Dividing a decimal by a natural number. "Dividing decimals by natural numbers
Lesson: "Dividing a decimal fraction by a natural number"
Mathematic teacher
Starodubtseva Elena Alekseevna
Kursk, 2015
Lesson topic: "Dividing a decimal fraction by a natural number"
Lesson type :
Lesson for studying new material on the topic "Dividing a decimal fraction by a natural number."
Goals:
Educational:
study and work out the algorithm for solving examples on the topic "Dividing a decimal fraction by a natural number."
Developing:
develop attention, logical thinking, to intensify mental activity through the use of information technology, to establish interdisciplinary connections between mathematics and geography.
Educational:
to instill an interest in mathematics, to cultivate a sense of responsibility, collectivism, diligence, accuracy, develop common culture personality, environmental education.
Forms of organization learning activities : collective, group, individual.
Equipment : computer, projector, interactive whiteboard.
Didactic support of the lesson : presentation “Dividing a decimal fraction by a natural number””, an excerpt from the film “Lake Baikal” , ropes on each desk, measuring devices, multi-colored estimates.
During the classes .
Teacher:
Hello guys! Greet your desk mate and guests with a smile!
Emotional mood for the lesson.
Children, are you warm? (Yes!)
Has the bell rung already? (Yes!)
Has the lesson just started? (Yes!)
Do you want to study? (Yes!)
So everyone can sit down!
I wish you Have a good mood and activity in the classroom.
Lesson motivation. slide 1
Who doesn't study
He doesn't notice anything.
Who doesn't notice
He is always whining and bored.
Poet R. Sef
- And so that you guys don’t get bored in the lesson, everyone should take an active part. In this lesson, we will be given the right to make many discoveries.
Oral work Cards
Exercise. slide 2-4
1. If you are on these numbers
Fix your gaze with attention
You will find a pattern
And keep on numbersrow:
a) 1.2; 1.8; 2.4; 3…3,6; 4,2
b) 9.6; 8.9; 8.2; 7.5…6,8; 6,1
c) 0.9; 1.8; 3.6; 7.2…14,4; 28,8
2. Follow these steps:
2,5 – 1,6 0,9
2,7 + 1,6 4,3
0,55 + 0,45 1
4 – 0,8 3,2
4,71 *10 47,1
1,6 * 5 8
1,2 *3 3,6
3,2 *100 320
0,3 * 2 0,6
The first examples are related to adding and subtracting decimals. Recall the rule: To add (subtract) decimal fractions, you need:
equalize in these fractions the number of decimal places;
write them one under the other so that the comma is written under the comma;
perform addition (subtraction), ignoring the comma;
put a comma under the comma in the given fractions in the answer.
The following examples are related to the rule for multiplying a decimal by a natural number: To multiply decimal to a natural number, you need:
1) multiply it by this number, ignoring the comma,
2) in the resulting product, separate with a comma as many digits on the right as there are separated by a comma in a decimal fraction.
To multiply a decimal fraction by 10,100,1000, etc., it is necessary to move the comma in this fraction to as many digits to the right as there are zeros in the multiplier after one.
3. Perform and division:
2,15:10 = 0,215 11,3: 100 = 0,113 16,8:10= 1,68 23,7:1000= 0,0237
Teacher:
Take a close look at the pictures of the lake on slide 5. This lake is close to the heart of every Russian person, is the pearl of Russia. What is this lake? Yes, this is Lake Baikal.
(There is an excerpt from the film about Lake Baikal) on the 2,13 stop
What is the nature of Lake Baikal?
What did you see in this film?
Very often, when people travel along Lake Baikal, they cannot do without a rope, as there are mountains along the banks.
Laboratory work. Explanation of new material. slide 6
Teacher:
There are strings on your tables, you work in pairs. Measure the length of the rope in millimeters and write down the result in a notebook.
You could get different measurement results, let's agree that the length of the rope is 116mm.
Very often it is necessary to divide the rope into parts.
How can a rope be divided into four equal parts without measuring instruments? The rope can be folded in half, and then in half again.
Let's do the division:
116:4=29(mm)
We have divided a natural number by a natural number.
Let's try to write the division in a column.
(on the board the division is written in a column - in detail.)
A task.The length of the rope is 11.6 cm. How to divide the rope into four
equal parts? Slide 7
Can we divide a decimal fraction by a natural number?
