Division of numbers with different signs. Division of numbers with different signs, rule, examples

Abstract lesson

Pedagogy and didactics

Organizational moment Teacher: Hello Sit down. Check homework The teacher includes the projector with a slide of homework at which the criteria for evaluating the work of the work teacher are also reflected: Change notebooks. Pupils are checked by Teacher's responses: Evaluation Criterion: Everything is decided correctly put five one minus four twitties minus three in all other cases two. Oral work Table with the rule of signs on the magnetic board Teacher: Repeat the rule of signs to multiply attention to the magnetic board.

Abstract lesson mathematics

Topic: "The division of numbers with different signs».

Class 6.

Tutorial: Mauraravin and Muravin

Date: 15. 02. 2010

Lesson number: 3

Kurgan 2010.

Objectives lesson:

1. Educational: teach to divide numbers with different signs.

2. Developing: develop thinking and personal work skills.

3. Educational: to form a culture of mathematical letter.

Equipment:

1. Presentation

2. Wall table "Rules of signs"

3. Cards for oral work

4. Cards for independent work

Lesson plan:

I. . Organizational moment (1min)

II. . Checking homework (2 min)

III . Oral work (3 min)

IV . Independent work (5 min)

V. . Study of a new material (15 min)

VI . Consolidation of studied (12 min)

VII . Dacha homework (1 min)

VIII. . Total lesson (1 min)

During the classes:

I. Organizing time

Teacher: Hello, sit down. Open the notebook, write down the number: February 15, theme of the lesson: "The division of numbers with different signs", cool job.

Today at the lesson we continue to get acquainted with actions above the numbers with different signs. You will find out that you can share not only positive, but also negative numbers.

II. Checking homework

(A teacher includes a projector with a slide of homework, which also reflects the criteria for evaluating work)

Teacher: Change notebooks. Attention to the slide. Rooms were asked: 515 (A, B, B, D), 517 (B, D). Check the task execution correctness, deposit the answers. I put the red pencil about the answer, if the task is solved correctly and "-", if an error is made.

(students are responsible for answers)

Teacher: Evaluation Criterion: Everything is decided right - put five, one minus - four, two or three minuses - three, in all other cases - two. Next to the name of the surname verified. Retriend the notebook to the neighbor.

III. Oral work

(Table with the rule of signs on the magnetic board)

Teacher: We repeat the rule of signs for multiplication, attention to the magnetic board.

Same signs	On +.
		On the -
	Different signs	On the -
		On +.

Teacher: We consider orally.

(Teacher raises cards with tasks)

Masha: 75 × (-1) \u003d -75

Teacher: Explain the choice of sign.

Masha: Rule of signs for multiplication: "Plus for minus - it turns out minus."

Valera: -36 × 2 \u003d -72

Teacher: How much did Sasha work out?

Sasha: -72

Teacher: Why did the minus sign?

Sasha: rule of signs for multiplication: "Minus on plus - it turns out minus."

Nina: 0.9 × (-3) \u003d -2.7

Anton: -2.1 × (-5) \u003d 10.5

× 5.

Gene: × 5 \u003d 1

× (-3)

Lida: × (-3) \u003d 1

Ira: the denominator is equal to the nul. It is impossible to share on zero.

Teacher: Well done! Well worked orally, now we work independently on cards.

IV. Independent work

(Before the lesson, the teacher distributes cards with tasks for independent work and leaves for answers)

Teacher: Do you have a sheet on your table. In the left corner, write the surname, in the middle of the number of the option, decide in any order, the tasks to rewrite, everyone will receive an assessment. Do not forget the rule of signs.

