Outline of a lesson in algebra (grade 6) on the topic: "Mutually reciprocal numbers." Reverse numbers

Due to the fact that in almost all modern schools there is necessary equipment, in order to show children videos and various electronic learning resources during lessons, it becomes possible to better interest students in a particular subject or in a particular topic. As a result, student performance and the overall ranking of the school increase.

It's no secret that visual demonstration during the lesson helps to better memorize and assimilate definitions, tasks and theory. If this is accompanied by vocalization, then both visual and auditory memory work for the student. Therefore, video tutorials are considered one of the most effective materials for training.

There are a number of rules and requirements that video lessons must meet in order to be as effective and useful as possible for students of the appropriate age. The background and color of the text should be chosen appropriately, the font size should not be too small so that the text could be read by students with poor vision, however, and not too large to irritate the eyes and create inconvenience, etc. Particular attention is paid to illustrations - they should be contained in moderation and not distract from the main theme.

The video tutorial "Reciprocal Numbers" is an excellent example of such a learning resource. Thanks to him, a 6th grade student can fully understand what mutually inverse numbers are, how to recognize them and how to work with them.

The lesson starts with simple example, in which two fractions 8/15 and 15/8 are multiplied by each other. It becomes possible to remember the rule by which, as was studied earlier, fractions should be multiplied. That is, in the numerator, the product of the numerators should be written, and in the denominator, the product of the denominators. As a result of the reduction, which is also worth remembering, one is obtained.

After this example, the announcer gives a generalized definition, which is displayed in parallel on the screen. It says that the numbers that, when multiplied by each other, result in one, are called mutually inverse. The definition is very easy to remember, but it will become more confident in memory if you give some examples.

On the screen, after defining the concept of mutually inverse numbers, a series of products of numbers is displayed, which, as a result, give one.

To give a generalized example that will not depend on certain numerical values, the variables a and b are used, which are different from 0. Why? After all, schoolchildren in grade 6 should know perfectly well that the denominator of any fraction cannot be zero, and in order to show mutually reciprocal numbers, one cannot do without the location of these values ​​in the denominator.

After displaying this formula and commenting on it, the speaker begins to consider the first task. The bottom line is that it is necessary to find the inverse of the given mixed fraction... To solve it, the fraction is written in the wrong form, and the numerator and denominator are swapped. The result is the answer. The student can independently check it, using the definition of mutually reciprocal numbers.

The video tutorial is not limited to this example. Following the previous one, another task is displayed on the screen, in which it is necessary to find the product of three fractions. If the student is attentive, he will find that two of these fractions are reciprocal numbers, therefore, their product will be equal to one. Based on the property of multiplication, one can first of all multiply mutually reciprocal fractions, and lastly, multiply the result, that is, 1, by the first fraction. The announcer explains in detail, demonstrating the entire process from start to finish on the screen, step by step. Finally, a theoretical generalized explanation of the multiplication property is given, which was used to solve the example.

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

Inverse - or mutually inverse - numbers are a pair of numbers that, when multiplied, give 1. In the very general view the inverse are numbers. Characteristic special case reciprocal numbers - a pair. The inverse are, say, numbers; ...

How to find the reciprocal

Rule: you need to divide 1 (one) by a given number.

Example # 1.

Given the number 8. Its reverse is 1: 8 or (the second option is preferable, because such a notation is mathematically more correct).

When looking for the reciprocal of common fraction, then dividing it by 1 is not very convenient, since the recording turns out to be cumbersome. In this case, it is much easier to do otherwise: the fraction is simply inverted, changing the places of the numerator and denominator. If a correct fraction is given, then after turning over, the fraction is incorrect, i.e. one from which you can select a whole part. To do it or not, it is necessary to decide in each case separately. So, if you have to perform some actions with the resulting inverted fraction (for example, multiplication or division), then you should not select the whole part. If the resulting fraction is the final result, then it is possible that the selection of the whole part is desirable.

Example No. 2.

A fraction is given. Back to her:.

If you want to find the reciprocal of decimal, then you should use the first rule (dividing 1 by a number). In this situation, you can act in one of 2 ways. The first is to simply divide 1 by that number per column. The second is to form a fraction of 1 in the numerator and a decimal fraction in the denominator, and then multiply the numerator and denominator by 10, 100, or another number consisting of 1 and as many zeros as you need to get rid of the decimal point in the denominator. The result will be an ordinary fraction, which is the result. If necessary, you may need to shorten it, extract an entire part from it, or convert it to decimal form.

Example No. 3.

Given the number 0.82. The inverse number to it is: ... Now we will reduce the fraction and select the whole part:.