Let's translate the numbers 116 mm and 29 mm into centimeters.
How many in 1 cm mm? 1cm = 10mm.
11.6: 4=2.9 (cm)
There was a division of natural numbers, and there was a division of a decimal fraction by a natural number.
How are these rules different?
When dividing a decimal fraction by a natural number important role the setting of the comma plays, it is set when the division of the integer part ends.
Questions: Slide 8
What is the topic of our today's lesson?
And what are our goals?
Today in class I want to: Slide 9
To know….
Learn to…..
Understand…….
Lesson topic: Division of decimals by integers Slide 10
Targets and goals:
Learn the rule of dividing decimal fractions by natural numbers.
Learn how to divide decimals by natural numbers.
Guys! Who among you can come up with the rule? slide 11
To divide a decimal by a natural number:
divide the fraction by this number, ignoring the comma;
2) put a comma in the private when the division of the whole part ends.
If the whole part less divisor, then the quotient starts from zero integers:
Comma Poem: Slide 12
The sun rises,
the night has disappeared
A comma is not averse to stand up.
You will share the whole part -
Comma do not let the abyss
Put it and part later
Fractional Delhi with difficulty
Because it's easy
You will never share!
Consolidation of new material. slide 13
Let's practice this rule with examples:
Calculate orally:
7,6: 2 = 3,8 0,8: 4 = 0,2
1,4: 7 = 0,2 1,8: 4 = 0,45
6,3: 3 = 2,1 3,9: 3 = 1,3
Solving and writing examples from the textbook
The second part of the rule (if the integer part is less than the divisor).
Imagine a fraction142 as a decimal. (28,4 )
Fizminutka
Let's look at the next slide. It depicts the indigenous inhabitants of Lake Baikal - fur seals.
Task number 1. slide 15
The world's fresh water reserves are 115 million tons (0.115 billion tons). One fifth of the world's fresh water reserves are located in Lake Baikal. How many billion tons of fresh water is contained in Lake Baikal?
To solve this problem, you need to find one-fifth of the number 0.115.
0.115:5=0.023 (billion tons)
Answer: 0.023 billion tons.
If we consider the following slide 16, we will see that Lake Baikal does not look like a calm lake, but resembles the sea. This happens because Lake Baikal is the deepest lake in the world.
The depth of Lake Baikal is 1642 meters.
Task number 2. Slide 17
At one of the islands, the depth of Lake Baikal is 1.61 km, and the depth of Lake Ladoga is 7 times less. Find the depth of Lake Ladoga.
1.61:7=0.23(km)=230(m)
Answer: 230 meters.
Independent work. Slide 18
Follow the steps, select a letter and get the name of a fish that is found only in Lake Baikal.
72.8: 8 = 9.1 0.03 - b
5.1:17 = 0.3 5.3 - y
26.5:5 \u003d 5.3 9.1 - about
1.6: 8 \u003d 0.2 0, 2 - l
0.48: 16 = 0.03 0.3 - m
This fish is called omul, it is found only in Lake Baikal, it is an unusually tender and pleasant-tasting fish, and whitefish, sturgeon, and grayling are also found in the lake.
Mysteries of Lake Baikal Slide 19
Today you are fifth-graders, but in the future, maybe some of you will have to unravel the mysteries of Lake Baikal. Every year, as soon as ice appears on the lake, you can see circles of various sizes on its surface. You can see it on the slide. There are many versions of this riddle: aliens draw them on the ice, underwater currents influence this phenomenon, the composition of the water allows you to make drawings .. But so far the nature of this phenomenon has not been solved.
Environmental problems
There is a big environmental problem associated with Lake Baikal. A pulp and paper mill was built on it, residents pollute the shores of the lake when they come to rest.
Slide 20
King among other lakes,
In the kingdom of the sun, forests, mountains,
Baikal rules
to drink everyone, to feed would be glad
But people don't understand
That Baikal will be a desert,
The mighty king dies
The forest is not the one that is old,
And in the crystal waters
Dirt is drained and waste,
Dying fish, beast and bird
Vodka is poisoned ... ..
Told me about it
Glorious king of lakes Baikal.
He asked you guys
Help him now!
And you, when you come to the lakes, do you always clean up after yourself, put the shores in order. After all, we have many beautiful lakes!
Homework Slide 2 1
* Using any map (knowing its scale), determine the length and width of Lake Baikal.