Option 1 1) - 5 × 6; 2) - 1 × (-7); 3) - 11 × 0; 4) 0.2 × (-8); 5) 12 × (-0.2); 6) - 2.5 × 0.4; 7) 1.2 × (-14); 8) -9.8 × (-10) 9) -1 × (-12) × (-0,5)	Option 2. 1) 4 × (-7); 2) - 1 × 6; 3) 0 × (-13); 4) 0.3 × (-6); 5) 11 × (-0.1); 6) - 2.4 × 0.2; 7) 1.2 × (-14); 8) -9.8 × (-10) 9) -1 × (-14) × (-0.2)
Solution 1 option 1) - 5 × 6 \u003d -30 2) - 1 × (-7) \u003d 7 3) - 11 × 0 \u003d 0 4) 0.2 × (-8) \u003d - 1.6 5) 12 × (-0.2) \u003d - 2.4 6) - 2.5 × 0.4 \u003d -1 7) 1.2 × (-14) \u003d - 16.8 8) -9.8 × (-10) \u003d 98 9) -1 × (-12) × (-0,5) \u003d 12 × (-0,5) \u003d - 6	Solution 2 option 1) 4 × (-7) \u003d - 28 2) - 1 × 6 \u003d -6 3) 0 × (-13) \u003d 0 4) 0.3 × (-6) \u003d 1.8 5) 11 × (-0.1) \u003d - 1.1 6) - 2.4 × 0.2 \u003d -0.48 7) 1.2 × (-14) \u003d - 16.8 8) -9.8 × (-10) \u003d 98 9) -1 × (-14) × (-0.2) \u003d 14 × (-0.2) \u003d - 2.8
Answers 1 option	Answers 2 options
1) -30 2) 7 3) 0 4) -1,6 5) -2,4 6) -1 7) -16,8 8) 98 9) -6	1) -28 2) - 6 3) 0 4) -1,8 5) -1,1 6) - 0,48 7) -16,8 8) 98 9) -2,8

Teacher: Finish work - cards and leaflets. At the expense, three works are not accepted. Once or two or three - all works are handed over.

V. Studying a new material

Teacher: Go to the study of a new material. You already know how to multiply positive and negative numbers, at today's lesson you will learn how to split numbers with different signs.

a: B.

I write on the board, you are in the notebook.

Now the same expression in the form of a fraction

Teacher: division We replaced multiplication. Write down and highlight the color

Teacher: Write down the two examples of replacing division by multiplication.

(pause)

Teacher: We read your examples, please, Anton.

Anton: \u003d.

Teacher: Right - write down the example of Anton, read the second example.

Anton: - \u003d;

Teacher: True - write, reads your examples.

Light: -11: 5 \u003d

Teacher: True, the second example.

Light: \u003d.

Teacher: Well done.

Teacher: Write down in Tetradi 5: (-7). How to write this expression using multiplication?

Anya: 5: (-7) \u003d

Teacher: True. Record

5: (-7) = = - = -

Note that when dividing a plus for minus gives minus.

On the -

Record -3: 8 \u003d \u003d -.

When dividing minus on plus it turns out minus.

On +.

Next example:

4: (-5) = = =

When dividing minus for minus it turns out a plus.

On the -

(the teacher highlights the table to the table the rule of signs for division)

Teacher: Carefully look at the table and find the difference from the table of signs of signs for multiplication.

Katya: There are no differences, the tables are the same.

Teacher: True. The rule of signs for division is exactly the same as for multiplication.

Same signs	On +.
		On the -
	Different signs	On the -
		On +.

Teacher: Relieve yourself in the notebook a table of signs for division, highlight the color signs, remember.

Teacher: numbers and reverse. Find their work.

- (-8) = = 1

These numbers in the work give a unit.

Consider the number A and

Select color:

The numbers that give a unit are called mutually.

Teacher: Let us give an example of conjunctional numbers. and 2 - Missile? Check:

We write another example

Teacher: Will Missing Numbers and 3?

Katya: and 3 are not reciprocal, since their product is -1.

Teacher: Come up and write 3 examples of mutually perfection numbers and write down in a notebook.

(pause)

Teacher: We read your examples on the chain, starting with the last part of the third row. Vasya, please.

Vasya: and 4.

Teacher: Why?

Vasya: the work is equal to one.

Anya: and -7.

Pasha: and -3.

Anton: and 3.

Teacher: Well done. Enough. Mutroofing numbers - the number giving a unit.

