Essay on rational numbers. Rational numbers, definition, examples

Rational numbers are numbers of the form , where
is an integer, and - natural. The set of rational numbers is denoted by the letter . In this case, the relation
, because any integer
can be represented as . Thus, we can say that rational numbers are all integers, as well as positive and negative ordinary fractions.

Decimals - these are ordinary fractions, in which the denominator is one with zeros, that is, 10; 100; 1000 etc. Decimals are written without denominators. First, the integer part of the number is written, a comma is placed to the right of it; the first digit after the decimal point means the number of tenths, the second - hundredths, the third - thousandths, etc. The numbers after the decimal point are called decimal places.

Endless A decimal fraction is called, which has an infinite number of digits after the decimal point.

Every rational number can be represented as finite or infinite decimal fraction. This is achieved by dividing the numerator by the denominator.

An infinite decimal is called periodical , if it has, starting from a certain place, one digit or a group of digits is repeated, immediately following one after another. A repeated digit or group of digits is called a period and is written in brackets. For example, .

The converse is also true: any infinite decimal periodic fraction can be represented as an ordinary fraction.

We list some information about periodic fractions.

1. If the period of a fraction begins immediately after the decimal point, then the fraction is called purely periodic , if not immediately after the decimal point - mixed periodic .

For example, 1,(58) is a purely periodic fraction, and 2,4(67) is a mixed periodic fraction.

2. If an irreducible fraction is such that the decomposition of its denominator into prime factors contains only the numbers 2 and 5, then the notation of the number in the form of a decimal fraction represents the final decimal fraction; if there are other prime factors in the indicated decomposition, then an infinite decimal periodic fraction will be obtained.

3. If an irreducible fraction is such that the decomposition of its denominator into prime factors does not contain the numbers 2 and 5, then the record of the number in the form of a decimal fraction is a purely periodic decimal fraction; if in the specified expansion, along with other prime factors, there are 2 or 5, then you get a mixed-periodic decimal fraction.

4. A periodic fraction can have a period of any length, that is, contain any number of digits.

1.3. Irrational numbers

Irrational number is called an infinite decimal non-periodic fraction .

Examples of irrational numbers are the roots of natural numbers that are not squares of natural numbers. For example,
,
. Numbers are irrational
;
. The set of irrational numbers is denoted by the letter .

Example 1.10. Prove that
is an irrational number.

Solution. Let's pretend that
– rational number. Obviously, it is not a whole, and therefore
, where
and is an irreducible fraction; means the numbers
and mutually simple. Because
, then
, that is
.

Integers

Natural numbers definition are positive integers. Natural numbers are used to count objects and for many other purposes. Here are the numbers:

This is a natural series of numbers.
Zero natural number? No, zero is not a natural number.
How many natural numbers are there? There is an infinite set of natural numbers.
What is the smallest natural number? One is the smallest natural number.
What is the largest natural number? It cannot be specified, because there is an infinite set of natural numbers.

The sum of natural numbers is a natural number. So, the addition of natural numbers a and b:

The product of natural numbers is a natural number. So, the product of natural numbers a and b:

c is always a natural number.

Difference of natural numbers There is not always a natural number. If the minuend is greater than the subtrahend, then the difference of natural numbers is a natural number, otherwise it is not.

The quotient of natural numbers There is not always a natural number. If for natural numbers a and b

where c is a natural number, it means that a is evenly divisible by b. In this example, a is the dividend, b is the divisor, c is the quotient.

The divisor of a natural number is the natural number by which the first number is evenly divisible.

Every natural number is divisible by 1 and itself.

Simple natural numbers are only divisible by 1 and themselves. Here we mean divided completely. Example, numbers 2; 3; 5; 7 is only divisible by 1 and itself. These are simple natural numbers.

One is not considered a prime number.

The numbers that more than one and which are not simple are called composite. Examples of composite numbers:

One is not considered a composite number.

The set of natural numbers is one, prime numbers and composite numbers.

The set of natural numbers is denoted by the Latin letter N.

Properties of addition and multiplication of natural numbers:

commutative property of addition

associative property of addition

(a + b) + c = a + (b + c);

commutative property of multiplication

associative property of multiplication

(ab)c = a(bc);

distributive property of multiplication

A (b + c) = ab + ac;

Whole numbers

Integers are natural numbers, zero and the opposite of natural numbers.

Numbers that are opposite to natural numbers are integers. negative numbers, for example:

1; -2; -3; -4;...

