Mutually reverse numbers, finding the reverse number. Plan-abstract lesson on algebra (grade 6) on the topic: "Mutual reverse numbers"

Let us give a definition and give examples of mutually reverse numbers. Consider how to find a number inverse to a natural number and an inverse of ordinary fraction. In addition, we write and prove inequality, reflecting the property of mutually reverse numbers.

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Mutually reverse numbers. Definition

Definition. Mutually reverse numbers

Mutually reverse numbers are such numbers whose product gives one.

If a · b \u003d 1, then we can say that the number A is back to the number B, as well as the number B back to the number a.

The easiest example of mutually reverse numbers is two units. Indeed, 1 · 1 \u003d 1, therefore a \u003d 1 and b \u003d 1 - mutually reverse numbers. Another example is the numbers 3 and 1 3, - 2 3 and - 3 2, 6 13 and 13 6, Log 3 17 and Log 17 3. The product of any pair of the above numbers is equal to one. If this condition is not performed, such as numbers 2 and 2 3, the numbers are not mutually reverse.

The definition of mutually reverse numbers is valid for any numbers - natural, integers, valid and complex.

How to find the number in the reverse

Consider general. If the initial number is equal to a, the inverse number is recorded as 1 a, or a - 1. Indeed, A · 1 A \u003d A · A - 1 \u003d 1.

For natural numbers and ordinary fractions, find the opposent number is quite simple. It can be said, even obvious. In case of finding a number, inverse irrational or integrated number, you will have to make a number of calculations.

Consider the most common cases of location inverse.

Number, reverse of ordinary fraction

Obviously, the number, the inverse of the ordinary fraction A B is a fraction of b a. So, to find the opposite fraction number, the fraction you need to just turn over. That is, change the numerator and denominator in places.

According to this rule, write the reverse of any ordinary fraction. The number can be almost immediately. So, for the fraction 28 57 in the premises will be shot 57 28, and for the fraction 789 256 - the number 256 789.

Number inverse to a natural number

Find the number inverse to any natural number, you can also, like the number, reverse fraction. It is enough to represent the natural number A as an ordinary fraction A 1. Then the number 1 A feeder will be the number 1 a. For natural Number 3 inversely, the number will be fraction 1 3, for the number 666 the reverse number is 1,666, and so on.

Special attention should be paid to one, as it is singular, the reverse number for which is equal to himself.

Other pairs of mutually reverse numbers, where both components are equal, does not exist.

Number, reverse mixed number

Mixed number we have a view A b c. To find the opposite number to him, it is necessary mixed number Present in the Side of the wrong fraction, and already for the resulting fraction, pick up the opposite.

For example, we find the opposite number for 7 2 5. First, imagine 7 2 5 in the form of incorrect fraction: 7 2 5 \u003d 7 · 5 + 2 5 \u003d 37 5.

For incorrect fraction 37 5, it will turn 5 37 in the premises.

Number, reverse decimal fraction

The decimal fraction can also be represented as an ordinary fraction. Finding back decimal fractions The numbers are reduced to the representation of the decimal fraction in the form of an ordinary fraction and finding the reverse number for it.

For example, there is a fraction 5, 128. Find the opposite number. First we translate the decimal fraction in the ordinary: 5, 128 \u003d 5 128 1000 \u003d 5 32 250 \u003d 5 16 125 \u003d 641 125. For the resulting fraction in the premises will be shot 125 641.

Consider another example.

Example. Finding a number, reverse decimal fraction

We find the reverse number for the periodic decimal fraction 2, (18).

We translate the decimal fraction to ordinary:

2, 18 \u003d 2 + 18 · 10 - 2 + 18 · 10 - 4 +. . . \u003d 2 + 18 · 10 - 2 1 - 10 - 2 \u003d 2 + 18 99 \u003d 2 + 2 11 \u003d 24 11

After translation, we can easily burn the reverse number for the fraction 24 11. This number, obviously, will be 11 24.