VI . Fastening learned

Teacher: orally in a chain decide and comment on №520 - you need to replace the division by multiplication and explain the sign, start from the first part of the first row, please, Vova, under the letter "A".

Vova: 6: 3 \u003d 6 \u003d 2 plus on plus gives plus

Katya: 63: (-3) \u003d 63 -63 \u003d - 21 plus per minus gives minus.

Teacher: the following examples under the letters "g" and "d" with back side The boards solve Peter and Masha, the rest in the notebooks.

(pause)

Teacher: Attention to the board. Check.

Petya: -23: (-) \u003d -23 \u003d 232 \u003d 46

Teacher: Explain the choice of sign.

Petya: According to the rule: minus for minus gives plus.

Masha: -: \u003d - \u003d - \u003d -1.5

Teacher: Why a minus sign?

Masha: Minus on Plus gives minus.

Teacher: Decide # 521. Deciding with an explanation at the board will go, Anton. Please Anton under the letter "A". All others in the notebook.

Anton: -: \u003d - \u003d - \u003d - \u003d -2

Teacher: I got another sign, and you?

Katya: The sign is correct, as according to the rule: minus on plus gives minus.

Teacher: Well done, sit down. The following example decides on the back side of the Lena Board. We work yourself.

(pause)

Teacher: Lena, explain how to solve.

Lena: -: \u003d - \u003d \u003d \u003d

Teacher: Thank you, Lena, sit down. Under the letters "B" and "G" decide on their own, someone commetes one at the end of the decision.

(pause)

Teacher: Kostya, please, you word.

Kostya: -: \u003d -: 0. It is impossible to share for zero.

1: (-) = -1)= 1 = 3

Teacher: Kostya, why precisely, plus?

Kostya: Minus for minus gives a plus.

VII . Dacha homework

Teacher: Home Task on Side Board # 521 (D, E), 522 (D, E). Do not forget the rule of signs. Learn definitions.

VIII. Total lesson

Teacher: Today we learned to divide the numbers with different signs, we repeated the rule of signs for multiplication, checked his justice to divide and met with mutually digested numbers. Katya, what numbers are called Missile Order?

Katya: Mutually called a couple of numbers, giving a unit.

Teacher: Thank you, Katya. For work in the lesson, evaluations are obtained:

Anton - Five, Katya - Five, Lights - Five.

In addition to these estimates, everyone will receive estimates for independent work, results you will learn in the next lesson.

Attachment 1.

Slide S. homework and evaluation criterion

№515

a) 2 ⋅ (0.2 + 1) \u003d 2 ⋅ 1,2 \u003d 2.4

b) 0.8 ⋅ (27 - 29) \u003d 0.8 ⋅ (-2) \u003d -1.6

c) (99.9 - 100.9) ⋅ (-1.7 - 0.3) \u003d -1 ⋅ (-2) \u003d 2

d) (2009-2000) ⋅ (-0.8) ⋅ (2,4 - 5,8) \u003d 9 ⋅ (-0.8) ⋅ (-3.4) \u003d 24,48

№517

Evaluation Criteria:

everything is decided right - put five,

one minus - four,

two or three minuses - three

in all other cases - two.

Appendix 2.

Homework.

№521

e) -: \u003d - \u003d - \u003d - \u003d -15

e) -: (- \u003d - \u003d \u003d \u003d 84

№522

e): (\u003d: (- \u003d - \u003d - \u003d - \u003d 20

e) -: (- \u003d -: (- \u003d -: 0 - You can not divide on zero!

Appendix 3.

The design of the board.

Same signs	On +.
		On the -
	Different signs	On the -
		On +.

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Now let's figure it out with multiplication and division.

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

Let's consider such a case. Three people climbed into debt, and every 4 dollar debt. What is the total debt? In order to find it, it is necessary to fold all three debt: 4 dollars + 4 dollars + 4 dollars \u003d 12 dollars. We decided that the addition of three numbers 4 is indicated as 3 × 4. Since in this case we are talking about the debt, before 4 there is a sign "-". We know that the total debt is equal to 12 dollars, so now our task has the form 3x (-4) \u003d - 12.

