12 is a rational number. Definition and examples of rational numbers

) are numbers with positive or negative sign(integers and fractions) and zero. A more precise concept of rational numbers sounds like this:

Rational number- the number that is represented ordinary fraction m/n, where the numerator m are integers, and the denominator n- integers, for example 2/3.

Infinite non-periodic fractions are NOT included in the set of rational numbers.

a/b, Where a∈ Z (a belongs to integers), b∈ N (b belongs to natural numbers).

Using rational numbers in real life.

IN real life the set of rational numbers is used to count the parts of some integer divisible objects, For example, cakes or other foods that are cut into pieces before consumption, or for roughly estimating the spatial relationships of extended objects.

Properties of rational numbers.

Basic properties of rational numbers.

1. Orderliness a And b there is a rule that allows you to unambiguously identify 1 and only one of 3 relations between them: “<», «>" or "=". This rule is - ordering rule and formulate it like this:

• 2 positive numbers a=m a /n a And b=m b /n b are related by the same relationship as 2 integers m a⋅ n b And m b⋅ n a;
• 2 negative numbers a And b are related by the same ratio as 2 positive numbers |b| And |a|;
• When a positive and b- negative, then a>b.

∀ a,b∈ Q(a ∨ a>b∨ a=b)

2. Addition operation. For all rational numbers a And b There is summation rule, which assigns them a certain rational number c. Moreover, the number itself c- This sum numbers a And b and it is denoted as (a+b) summation.

Summation Rule looks like that:

m a/n a + m b/n b =(m a⋅ n b + m b⋅ n a)/(n a⋅ n b).

∀ a,b∈ Q∃ !(a+b)∈ Q

3. Multiplication operation. For all rational numbers a And b There is multiplication rule, it associates them with a certain rational number c. The number c is called work numbers a And b and denote (a⋅b), and the process of finding this number is called multiplication.

Multiplication rule looks like that: m a n a⋅ m b n b =m a⋅ m b n a⋅ n b.

∀a,b∈Q ∃(a⋅b)∈Q

4. Transitivity of the order relation. For any three rational numbers a, b And c If a less b And b less c, That a less c, and if a equals b And b equals c, That a equals c.

∀ a,b,c∈ Q(a ∧ b ⇒ a ∧ (a = b∧ b = c⇒ a = c)

5. Commutativity of addition. Changing the places of the rational terms does not change the sum.

∀ a,b∈ Q a+b=b+a

6. Addition associativity. The order in which 3 rational numbers are added does not affect the result.

∀ a,b,c∈ Q (a+b)+c=a+(b+c)

7. Presence of zero. There is a rational number 0, it preserves every other rational number when added.

∃ 0 ∈ Q∀ a∈ Q a+0=a

8. Availability opposite numbers . Any rational number has an opposite rational number, and when they are added, the result is 0.

∀ a∈ Q∃ (−a)∈ Q a+(−a)=0

9. Commutativity of multiplication. Changing the places of rational factors does not change the product.

∀ a,b∈ Q a⋅ b=b⋅ a

10. Associativity of multiplication. The order in which 3 rational numbers are multiplied has no effect on the result.

∀ a,b,c∈ Q(a⋅ b)⋅ c=a⋅ (b⋅ c)

11. Unit availability. There is a rational number 1, it preserves every other rational number in the process of multiplication.

∃ 1 ∈ Q∀ a∈ Q a⋅ 1=a

12. Presence of reciprocal numbers. Every rational number other than zero has an inverse rational number, multiplying by which we get 1 .

∀ a∈ Q∃ a−1∈ Q a⋅ a−1=1

13. Distributivity of multiplication relative to addition. The multiplication operation is related to addition using the distributive law:

∀ a,b,c∈ Q(a+b)⋅ c=a⋅ c+b⋅ c

14. Relationship between the order relation and the addition operation. To the left and right side rational inequality add the same rational number.

∀ a,b,c∈ Q a ⇒ a+c

15. Relationship between the order relation and the multiplication operation. The left and right sides of a rational inequality can be multiplied by the same non-negative rational number.

∀ a,b,c∈ Q c>0∧ a ⇒ a⋅ c ⋅ c

16. Axiom of Archimedes. Whatever the rational number a, it is easy to take so many units that their sum will be greater a.

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 given number 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 the integers 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, one can always imagine 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's 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 finite or infinite periodic decimal.

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 Square root from natural number is a rational number only in those cases when the root contains 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 in this case It is easy to see 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 one 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.

Rational numbers

Quarters

1. Orderliness. a And b there is a rule that allows one to uniquely identify one and only one of three relationships between them: “< », « >" 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, but b- negative, then a > b. src="/pictures/wiki/files/57/94586b8b651318d46a00db5413cf6c15.png" border="0">

   

   Adding Fractions

2. Addition operation. For any rational numbers a And b there is a so-called summation rule c. Moreover, the number itself c called amount numbers a And b and is denoted by , and the process of finding such a number is called summation. The summation rule has the following form: .
3. Multiplication operation. For any rational numbers a And b there is a so-called multiplication rule, which assigns them some rational number c. Moreover, the number itself c called work numbers a And b and is denoted by , and the process of finding such a number is also called multiplication. The multiplication rule looks like this: .
4. Transitivity of the order relation. For any triple of rational numbers a , b And c If a less b And b less c, That a less c, and if a equals b And b equals c, That a equals c. 6435">Commutativity of addition. Changing the places of rational terms does not change the sum.
5. Associativity of addition. The order in which three rational numbers are added does not affect the result.
6. Presence of zero. There is a rational number 0 that preserves every other rational number when added.
7. The presence of opposite numbers. Any rational number has an opposite rational number, which when added to gives 0.
8. Commutativity of multiplication. Changing the places of rational factors does not change the product.
9. Associativity of multiplication. The order in which three rational numbers are multiplied does not affect the result.
10. Availability of unit. There is a rational number 1 that preserves every other rational number when multiplied.
11. Presence of reciprocal numbers. Any rational number has an inverse rational number, which when multiplied by gives 1.
12. Distributivity of multiplication relative to addition. The multiplication operation is coordinated with the addition operation through the distribution law:
13. 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">
14. Axiom of Archimedes. Whatever the rational number a, you can take so many units that their sum exceeds a. src="/pictures/wiki/files/55/70c78823302483b6901ad39f68949086.png" border="0">

