Topic of the open lesson: "Multiplication of negative and positive numbers"
Date of: 03/17/2017
Teacher: Kuts V.V.
Class: 6 g
The purpose and objectives of the lesson:
introduce rules for multiplying two negative numbers and numbers with different signs;
to promote the development of mathematical speech, working memory, voluntary attention, visual-effective thinking;
formation of internal processes of intellectual, personal, emotional development.
to cultivate a culture of behavior in frontal work, individual and group work.
Lesson type: lesson of primary presentation of new knowledge
Forms of study: 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 of number, 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, fix the rules for multiplying decimal and ordinary fractions.
Regulatory - be able to determine and formulate the goal in the lesson with the help of a teacher; pronounce the sequence of actions in the lesson; work according to a collective plan; evaluate the correctness of the action. Plan your action in accordance with the task; make the necessary adjustments to the action after its completion based on its assessment and taking into account the mistakes made; express your guess.Communicative - be able to formulate their thoughts orally; listen and understand the speech of others; jointly agree on the rules of behavior and communication at school and follow them.
Cognitive - to be able to navigate in their system of knowledge, to distinguish new knowledge from already known with the help of a teacher; acquire 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 for learning new things;
Formation of communicative competence in the process of communication and cooperation with peers in educational activities;
To be able to carry out self-assessment based on the criterion of success of educational activities; focus on learning success.
During the classes
Structural elements of the lesson
Didactic tasks
Projected teacher activity
Projected student activity
Result
1. Organizational moment
Motivation for successful activity
Check readiness for the lesson.
- Good afternoon guys! Have a seat! Check if you have everything ready for the lesson: notebook and textbook, diary and writing materials.
I am glad to see you at the lesson today in a good mood.
Look into each other's eyes, smile, wish your comrade a good working mood with your eyes.
I also wish you good work today.
Guys, the motto of 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 gusto.”
Guys, who will tell me what it means to absorb knowledge with an appetite?
So today we will absorb knowledge with great pleasure at the lesson, because they will be useful to us in the future.
Therefore, we rather open notebooks and write down the number, cool work.
Emotional mood
- With interest, with pleasure.
Ready to start the lesson
Positive motivation to learn a new topic
2. Activation of cognitive activity
Prepare them to learn new knowledge and ways of doing things.
Organize a face-to-face survey on the material covered.
Guys, who will tell me what is the most important skill in mathematics? ( Check). Right.
So I'll test you now, how well you can count.
We will now do a math exercise.
We work as usual, we count orally, and write down the answer in writing. I give you 1 min.
5,2-6,7=-1,52,9+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 stomp your feet.
Well done boys.
Tell me, what actions did we perform with numbers?
What rule did we use when counting?
Formulate these rules.
Answer questions by solving small examples.
Addition and subtraction.
Adding numbers with different signs, adding numbers with negative signs, and subtracting positive and negative numbers.
The readiness of students to formulate a problematic issue, to find ways to solve the problem.
3. Motivation for setting the topic and purpose of the lesson
Encourage students to set the topic and purpose of the lesson.
Organize work in pairs.
Well, it's time to move on to the study of new material, but first, let's repeat the material of the previous lessons. A mathematical crossword puzzle will help us with this.
But this crossword puzzle is not ordinary, it contains a keyword that will tell us the topic of today's lesson.
The crossword puzzle lies 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 start solving the crossword puzzle, I give you 1.5 minutes. Whoever does everything, put your pens so I can see.
(Attachment 1)
1. What numbers are used in counting?
2. The distance from the origin to any point is called?
3. Are the numbers that are represented by a fraction called?
4. Are two numbers that differ from each other only in signs called?
5. What numbers lie to the right of zero on the coordinate line?
6. Natural numbers, their 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 whole 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 back to our crossword puzzle. At the very beginning, I said that it contained a word that would tell us the topic of the lesson.
So what is the topic of our lesson?
