Mathematics. Algebra. Geometry. Trigonometry

ALGEBRA: Numbers

2.2. Integers and rational numbers. Interest

Ordinary fractions.

Common fraction

is a number of the form , where m and n are natural numbers. The number m is called numerator of the fraction, n- denominator. If n = 1, then the fraction has the form , but more often they write simply m, i.e. Any natural number can be represented as a common fraction with a denominator of 1.

The fraction is called correct if its numerator is less than its denominator, and wrong if its numerator is greater than or equal to the denominator. Every improper fraction can be represented as the sum of a natural number and a proper fraction (or as a natural number if m is a multiple of n).

It is customary to write the sum of a natural number and a proper fraction without the addition sign, i.e. instead of writing . A number written in this form is called mixed number. It consists of an integer and a fractional part.

Equality of fractions. Reducing fractions.

Two fractions are counted equal if ad = bc. From the definition of equality it follows that

= , because . The main property of a fraction:If the numerator and denominator of a fraction are multiplied or divided by the same natural number, you get a fraction equal to the given one. Using the basic property of a fraction, it is sometimes possible to replace a given fraction with another whose numerator and denominator are less than the data. This replacement is called reduction fractions If the numerator and denominator are mutually prime numbers, then reduction is not possible and such a fraction is called irreducible.

Arithmetic operations on ordinary fractions.

Let two fractions be given and

, . You can replace these fractions with others equal to them, so that the resulting fractions have the same denominators. This transformation is called bringing fractions to a common denominator. Usually they try to reduce fractions to lowest common denominator, which is equal to N.O.K.().

1.Addition ordinary fractions is done like this:

A) if the denominators are the same, then the numerators are added and leave the same denominator:;

2. Subtraction ordinary fractions is performed as follows:

A) if the denominators are the same, then

b) if the denominators of the fractions are different, then the fractions are first reduced to the lowest common denominator, and then rule a) is applied.

3. Multiplicationordinary fractions is performed as follows:

4. The division of ordinary fractions is performed as follows:

.

Decimals. Converting a decimal fraction to a common fraction.

A decimal is another form of writing a fraction with a denominator. For example, . If the denominator of a fraction is factored only by 2 and 5, then the fraction can be written as a decimal; If the fraction is irreducible and the decomposition of its denominator into prime factors includes other prime factors, then this fraction cannot be written as a decimal.

In a decimal fraction, you can add and discard zeros on the right - you get a fraction equal to it.

A fraction that has an infinite number of decimal places is called infinite decimal fraction.

Theorem 10.

Any common fraction can be represented as an infinite decimal fraction.

A sequentially repeating group of digits (minimum) after the decimal point in the decimal notation of a number is called a period, and an infinite decimal fraction having a period is called periodic.

Let it be given by a periodic decimal fraction: , where - m-digit number, then

, YU
YU - formula for converting a periodic decimal fraction into an ordinary fraction.

Interest.

Among decimal fractions, the most commonly used fraction is 0.01, which is called percentage and is denoted by 1

%. So 1% = 0.01; 25% = 0.25; 450% = 4.5, etc.

EXAMPLE A worker had to produce 60 parts per shift. At the end of the working day it turned out that he had completed 125

% tasks. How many parts did the worker make?

Solution: 1) 125

% = 1,25

2)60H 1.25 = 75.

ANSWER: 75 parts.

Coordinate line.

Let's take a straight line l, mark a point O on it, which we will take as the origin, set the direction and the unit segment. In this case they say that given coordinate line. Each natural number or fraction corresponds to one point on the line l. If a point M of a line l corresponds to a certain number r, then this number is called coordinate point M and is denoted by M(r). The numbers a and -a are called opposite. The numbers that correspond to points located on a coordinate line in a given direction are called positive; numbers that correspond to points located on a coordinate line in the direction opposite to a given one are called negative. The number 0 is considered neither positive nor negative. Point O, corresponding to the number 0, separates points with positive coordinates from points with negative coordinates on the coordinate line.

A given direction on a coordinate line is called positive(usually it goes to the right), and the direction opposite to the given one is negative

.

Integers and rational numbers.

The natural numbers 1, 2, 3, ... are also called positive integers. The numbers -1, -2, -3, ..., opposite the natural numbers, are called negative integers. The number 0 is also an integer. Whole numbers- natural numbers, their opposites and 0.