Lesson summary:
- Today at the lesson: slide 22
I found out……
I learned…..
I understand…..
Today in the lesson we made many discoveries: we studied the rule for dividing a decimal fraction by a natural number (repeat the rule), learned the name of a fish that is found only on Lake Baikal, learned that Lake Baikal is the deepest lake in the world, and it is fraught with a lot of unsolved mysteries.
Parable:
A wise man was walking, and three people were walking towards him, who were carrying carts with stones for construction under the hot sun. The sage stopped and asked each one a question. He asked the first one: “What did you do all day?”. And he replied with a grin that he had been carrying cursed stones all day. The sage asked the second: “What did you do all day?”, And he answered: “I did my job conscientiously.” And the third smiled, his face lit up with joy and pleasure: “And I took part in the construction of the temple!”.
Guys! Let's try to evaluate each of our work for the lesson.
slide 23
The children write their grades on the board. The song "Sacred Baikal" sounds.
Thank each other for Good work applause.
Goodbye! The lesson is over.
A fraction is one or more parts of a whole, which is usually taken as a unit (1). As with natural numbers, you can perform all basic arithmetic operations with fractions (addition, subtraction, division, multiplication), for this you need to know the features of working with fractions and distinguish between their types. There are several types of fractions: decimal and ordinary, or simple. Each type of fractions has its own specifics, but once you have thoroughly figured out how to deal with them once, you will be able to solve any examples with fractions, since you will know the basic principles for performing arithmetic calculations with fractions. Let's look at examples of how to divide a fraction by an integer using different types fractions.
How to divide a fraction by a natural number?Ordinary or simple fractions are called, written in the form of such a ratio of numbers, in which the dividend (numerator) is indicated at the top of the fraction, and the divisor (denominator) of the fraction is indicated below. How to divide such a fraction by an integer? Let's look at an example! Let's say we need to divide 8/12 by 2.
To do this, we must perform a series of actions:
Thus, if we are faced with the task of dividing a fraction by an integer, the solution scheme will look something like this: 
Similarly, you can divide any ordinary (simple) fraction by an integer.
How to divide a decimal by an integer?
A decimal fraction is a fraction that is obtained by dividing a unit into ten, a thousand, and so on parts. Arithmetic operations with decimal fractions are quite simple.
Consider an example of how to divide a fraction by an integer. Let's say we need to divide the decimal fraction 0.925 by the natural number 5.
Summing up, we will focus on two main points that are important when performing the operation of dividing decimal fractions by an integer: - to divide a decimal fraction by a natural number, division into a column is used;
- a comma is placed in the private when the division of the integer part of the dividend is completed.
The rule for dividing decimal fractions by natural numbers.
Four identical toys cost 921 rubles 20 kopecks in total. How much does one toy cost (see Fig. 1)?
Rice. 1. Illustration for the problem
Solution
To find the cost of one toy, you need to divide this amount by four. Let's convert the amount to kopecks:
Answer: the cost of one toy is 23,030 kopecks, that is, 230 rubles 30 kopecks, or 230.3 rubles.
You can solve this problem without converting rubles into kopecks, that is, divide the decimal fraction by a natural number:.
To divide a decimal fraction by a natural number, you need to divide the fraction by this number, as natural numbers are divided, and put in a private comma when the division of the whole part is over.
We divide in a column as we divide natural numbers. After we demolish the number 2 (the number of tenths is the first digit after the decimal point in the record of the dividend 921.20), put a comma in the quotient and continue the division:
Answer: 230.3 rubles.
We divide in a column as we divide natural numbers. After we take down the number 6 (the number of tenths is the number after the decimal point in the record of the dividend 437.6), put a comma in the quotient and continue the division:
If the dividend is less than the divisor, then the quotient will start from zero.
1 is not divisible by 19, so we put zero in the quotient. The division of the integer part is over, in the private we put a comma. We demolish 7. 17 is not divisible by 19, in private we write zero. We demolish 6 and continue the division:
We divide as we divide natural numbers. In the quotient, we put a comma as soon as we take down 8 - the first digit after the decimal point in the dividend 74.8. Let's continue the division. When subtracting, we get 8, but the division is not over. We know that zeros can be added at the end of a decimal fraction - this will not change the value of the fraction. We assign zero and divide 80 by 10. We get 8 - the division is over.