And what are we going to multiply today?
Let's think, for this we recall the types of numbers that we already know.
Let's think about what numbers we already know how to multiply?
What numbers will we learn to multiply today?
Write in your notebook the topic of the lesson: "Multiplying positive and negative numbers."
So, guys, figured out what we will talk about today in the lesson.
Tell me, please, 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.
Work in pairs, set the topic and purpose of the lesson.
1.Natural
2.Module
3. Rational
4.Opposite
5.Positive
6. Whole
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.
Secondly, when we get the rule, then what should we do? (learn to apply it when solving examples).
4. Learning new knowledge and ways of acting
Acquire new knowledge on the topic.
-Organize work in groups (learning new material)
- Now, in order to achieve our goal, we will proceed to the first task, we will derive a 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 explore 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.
We repeated in our heads how we should work in a group. If someone forgot, then the rules are in front of you on the screen.
The purpose of your research work: Exploring the tasks, gradually deduce the rule "Multiplication of negative and positive numbers" in task No. 2, in task No. 1 you have 4 tasks in total. And in order to solve these problems, our thermometer will help you, each group has one.
All entries are made on a piece of paper.
Once the group has a solution for the first problem, you show it on the board.
You are given 5-7 minutes to work.
(Annex 2 )
Work in groups (fill in the table, conduct research)
Rules for working in groups.Working in groups is very easy
Know five rules to follow:
first: do not interrupt,
when he tells
friend, there should be silence around;
second: do not shout loudly,
and give arguments;
and the third rule is simply:
decide what is important to you;
fourthly: it is not enough to know orally
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.Fizminutka
To establish the correctness of assimilation of new material at this stage, to identify misconceptions and their correction
Okay, I put all your answers in a table, now let's look at each line in our table (see presentation)
What conclusions can we draw from the study of the table.
1 line. What numbers are we multiplying? What number is the answer?
2 line. 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 to 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)
Multiplying positive numbers always results in a positive number.
Multiplying a negative number by a positive number always results in a negative number.
Multiplying negative numbers always results in a positive number.
Multiplying a positive number by a negative number results in a negative number.
To multiply two numbers with different signs,multiply 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 perform physical exercises, reinforcing the rules.
Prevent fatigue
7.Primary fixing of new material
To master the ability to apply the acquired knowledge in practice.
Organize frontal and independent work on the material covered.
We will fix the rules, and we will tell each other in pairs these same rules. I give you a minute for this.
Tell me, can we now move on to solving examples? Yes we can.
We open page 192 No. 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 have 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 is equal to the number on the roof of the house.
Solve examples using the knowledge gained
Raise your hands who did not have mistakes, well done ....
Active actions of students to apply knowledge in life.
9. Reflection (outcome of the lesson, assessment of the results of students' activities)
Provide students with reflection, i.e. their evaluation of their activities
Organize a lesson summary
Our lesson has come to an end, let's summarize.
Let's revisit the topic of our lesson, shall we? What was our goal? - 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. (Give me half a minute)
Well, guys, are you ready to show how you worked in class today? So, we raise and, I also raise a smiley for you.
I am very pleased with you today at the lesson! I see that everyone understood the material. Guys, you are great!
Lesson over, thanks for reading!
Answer questions and evaluate your work
Yes, we have.
The openness of students to the transfer and understanding of their actions, to identify positive and negative aspects of the lesson
10 .Homework Information
Provide an understanding of the purpose, content and methods of doing homework
Provides understanding of the purpose of homework.
Homework:
1.
Learn the rules of multiplication
2. No. 1121 (3rd column).
3.Creative task: compose a test of 5 multiple-choice questions.
Write down homework, trying to comprehend and understand.
Implementation 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
In this article, we will give a definition of dividing a negative number by a negative one, formulate and justify the rule, give examples of dividing negative numbers and analyze the course of their solution.