Whole numbers and fractions (positive and negative) make up the set rational numbers.

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2 MAIN WAVE 2013 CENTER URAL SIBERIA EAST: fractions percentages rational numbers Theory: Set of rational numbers 1 1 ~ HOD ge N Z Basic property 0 0. Proportion is the equality of two ratios. Property: consequences Scheme of directly proportional dependence. Basic properties 1. Order: 0; 0 ; Addition operation: ; HOK 3. Operation of multiplication and division: 4. Transitivity of order relation: 5. Commutativity: 6. Associativity: 7. Distributivity: 8. Presence of zero: Presence of opposite numbers: Presence of one: Presence of reciprocal numbers: R R. 12. Connection of order relation with the addition operation. The same rational number can be added to the left and right sides of a rational inequality. 2 B1

3 13. Connection of the order relation with the multiplication operation. The left and right sides of a rational inequality can be multiplied by the same positive rational number. Axiom of Archimedes. Whatever the rational number, you can take so many units that their sum exceeds a. N k Rational inequalities of the same sign can be added term by term. Any rational fraction can be converted to its equivalent decimal fraction by dividing the numerator by the denominator. 1 remainder may turn out to be equal to zero and the quotient will be expressed as a finite decimal fraction, for example 3:4 = zero in the remainder will never work out since the remainders will be endlessly repeated and the quotient will be expressed as an infinite periodic decimal fraction. For example 2:3=0666 =06 7:13= = :15=21333 = ? Interest. The hundredth part of a number is called its percentage. Three types of problems involving percentages A 100% 1. Finding the percentage of a given number A p% x. x p% 100% To find p% of the number “A” you need to find 1% of “A” A: 100% and multiply by p%. 2. Finding a number from another number and its value as a percentage of the desired number. x 100% 100% x. p% p% To find a number for a given value “a” its p% you need to find 1% of the desired number by dividing the given value “a” by p% and multiply the result by 100% A 100% 3. Finding the percentage of numbers. 100% x% x% A You need to find the ratio of the number “a” to the number “A” and multiply by 100%. 3

4 CENTER Option 1;8. One tablet of the drug weighs 70 mg and contains 4% of the active substance. Does the doctor prescribe 105 mg of the active substance for a child under 6 months of age and weighing 8 kg per day? Option 2. One tablet of the drug weighs 20 mg and contains 5% of the active substance. Does the doctor prescribe 04 mg of the active substance for a child under 6 months of age for each child aged three months and weighing 5 kg per day? Option 3. One tablet of the drug weighs 20 mg and contains 5% of the active substance. Does the doctor prescribe 1 mg of the active substance for a child under 6 months of age and weighing 7 kg per day? Option 4;5. One tablet of the drug weighs 20 mg and contains 9% of the active substance. Does the doctor prescribe 135 mg of the active substance for a child under 6 months of age and weighing 8 kg for each child aged four months and weighing 8 kg per day? Option 6. One tablet of the drug weighs 30 mg and contains 5% of the active substance. Does the doctor prescribe 075 mg of the active substance for a child under 6 months of age per 5 months and weighing 8 kg per day? Option 7. One tablet of the drug weighs 40 mg and contains 5% of the active substance. Does the doctor prescribe 125 mg of the active substance for a child under 6 months of age and weighing 8 kg per day for each child aged three months and weighing 8 kg? Note that the eight options are made up of six problems with different numerical data but the same content. The necessary information for the calculation was written down in the table: Weight of one Percentage content Options Recipe mg Weight of the child kg tablets mg of the active substance % 1 and and Solution of option 1. Idea: The percentage of the active substance in one tablet is known, which means you can find the corresponding amount of the substance in mg. Knowing the child’s weight and the dosage of the active substance per 1 kg of weight, you can find the daily dose of the active substance. Then the number of tablets is the quotient of the daily norm of the active substance divided by the amount of active substance in one tablet. Actions: 1. Determine the amount of active substance in one tablet. Let’s make a proportion: take the weight of one tablet 70 mg as 100% and 4% of this weight will be x mg the amount of active substance in one tablet. Let us write this proportion schematically. From here we find the unknown term of the proportion. To do this, you need to multiply x 4% the known terms of one diagonal and divide by the known term of the other diagonal: 70 4% x 28 mg. 100% 4