To divide a decimal fraction by 10, 100, 1000, etc., you need to move the comma in this fraction as many digits to the left as there are zeros after one in the divisor.
In this lesson, we learned how to divide a decimal fraction by a natural number. We considered a variant with an ordinary natural number, as well as a variant in which division by a bit unit occurs (10, 100, 1000, etc.).
Solve the equations:
To find an unknown divisor, you need to divide the dividend by the quotient. That is .
We divide into a column. After we demolish the number 4 (the number of tenths is the first digit after the decimal point in the record of the dividend 134.4), put a comma in the quotient and continue the division:
Find the first digit of the quotient (the result of division). To do this, divide the first digit of the dividend by the divisor. Write the result under the divisor.
- In our example, the first digit of the dividend is 3. Divide 3 by 12. Since 3 is less than 12, then the result of the division will be 0. Write 0 under the divisor - this is the first digit of the quotient.
Multiply the result by the divisor. Write the result of the multiplication under the first digit of the dividend, since this is the number you just divided by the divisor.
- In our example, 0 × 12 = 0, so write 0 under 3.
Subtract the result of the multiplication from the first digit of the dividend. Write your answer on a new line.
- In our example: 3 - 0 = 3. Write 3 directly below 0.
Move down the second digit of the dividend. To do this, write down next digit divisible next to the result of the subtraction.
- In our example, the dividend is 30. The second digit of the dividend is 0. Move it down by writing 0 next to 3 (the result of the subtraction). You will get the number 30.
Divide the result by a divisor. You will find the second digit of the private. To do this, divide the number on the bottom line by the divisor.
- In our example, divide 30 by 12. 30 ÷ 12 = 2 plus some remainder (because 12 x 2 = 24). Write 2 after 0 under the divisor - this is the second digit of the quotient.
- If you cannot find a suitable digit, iterate over the digits until the result of multiplying any digit by a divisor is less than and closest to the number located last in the column. In our example, consider the number 3. Multiply it by the divisor: 12 x 3 = 36. Since 36 is greater than 30, the number 3 is not suitable. Now consider the number 2. 12 x 2 = 24. 24 is less than 30, so the number 2 is the correct solution.
Repeat the steps above to find the next digit. The described algorithm is used in any long division problem.
- Multiply the second quotient by the divisor: 2 x 12 = 24.
- Write the result of multiplication (24) under last number in column (30).
- Subtract the smaller number from the larger one. In our example: 30 - 24 = 6. Write the result (6) on a new line.
If there are digits left in the dividend that can be moved down, continue the calculation process. Otherwise, proceed to the next step.
- In our example, you moved down the last digit of the dividend (0). So move on to the next step.
If necessary, use a decimal point to expand the dividend. If the dividend is evenly divisible by the divisor, then on the last line you will get the number 0. This means that the problem is solved, and the answer (in the form of an integer) is written under the divisor. But if any digit other than 0 is at the very bottom of the column, you need to expand the dividend by putting a decimal point and assigning 0. Recall that this does not change the value of the dividend.
- In our example, the number 6 is on the last line. Therefore, to the right of 30 (dividend), write a decimal point, and then write 0. Also put a decimal point after the quotient digits found, which you write under the divisor (do not write anything after this comma yet!) .
Repeat the above steps to find the next digit. The main thing is not to forget to put a decimal point both after the dividend and after the found digits of the private. The rest of the process is similar to the process described above.
- In our example, move down the 0 (which you wrote after the decimal point). You will get the number 60. Now divide this number by the divisor: 60 ÷ 12 = 5. Write 5 after the 2 (and after the decimal point) below the divisor. This is the third digit of the quotient. So the final answer is 2.5 (the zero in front of the 2 can be ignored).
Consider examples of dividing decimals in this light.
Example.
Divide decimal 1.2 by decimal 0.48.
Solution.
Answer:
1,2:0,48=2,5 .
Example.
Divide the periodic decimal 0.(504) by the decimal 0.56 .
Solution.
Let's translate the periodic decimal fraction into an ordinary:. We also translate the final decimal fraction 0.56 into an ordinary one, we have 0.56 \u003d 56/100. Now we can move from dividing the original decimals to dividing ordinary fractions and finish the calculations: .
Let's translate the resulting ordinary fraction into a decimal fraction by dividing the numerator by the denominator in a column:
Answer:
0,(504):0,56=0,(900) .