How to check if two numbers are reciprocal

The principle of verification is based on the definition of reciprocal numbers. That is, in order to make sure that the numbers are inverse to each other, you need to multiply them. If the result is one, then the numbers are mutually inverse.

Example No. 4.

The numbers 0,125 and 8 are given. Are they inverse?

Examination. It is necessary to find the product of 0.125 and 8. For clarity, we present these numbers in the form of ordinary fractions: (we will reduce the 1st fraction by 125). Conclusion: the numbers 0.125 and 8 are inverse.

Reverse number properties

Property number 1

The inverse exists for any number other than 0.

This restriction is due to the fact that you cannot divide by 0, and when determining the reciprocal of zero, you just have to move it to the denominator, i.e. actually divide by it.

Property number 2

The sum of a pair of reciprocal numbers is always at least 2.

Mathematically, this property can be expressed by the inequality:.

Property number 3

Multiplying a number by two reciprocal numbers is equivalent to multiplying by one. Let us express this property mathematically:.

Example No. 5.

Find the value of the expression: 3.4 · 0.125 · 8. Since the numbers 0.125 and 8 are inverse (see Example # 4), there is no need to multiply 3.4 by 0.125 and then by 8. So the answer here is 3.4.

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

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Mutually reciprocal numbers. Definition

Definition. Reciprocal numbers

Mutually reciprocal numbers are numbers whose product gives one.

If a · b = 1, then we can say that the number a is inverse to the number b, just as the number b is inverse to the number a.

The simplest example of mutually inverse numbers is two ones. Indeed, 1 · 1 = 1, therefore a = 1 and b = 1 are mutually inverse 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 met, as, for example, for the numbers 2 and 2 3, then the numbers are not mutually inverse.

The definition of mutually reciprocal numbers is valid for any numbers - natural, integer, real and complex.

How to find the inverse of a given number

Consider general case... If the original number is a, then its inverse will be written as 1 a, or a - 1. Indeed, a 1 a = a a - 1 = 1.

Finding the reciprocal is pretty easy for natural numbers and fractions. One might even say it is obvious. In the case of finding the inverse of an irrational or complex number, you will have to make a number of calculations.

Let's consider the most common cases of finding the reciprocal number in practice.

The reciprocal of an ordinary fraction

Obviously, the reciprocal of the ordinary fraction a b is the fraction b a. So, to find the reciprocal of a number, you just need to flip the fraction. That is, swap the numerator and denominator.

According to this rule, you can write the reciprocal of any ordinary fraction almost immediately. So, for the fraction 28 57, the reciprocal will be the fraction 57 28, and for the fraction 789 256 - the number 256 789.

The inverse of a natural number

You can find the inverse of any natural number in the same way as the inverse of a fraction. It is enough to represent the natural number a as an ordinary fraction a 1. Then the number 1 a will be its inverse. For natural number 3, its reciprocal is the fraction 1 3, for 666, the reciprocal is 1 666, and so on.

Particular attention should be paid to the unit, since it is singular for which the reciprocal is equal to itself.

There are no other pairs of mutually reciprocal numbers, where both components are equal.

The inverse of the mixed number

The mixed number is a b c. To find its inverse, you need mixed number present in the side of an improper fraction, and already for the resulting fraction, choose the reciprocal.

For example, find the reciprocal of 7 2 5. First, imagine 7 2 5 as an improper fraction: 7 2 5 = 7 5 + 2 5 = 37 5.

For an improper fraction 37 5, the reciprocal is 5 37.

The reciprocal of the decimal fraction

A decimal can also be represented as a fraction. Finding the reciprocal of a number is reduced to representing the decimal as an ordinary fraction and finding the reciprocal for it.

For example, there is a fraction 5, 128. Let's find its inverse number. First, we convert the decimal fraction to an ordinary one: 5, 128 = 5 128 1000 = 5 32 250 = 5 16 125 = 641 125. For the resulting fraction, the reciprocal is the fraction 125 641.

Let's take another example.

Example. Finding the reciprocal of a decimal fraction

Find the reciprocal for the periodic decimal fraction 2, (18).

We convert a decimal fraction to an ordinary one:

2, 18 = 2 + 18 10 - 2 + 18 10 - 4 +. ... ... = 2 + 18 10 - 2 1 - 10 - 2 = 2 + 18 99 = 2 + 2 11 = 24 11

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

For an infinite and non-periodic decimal fraction, the reciprocal is written as a fraction and a unit in the numerator and the fraction itself in the denominator. For example, for the infinite fraction 3, 6025635789. ... ... the reciprocal will be 1 3, 6025635789. ... ... ...