We will get the same result if, by the condition of the task, each of the four people has a debt of 3 dollars. In other words, (+4) x (-3) \u003d - 12. And since the order of the factors does not matter, we obtain (-4) x (+3) \u003d - 12 and (+4) x (-3) \u003d - 12.

Let's summarize the results. With multiplying one positive and one negative number, the result will always be a negative number. The numerical amount of the answer will be the same as in the case of positive numbers. Work (+4) x (+3) \u003d + 12. The presence of a sign "-" affects only a sign, but does not affect the numerical value.

And how to multiply two negative numbers?

Unfortunately, this topic is very difficult to come up with a suitable example from life. It is easy to imagine a debt in the amount of 3 or 4 dollars, but it is absolutely impossible to imagine -4 or -3 people who climbed into debt.

Perhaps we will go different ways. In multiplication, when a sign of one of the multipliers changes the sign of the work. If we change signs from both multipliers, we must change twice sign of workFirst, with a positive on a negative, and then on the contrary, with a negative on a positive, that is, the work will have an initial sign.

Consequently, it is quite logical, although it is slightly strange that (-3) x (-4) \u003d + 12.

Sign position When multiply changed in this way:

• positive number x positive number \u003d positive number;
• a negative number x positive number \u003d negative number;
• positive number x Negative number \u003d negative number;
• negative number x negative number \u003d positive number.

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

The same rule is valid for the opposite multiplication - for.

You can easily make sure by spending inverse operations multiplication. If in each of the examples given above, you multiply the private per divider, then get divisible, and make sure it has the same sign, for example (-3) x (-4) \u003d (+ 12).

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Grade 6 division

Theme lesson:Multiplying positive and negative numbers. 6th grade
Objectives lesson : To organize joint activities in the process of which students offer their versions, they learn to competently formulate, listen.

Tasks:

• 
  Organize joint activities aimed at the subject: to bring the rules for multiplying positive and negative numbers;
• 
  Create conditions for the development of skills to compare, detect patterns, summarize, teach thinking, express your opinion;
• 
  Training student search different ways and methods of solving practical problems;
• 
  Organize the reflection of joint activities.

During the classes:

I. Immersion in a problem situation.

Greeting students.

"He lived in the light of the rich, very rich rich, the richest on earth, but it all seemed to him that he was still not rich enough.
And once he came to this very rich rich, the poorest poorness in the world and said:
- Oh Lord! The radiance of your treasures blind eyes. Still, I have a way to multiply your wealth. And at the same time and its own.
The rich is straight out of greed:
- What are you standing? Multiply rather!
- And you won't be on me in the offense? - Poorly asked the poor man.
- Yes, you! After all, you want to multiply my wealth!
- Of course, multiply, - confirmed the poor man.
- So multiply, and the case with the end! - shouted the rich, losing patience.
"To be in yours," the one replied. - One two Three! Ready!
Bogach rushed to his chests Yes, how to shout:
- What did you run, unfit?! You ruined me! Where is my gold? Where are diamonds? Where is pearls?
- Were you, now they have them, "said the poor man." After all, you asked me to multiply me! I multiplied. "

II. Creating a problem situation.

• 
  What do you think, why did it happen?
• 
  What action with numbers need to know what to answer this question? (multiplication)
• 
  Do you know how multiplication of numbers is performed? (Natural and fractional positive, yes)
• 
  Then what is the task of our today's lesson, what would you like to know? (How to multiply positive and negative numbers)
• 
  And what numbers can still multiply? (negative)
• 
  So, the topic of our lesson: "multiplying positive and negative numbers."

Please remember what methods we used when you derive the rules for addition and subtract positive and negative numbers and offer your versions as we get the multiplication rules of numbers.

III. Working with versions of children.

Versions are fixed on the board and in notebooks.

1. 
   Use the thermometer and consider multiplication by the example of temperature change.
2. 
   Multiplication is replaced by adding.

I offer my version:

3. Covering by to signify the word "friend" - a positive number, and the word "enemy" is negative, you can get an interesting rule of multiplication of numbers.
IV. Work on the substantiation of versions in groups.