5 2. Determine the amount of the active substance prescribed by the doctor according to the prescription, taking into account the child’s weight. The dose of the substance must be multiplied by the child’s weight: mg. This means that the child needs to take 84 mg of the active substance per day. Determine the number of tablets containing 84 mg of the active substance. 3 tab. 28 Answer 3. Other options are solved similarly. IN URAL Option 1;5. In the apartment where Anastasia lives, a cold water meter is installed. On September 1, the meter showed a consumption of 122 cubic meters of water, and on October 1, 142 cubic meters. What amount should Anastasia pay for cold water in September if the price of 1 cubic meter of cold water is 9 rubles 90 kopecks? Give your answer in rubles. Option 2. In the apartment where Maxim lives, a cold water meter is installed. On February 1, the meter showed a consumption of 129 cubic meters of water, and on March 1, 140 cubic meters. What amount should Maxim pay for cold water in February if the price of 1 cubic meter of cold water is 10 rubles 60 kopecks? Give your answer in rubles. Option 3. In the apartment where Alexey lives, a cold water meter is installed. On June 1, the meter showed a consumption of 151 cubic meters of water, and on July 1, 165 cubic meters. What amount should Alexey pay for cold water in March if the price of 1 cubic meter of cold water is 20 rubles. 80 kopecks? Give your answer in rubles. Option 4. In the apartment where Asya lives, a hot water meter is installed. On May 1, the meter showed a consumption of 84 cubic meters of water, and on June 1, 965 cubic meters. What amount should Anastasia pay for hot water in January if the price of 1 cubic meter of hot water is 72 rubles. 60 kopecks? Give your answer in rubles. Option 6;8. In the apartment where Anfisa lives there is a hot water meter installed. On September 1, the meter showed a consumption of 239 cubic meters of water, and on October 1, 349 cubic meters. What amount should Anfisa pay for hot water in September if the price of 1 cubic meter of hot water is 78 rubles. 60 kopecks? Give your answer in rubles. Option 7. In the apartment where Alla lives, a hot water meter is installed. On July 1, the meter showed a consumption of 772 cubic meters of water, and on August 1, 797 cubic meters. What amount should Alla pay for hot water in July if the price of 1 cubic meter of hot water is 144 rubles. 80 kopecks? Give your answer in rubles. The URAL region solved the problem of paying for water consumption using a meter. The numerical data for the calculation of the options was entered into the table: Vari Meter readings at the beginning Meter readings at the beginning Price 1 cubic meter ante of the calendar month cubic meter of the next calendar month cubic meter 1 and ruble 90 kopecks ruble 60 kopecks ruble 80 kopecks ruble 60 kopecks 6 and ruble 60 kopecks ruble 80 kopecks Solution to option 1. Idea: The meter readings are known at the beginning of the calendar month cubic meters and at the beginning of the next calendar month cubic meters. This means you can find out the monthly water consumption to be paid. Knowing the number of cubic meters of water consumed and the price of one cubic meter of water, you can find the amount you need to pay for this water. 5