For an infinite and non-periodic decimal fraction, the inverse number is written in the form of a fraction and unit in the numerator and the fraction itself in the denominator. For example, for infinite fraction 3, 6025635789. . . The reverse number will be viewed 1 3, 6025635789. . . .

Similar to both irrational numberscorresponding to non-periodic infinite fractions, reverse numbers are recorded in the form of fractional expressions.

For example, the inverse number for π + 3 3 80 will be 80 π + 3 3, and for the number of 8 + e 2 + e, there will be a fraction of 1 8 + e 2 + e.

Mutually reverse numbers with roots

If the form of two numbers is different from A and 1 A, it is not always possible to easily determine whether the numbers are mutually reverse. This is especially true for numbers that have a root sign in their record, since the root is usually made to get rid of the denominator.

Turn to practice.

Answer the question: are mutually reverse numbers 4 - 2 3 and 1 + 3 2.

To find out whether the numbers are mutually reverse, we calculate their work.

4 - 2 3 · 1 + 3 2 \u003d 4 - 2 3 + 2 3 - 3 \u003d 1

The work is equal to one, it means that the numbers are mutually reversed.

Consider another example.

Example. Mutually reverse numbers with roots

Record the number, the reverse number 5 3 + 1.

You can immediately write down that the reverse number is the fraction 1 5 3 + 1. However, as we have already spoken, it is customary to get rid of the root in the denominator. To make it multiply the numerator and the denominator on 25 3 - 5 3 + 1. We get:

1 5 3 + 1 \u003d 25 3 - 5 3 + 1 5 3 + 1 · 25 3 - 5 3 + 1 \u003d 25 3 - 5 3 + 1 5 3 3 + 1 3 \u003d 25 3 - 5 3 + 1 6

Mutually reverse numbers with degrees

Suppose there is a number equal to some extent number a. In other words, the number A erected into the degree of n. The inverse number A n will be the number a - n. Check it. Indeed: a n · a - n \u003d a n 1 · 1 a n \u003d 1.

Example. Mutually reverse numbers with degrees

Find the reverse number for 5 - 3 + 4.

According to the above, the desired number is 5 - - 3 + 4 \u003d 5 3 - 4

Mutually reverse numbers with logarithms

For the logarithm number a for the base B reverse is the number equal Logarithm Numbers B based on a.

log A B and Log B A - mutually reverse numbers.

Check it. From the properties of the logarithm it follows that Log A B \u003d 1 Log B A, which means Log A B · Log B a.

Example. Mutually reverse numbers with logarithms

Find a number, reverse Log 3 5 - 2 3.

The inverse logarithm of numbers 3 for the base 3 5 - 2 will be the logarithm of 3 5 - 2 for the base 3.

Number inverse integrated number

As noted earlier, the definition of mutually reverse numbers is fair not only for real numbers, but also for complex.

Typically, complex numbers are represented in the algebraic form z \u003d x + i y. The number, inverse this, will fraction

1 x + i y. For convenience, it is possible to reduce this expression, multiplying the numerator and denominator on x - i y.

Example. Number inverse integrated number

Let there be a complex number z \u003d 4 + i. We find the number inverse to him.

The number, the reverse z \u003d 4 + i, will be 1 4 + i.

Multiply the numerator and denominator at 4 - I and get:

1 4 + i \u003d 4 - i 4 + i 4 - i \u003d 4 - i 4 2 - i 2 \u003d 4 - i 16 - (- 1) \u003d 4 - i 17.

In addition to the algebraic form, a complex number can be represented in trigonometric or indicative form as follows:

z \u003d r · cos φ + i · sin φ

z \u003d r · e i · φ

Accordingly, the reverse number will look:

1 r cos (- φ) + i · sin (- φ)

Make sure that:

r · cos φ + i · sin φ · 1 r cos (- φ) + i · sin (- φ) \u003d rr cos 2 φ + sin 2 φ \u003d 1 r · ei · φ · 1 rei · (- φ) \u003d RRE 0 \u003d 1

Consider examples with the presentation of complex numbers in trigonometric and indicative form.

