up. Recordings are made on the board. An example diagram is on slide 13.

2) Performing actions according to the scheme.

We solve from textbook No. 1121 (b, c, i, k, p, p). We carry out the solution in accordance with the drawn up scheme. Each example is explained by one of the students. At the same time, the solution is shown on slide No. 14.

3) Work in pairs.

Task on slide number 15.

Students work on options. First, the student of option 1 decides and explains the solution to option 2, the student of option 2 listens carefully, helps and corrects if necessary, and then the students switch roles.

Additional task for those couples who finish work earlier: No. 1125.

Upon completion of the work, verification is carried out according to the finished solution, placed on slide No. 15 (animation is used).

If many managed to solve No. 1125, then it is concluded that the sign of the number has changed when multiplied by (? 1).

4) Psychological relief.

5) Independent work.

Independent work - text on slide No. 17. After completing the work - self-checking on the finished solution (slide No. 17 - animation, hyperlink to slide No. 18).

VI. Checking the level of assimilation of the studied material. Reflection.

Students take a test. On the same sheet, they evaluate their work in the lesson by filling out the table.

Test “Multiplication rule”. Option 1.

1) –13 * 5

A. -75. B. - 65. V. 65. G. 650.

2) –5 * (–33)

A. 165. B. -165. W. 350 G. -265.

3) –18 * (–9)

A. -162. B. 180. V. 162. D. 172.

4) –7 * (–11) * (–1)

A. 77. B. 0. C.–77. G. 72.

Test “Multiplication rule”. Option 2.

A. 84. B. 74. C. -84. G. 90.

2) –15 * (–6)

A. 80. B. -90. V. 60. G. 90.

A. 115. B. -165. V. 165. G. 0.

4) –6 * (–12) * (–1)

A. 60. B. -72. V. 72. G. 54.

VII. Homework.

P. 35, rules, No. 1143 (a - h), No. 1145 (c).

Literature.

1) Vilenkin N.Ya., Zhokhov V.I., Chesnokov A.S., Shvartsburd S.I. “Mathematics 6. Textbook for educational institutions”, - M: “Mnemosyne”, 2013.

2) Chesnokov A.S., Neshkov K.I. “Didactic materials in mathematics for grade 6”, M: “Prosveshchenie”, 2013.

3) Nikolsky S.M. and others. "Arithmetic 6": a textbook for educational institutions, M: "Prosveshchenie", 2010.

4) Ershova A.P., Goloborodko V.V. “Independent and test papers Mathematics for Grade 6. M: "Ileksa", 2010.

5) “365 tasks for ingenuity”, compiled by G. Golubkova, M: “AST-PRESS”, 2006.

6) “Big Encyclopedia Cyril and Methodius 2010”, 3 CD.

Strengthen the ability to multiply integers, ordinary and decimal fractions;

Learn to multiply positive and negative numbers;

Develop the ability to work in groups

Develop curiosity, interest in mathematics; the ability to think and speak on a topic.

Equipment: models of thermometers and houses, cards for oral account and verification work, a poster with the rules of signs in multiplication.

During the classes

Motivation

Teacher . Today we begin to study new topic. We are going to build a new house. Tell me, what determines the strength of the house?

[From the foundation.]

Now let's check what our foundation is, that is, the strength of our knowledge. I didn't tell you the topic of the lesson. It is coded, that is, hidden in the task for oral counting. Be attentive and observant. Here are cards with examples. By solving them and matching the letter to the answer, you will find out the name of the topic of the lesson.

[MULTIPLICATION]

Teacher. So that word is multiplication. But we are already familiar with multiplication. Why do we need to study it? What numbers have you recently met?

[With positive and negative.]

Can we multiply them? Therefore, the topic of the lesson will be "Multiplication of positive and negative numbers."

You quickly and correctly solved the examples. A good foundation has been laid. ( Teacher on model house« lays» foundation.) I think that the house will be durable.

Exploring a new topic

Teacher . Now let's build walls. They connect the floor and the roof, that is, the old theme with the new one. Now you will work in groups. Each group will be given a problem to solve together and then explain the solution to the class.

1st group

The air temperature drops by 2° every hour. Now the thermometer shows zero degrees. What temperature will it show after 3 hours?

Group decision. Since the temperature is now 0 and for every hour the temperature drops by 2°, it is obvious that after 3 hours the temperature will be -6°. Let us denote the temperature decrease as –2°, and the time as +3 hours. Then we can assume that (–2) 3 = –6.

Teacher . And what happens if I rearrange the factors, that is, 3 (–2)?

Students. The answer is the same: -6, since the commutative property of multiplication is used.

2nd group

The air temperature drops by 2° every hour. Now the thermometer shows zero degrees. What air temperature did the thermometer show 3 hours ago?

Group decision. Since the temperature dropped by 2° every hour, and now it is 0, it is obvious that 3 hours ago it was +6°. Let us denote the decrease in temperature by -2°, and the elapsed time by -3 hours. Then we can assume that (–2) (–3) = 6.

Teacher . You don't know how to multiply positive and negative numbers yet. But they solved problems where it was necessary to multiply such numbers. Try yourself to derive the rules for multiplying positive and negative numbers, two negative numbers. ( The students are trying to figure out the rule.) Good. Now let's open the textbooks and read the rules for multiplying positive and negative numbers. Compare your rule with what is written in the textbook.