Division of negative numbers. rule
Recall what the essence of the division operation is. This action is a finding of an unknown multiplier by a known product and a known other multiplier. A number c is called a quotient from the division of numbers a and b if the product c · b = a is true. In this case, a ÷ b = c .
Rule for dividing negative numbers
The quotient of dividing one negative number by another negative number is equal to the quotient of dividing the modules of these numbers.
Let a and b be negative numbers. Then
a ÷ b = a ÷ b .
This rule reduces the division of two negative numbers to the division of positive numbers. It is valid 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 says: 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 that reduces division to multiplication can also be applied to division of numbers with different signs.
The equality a ÷ b = a b - 1 can be proved using the multiplication property of real numbers and the definition of reciprocal numbers. Let's write down 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 of dividing a number by the number b.
Let's move on to examples.
Let's start with simple cases, moving on to more complex ones.
Example 1. How to divide negative numbers
Divide - 18 by - 3 .
The divisor and dividend modules are 3 and 18 respectively. Let's write:
18 ÷ - 3 = - 18 ÷ - 3 = 18 ÷ 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 the reverse number.
5 ÷ - 2 = - 5 - 1 2 = 5 1 2 = 5 2 = 2 1 2 .
When dividing fractional rational numbers, it is most convenient to represent them as ordinary fractions. However, you can also divide trailing decimals.
Example 3. How to divide negative numbers
Divide - 0.004 by - 0.25 .
First, we write down the modules of these numbers: 0 , 004 and 0 , 25 .
Now you can choose one of two methods:
- Separate decimal fractions with a column.
- Go to ordinary fractions and perform division.
Let's take a look at both methods.
1. Performing the division of decimal fractions by a column, move the comma two digits to the right.
Answer: - 0, 004 ÷ 0, 25 = 0, 016
2. Now we give a solution with the translation of decimal fractions into 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 obtained are the same.
In conclusion, we note that if the dividend and divisor are irrational numbers and are given in terms of roots, powers, logarithms, etc., the result of division is written as a numeric expression, the approximate value of which is calculated if necessary.
Example 4. How to divide negative numbers
Calculate the quotient of 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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Task 1. A point moves in a straight line from left to right with a speed of 4 dm. per second and is currently passing through point A. Where will the moving point be after 5 seconds?
It is easy to figure out that the point will be at 20 dm. to the right of A. Let's write the solution of this problem in relative numbers. To do this, we agree on the following signs:
1) the speed to the right will be denoted by the sign +, and to the left by the sign -, 2) the distance of the moving point from A to the right will be denoted by the sign + and to the left by the sign -, 3) the time interval after the present moment by the sign + and up to the present moment by the sign -. In our problem, the following numbers are given: speed = + 4 dm. per second, time \u003d + 5 seconds and it turned out, as they figured out arithmetically, the number + 20 dm., Expressing the distance of the moving point from A after 5 seconds. By the meaning of the problem, we see that it refers to multiplication. Therefore, it is convenient to write the solution of the problem:
(+ 4) ∙ (+ 5) = + 20.
Task 2. A point moves in a straight line from left to right with a speed of 4 dm. per second and is currently passing through point A. Where was this point 5 seconds ago?
The answer is clear: the point was to the left of A at a distance of 20 dm.
The solution is convenient, according to the conditions regarding signs, and, bearing in mind that the meaning of the problem has not changed, write it down as follows:
(+ 4) ∙ (– 5) = – 20.
Task 3. A point moves in a straight line from right to left with a speed of 4 dm. per second and is currently passing through point A. Where will the moving point be after 5 seconds?
The answer is clear: 20 dm. to the left of A. Therefore, under the same sign conditions, we can write the solution to this problem as follows:
(– 4) ∙ (+ 5) = – 20.
Task 4. A point moves in a straight line from right to left with a speed of 4 dm. per second and is currently passing through point A. Where was the moving point 5 seconds ago?