The set of integers is denoted by the Latin letter Z.

Rational numbers

Rational numbers are integers and fractions.

Any rational number can be represented as a periodic fraction. Examples:

1,(0); 3,(6); 0,(0);...

It can be seen from the examples that any integer is a periodic fraction with a period of zero.

Any rational number can be represented as a fraction m/n, where m integer,n natural number. Let's represent the number 3,(6) from the previous example as such a fraction.

Definition of rational numbers:

A rational number is a number that can be represented as a fraction. The numerator of such a fraction belongs to the set of integers, and the denominator belongs to the set of natural numbers.

Why are numbers called rational?

In Latin "ratio" (ratio) means ratio. Rational numbers can be represented as a ratio, i.e. in other words, as a fraction.

Rational number example

The number 2/3 is a rational number. Why? This number is represented as a fraction, the numerator of which belongs to the set of integers, and the denominator belongs to the set of natural numbers.

For more examples of rational numbers, see the article.

Equal rational numbers

Different fractions can represent a single rational number.

Consider the rational number 3/5. This rational number is equal to

Let's reduce the numerator and denominator by common factor 2:

We got the fraction 3/5, which means that

Number- the most important mathematical concept that has changed over the centuries.

The first ideas about the number arose from counting people, animals, fruits, various products, etc. The result is natural numbers: 1, 2, 3, 4, ...

Historically, the first extension of the concept of number is the addition of fractional numbers to a natural number.

Shot called a part (share) of a unit or several equal parts of it.

Designated: , where m,n- whole numbers;

Fractions with denominator 10 n, where n is an integer, they are called decimal: .

Among decimal fractions, a special place is occupied by periodic fractions: - pure periodic fraction, - mixed periodic fraction.

Further expansion of the concept of number is already caused by the development of mathematics itself (algebra). Descartes in the 17th century introduces the concept negative number.

Numbers whole (positive and negative), fractional (positive and negative) and zero are called rational numbers. Any rational number can be written as a finite and periodic fraction.

To study continuously changing variables, it turned out to be necessary to expand the concept of number - the introduction of real (real) numbers - by adding irrational numbers to rational numbers: irrational numbers are infinite decimal non-periodic fractions.

Irrational numbers appeared when measuring incommensurable segments (side and diagonal of a square), in algebra - when extracting roots, an example of a transcendental, irrational number is π, e .

Numbers natural(1, 2, 3,...), whole(..., –3, –2, –1, 0, 1, 2, 3,...), rational(represented as a fraction) and irrational(not representable as a fraction ) form a set real (real) numbers.

Separately in mathematics, complex numbers are distinguished.

Complex numbers arise in connection with the problem of solving squares for the case D< 0 (здесь D is the discriminant of the quadratic equation). For a long time, these numbers did not find physical use, which is why they were called "imaginary" numbers. However, they are now widely used in various areas physics and technology: electrical engineering, hydro- and aerodynamics, elasticity theory, etc.

Complex numbers are written as: z= a+ bi. Here a and b – real numbers, a i – imaginary unit.e. i 2 = -one. Number a called abscissa, a b-ordinate complex number a+ bi. Two complex numbers a+ bi and a-bi called conjugate complex numbers.

Properties:

1. Real number a can also be written as a complex number: a+ 0i or a - 0i. For example 5 + 0 i and 5 - 0 i mean the same number 5 .

2. Complex number 0 + bi called purely imaginary number. Recording bi means the same as 0 + bi.

3. Two complex numbers a+ bi and c+ di are considered equal if a= c and b= d. Otherwise, the complex numbers are not equal.

Actions:

Addition. The sum of complex numbers a+ bi and c+ di is called a complex number ( a+ c) + (b+ d)i. In this way, when adding complex numbers, their abscissas and ordinates are added separately.

Subtraction. The difference between two complex numbers a+ bi(reduced) and c+ di(subtracted) is called a complex number ( a-c) + (b-d)i. In this way, when subtracting two complex numbers, their abscissas and ordinates are subtracted separately.

Multiplication. The product of complex numbers a+ bi and c+ di is called a complex number.

(ac-bd) + (ad+ bc)i. This definition stems from two requirements:

1) numbers a+ bi and c+ di must multiply like algebraic binomials,

2) number i has the main property: i 2 = –1.

EXAMPLE ( a + bi)(a-bi)= a 2 +b 2 . Consequently, workof two conjugate complex numbers is equal to a positive real number.

