Rational numbers, definition, examples.

15.10.2019

In this article, we will begin to study rational numbers. Here we give definitions of rational numbers, give the necessary explanations and give examples of rational numbers. After that, 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 subsection we give several definitions of rational numbers. Despite the differences in wording, all these definitions have the same meaning: rational numbers unite integers and fractional numbers, just as integers unite 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.

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 or infinite periodic fraction. This is so because the specified decimal fractions are converted to ordinary fractions. For example, , and 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. The numbers 4, 903, 100,321 are rational numbers, since they are natural numbers. The integers 58 , −72 , 0 , −833 333 333 are also examples of rational numbers. Ordinary 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 call 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 bar of a fraction as a sign of division, then from the properties of dividing integers and the rules for dividing integers, the following equalities follow and . Thus, which 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 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, the numbers 5 , 0 , −13 , are examples of rational numbers because they can be written as the following decimals 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;
each 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.

Is this number rational?

In the previous paragraph, we found out that any natural number, any integer, any ordinary fraction, any mixed number, any final decimal fraction, and also any periodic decimal fraction is a rational number. This knowledge allows us to "recognize" rational numbers from the set of written numbers.

But what if the number is given as some , or as , etc., how to answer the question, is the given number rational? In many cases, it is very difficult to answer it. Let us point out some directions for the course of thought.

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

Sometimes, after simplification of expressions and a more complex form, 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 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 textbook on algebra for grade 8, indicated below in the list of references). It has also been proven that Square root from a natural number is a rational number only in those cases when the root 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 natural numbers.

Is the number rational or not? In this case, it is easy to see that, therefore, this number is rational. Is the number rational? It is proved 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 contradiction method allows us to prove that the logarithms of some numbers are not rational numbers for some reason. For example, let's prove that - is not a rational number.

Assume the opposite, that is, suppose that is a rational number and can be written as an ordinary fraction m/n. Then and give the following equalities: . The last equality is impossible, since on its left side there is odd number 5 n , and on the right side there is an even number 2 m . Therefore, our assumption is wrong, thus is not a rational number.

In conclusion, it is worth emphasizing that when clarifying the rationality or irrationality of numbers, one should refrain from sudden conclusions.

For example, one should not immediately assert that the product of irrational numbers π and e is an irrational number, this is “as if obvious”, but not proven. This raises the question: “Why would the 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 the numbers and many other numbers are rational or not. For example, there are irrational numbers whose irrational power is a rational number. To illustrate, let's give 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.

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.

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.

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 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

Similar information.

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

Reduce the numerator and denominator by a common factor of 2:

6	=	2 * 3	=	3
10	=	2 * 5	=	5

We got the fraction 3/5, which means that

Rational numbers

quarters

Orderliness. a And b there is a rule that allows you to uniquely identify between them one and only one of the three relations: “< », « >' or ' = '. This rule is called ordering rule and is formulated as follows: two non-negative numbers and are related by the same relation as two integers and ; two non-positive numbers a And b are related by the same relation as two non-negative numbers and ; if suddenly a non-negative, and b- negative, then a > b. src="/pictures/wiki/files/57/94586b8b651318d46a00db5413cf6c15.png" border="0">
summation of fractions
addition operation. For any rational numbers a And b there is a so-called summation rule c. However, the number itself c called sum numbers a And b and is denoted , and the process of finding such a number is called summation. The summation rule has the following form: .
multiplication operation. For any rational numbers a And b there is a so-called multiplication rule, which puts them in correspondence with some rational number c. However, the number itself c called work numbers a And b and is denoted , and the process of finding such a number is also called multiplication. The multiplication rule is as follows: .
Transitivity of the order relation. For any triple of rational numbers a , b And c if a less b And b less c, then a less c, and if a equals b And b equals c, then a equals c. 6435">Commutativity of addition. The sum does not change from changing the places of rational terms.
Associativity of addition. The order in which three rational numbers are added does not affect the result.
The presence of zero. There is a rational number 0 that preserves every other rational number when summed.
The presence of opposite numbers. Any rational number has an opposite rational number, which, when summed, gives 0.
Commutativity of multiplication. By changing the places of rational factors, the product does not change.
Associativity of multiplication. The order in which three rational numbers are multiplied does not affect the result.
The presence of a unit. There is a rational number 1 that preserves every other rational number when multiplied.
The presence of reciprocals. Any rational number has an inverse rational number, which, when multiplied, gives 1.
Distributivity of multiplication with respect to addition. The multiplication operation is consistent with the addition operation through the distribution law:
Connection of the order relation with the operation of addition. The same rational number can be added to the left and right sides of a rational inequality. /pictures/wiki/files/51/358b88fcdff63378040f8d9ab9ba5048.png" border="0">
Axiom of Archimedes. Whatever the rational number a, you can take so many units that their sum will exceed a. src="/pictures/wiki/files/55/70c78823302483b6901ad39f68949086.png" border="0">

Additional properties

All other properties inherent in rational numbers are not singled out as basic ones, because, generally speaking, they are no longer based directly on the properties of integers, but can be proved on the basis of the given basic properties or directly by the definition of some mathematical object. There are a lot of such additional properties. It makes sense here to cite just a few of them.