Now work in groups, consider the version you take on the examples and be sure to conclude, i.e. Try to formulate the multiplication rule of numbers.

V. Submission of versions check results.
1. Task 1.. The air temperature drops every hour to 2 degrees. Now the thermometer shows zero degrees. What temperature he will show after 3 hours.

(- 2) · 3 \u003d - 6

Task 2.The air temperature drops every hour to 2 degrees. Now the thermometer shows zero degrees. What temperature he showed 3 hours ago.

(- 2) · (-3) \u003d 6

2. Example 1.(- 2) · 3 \u003d (- 2) + (- 2) + (- 2) \u003d - (2 + 2 + 2) \u003d - 6

Example 2.(- 2) · (-3) – addition is not replaced , but if (- 2) · 3 \u003d - 6, then

(- 2) · (-3) - 6

since 3 and - 3 opposite numbers, the result will be the opposite,

so (- 2) · (-3) \u003d 6
3. My friend's friend is my friend

(+ X) · (+ x) \u003d (+ x)

My enemy friend is my enemy

(+ X) · (-X) \u003d (-X)

The enemy of my friend is my enemy

(- x) · (+ x) \u003d (- x)

The enemy of my enemy is my friend

(- x) · (- x) \u003d (+ x)

Conclusions: 1) The product of two numbers of one sign is positive, and the product of two numbers with different signs is negative;
2) To find the module of the work, you need to multiply the womb modules.

Vi. Comparison of the resulting result with scientific.

"Thus, we received the rules for multiplying positive and negative numbers.

- Open the tutorial, read the rules, compare them with those that we brought themselves, draw out how to multiply two negative numbers, how to multiply two numbers with different signs:

1. To establish what signs have multipliers.

2. Set the result sign.

3. Find the work module.

- Let's go back to the fairy tale that you heard at the beginning of the lesson. Can you answer the question now why rich was lost his wealth, for what number the poor man multiplied the richness of rich?
- And now task for all groups: Determine the mark of the work and calculate.
a) (-7) · (-5) · 2 \u003d 70

(-4) · (-10) · 8 \u003d 320

b) (-2) · (-3) · (-4) \u003d - 24

(-1,2) · (-2) · (-12) \u003d - 28.8

c) (-1) · (-2) · (-5) · (-15) · 2 \u003d 300
- What conclusion can be made regarding the sign of the work, where is the one (odd) number of negative multipliers?

Output: 1. If the number of negative multipliers is odd, then the product is a negative number.
2. If the number of negative multipliers is something, then the product is a number positive.
VII.reflexia

- And now let's try to understand that everyone gave today's lesson. Whether it was interesting to you today. Let's listen to experts:

1. How did the group work famous?

2. Did you put forward the versions in the group?

3. Are all members of the group participated in reflections and solving problems?

4. Which of the group members was more active?

5. Who did not participate in the work of the group?

6. Which and what marks can be evaluated in the group?

Homework: p.35 Rules

№ 1143 №1148.

Cards for independent work

Option 1 1. Calculate: a) (-5) ∙ (-1) d) -0.6 ∙ (-2) g) -2,5: (-0.05) h) -81: (-0.9) 2. Perform actions: 8 ∙ (-3 + 12) : 36 + 2 5 ∙ 3,7 - 4 ∙ 3,7	Multiplication and division of positive and negative numbers. Option 2. 1. Calculate: d) -11 ∙ (-2) d) 0.8 ∙ (-4) g) -3,6: (-0,6) 2. Perform actions: 9 ∙ (-7 + 12) : 15 + 4 3. Calculate the most rational way: - 2 ∙ 3,5 - 7 ∙ 3,5
Multiplication and division of positive and negative numbers. Option 3. 1. Calculate: a) (-9) ∙ (-1) d) -0.8 ∙ (-4) g) -2.8: 0.07 h) -36: (-0.9) 2. Perform actions: 6 ∙ (-5 + 21) : 32 + 3 3. Calculate the most rational way 7,8 ∙ 2 - 7,8 ∙ 8	Multiplication and division of positive and negative numbers. Option 4. 1. Calculate: d) 0.6 ∙ (-4) g) -3.2: (-0.08) 2. Perform actions: 8 ∙ (-7 + 23) : 64 + 3 3. Calculate the most rational way 5,9 ∙ 3 - 5,9 ∙ 7

Mathematics lesson in grade 6.