Similarly for irrational numbers, corresponding to non-periodic infinite fractions, reciprocal numbers are written in the form of fractional expressions.

For example, the reciprocal for π + 3 3 80 is 80 π + 3 3, and for the number 8 + e 2 + e, the reciprocal is the fraction 1 8 + e 2 + e.

Reciprocal numbers with roots

If the form of two numbers is different from a and 1 a, then it is not always easy to determine whether the numbers are mutually inverse. This is especially true for numbers that have a root sign in their notation, since it is usually customary to get rid of the root in the denominator.

Let's turn to practice.

Let's answer the question: are the numbers 4 - 2 3 and 1 + 3 2 mutually inverse?

To find out if the numbers are mutually inverse, let's calculate their product.

4 - 2 3 1 + 3 2 = 4 - 2 3 + 2 3 - 3 = 1

The product is equal to one, which means that the numbers are mutually inverse.

Let's take another example.

Example. Reciprocal numbers with roots

Write down the reciprocal of 5 3 + 1.

You can immediately write down that the reciprocal is equal to the fraction 1 5 3 + 1. However, as we have already said, it is customary to get rid of the root in the denominator. To do this, multiply the numerator and denominator by 25 3 - 5 3 + 1. We get:

1 5 3 + 1 = 25 3 - 5 3 + 1 5 3 + 1 25 3 - 5 3 + 1 = 25 3 - 5 3 + 1 5 3 3 + 1 3 = 25 3 - 5 3 + 1 6

Reciprocal numbers with powers

Let's say there is a number equal to some power of the number a. In other words, the number a raised to the power n. The inverse of a n will be a - n. Let's check it out. Indeed: a n a - n = a n 1 1 a n = 1.

Example. Reciprocal numbers with powers

Find the reciprocal of 5 - 3 + 4.

According to the above, the required number is 5 - - 3 + 4 = 5 3 - 4

Reciprocal numbers with logarithms

For the logarithm of a to base b, the inverse is the number, equal to logarithm numbers b in base a.

log a b and log b a are mutually inverse numbers.

Let's check it out. It follows from the properties of the logarithm that log a b = 1 log b a, so log a b log b a.

Example. Reciprocal numbers with logarithms

Find the reciprocal of log 3 5 - 2 3.

The reciprocal of the logarithm base of 3 5 - 2 is the logarithm of the number 3 5 - 2 to the base 3.

The inverse of a complex number

As noted earlier, the definition of mutually inverse numbers is valid not only for real numbers, but also for complex ones.

Usually complex numbers are represented in algebraic form z = x + i y. The inverse of the given number is the fraction

1 x + i y. For convenience, you can shorten this expression by multiplying the numerator and denominator by x - i y.

Example. The inverse of a complex number

Let there be a complex number z = 4 + i. Let's find the inverse of it.

The inverse of z = 4 + i will be equal to 1 4 + i.

Multiply the numerator and denominator by 4 - i and get:

1 4 + i = 4 - i 4 + i 4 - i = 4 - i 4 2 - i 2 = 4 - i 16 - (- 1) = 4 - i 17.

Besides the algebraic form, the complex number can be expressed in trigonometric or exponential form as follows:

z = r cos φ + i sin φ

z = r e i φ

Accordingly, the inverse number will be:

1 r cos (- φ) + i sin (- φ)

Let's make sure of this:

r cos φ + i sin φ 1 r cos (- φ) + i sin (- φ) = rr cos 2 φ + sin 2 φ = 1 r ei φ 1 rei (- φ) = rre 0 = 1

Consider examples with the representation of complex numbers in trigonometric and exponential forms.

Find the reciprocal of 2 3 cos π 6 + i sin π 6.

Taking into account that r = 2 3, φ = π 6, we write the inverse number

3 2 cos - π 6 + i sin - π 6

Example. Find the inverse of a complex number

What is the inverse of 2 · e i · - 2 π 5.

Answer: 1 2 e i 2 π 5

The sum of mutually reciprocal numbers. Inequality

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

Sum of reciprocal numbers

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

Let us present the proof of the theorem. As you know, for any positive numbers a and b, the arithmetic mean is greater than or equal to the geometric mean. This can be written as an inequality:

a + b 2 ≥ a b

If instead of the number b we take the inverse of a, the inequality takes the form:

a + 1 a 2 ≥ a 1 a a + 1 a ≥ 2

Q.E.D.

Let's give a practical example to illustrate this property.

Example. Find the sum of mutually reciprocal numbers

Calculate the sum of the numbers 2 3 and its inverse.

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

As the theorem says, the resulting number is greater than two.

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