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Set countability

Numbering of rational numbers

To estimate the number of rational numbers, you need to find the cardinality of their set. It is easy to prove that the set of rational numbers is countable. To do this, it suffices to give an algorithm that enumerates rational numbers, that is, establishes a bijection between the sets of rational and natural numbers.

The simplest of these algorithms is as follows. An endless table is being compiled ordinary fractions, on each i-th line in each j th column of which is a fraction. For definiteness, it is assumed that the rows and columns of this table are numbered from one. Table cells are denoted , where i- the row number of the table in which the cell is located, and j- column number.

The resulting table is managed by a "snake" according to the following formal algorithm.

These rules are searched from top to bottom and the next position is selected by the first match.

In the process of such a bypass, each new rational number is assigned to the next natural number. That is, fractions 1 / 1 are assigned the number 1, fractions 2 / 1 - the number 2, etc. It should be noted that only irreducible fractions are numbered. The formal sign of irreducibility is the equality to unity of the greatest common divisor of the numerator and denominator of the fraction.

Following this algorithm, one can enumerate all positive rational numbers. This means that the set of positive rational numbers is countable. It is easy to establish a bijection between the sets of positive and negative rational numbers, simply by assigning to each rational number its opposite. That. the set of negative rational numbers is also countable. Their union is also countable by the property of countable sets. The set of rational numbers is also countable as the union of a countable set with a finite one.

The statement about the countability of the set of rational numbers may cause some bewilderment, since at first glance one gets the impression that it is much larger than the set of natural numbers. In fact, this is not the case, and there are enough natural numbers to enumerate all rational ones.

Insufficiency of rational numbers

The hypotenuse of such a triangle is not expressed by any rational number

Rational numbers of the form 1 / n at large n arbitrarily small quantities can be measured. This fact creates a deceptive impression that rational numbers can measure any geometric distances in general. It is easy to show that this is not true.

Notes

Literature

I. Kushnir. Handbook of mathematics for schoolchildren. - Kyiv: ASTARTA, 1998. - 520 p.
P. S. Alexandrov. Introduction to set theory and general topology. - M.: head. ed. Phys.-Math. lit. ed. "Science", 1977
I. L. Khmelnitsky. Introduction to the theory of algebraic systems

Links

Wikimedia Foundation. 2010 .

As we have seen, the set of natural numbers

is closed under addition and multiplication, and the set of integers

closed under addition, multiplication and subtraction. However, none of these sets is closed under division, since division of integers can lead to fractions, as in the cases of 4/3, 7/6, -2/5, and so on. The set of all such fractions forms the set of rational numbers. Thus, a rational number (rational fraction) is a number that can be represented as , where a and d are integers, and d is not equal to zero. Let us make some remarks about this definition.

1) We required that d be different from zero. This requirement (mathematically written as the inequality ) is necessary because here d is a divisor. Consider the following examples:

Case 1. .

Case 2. .

In case 1, d is a divisor in the sense of the previous chapter, i.e., 7 is an exact divisor of 21. In case 2, d is still a divisor, but in a different sense, since 7 is not an exact divisor of 25.

If 25 is called a divisible and 7 a divisor, then we get the quotient 3 and the remainder 4. So, the word divisor is used here in a more general sense and applies to more cases than in Ch. I. However, in cases like Case 1, the concept of a divisor introduced in Ch. I; therefore it is necessary, as in Chap. I, exclude the possibility d = 0.

2) Note that, while the expressions rational number and rational fraction are synonymous, the word fraction itself is used to refer to any algebraic expression consisting of a numerator and a denominator, such as, for example,

3) The definition of a rational number includes the expression “a number that can be represented as , where a and d are integers and . Why can’t it be replaced by the expression “a number of the form where a and d are integers and The reason for this is the fact that there are infinitely many ways to express the same fraction (for example, 2/3 can also be written as 4/6, 6 /9, or or 213/33, or etc.), and it is desirable for us that our definition of a rational number does not depend on a particular way of expressing it.

A fraction is defined in such a way that its value does not change when the numerator and denominator are multiplied by the same number. However, it is not always possible to tell just by looking at a given fraction whether it is rational or not. Consider, for example, the numbers

None of them in the notation we have chosen has the form , where a and d are integers.

We can, however, perform a series of arithmetic transformations on the first fraction and get

Thus, we arrive at a fraction equal to the original fraction for which . The number is therefore rational, but it would not be rational if the definition of a rational number required that the number be of the form a/b, where a and b are integers. In the case of a conversion fraction

lead to a number. In later chapters, we will learn that a number cannot be represented as a ratio of two integers and therefore is not rational, or is said to be irrational.

4) Note that every integer is rational. As we have just seen, this is true in the case of the number 2. In the general case of arbitrary integers, one can similarly assign a denominator equal to 1 to each of them and obtain their representation as rational fractions.