Division. Divide a complex number a+ bi(divisible) to another c+ di (divider) - means to find the third number e+ fi(chat), which, when multiplied by a divisor c+ di, which results in the dividend a+ bi. If the divisor is not zero, division is always possible.

EXAMPLE Find (8+ i) : (2 – 3i) .

Solution. Let's rewrite this ratio as a fraction:

Multiplying its numerator and denominator by 2 + 3 i and doing all the transformations, we get:

Task 1: Add, subtract, multiply and divide z 1 to z 2

Extracting the square root: Solve the equation x 2 = -a. To solve this equation we are forced to use a new type of numbers - imaginary numbers . In this way, imaginary the number is called whose second power is a negative number. According to this definition of imaginary numbers, we can define and imaginary unit:

Then for the equation x 2 = - 25 we get two imaginary root:

Task 2: Solve the equation:

1) x 2 = – 36; 2) x 2 = – 49; 3) x 2 = – 121

Geometric representation of complex numbers. Real numbers are represented by points on the number line:

Here is the point A means number -3, dot B is the number 2, and O-zero. In contrast, complex numbers are represented by points on the coordinate plane. For this, we choose rectangular (Cartesian) coordinates with the same scales on both axes. Then the complex number a+ bi will be represented by a dot P with abscissaa and ordinateb. This coordinate system is called complex plane .

module complex number is called the length of the vector OP, depicting a complex number on the coordinate ( comprehensive) plane. Complex number modulus a+ bi denoted by | a+ bi| or) letter r and is equal to:

Conjugate complex numbers have the same modulus.

The rules for drawing up a drawing are almost the same as for a drawing in a Cartesian coordinate system. Along the axes, you need to set the dimension, note:

e
unit along the real axis; Rez

imaginary unit along the imaginary axis. im z

Task 3. Construct the following complex numbers on the complex plane: , , , , , , ,

1. Numbers are exact and approximate. The numbers we encounter in practice are of two kinds. Some give the true value of the quantity, others only approximate. The first is called exact, the second - approximate. Most often it is convenient to use an approximate number instead of an exact number, especially since in many cases the exact number cannot be found at all.

So, if they say that there are 29 students in the class, then the number 29 is exact. If they say that the distance from Moscow to Kyiv is 960 km, then here the number 960 is approximate, since, on the one hand, our measuring instruments are not absolutely accurate, on the other hand, the cities themselves have some extent.

The result of operations with approximate numbers is also an approximate number. By performing some operations on exact numbers (dividing, extracting the root), you can also get approximate numbers.

The theory of approximate calculations allows:

1) knowing the degree of accuracy of the data, assess the degree of accuracy of the results;

2) take data with an appropriate degree of accuracy, sufficient to ensure the required accuracy of the result;

3) rationalize the calculation process, freeing it from those calculations that will not affect the accuracy of the result.

2. Rounding. One source of approximate numbers is rounding. Round off both approximate and exact numbers.

Rounding a given number to some of its digits is the replacement of it with a new number, which is obtained from the given one by discarding all of its digits written to the right of the digit of this digit, or by replacing them with zeros. These zeros are usually underlined or written smaller. To ensure the greatest proximity of the rounded number to the rounded one, the following rules should be used: in order to round the number to one of a certain digit, you must discard all the digits after the digit of this digit, and replace them with zeros in the whole number. This takes into account the following:

1) if the first (left) of the discarded digits is less than 5, then the last remaining digit is not changed (rounding down);

2) if the first discarded digit is greater than 5 or equal to 5, then the last remaining digit is increased by one (rounding up).

Let's show this with examples. Round up:

a) up to tenths of 12.34;

b) up to hundredths of 3.2465; 1038.785;

c) up to thousandths of 3.4335.

d) up to 12375 thousand; 320729.

a) 12.34 ≈ 12.3;

b) 3.2465 ≈ 3.25; 1038.785 ≈ 1038.79;

c) 3.4335 ≈ 3.434.

d) 12375 ≈ 12,000; 320729 ≈ 321000.

3. Absolute and relative errors. The difference between the exact number and its approximate value is called the absolute error of the approximate number. For example, if the exact number 1.214 is rounded to tenths, we get an approximate number of 1.2. AT this case the absolute error of the approximate number 1.2 is 1.214 - 1.2, i.e. 0.014.