We find the number inverse for 2 3 cos π 6 + i · sin π 6.

Considering that R \u003d 2 3, φ \u003d π 6, write the opposite number

3 2 COS - π 6 + i · sin - π 6

Example. Find the number in the integrated number

What number will be reverse for 2 · E i · - 2 π 5.

Answer: 1 2 · E i 2 π 5

The sum of mutually reverse numbers. Inequality

There is a theorem on the sum of two mutually reverse numbers.

Amount of mutually reverse numbers

The sum of two positive and mutually reverse numbers is always greater than or equal to 2.

We give proof of the theorem. As is known, for any positive numbers a and b, arithmetic averages are greater than or equal to the average geometric. This can be written in the form of inequality:

a + B 2 ≥ A · B

If instead of the number B take the number, reverse A, the inequality will take the form:

a + 1 A 2 ≥ A · 1 A A + 1 A ≥ 2

Q.E.D.

Let us give a practical example illustrating this property.

Example. Find the sum of mutually reverse numbers

I calculate the sum of numbers 2 3 and the number opposite it.

2 3 + 3 2 = 4 + 9 6 = 13 6 = 2 1 6

As the theorem says, the number obtained is more than two.

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MOU "PARKSKAYA OOSH №2 them. DI. Mishchenko

Mathematics lesson in the 6th grade on the topic

"Mutually reverse numbers"

Held teacher

mathematics and informatics

I qualifying category

Balan V.M.

Parkans 2011

P.S. Due to the limitations on MAX file size (no more than 3mb), the presentation is divided into 2 parts. You must consistently copy slides into one presentation.

Mathematics lesson in the 6th grade on the topic "mutually reverse numbers"

Purpose:

1. Enter the concept of mutually reverse numbers.
2. Learn to determine the pairs of mutually reverse numbers.
3. Repeat multiplication and reduction of fractions.

Type of lesson : Studying and primary consolidation of new knowledge.

Equipment:

• computers;
• signal cards;
• workbooks, notebooks, tutorial;
• drawing accessories;
• presentation to the lesson (seeapplication ).

Individual task:message about one.

During the classes

1. Organizational moment.(3 minutes)

Hello guys sit down! Let's start our lesson! Today you will need attention, concentration and, of course, discipline.(Slide 1. )

I took the word to the epigraph to today:

It is often said that the figures control the world;

at least there is no doubt

that the numbers show how it is controlled.

And funny men are in a hurry to help me: pencil and self-celkin. They will help me to spend this lesson.(Slide 2. )

The first task from the pencil is to solve anagram. (Slide 3. )

Let's remember how anagram is? (Anagram - permutation in the word letters, forming another word. For example, "Ropot" - "ax").

(Children answer what angrum is and solve words.)

Well done! The theme of today's lesson: "Mutually reverse numbers."

Open a notebook, write the number, class work and theme lesson. (Slide 4. )

Guys, tell me, please, what should you learn today at the lesson?

(Children call the purpose of the lesson.)

The purpose of our lesson:

• Find out what numbers are called mutually reverse.
• Learn to find pairs of mutually reverse numbers.
• Repeat the rule of multiplication and reduction of fractions.
• Develop logical thinking of students.

2. We work orally.(3 minutes)

We repeat the rule of multiplication of fractions. (Slide 5. )

Task from Samodelkin (children read examples and perform multiplication):

What rule did we use?

Pencil prepared the task more complicated (Slide 6. ):

What is this work?

Guys, we repeated the actions of multiplication and reduction of fractions, without which it is not necessary when studying a new topic.

3. Explanation of the new material. (15 minutes) ( Slide 7. )

1. Take the shot 8/17, put instead of the numerator - the denominator and vice versa. It turns out the shot 17/8.

We write: Scroll 17/8 is called the back to the fraction 8/17.

Attention! Return to the fraction M / N is called fraction N / m. (Slide 8. )

Guys, how are you still getting from this fraction back to it?(Children answer.)