Division of numbers with different signs.

Purpose: Teach students to share numbers with different signs.

Educational: Teach children to share numbers with different signs;

Developing: Develop cognitive interest, by using historical material;

Raising: Learn the correct record of the division of numbers with different signs.

During the classes:

1) Checking homework.

2) actualization of knowledge.

3) study of a new material.

4) Fixing the material passed.

5) Recording homework.

6) Summing up the lesson.

I. . Check your homework.

Teacher: Do you have any questions about your homework?

If there are no questions, then one or two people go to the board, three more people receive a card.

Card.

II. . Actualization of knowledge.

Find the value of the expression.

1. – 0,4 * (- 2,5)

Solve equation:
1) x * 47 \u003d 141

III . Studying a new material

Let the following equation.

What is called root?

How to find the root of this equation?

Can we divide the number of different signs?

To multiply 25, what happened - 125 (-5).

Let's check

5 * 25 \u003d -125, i.e. -125: 25 \u003d -5

From here, please make a conclusion how to divide the number of different signs?

The rule form pupils.

I decide another equation.

Can we share negative numbers?

What you need to multiply -14, what happened -42 (3)

Those. -42: (-14) \u003d 3

Let us bring the rule to divide numbers of one sign.

The rule form pupils.

Look at what rule you are offered in the textbook. (p. 36)

IV . Fastening the material passed.

It is known that natural items have arisen when with a matter of items. Human need andvämer IT values about liaison,
that R. measurement resultsnot always expressed by integer
numbers with the fragmentation of the set of natural numbers.
Zero and fractional numbers were introduced. The process of historic who
develop a japanese aids did not end. However, N.
always the impetus to the worst of the concept of the number were tough practicalthe needs of people. It happened so
that the tasks of mathematics itself demanded expansion of the concept
numbers.

This was the case with the connation of negative
numbers.

Let's remember with you when we needed negative numbers? (when subtracting from a smaller larger.)

For the production of computing mathematics of that time used
counting board on which h isla depictedvia
accounting sticks. Since the signs + and-WTO time is not yet
it was, red chopsticks and soldered positive
the numbers, the negative and the same-covers of black. Negative numbers have known for a long time with the words that mean "debt", "shortage".

At the slide, you are now seeing the ancient counting boards of Rial, Greeks and Chinese.

Even VV II century. Winera positive interpreted as property, aestricative - as
, debt. Odra China, only the rules of addition were known
iW attachment positiveizothricative numbers; regulations
multiplication of camees were not applied.

On the slide 8.

In ancient India, Bhaskar Mathematics (XII century) expressed the rules
multiplying the selection of the followingimage: "Work d vukh property ortwo debts have property; The product of property for debt is a loss. The same rule takes place and
when dividing. "

For a long time Negative numbers caused disapproval. Did not approve of them for a long time of the Evropia Mathematics, because
that the interpretation of "Property -Dong" caused bewilderment and
doubt. In fact, you can "fold" or "deduct"
idolgi's property, but what real meaning may have multiplication or " decision »Propertyfor debt?

That is why the launched work won their place ematika Negativenumbers.

And only in the 17th century in Europe, negative numbers are firmly entered into mathematics.

Let's go back to the menu (slide 2). And perform gymnastics for the eyes. Each item is made in the form of any figure, now in turn only through the eyes, circle each one of them first clockwise, then counterclockwise.

Each of you have a table, fill it out.

b.

0 , 48

b.