But in most cases exact value considered value is unknown, but only approximate. Then the absolute error is also unknown. In these cases, indicate the limit that it does not exceed. This number is called the marginal absolute error. They say that the exact value of a number is equal to its approximate value with an error less than the boundary error. For example, the number 23.71 is the approximate value of the number 23.7125 with an accuracy of 0.01, since the absolute approximation error is 0.0025 and less than 0.01. Here the boundary absolute error is equal to 0.01 * .

Boundary absolute error of the approximate number a denoted by the symbol Δ a. Recording

x≈a(±Δ a)

should be understood as follows: the exact value of the quantity x is in between a– Δ a and a+ Δ a, which are called the lower and upper bounds, respectively. X and denote NG x VG X.

For example, if x≈ 2.3 (±0.1), then 2.2<x< 2,4.

Conversely, if 7.3< X< 7,4, тоX≈ 7.35 (±0.05). Absolute or marginal absolute error does not characterize the quality of the measurement. The same absolute error can be considered significant and insignificant, depending on the number that expresses the measured value. For example, if we measure the distance between two cities with an accuracy of one kilometer, then such accuracy is quite sufficient for this change, while at the same time, when measuring the distance between two houses on the same street, such accuracy will be unacceptable. Therefore, the accuracy of the approximate value of a quantity depends not only on the magnitude of the absolute error, but also on the value of the measured quantity. Therefore, the measure of accuracy is the relative error.

Relative error is the ratio of the absolute error to the value of the approximate number. The ratio of the boundary absolute error to the approximate number is called the boundary relative error; denote it like this: Relative and boundary relative errors are usually expressed as a percentage. For example, if measurements show that the distance X between two points is more than 12.3 km, but less than 12.7 km, then the arithmetic mean of these two numbers is taken as an approximate value, i.e. their half-sum, then the boundary absolute error is equal to the half-difference of these numbers. In this case X≈ 12.5 (±0.2). Here, the boundary absolute error is 0.2 km, and the boundary relative

In this subsection we give several definitions of rational numbers. Despite the differences in wording, all these definitions have the same meaning: rational numbers combine integers and fractional numbers, just as integers combine natural numbers, their opposite numbers, 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 as the most natural.

Definition.

Rational numbers are numbers that can be written as a positive common fraction, a negative common fraction, or the number zero.

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

any natural number n. Indeed, any natural number can be represented as an ordinary fraction, for example, 3=3/1 .

· Any integer, in particular, the number zero. Indeed, any integer can be written as either a positive common fraction, or as a negative common fraction, or as zero. For example, 26=26/1 , .

Any ordinary fraction (positive or negative). This is directly stated by the given definition of rational numbers.

· Any mixed number. Indeed, it is always possible to represent a mixed number as an improper common fraction. For example, and.

· Any finite decimal fraction or infinite periodic fraction. This is so because the specified decimal fractions are converted to ordinary fractions. For example, a 0,(3)=1/3 .

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

Now we can easily bring examples of rational numbers. Numbers 4 ,903 , 100 321 are rational numbers, since they are natural numbers. Whole numbers 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.

The above examples show 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 shorter form.

Definition.

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

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

We give examples of rational numbers based on this definition. 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 also 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 any 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, numbers 5 , 0 , −13 , are examples of rational numbers, since they can be written as the following decimal fractions 5,0 , 0,0 ,−13,0 , 0,8 and −7,(18) .

We finish the theory of this section with the following statements:

integer and fractional numbers (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 is a rational number;

Every rational number can be represented as a finite or infinite periodic decimal fraction, and each such fraction represents some rational number.

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The addition of positive rational numbers is commutative and associative,

("a, b н Q +) a + b= b + a;

("a, b, c н Q +) (a + b)+ c = a + (b+ c)

Before formulating the definition of multiplication of positive rational numbers, consider the following problem: it is known that the length of the segment X is expressed as a fraction at unit length E, and the length of the unit segment is measured using the unit E 1 and is expressed as a fraction. How to find the number that will represent the length of the segment X, if you measure it using the unit of length E 1?

Since X=E, then nX=mE, and from the fact that E =E 1 it follows that qE=pE 1 . We multiply the first equality obtained by q, and the second by m. Then (nq)X \u003d (mq)E and (mq)E \u003d (mp)E 1, whence (nq)X \u003d (mp)E 1. This equality shows that the length of the segment x at unit length is expressed as a fraction, and hence , =, i.e. multiplication of fractions is associated with the transition from one unit of length to another when measuring the length of the same segment.