2. Task from Samodelkin:

Name the fraction, the inverse one.(Children call.)

About such fractions say they are reverse to each other! (Slide 9. )

What then can be said about the fraction 8/17 and 17/8?

Answer: Reverse to each other (write).

3. What happens if you multiply two fractions, reverse to each other?

(Working with slides. (Slide 10. ))

Guys! Look and tell me what can be equal to M and N?

I repeat once again that the work of any fraction inverse to each other is 1.Slide 11. )

4. It turns out that one is a magical number!

And what do we know about one?

Interesting judgments about the world of numbers reached us through the century from the Pythagorean school, which we will tell us the bobbing Nadia (a small message).

5. We stopped on the fact that the work of any numbers back to each other is 1.

What are these numbers called?(Definition.)

Let's check whether the fractions are mutually reverse numbers: 1.25 and 0.8. (Slide 12. )

It can be checked in another way whether the numbers are mutually reverse (2 ways).

Let's guys make a conclusion:

How to check whether the numbers are mutually reverse?(Children answer.)

6. Now consider several examples to find mutually reverse numbers (we consider two examples). (Slide 13)

4. Fastening. (10 minutes)

1. Working with warning cards. You have signal cards on your table. (Slide 14)

Red - no. Green - yes.

(Last example 0.2 and 5.)

Well done! Know how to determine the couples of mutually reverse numbers.

2. Attention on the screen! - We work orally. (Slide 15)

Find an unknown number (solve the equation, last 1/3 x \u003d 1).

ATTENTION QUESTION: When are two numbers in the work give 1?(Children answer.)

5. Physical traffic.(2 minutes)

And now distracted from the screen - a little rest!

1. Close your eyes, very much climb, dramatically open your eyes. Do it 4 times.
2. Hold your head straight, eyes raised up, lowered down, looked left, looked to the right (4 times).
3. Fuck your head back, drop forward so that the chin leaves in the chest (2 times).

6. Continue the fixation of the new material [3], [4]. (5 minutes)

Rested, and now fixing a new material.

In the textbook number 563, No. 564 - at the board. (Slide 16)

7. The result of the lesson, homework. (3 minutes)

Our lesson comes to an end. Tell me, guys, what's new we learned today at the lesson?

1. How to get back to each other?
2. What numbers are called mutually reverse?
3. How to find the intelligent number to a mixed number, to decimal fraction?

Did we fulfill the purpose of the lesson?

Let's open the diaries, write homework: №591 (a), 592 (A, B), 595 (a), paragraph 16.

And now, I ask you to unravel you this rebus (if time remains).

Thank you for the lesson! (Slide 17)

Literature:

1. Mathematics 5-6: Textbook-interlocutor. L.N. Chevrin, A.G. Hein, I.O. Koryakov, M.V. Volkov, - M.: Enlightenment, 1989.
2. Mathematics Grade 6: Pounding plans for the textbook N.Ya. Vilenkin, V.I. Johova. L.A. Tapilina, T.L. Afanasyev. - Volgograd: Teacher, 2006.
3. Mathematics: textbook Grade 6. N.Ya.Vilekin, V.I. Zhokhov, A.S. Chesnokov, S.I. Schwarzburg.- M.: Mnemozina, 1997.
4. Pencil journey and selfklin. Y. Friends. - M.: Dragonfly Press, 2003.

Preview:

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Signatures for slides:

1 "It is often said that the figures are controlled by the world; At least there is no doubt that the numbers show how controlled »Johann Wolfgang Goethe

3 To learn the topic of today's lesson, you need to solve anagram! 1) Ichlas of Numbers 2) dorb fraction 3) Are youborbor inverse 4) Inomes mutually solved? And now remove the word too much, the rest are in the desired order!

4 mutually reverse numbers

5 Multiplication of fractions Calculate orally: Well done!

6 And now the task is more complicated! Calculate: Well done!

1 What happens if you multiply two fractions, reverse to each other? Let's see (write with me): Attention! The work of fractions, reverse to each other, equal to one! And what do we know about one? Remember!