The principle of dividing infinite non-periodic decimal fractions differs from the principle of dividing finite and periodic decimal fractions, since non-repeating decimal fractions cannot be converted to ordinary fractions. The division of infinite non-periodic decimal fractions is reduced to the division of finite decimal fractions, for which it is carried out rounding numbers up to a certain level. Moreover, if one of the numbers with which the division is carried out is a final or periodic decimal fraction, then it is also rounded to the same digit as the non-periodic decimal fraction.
Example.
Divide the infinite non-recurring decimal 0.779... by the final decimal 1.5602.
Solution.
First, you need to round the decimal fractions in order to go from dividing an infinite non-repeating decimal fraction to dividing finite decimal fractions. We can round to hundredths: 0.779…≈0.78 and 1.5602≈1.56. Thus, 0.779…:1.5602≈0.78:1.56= 78/100:156/100=78/100 100/156= 78/156=1/2=0,5 .
Answer:
0,779…:1,5602≈0,5 .
Dividing a natural number by a decimal fraction and vice versa
The essence of the approach to dividing a natural number by a decimal fraction and to dividing a decimal fraction by a natural number is no different from the essence of dividing decimal fractions. That is, finite and periodic fractions are replaced by ordinary fractions, and infinite non-periodic fractions are rounded.
To illustrate, consider the example of dividing a decimal fraction by a natural number.
Example.
Divide the decimal fraction 25.5 by the natural number 45.
Solution.
Replacing the decimal fraction 25.5 with an ordinary fraction 255/10=51/2, division is reduced to dividing an ordinary fraction by a natural number: . The resulting fraction in decimal notation is 0.5(6) .
Answer:
25,5:45=0,5(6) .
Division of a decimal fraction by a natural number by a column
Division of final decimal fractions by natural numbers is conveniently carried out by a column by analogy with division by a column of natural numbers. Here is the division rule.
To divide a decimal by a natural number by a column, necessary:
- add a few digits to the right in the divisible decimal fraction 0, (during the division, if necessary, you can add any number of zeros, but these zeros may not be needed);
- perform division by a column of a decimal fraction by a natural number according to all the rules for dividing by a column of natural numbers, but when the division of the integer part of the decimal fraction is completed, then in the private one you need to put a comma and continue the division.
Let's say right away that as a result of dividing a finite decimal fraction by a natural number, either a final decimal fraction or an infinite periodic decimal fraction can be obtained. Indeed, after the division of all decimal places of the divisible fraction other than 0, we can get either a remainder 0, and we will get a final decimal fraction, or the remainder will begin to repeat periodically, and we will get a periodic decimal fraction.
Let's deal with all the intricacies of dividing decimal fractions into natural numbers by a column when solving examples.
Example.
Divide the decimal 65.14 by 4 .
Solution.
Let's perform the division of a decimal fraction by a natural number by a column. Let's add a pair of zeros to the right in the record of the fraction 65.14, while we get the decimal fraction equal to it 65.1400 (see equal and unequal decimal fractions). Now you can start dividing the integer part of the decimal fraction 65.1400 by a natural number 4 by a column: 
This completes the division of the integer part of the decimal fraction. Here in private you need to put a decimal point and continue the division: 
We have come to a remainder of 0, at this stage the division by a column ends. As a result, we have 65.14:4=16.285.
Answer:
65,14:4=16,285 .
Example.
Divide 164.5 by 27.
Solution.
Let's divide a decimal fraction by a natural number by a column. After dividing the integer part, we get the following picture: 
Now we put a comma in private and continue the division with a column: 
Now it is clearly seen that the remnants of 25, 7 and 16 have begun to repeat, while the numbers 9, 2 and 5 are repeated in the quotient. So dividing the decimal 164.5 by 27 gives us the periodic decimal 6.0(925) .
Answer:
164,5:27=6,0(925) .
Division of decimal fractions by a column
The division of a decimal fraction by a decimal fraction can be reduced to dividing a decimal fraction by a natural number by a column. To do this, the dividend and the divisor must be multiplied by such a number 10, or 100, or 1000, etc., so that the divisor becomes a natural number, and then divide by a natural number by a column. We can do this due to the properties of division and multiplication, since a:b=(a 10):(b 10) , a:b=(a 100):(b 100) and so on.