6 Actions: Determine water consumption for the month Determine the amount to be paid for consumed water for the month p Answer 198. The remaining options are solved in the same way. TO SIBERIA Option 1. 1 kilowatt-hour of electricity costs 1 ruble 40 kopecks. The electricity meter showed kilowatt-hours on June 1 and showed kilowatt-hours on July 1. How much do you need to pay for electricity for June? Give your answer in rubles. Option 2. 1 kilowatt-hour of electricity costs 1 ruble 20 kopecks. The electricity meter on November 1 showed 669 kilowatt-hours and on December 1 showed 846 kilowatt-hours. How much should I pay for electricity for November? Give your answer in rubles. Option 3. 1 kilowatt-hour of electricity costs 2 rubles 40 kopecks. The electricity meter showed kilowatt-hours on October 1 and kilowatt-hours on November 1st. How much should I pay for electricity in October? Give your answer in rubles. Option 4;5. 1 kilowatt-hour of electricity costs 2 rubles 50 kopecks. The electricity meter on January 1 showed kilowatt-hours and on February 1 it showed kilowatt-hours. How much should I pay for electricity in January? Give your answer in rubles. Option 6. 1 kilowatt-hour of electricity costs 1 ruble 30 kopecks. The electricity meter showed kilowatt-hours on September 1 and showed kilowatt-hours on October 1. How much should I pay for electricity for September? Give your answer in rubles. Option 7;8. 1 kilowatt-hour of electricity costs 1 ruble 70 kopecks. On April 1, the electricity meter showed kilowatt-hours and on May 1st it showed kilowatt-hours. How much should I pay for electricity for April? Give your answer in rubles. The SIBERIA region solved the problem of paying for electricity consumption by meter. Numerical data for calculation according to the options were entered into the table: Options Meter readings at the beginning of the calendar month kWh Meter readings at the beginning of the next calendar month kWh Cost of 1 kilowatt-hour ruble 40 kopecks ruble 20 kopecks ruble 40 kopecks 4 and ruble 50 kopecks ruble 30 7 kopecks and 70 kopecks ruble Solution to option 1. Idea: The meter readings at the beginning of the kilowatt-hour calendar month and at the beginning of the next kilowatt-hour calendar month are known. This means you can find out the monthly electricity consumption to be paid. Knowing the number of kilowatt-hours of electricity consumed and the price of one kilowatt-hour, you can find the amount you need to pay for this electricity. Actions: Determine electricity consumption for the month Determine the amount to pay for consumed electricity for the month. 6

7 p Answer The remaining options are solved similarly. TO THE EAST Option1;5;8. In the apartment where Ekaterina lives, a cold water meter is installed. On September 1, the meter showed a consumption of 189 cubic meters of water, and on October 1, 204 cubic meters. What amount should Ekaterina pay for cold water in September if the price of 1 cubic meter of cold water is 16 rubles 90 kopecks? Give your answer in rubles. Option 2. In the apartment where Valery lives, a cold water meter is installed. On March 1, the meter showed a consumption of 182 cubic meters of water, and on April 1, 192 cubic meters. What amount should Valery pay for cold water in March if the price of 1 cubic meter of cold water is 23 rubles. 10 kopecks? Give your answer in rubles. Option 3. In the apartment where Marina lives, a cold water meter is installed. On July 1, the meter showed a consumption of 120 cubic meters of water, and on August 1, 131 cubic meters. What amount should Marina pay for cold water in July if the price of 1 cubic meter of cold water is 20 rubles 60 kopecks? Give your answer in rubles. Option 4. In the apartment where Egor lives, a hot water meter is installed. On November 1, the meter showed a consumption of 879 cubic meters of water, and on December 1, 969 cubic meters. What amount should Egor pay for hot water in November if the price of 1 cubic meter of hot water is 108 rubles. 20 kopecks? Give your answer in rubles. Option 6. In the apartment where Mikhail lives, a hot water meter is installed. On March 1, the meter showed a consumption of 708 cubic meters of water, and on April 1, 828 cubic meters. What amount should Mikhail pay for hot water in March if the price of 1 cubic meter of hot water is 72 rubles. 20 kopecks? Give your answer in rubles. Option 7. In the apartment where Anastasia lives, a hot water meter is installed. On January 1, the meter showed a consumption of 894 cubic meters of water, and on February 1, 919 cubic meters. What amount should Anastasia pay for hot water in January if the price of 1 cubic meter of hot water is 103 rubles. 60 kopecks? Give your answer in rubles. The tasks of the VOSTOK region coincided with the tasks of the URAL region with a difference in numerical data. Options Meter readings at the beginning of a calendar month, cubic meters Meter readings at the beginning of the next calendar month, cubic meters Price 1 cubic meter 1 and 5 and ruble 90 kopecks ruble 10 kopecks ruble 60 kopecks ruble 20 kopecks ruble 20 kopecks ruble 60 kopecks Therefore, the idea of ​​the solution and the actions will be similar to those discussed earlier for the URAL region. IN

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An ordinary fraction is a number of the form where the type is natural numbers, for example The number is called the numerator of the fraction, the denominator. In particular, maybe in this case the fraction has the form, but more often it is written simply. This means that any natural number can be represented as an ordinary fraction with a denominator of 1. Notation - another version of notation

Common fractions are divided into proper and improper

fractions A fraction is called proper if its numerator is less than its denominator, and improper if its numerator is greater than or equal to the denominator.