2 Two numbers, the product of which is equal to one, is called mutually reverse numbers check whether the fractions are mutually reverse numbers: 1.25 and 0.8 We write them in the form of ordinary fractions: Mutually reverse numbers otherwise, you can check multiplication:

3 We prove that the opposite number is to the number of 0.75. We write:, and inverse to it will find the number back to the number write a mixed number as an incorrect fraction: to this number the opposite

4 We work with signal cards Yes No Are the numbers are mutually reverse?

5 Work orally: Find an unknown number:

6 We work in notebooks. Tutorial page 8 9 №5 63

7 Thanks for the lesson?

Preview:

Analysis

mathematics lesson in grade 6

MOU "PARKAN OOSH№2 them. D.I.Mishchenko "

Teacher Balan V.M.

The subject of the lesson: "Mutually reverse numbers."

The lesson is built with a support for previous lessons, students' knowledge was checked by various methods in order to find out how students learned the previous material, and how this lesson will "work" in the following lessons.

The lesson stages are logically traced, a smooth transition from one to another. You can trace the integrity and completion of the lesson. The assimilation of the new material was alone through the creation problem situation And her decision. I believe that the selected structure of the lesson is rational, since it allows you to implement all the goals and objectives in the complex.

Currently, the use of ICT is very actively used in the lessons, so Balan V.M. Applied multimedia for greater clarity.

The lesson was conducted in grade 6, where the level of performance, cognitive interest And the memory is not very high, there are also such guys who have gaps in actual knowledge. Therefore, at all stages of the lesson were used various methods Activation of students, which did not allow them to be tired of the monotony of the material.

Slides with ready-made answers for self-test, mutual tests were used to verify and evaluate students' knowledge.

In the process of lesson, the teacher sought to intensify the mental activity of students using the following techniques and methods: angrum at the beginning of the lesson, conversation, student story "what do we know about unity? ", clarity, work with signal cards.

Thus, I think that the lesson is creative, represents holistic system. The objectives set in the lesson are achieved.

Mathematics teacher I category / Kurteva F.I. /

Reverse - or mutually reverse - numbers are called a couple of numbers that give 1. in multiplies. general Reverse are numbers. Characteristic private case Mutually reverse numbers - steam. Reverse are, say, numbers; .

How to find the opposite

Rule: need 1 (unit) to divide on this number.

Example number 1.

The number 8 is given. The opposite is 1: 8 or (the second option is preferable, because such a record is mathematically more correct).

When the rear number is searched for an ordinary fraction, then it is not very convenient to share it, because Record is obtained by cumbersome. In this case, it is much easier to come differently: the fraction simply turn over, by changing the nipple and denominator by places. If the correct fraction is given, then after turning it turns out the fraction is incorrect, i.e. Such from which it is possible to allocate the whole part. Do this or not, it is necessary to decide in each particular case. So, if an inverted fraction with the resulting fraquence then will have to make some actions (for example, multiplication or division), then it is not worth allocating the whole part. If the resulting fraction is the end result, then it is possible that the allocation of the whole part is preferably.

Example number 2.

Dana fraction. Return to her :.

If you want to find the countdown to the decimal fraction, you should use the first rule (division 1 to the number). In this situation, you can act in one of 2 ways. The first is to simply divide 1 on this number in the column. The second is to form a fraction of 1 in a numerator and a decimal fraction in a denominator, and then multiply a numerator and a denominator for 10, 100 or another number consisting of 1 and such a number of zeros, which is necessary to get rid of the decimal point in the denominator. As a result, an ordinary fraction will be obtained, which is the result. If necessary, it may be necessary to reduce, allocate a whole part from it or translate into a decimal form.

Example number 3.

The number is 0.82. The opposite number to it is: . Now will reduce the fraction and highlight the whole part :.

How to check whether two numbers are reverse

The principle of verification is based on the definition of reverse numbers. That is, in order to make sure that the numbers are back to each other, you need to multiply them. If a unit is obtained, it means that the numbers are mutually reverse.