In other words, to divide a ending decimal by a ending decimal, need:
- in the dividend and divisor, move the comma to the right by as many characters as there are after the decimal point in the divisor, if at the same time there are not enough characters in the dividend to move the comma, then you need to add the required number of zeros to the right;
- after that, carry out the division by a column of a decimal fraction by a natural number.
Consider, when solving an example, the application of this rule for dividing by a decimal fraction.
Example.
Do column division 7.287 by 2.1.
Solution.
Let's move the comma in these decimal fractions one digit to the right, this will allow us to go from dividing the decimal fraction 7.287 by the decimal fraction 2.1 to dividing the decimal fraction 72.87 by the natural number 21. Let's divide by a column: 
Answer:
7,287:2,1=3,47 .
Example.
Divide decimal 16.3 by decimal 0.021.
Solution.
Move the comma in the dividend and divisor to the right by 3 digits. Obviously, there are not enough digits in the divisor to carry the comma, so let's add the required number of zeros to the right. Now let's divide the column of the fraction 16300.0 by the natural number 21: 
From this moment, the remainders 4, 19, 1, 10, 16 and 13 begin to repeat, which means that the numbers 1, 9, 0, 4, 7 and 6 in the quotient will also repeat. As a result, we get a periodic decimal fraction 776,(190476) .
Answer:
16,3:0,021=776,(190476) .
Note that the voiced rule allows you to divide a natural number by a final decimal fraction by a column.
Example.
Divide the natural number 3 by the decimal fraction 5.4.
Solution.
After moving the comma 1 digit to the right, we come to dividing the number 30.0 by 54. Let's divide by a column: 
.
This rule can also be applied when dividing infinite decimal fractions by 10, 100, .... For example, 3,(56):1000=0.003(56) and 593.374…:100=5.93374… .
Dividing decimals by 0.1, 0.01, 0.001, etc.
Since 0.1 \u003d 1/10, 0.01 \u003d 1/100, etc., then from the rule of division by an ordinary fraction it follows that dividing a decimal fraction by 0.1, 0.01, 0.001, etc. . it's like multiplying the given decimal by 10 , 100 , 1000 , etc. respectively.
In other words, to divide a decimal fraction by 0.1, 0.01, ... you need to move the comma to the right by 1, 2, 3, ... digits, and if there are not enough digits in the decimal fraction to move the comma, then you need to add the required number to the right zeros.
For example, 5.739:0.1=57.39 and 0.21:0.00001=21,000 .
The same rule can be applied when dividing infinite decimals by 0.1, 0.01, 0.001, etc. In this case, you should be very careful with the division of periodic fractions, so as not to be mistaken with the period of the fraction, which is obtained as a result of division. For example, 7.5(716):0.01=757,(167) , since after moving the comma in the decimal fraction record 7.5716716716… two digits to the right, we have the record 757.167167…. With infinite non-periodic decimals, everything is simpler: 394,38283…:0,001=394382,83… .
Dividing a fraction or mixed number by a decimal and vice versa
Dividing a common fraction or a mixed number by a finite or recurring decimal, or dividing a finite or recurring decimal by a common fraction or mixed number is reduced to the division of ordinary fractions. To do this, decimal fractions are replaced by the corresponding ordinary fractions, and the mixed number is represented as an improper fraction.
When dividing an infinite non-periodic decimal fraction by an ordinary fraction or a mixed number and vice versa, one should proceed to the division of decimal fractions, replacing the ordinary fraction or mixed number with the corresponding decimal fraction.
Bibliography.
- Maths: studies. for 5 cells. general education institutions / N. Ya. Vilenkin, V. I. Zhokhov, A. S. Chesnokov, S. I. Shvartsburd. - 21st ed., erased. - M.: Mnemosyne, 2007. - 280 p.: ill. ISBN 5-346-00699-0.
- Maths. Grade 6: textbook. for general education institutions / [N. Ya. Vilenkin and others]. - 22nd ed., Rev. - M.: Mnemosyne, 2008. - 288 p.: ill. ISBN 978-5-346-00897-2.
- Algebra: textbook for 8 cells. general education institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova]; ed. S. A. Telyakovsky. - 16th ed. - M. : Education, 2008. - 271 p. : ill. - ISBN 978-5-09-019243-9.
- Gusev V. A., Mordkovich A. G. Mathematics (a manual for applicants to technical schools): Proc. allowance.- M.; Higher school, 1984.-351 p., ill.