Definition. If a positive number a is represented by a fraction, and a positive rational number b is a fraction, then their product is the number a b, which is represented by a fraction.

Multiplication of positive rational numbers commutative, associative, and distributive with respect to addition and subtraction. The proof of these properties is based on the definition of multiplication and addition of positive rational numbers, as well as on the corresponding properties of addition and multiplication of natural numbers.

46. ​​As you know subtraction is the opposite of addition.

If a a and b - positive numbers, then subtracting the number b from the number a means finding a number c that, when added to the number b, gives the number a.
a - b = c or c + b = a
The definition of subtraction holds true for all rational numbers. That is, the subtraction of positive and negative numbers can be replaced by addition.
To subtract another from one number, you need to add the opposite number to the minuend.
Or, in another way, we can say that the subtraction of the number b is the same addition, but with the number opposite to the number b.
a - b = a + (- b)
Example.
6 - 8 = 6 + (- 8) = - 2
Example.
0 - 2 = 0 + (- 2) = - 2
It is worth remembering the expressions below.
0 - a = - a
a - 0 = a
a - a = 0

Rules for subtracting negative numbers
The subtraction of the number b is the addition with the number opposite to the number b.
This rule is preserved not only when subtracting a smaller number from a larger number, but also allows you to subtract a larger number from a smaller number, that is, you can always find the difference between two numbers.
The difference can be a positive number, a negative number, or zero.
Examples of subtracting negative and positive numbers.
- 3 - (+ 4) = - 3 + (- 4) = - 7
- 6 - (- 7) = - 6 + (+ 7) = 1
5 - (- 3) = 5 + (+ 3) = 8
It is convenient to remember the sign rule, which allows you to reduce the number of brackets.
The plus sign does not change the sign of the number, so if there is a plus in front of the bracket, the sign in the brackets does not change.
+ (+ a) = + a
+ (- a) = - a
The minus sign in front of the brackets reverses the sign of the number in the brackets.
- (+ a) = - a
- (- a) = + a
It can be seen from the equalities that if there are identical signs before and inside the brackets, then we get “+”, and if the signs are different, then we get “-”.
(- 6) + (+ 2) - (- 10) - (- 1) + (- 7) = - 6 + 2 + 10 + 1 - 7 = - 13 + 13 = 0
The rule of signs is also preserved if there is not one number in brackets, but an algebraic sum of numbers.
a - (- b + c) + (d - k + n) = a + b - c + d - k + n
Please note that if there are several numbers in brackets and there is a minus sign in front of the brackets, then the signs in front of all the numbers in these brackets must change.
To remember the rule of signs, you can make a table for determining the signs of a number.
Sign rule for numbers + (+) = + + (-) = -
- (-) = + - (+) = -
Or learn a simple rule.
Two negatives make an affirmative,
Plus times minus equals minus.

Rules for dividing negative numbers.
To find the modulus of the quotient, you need to divide the modulus of the dividend by the modulus of the divisor.
So, to divide two numbers with the same signs, you need:

Divide the modulus of the dividend by the modulus of the divisor;

Put a "+" sign in front of the result.

Examples of dividing numbers with different signs:

You can also use the following table to determine the quotient sign.
The rule of signs when dividing
+ : (+) = + + : (-) = -
- : (-) = + - : (+) = -

When calculating "long" expressions, in which only multiplication and division appear, it is very convenient to use the sign rule. For example, to calculate a fraction
You can pay attention that in the numerator there are 2 "minus" signs, which, when multiplied, will give a "plus". There are also three minus signs in the denominator, which, when multiplied, will give a minus. Therefore, in the end, the result will be with a minus sign.
Fraction reduction (further actions with modules of numbers) is performed in the same way as before:
The quotient of dividing zero by a non-zero number is zero.
0: a = 0, a ≠ 0
Do NOT divide by zero!
All previously known rules for dividing by one also apply to the set of rational numbers.
a: 1 = a
a: (- 1) = - a
a: a = 1, where a is any rational number.
The dependencies between the results of multiplication and division, which are known for positive numbers, are also preserved for all rational numbers (except for the number zero):
if a × b = c; a = c: b; b = c: a;
if a: b = c; a = c × b; b=a:c
These dependencies are used to find the unknown factor, dividend and divisor (when solving equations), as well as to check the results of multiplication and division.
An example of finding the unknown.
x × (-5) = 10
x=10: (-5)
x=-2

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