Example number 4.

There are numbers 0.125 and 8. Are they back?

Check. It is necessary to find a product of 0.125 and 8. For clarity, we will imagine the number of numbers in the form of ordinary fractions: (will reduce the 1st fraction at 125). Conclusion: numbers 0.125 and 8 are reverse.

Properties of reverse numbers

Property №1

The opposite number exists for any number except 0.

This restriction is due to the fact that it is impossible to divide by 0, and when determining the reverse number for zero, it will just have to move to the denominator, i.e. To actually divide on it.

Property №2.

The sum of the pair of mutually reverse numbers is always no less than 2.

Mathematically, this property can be expressed in inequality :.

Property number 3.

Multiplication of a number of two mutual equivalent to multiplication by one. Express this property mathematically :.

Example number 5.

Find an expression value: 3.4 · 0.125 · 8. Since the numbers 0.125 and 8 are inverse (see example No. 4), then multiply 3.4 to 0.125 and then 8 is not necessary. So, the answer here will be 3.4.

Due to the fact that almost in all modern schools there is necessary equipmentSo that during the lessons to demonstrate children video and various electronic training resources, it is possible to better interest the students in that or another subject or in the other topic. As a result, students and school rating increase in general.

It's no secret that the visual demonstration during the lesson helps to remember and absorb definitions, tasks and theory. If it is accompanied by sounding, then the student employs both visual and hearing memory. Therefore, video tutorials are considered one of the most effective materials For learning.

There are a number of rules and requirements that video lessons must be configured to be as efficient and useful for students of the appropriate age. The background and color of the text must be selected accordingly, the font size should not be too small so that the text can read and poorly seen schoolchildren, however, and not too large to annoy the vision and create inconvenience, etc. Special attention is paid to the illustrations - they must be carried out in moderation and not distract from the main topic.

The video tutorial "mutually reverse numbers" is an excellent example of such a training resource. Thanks to him, a gradent grade 6 can fully understand what mutually reverse numbers are, how to recognize them and how to work with them.

The lesson begins S. simple examplein which two ordinary fractions 8/15 and 15/8 are multiplied by each other. There is an opportunity to recall the rule by which it was studied earlier, a fraction should be multiplied. That is, in the numerator it is necessary to record the product of numerals, and in the denominator - the product of the denominers. As a result of the reduction, which is also worth remembering, one turns out.

After this exampleThe speaker gives a generalized definition, which is displayed in parallel to the screen. It states that the numbers that when multiplying each other are given as a result of the unit, are called mutually reverse. The definition is remembered very simply, however, it will fix it more confidently in memory if you give some examples.

On the screen, after determining the concept of mutually reverse numbers, a number of works of numbers are derived, which as a result provide a unit.

To give a generalized example that will not depend on certain numeric values, A and B variables are used, which are different from 0. Why? After all, schoolchildren in the 6th grade should be perfectly aware that the denominator of any fraction cannot be equal to zero, and to show mutually reverse numbers, do not do without the location of these values \u200b\u200bin the denominator.

After the output of this formula and its comment, the announcer begins to consider the first task. The essence is that it is necessary to find the opposite mixed fraci. To solve it, the fraction is recorded in the wrong form, and the numerator and denominator are changed by places. The result obtained is the answer. A schoolboy can check him independently, using the definition of mutually reverse numbers.

The video language is not limited to this example. Next for the previous one, another task is displayed on the screen, in which it is necessary to find a product of three fractions. If the student shows attentiveness, he will find that two of these fractions are reverse numbers, therefore, their work will be equal to one. Relying the property of multiplication, you can first multiply mutually inverse fractions, and in the latter - multiply the result, i.e. 1, on the first fraction. The announcer explains in detail, step-by-step demonstrating the entire process on the screen from start to the end. Finally, a theoretical generalized explanation is given to the multiplication property, which was based on the solution of the example.

To secure certainly knowledge, it is worth trying to answer all the questions that will be bred at the end of the lesson.