Any improper fraction can be represented as the sum of a natural number and a proper fraction (or as a natural number, if the fraction is such that it is a multiple of e.g.

Example. Represent an improper fraction as the sum of a natural number and a proper fraction: a)

Solution a)

It is customary to write the sum of a natural number and a proper fraction without the addition sign, i.e. instead of writing instead of writing, a number written in this form is called a mixed number. It consists of two parts: integer and fractional. So, for the number 3 - the integer part is equal to 3, and the fractional part - Any improper fraction can be written as a mixed number (or as a natural number). The converse is also true: every mixed or natural number can be written as an improper fraction. For example, .

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(No. 2491) A ballpoint pen costs 20 rubles. What is the largest number of such pens that can be bought for 700 rubles after the price increases by 15%?

(No. 2503) The notebook costs 40 rubles. What is the largest number of such notebooks that can be purchased for 550 rubles after the price is reduced by 15%?

(No. 2513) The store purchases flower pots at a wholesale price of 100 rubles per piece. The trade margin is 15%. What is the largest number of such pots that can be bought in this store for 1300 rubles?

(No. 2595) A train ticket for an adult costs 550 rubles. The cost of a student ticket is 50% of the cost of an adult ticket. The group consists of 18 schoolchildren and 4 adults. How many rubles are tickets for the whole group?

(No. 2601) The price for an electric kettle was increased by 21% and amounted to 3,025 rubles. How many rubles did the product cost before the price increase?

(No. 2617) The T-shirt cost 800 rubles. After the price was reduced, it began to cost 680 rubles. By what percentage was the price of the T-shirt reduced?

(No. 6193) City N has 250,000 inhabitants. Among them, 15% are children and adolescents. Among adults, 35% do not work (pensioners, housewives, unemployed). How many adults work?

(No. 6235) The client took out a loan of 3,000 rubles from the bank. for a year at 12%. He must repay the loan by depositing the same amount of money into the bank every month in order to repay the entire amount borrowed along with interest after a year. How much should he deposit into the bank monthly?

(No. 24285) Income tax is 13% of wages. After withholding income tax, Maria Konstantinovna received 13,050 rubles. How many rubles is Maria Konstantinovna’s salary?

(No. 24261) Income tax is 13% of wages. Ivan Kuzmich's salary is 14,500 rubles. How many rubles will he receive after deducting income tax?

(No. 2587) The wholesale price of the textbook is 170 rubles. The retail price is 20% higher than the wholesale price. What is the largest number of such textbooks that can be purchased at a retail price of 7,000 rubles?

In this article we will begin to explore rational numbers. Here we will give definitions of rational numbers, give the necessary explanations and give examples of rational numbers. After this, we will focus on how to determine whether a given number is rational or not.

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Definition and examples of rational numbers

In this section we will give several definitions of rational numbers. Despite differences in wording, all of these definitions have the same meaning: rational numbers unite integers and fractions, just as integers unite natural numbers, their opposites, and the number zero. In other words, rational numbers generalize whole and fractional numbers.

Let's start with definitions of rational numbers, which is perceived most naturally.

From the stated definition it follows that a rational number is:

• Any natural number n. Indeed, you can represent any natural number as an ordinary fraction, for example, 3=3/1.
• Any integer, in particular the number zero. In fact, any integer can be written as either a positive fraction, a negative fraction, or zero. For example, 26=26/1, .
• Any common fraction (positive or negative). This is directly confirmed by the given definition of rational numbers.
• Any mixed number. Indeed, you can always represent a mixed number as an improper fraction. For example, and.
• Any finite decimal fraction or infinite periodic fraction. This is so due to the fact that the indicated decimal fractions are converted into ordinary fractions. For example, , and 0,(3)=1/3.

It is also clear that any infinite non-periodic decimal fraction is NOT a rational number, since it cannot be represented as a common fraction.

Now we can easily give examples of rational numbers. The numbers 4, 903, 100,321 are rational numbers because they are natural numbers. The integers 58, −72, 0, −833,333,333 are also examples of rational numbers. Common fractions 4/9, 99/3 are also examples of rational numbers. Rational numbers are also numbers.

From the above examples it is clear that there are both positive and negative rational numbers, and the rational number zero is neither positive nor negative.