The answer is clear: at a distance of 20 dm. to the right of A. Therefore, the solution to this problem should be written as follows:
(– 4) ∙ (– 5) = + 20.
The considered problems indicate how to extend the action of multiplication to relative numbers. We have in problems 4 cases of multiplication of numbers with all possible combinations of signs:
1) (+ 4) ∙ (+ 5) = + 20;
2) (+ 4) ∙ (– 5) = – 20;
3) (– 4) ∙ (+ 5) = – 20;
4) (– 4) ∙ (– 5) = + 20.
In all four cases, the absolute values of these numbers should be multiplied, the product has to put a + sign when the factors have the same signs (1st and 4th cases) and sign -, when the factors have different signs(cases 2 and 3).
From here we see that the product does not change from the permutation of the multiplicand and the multiplier.
Exercises.
Let's do one calculation example, which includes both addition and subtraction and multiplication.
In order not to confuse the order of actions, pay attention to the formula
Here the sum of the products of two pairs of numbers is written: therefore, first the number a is multiplied by the number b, then the number c is multiplied by the number d, and then the resulting products are added. Also in the formula
you must first multiply the number b by c and then subtract the resulting product from a.
If you wanted to add the product of numbers a and b to c and multiply the resulting sum by d, then you should write: (ab + c)d (compare with the formula ab + cd).
If it were necessary to multiply the difference of numbers a and b by c, then we would write (a - b)c (compare with the formula a - bc).
Therefore, we will establish in general that if the order of actions is not indicated by brackets, then we must first perform the multiplication, and then the addition or subtraction.
We proceed to the calculation of our expression: let's first perform the additions written inside all the small brackets, we get:
Now we need to perform the multiplication inside the square brackets and then subtract the resulting product from:
Now let's perform the actions inside the twisted brackets: first the multiplication and then the subtraction:
Now it remains to perform multiplication and subtraction:
16. The product of several factors. Let it be required to find
(–5) ∙ (+4) ∙ (–2) ∙ (–3) ∙ (+7) ∙ (–1) ∙ (+5).
Here it is necessary to multiply the first number by the second, the resulting product by the 3rd, and so on. It is not difficult to establish on the basis of the previous one that the absolute values of all numbers must be multiplied among themselves.
If all the factors were positive, then on the basis of the previous one we find that the product must also have a + sign. If any one factor were negative
e.g., (+2) ∙ (+3) ∙ (+4) ∙ (–1) ∙ (+5) ∙ (+6),
then the product of all the factors preceding it would give a + sign (in our example, (+2) ∙ (+3) ∙ (+4) = +24, from multiplying the resulting product by a negative number (in our example, +24 times -1) would get the sign of the new product -; multiplying it by the next positive factor (in our example -24 by +5), we again get a negative number; since all other factors are assumed to be positive, the sign of the product cannot change anymore.
If there were two negative factors, then, arguing as above, they would find that at first, until it reached the first negative factor, the product would be positive, from multiplying it by the first negative factor, the new product would turn out to be negative and such would be it and remained until we reach the second negative factor; then, from multiplying a negative number by a negative one, the new product would turn out to be positive, which will remain so in the future, if the other factors are positive.
If there were also a third negative factor, then the positive product obtained by multiplying it by this third negative factor would become negative; it would remain so if the other factors were all positive. But if there is also a fourth negative factor, then multiplying by it will make the product positive. Arguing in the same way, we find that in general:
To find out the sign of the product of several factors, you need to look at how many of these factors are negative: if there are none at all, or if there are an even number, then the product is positive: if there are an odd number of negative factors, then the product is negative.
So now we can easily find out that
(–5) ∙ (+4) ∙ (–2) ∙ (–3) ∙ (+7) ∙ (–1) ∙ (+5) = +4200.
(+3) ∙ (–2) ∙ (+7) ∙ (+3) ∙ (–5) ∙ (–1) = –630.