Teacher. As you saw when building the foundation, you have no problem multiplying natural and fractional numbers. Problems can arise when multiplying positive and negative numbers. Why?

Remember! When multiplying positive and negative numbers:

1) determine the sign;
2) find the product of modules.

Teacher . For multiplication signs, there are mnemonic rules that are very easy to remember. Briefly they are formulated as follows:

(In notebooks, students write down the rule of signs.)

Teacher . If we consider ourselves and our friends positive, and our enemies negative, then we can say this:

My friend's friend is my friend. My friend's enemy is my enemy. A friend of my enemy is my enemy. The enemy of my enemy is my friend.

Primary comprehension and application of the studied

Examples for oral solution on the board. Students say the rule:

–5 6;
–8 (–7);
9 (–3);
–45 0;
6 8.

Teacher . All clear? No questions? So the walls are built. ( The teacher puts up walls.) Now what are we building?

Consolidation.

(Four students are called to the board.)

Teacher. Is the roof ready?

(The teacher puts a roof on a model house.)

Verification work

Pupils complete the work in one version.

After completing the work, they exchange notebooks with their neighbor. The teacher reports the correct answers, and the students give marks to each other.

Summary of the lesson. Reflection

Teacher. What was our goal at the beginning of the lesson? Have you learned how to multiply positive and negative numbers? ( They repeat the rules.) As you saw in this lesson, each new topic is a house that needs to be built capitally, for years. Otherwise, all your buildings will collapse after a short time. Therefore, everything depends on you. I wish, guys, that luck always smiles at you, success in mastering knowledge.

In this lesson, we will review the rules for adding positive and negative numbers. We will also learn how to multiply numbers with different signs and learn the rules of signs for multiplication. Consider examples of multiplication of positive and negative numbers.

The property of multiplying by zero remains true in the case of negative numbers. Zero multiplied by any number is zero.

Bibliography

1. Vilenkin N.Ya., Zhokhov V.I., Chesnokov A.S., Shvartsburd S.I. Mathematics 6. - M.: Mnemosyne, 2012.
2. Merzlyak A.G., Polonsky V.V., Yakir M.S. Mathematics 6th grade. - Gymnasium. 2006.
3. Depman I.Ya., Vilenkin N.Ya. Behind the pages of a mathematics textbook. - M.: Enlightenment, 1989.
4. Rurukin A.N., Tchaikovsky I.V. Tasks for the course of mathematics grade 5-6. - M.: ZSh MEPhI, 2011.
5. Rurukin A.N., Sochilov S.V., Tchaikovsky K.G. Mathematics 5-6. A manual for students of the 6th grade of the MEPhI correspondence school. - M.: ZSh MEPhI, 2011.
6. Shevrin L.N., Gein A.G., Koryakov I.O., Volkov M.V. Mathematics: Interlocutor textbook for grades 5-6 high school. - M .: Education, Mathematics Teacher Library, 1989.

Homework

1. Internet portal Mnemonica.ru ().
2. Internet portal Youtube.com ().
3. Internet portal School-assistant.ru ().
4. Internet portal Bymath.net ().

Table 5

Table 6

With some stretch, the same explanation is suitable for the product 1-5, if we assume that the "sum" of a single

term is equal to this term. But the product 0 5 or (-3) 5 cannot be explained in this way: what does the sum of zero or minus three terms mean?

It is possible, however, to rearrange the factors

If we want the product not to change when the factors are rearranged - as it was for positive numbers - then we must thereby assume that

Now let's move on to the product (-3) (-5). What is it equal to: -15 or +15? Both options make sense. On the one hand, a minus in one factor already makes the product negative - all the more it should be negative if both factors are negative. On the other hand, in Table. 7 already has two minuses, but only one plus, and "fairly" (-3)-(-5) should be equal to +15. So what do you prefer?

Table 7

Of course, you will not be confused by such conversations: from a school mathematics course, you firmly learned that a minus by a minus gives a plus. But imagine that your younger brother or sister asks you: why? What is it - a teacher's whim, an indication of higher authorities, or a theorem that can be proven?

Usually, the rule for multiplying negative numbers is explained using examples like the one presented in Table. eight.

Table 8

It can be explained in another way. Let's write numbers in a row

Now let's write the same numbers multiplied by 3:

It is easy to see that each number is 3 more than the previous one. Now let's write the same numbers in reverse order(starting, for example, with 5 and 15):

At the same time, the number -15 turned out to be under the number -5, so 3 (-5) \u003d -15: plus by minus gives minus.

Now let's repeat the same procedure, multiplying the numbers 1,2,3,4,5... by -3 (we already know that a plus times a minus equals a minus):

Each next number of the bottom row is less than the previous one by 3. Let's write the numbers in reverse order

and continue:

The number -5 turned out to be 15, so (-3) (-5) = 15.

Perhaps these explanations would satisfy your younger brother or sister. But you have the right to ask how things really are and is it possible to prove that (-3) (-5) = 15?

The answer here is that it can be proven that (-3) (-5) must equal 15, if only we want the usual properties of addition, subtraction, and multiplication to remain true for all numbers, including negative ones. The outline of this proof is as follows.

Let us first prove that 3 (-5) = -15. What is -15? This is the opposite of 15, i.e. the number that adds up to 15 to 0. So we need to prove that