The above definition of rational numbers can be formulated in a more concise form.

Definition.

Rational numbers are numbers that can be written as a fraction z/n, where z is an integer and n is a natural number.

Let us prove that this definition of rational numbers is equivalent to the previous definition. We know that we can consider the line of a fraction as a sign of division, then from the properties of dividing integers and the rules for dividing integers, the validity of the following equalities follows and. Thus, that is the proof.

Let us give examples of rational numbers based on this definition. The numbers −5, 0, 3, and are rational numbers, since they can be written as fractions with an integer numerator and a natural denominator of the form and, respectively.

The definition of rational numbers can be given in the following formulation.

Definition.

Rational numbers are numbers that can be written as a finite or infinite periodic decimal fraction.

This definition is also equivalent to the first definition, since every ordinary fraction corresponds to a finite or periodic decimal fraction and vice versa, and any integer can be associated with a decimal fraction with zeros after the decimal point.

For example, the numbers 5, 0, −13, are examples of rational numbers because they can be written as the following decimal fractions 5.0, 0.0, −13.0, 0.8, and −7, (18).

Let’s finish the theory of this point with the following statements:

• integers and fractions (positive and negative) make up the set of rational numbers;
• every rational number can be represented as a fraction with an integer numerator and a natural denominator, and each such fraction represents a certain rational number;
• every rational number can be represented as a finite or infinite periodic decimal fraction, and each such fraction represents a rational number.

Is this number rational?

In the previous paragraph, we found out that any natural number, any integer, any ordinary fraction, any mixed number, any finite decimal fraction, as well as any periodic decimal fraction is a rational number. This knowledge allows us to “recognize” rational numbers from a set of written numbers.

But what if the number is given in the form of some , or as , etc., how to answer the question whether this number is rational? In many cases it is very difficult to answer. Let us indicate some directions of thought.

If a number is given as a numeric expression that contains only rational numbers and arithmetic signs (+, −, · and:), then the value of this expression is a rational number. This follows from how operations with rational numbers are defined. For example, after performing all the operations in the expression, we get the rational number 18.

Sometimes, after simplifying the expressions and making them more complex, it becomes possible to determine whether a given number is rational.

Let's go further. The number 2 is a rational number, since any natural number is rational. What about the number? Is it rational? It turns out that no, it is not a rational number, it is an irrational number (the proof of this fact by contradiction is given in the algebra textbook for grade 8, listed below in the list of references). It has also been proven that the square root of a natural number is a rational number only in those cases when under the root there is a number that is the perfect square of some natural number. For example, and are rational numbers, since 81 = 9 2 and 1 024 = 32 2, and the numbers and are not rational, since the numbers 7 and 199 are not perfect squares of natural numbers.

Is the number rational or not? In this case, it is easy to notice that, therefore, this number is rational. Is the number rational? It has been proven that the kth root of an integer is a rational number only if the number under the root sign is the kth power of some integer. Therefore, it is not a rational number, since there is no integer whose fifth power is 121.

The method by contradiction allows you to prove that the logarithms of some numbers are not rational numbers for some reason. For example, let us prove that - is not a rational number.

Let's assume the opposite, that is, let's say that is a rational number and can be written as an ordinary fraction m/n. Then we give the following equalities: . The last equality is impossible, since on the left side there is odd number 5 n, and on the right side is the even number 2 m. Therefore, our assumption is incorrect, thus not a rational number.

In conclusion, it is worth especially noting that when determining the rationality or irrationality of numbers, one should refrain from making sudden conclusions.

For example, you should not immediately assert that the product of the irrational numbers π and e is an irrational number; this is “seemingly obvious”, but not proven. This raises the question: “Why would a product be a rational number?” And why not, because you can give an example of irrational numbers, the product of which gives a rational number: .

It is also unknown whether numbers and many other numbers are rational or not. For example, there are irrational numbers whose irrational power is a rational number. For illustration, we present a degree of the form , the base of this degree and the exponent are not rational numbers, but , and 3 is a rational number.

Bibliography.

• Mathematics. 6th grade: educational. 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 8th grade. general education institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova]; edited by 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 those entering technical schools): Proc. allowance.- M.; Higher school, 1984.-351 p., ill.