Now it is easy to see that the sign of the product, as well as its absolute value, do not depend on the order of the factors.
It is convenient, when we are dealing with fractional numbers, to find the product immediately:
This is convenient because you do not have to do useless multiplications, since the previously obtained fractional expression is reduced as much as possible.
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.
Page navigation.
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.
Example.
Divide the negative number −35 by the positive number 7 .
Solution.
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: .
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 number that is 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 came to the same result.
Answer:
(−35):7=−5 .
Example.
Calculate the quotient 8:(−60) .
Solution.
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: .
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 a different notation (for example, in decimal).
Example.
Solution.
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.
Let's translate a mixed number into an ordinary fraction: 
, as well as
In this article, we will look at dividing positive numbers by negative numbers and vice versa. Let's give a detailed analysis of the rule for dividing numbers with different signs, and also give examples.
Rule for dividing numbers with different signs
The rule for integers with different signs, obtained in the article on the division of integers, is also valid for rational and real numbers. Let us give a more general formulation of this rule.
Rule for dividing numbers with different signs
When dividing a positive number by a negative one and vice versa, you need to divide the dividend modulus by the divisor modulus, and write the result with a minus sign.
In literal form, it looks like this:
a ÷ - b = - a ÷ b
A ÷ b = - a ÷ b .
Dividing numbers with different signs always results in a negative number. The considered rule, in fact, reduces the division of numbers with different signs to the division of positive numbers, since the modules of the dividend and divisor are positive.
Another equivalent mathematical formulation of this rule is:
a ÷ b = a b - 1
To divide the numbers a and bhaving different signs, you need to multiply the number a by the reciprocal of the number b, that is, b - 1. This formulation is applicable on the set of rational and real numbers, it allows you to go from division to multiplication.
Let us now consider how to apply the theory described above in practice.
How to divide numbers with different signs? Examples
Below we consider a few typical examples.
Example 1. How to divide numbers with different signs?
Divide - 35 by 7.
First, let's write the modules of the dividend and divisor:
35 = 35 , 7 = 7 .
Now let's separate the modules:
35 7 = 35 7 = 5 .
We add a minus sign in front of the result and get the answer:
Now let's use a different formulation of the rule and calculate the reciprocal of 7 .
Now let's do the multiplication:
35 1 7 = - - 35 1 7 = - 35 7 = - 5 .
Example 2. How to divide numbers with different signs?
If we divide fractional numbers with rational signs, the dividend and divisor must be represented as ordinary fractions.
Example 3. How to divide numbers with different signs?
Divide the mixed number - 3 3 22 by the decimal fraction 0 , (23) .
The modules of the dividend and the divisor are respectively 3 3 22 and 0 , (23) . Converting 3 3 22 to a common fraction, we get:
3 3 22 = 3 22 + 3 22 = 69 22 .
We can also represent the divisor as a common fraction:
0 , (23) = 0 , 23 + 0 , 0023 + 0 , 000023 = 0 , 23 1 - 0 , 01 = 0 , 23 0 , 99 = 23 99 .
Now we divide ordinary fractions, perform reductions and get the result:
69 22 ÷ 23 99 = - 69 22 99 23 = - 3 2 9 1 = - 27 2 = - 13 1 2 .
In conclusion, consider the case when the dividend and divisor are irrational numbers and are written as roots, logarithms, powers, etc.
In such a situation, the quotient is written as a numerical expression, which is simplified as much as possible. If necessary, its approximate value is calculated with the required accuracy.
Example 4. How to divide numbers with different signs?
Divide the numbers 5 7 and - 2 3 .
According to the rule for dividing numbers with different signs, we write the equality:
5 7 ÷ - 2 3 = - 5 7 ÷ - 2 3 = - 5 7 ÷ 2 3 = - 5 7 2 3 .
Let's get rid of the irrationality in the denominator and get the final answer:
5 7 2 3 = - 5 4 3 14 .
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