0 , 48

In 1881, he was found buried sample near Bakhshali (CE Vero-West India) Manuscript of an unknown author, which -
as believed, refers to the KV I -V III centuries. Wet P. amyatnik, writtenon the birch crust of the dwelling in crazy called "Bakhvashaly manuscript, "contains t akaya Task: (Slide 11)

"Of the four donors, the second gave half
more than the first, the third one is more than the second, the fourth four more than the third, ABSE was given together 132. How much did the first one? "

Solution: (Slide 12)

I doner - x

II sacrifice - 2x

III sacrifice- 3 * 2x 132

IV donor- 4 * 3 * 2x

X + 2x + 3 * 2x + 4 * 3 * 2x \u003d 132

X + 2x + 6x + 24x \u003d 132

In the same manuscript, a solution was proposed by the method of false, when it is assumed that the first donated - 1, the second-2, third-6 and fourth -24.

Together it turned out 33, which is 4 times less than 132. Next, the first sacrificed -4.

IV. Recording homework.

P. 36, No. 1172 (A-E), No. 1173 (A - B), No. 1175.

In this article, we will look at the division of positive numbers to negative and vice versa. Dadim detailed analysis The rules for dividing numbers with different signs, and also give examples.

The division rule of numbers with different signs

The rule for integers with different signs obtained in the article on dividing integers is also valid for rational and valid numbers. We give a more general formulation of this rule.

The division rule of numbers with different signs

When dividing a positive number on a negative and, on the contrary, you need to divide the divide module to the divider module, and write the result with a minus sign.

In an alphabent form it looks like this:

a ÷ - B \u003d - A ÷ B

A ÷ B \u003d - A ÷ b.

The result of dividing numbers with different signs is always a negative number. The rule considered, in fact, reduces the division of numbers with different signs to the division of positive numbers, since the divide and divider modules are positive.

Another equivalent mathematical formulation this rule It has the form:

a ÷ B \u003d A · b - 1

To divide the numbers A and B, having different signs, you need to multiply the number to the number reverse number b, that is, b - 1. This wording is applicable on a plurality of rational and valid numbers, it allows you to go from division to multiplication.

Consider now how to apply the theory described above in practice.

How to divide numbers with different signs? Examples

Below we consider several characteristic examples.

Example 1. How to divide numbers with different signs?

We divide - 35 to 7.

First, write the divide and divider modules:

35 = 35 , 7 = 7 .

Now we split the modules:

35 7 = 35 7 = 5 .

I add before the result of the minus sign and get the answer:

Now we use the other formulation of the rule and calculate the number, inverse 7.

Now spend multiplication:

35 · 1 7 \u003d - - 35 · 1 7 \u003d - 35 7 \u003d - 5.

Example 2. How to divide numbers with different signs?

If we divide fractional numbers with rational signs, a divide and divider must be represented as ordinary fractions.

Example 3. How to share numbers with different signs?

Slimming mixed number - 3 3 22 on decimal fraction 0 , (23) .

The dividery and divider modules are respectively 3 3 22 and 0, (23). Transferring 3 3 22 to an ordinary fraction, we get:

3 3 22 \u003d 3 · 22 + 3 22 \u003d 69 22.

The divider will also be presenting in the form of an ordinary fraction:

0 , (23) = 0 , 23 + 0 , 0023 + 0 , 000023 = 0 , 23 1 - 0 , 01 = 0 , 23 0 , 99 = 23 99 .

Now Delim. ordinary fractions, Perform reductions and get the result:

69 22 ÷ 23 99 \u003d - 69 22 · 99 23 \u003d - 3 2 · 9 1 \u003d - 27 2 \u003d - 13 1 2.

In conclusion, consider the case when delimi and divider are irrational numbers and recorded in the form of roots, logarithms, degrees, etc.

In such a situation, private is written in the form numerical expressionwhich is simplified if possible. If necessary, it is calculated its approximate value with the necessary accuracy.

Example 4. How to divide numbers with different signs?

We divide the numbers 5 7 and - 2 3.

According to the rules of dividing numbers with different signs, we will install the equality:

5 7 ÷ - 2 3 \u003d - 5 7 ÷ - 2 3 \u003d - 5 7 ÷ 2 3 \u003d - 5 7 · 2 3.

Get rid of irrationality in the denominator and get the final answer:

5 7 · 2 3 \u003d - 5 · 4 3 14.

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