Additional properties

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

Src="/pictures/wiki/files/48/0caf9ffdbc8d6264bc14397db34e8d72.png" border="0">

Countability of a set

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 is enough to give an algorithm that enumerates rational numbers, i.e., establishes a bijection between the sets of rational and natural numbers.

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

The resulting table is traversed using a “snake” according to the following formal algorithm.

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

In the process of such a traversal, each new rational number is associated with another natural number. That is, the fraction 1/1 is assigned to the number 1, the fraction 2/1 to the number 2, etc. It should be noted that only irreducible fractions are numbered. Formal sign irreducibility is the equality of the greatest common divisor of the numerator and denominator of the fraction to one.

Following this algorithm, we 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 by simply 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 confusion, since at first glance it seems that it is much more extensive than the set of natural numbers. In fact, this is not so and there are enough natural numbers to enumerate all rational ones.

Lack of rational numbers

The hypotenuse of such a triangle cannot be expressed by any rational number

Rational numbers of the form 1 / n at large n arbitrarily small quantities can be measured. This fact creates the misleading impression that rational numbers can be used to measure any geometric distances. 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.: chapter. ed. physics and mathematics lit. ed. "Science", 1977
• I. L. Khmelnitsky. Introduction to the theory of algebraic systems

Links

Wikimedia Foundation. 2010.

Integers

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

This is a natural series of numbers.
Is zero a natural number? No, zero is not a natural number.
How many natural numbers are there? There is an infinite number of natural numbers.
What is the smallest natural number? One is the smallest natural number.
What is the largest natural number? It is impossible to specify it, because there is an infinite number of natural numbers.

The sum of natural numbers is a natural number. So, adding 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 the natural numbers is a natural number, otherwise it is not.

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

where c is a natural number, this means that a is 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 a natural number by which the first number is divisible by a whole.

Every natural number is divisible by one and itself.

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

One is not considered a prime number.

Numbers that are greater than one and that are not prime are called composite numbers. Examples 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 the natural numbers, zero, and the opposites of the natural numbers.

The opposite of natural numbers are negative integers, for example:

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

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

Rational numbers

Rational numbers are whole numbers and fractions.

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

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

From the examples it is clear that any integer is a periodic fraction with period zero.

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

Definition of rational numbers

Rational numbers include:

• Natural numbers that can be represented as common fraction. For example, $7=\frac(7)(1)$.
• Whole numbers, including zero, which can be represented as a positive or negative fraction, or as zero. For example, $19=\frac(19)(1)$, $-23=-\frac(23)(1)$.
• Common fractions (positive or negative).
• Mixed numbers that can be represented as an improper fraction. For example, $3 \frac(11)(13)=\frac(33)(13)$ and $-2 \frac(4)(5)=-\frac(14)(5)$.
• A finite decimal and an infinite periodic fraction that can be represented as a fraction. For example, $-7.73=-\frac(773)(100)$, $7,(3)=-7 \frac(1)(3)=-\frac(22)(3)$.

Note 1

Note that an infinite non-periodic decimal fraction does not belong to rational numbers, because it cannot be represented as an ordinary fraction.

Example 1

The natural numbers $7, 670, 21\456$ are rational.

The integers $76, –76, 0, –555\666$ are rational.

Common fractions $\frac(7)(11)$, $\frac(555)(4)$, $-\frac(7)(11)$, $-\frac(100)(234)$ – rational numbers .

Thus, rational numbers are divided into positive and negative. The number zero is rational, but is neither a positive nor a negative rational number.

Let us formulate more short definition rational numbers.

Definition 3

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

The following conclusions can be drawn:

• positive and negative integers and fractions belong to the set of rational numbers;
• rational numbers can be represented as a fraction that has an integer numerator and a natural denominator and is a rational number;
• rational numbers can be represented as any periodic decimal fraction that is a rational number.

How to Determine If a Number is Rational

1. The number is specified as a numeric expression that consists only of rational numbers and arithmetic operations signs. In this case, the value of the expression will be a rational number.
2. The square root of a natural number is a rational number only if the root contains a number that is the perfect square of some natural number. For example, $\sqrt(9)$ and $\sqrt(121)$ are rational numbers, since $9=3^2$ and $121=11^2$.
3. The $n$th root of an integer is a rational number only if the number under the root sign is the $n$th power of some integer. For example, $\sqrt(8)$ is a rational number, because $8=2^3$.

On the number axis, rational numbers are densely distributed throughout: between every two rational numbers that are not equal to each other, at least one rational number can be located (hence, an infinite set of rational numbers). At the same time, the set of rational numbers is characterized by countable cardinality (that is, all elements of the set can be numbered). The ancient Greeks proved that there are numbers that cannot be written as a fraction. They showed that there is no rational number whose square is equal to $2$. Then rational numbers turned out to be insufficient to express all quantities, which later led to the appearance of real numbers. The set of rational numbers, unlike real numbers, is zero-dimensional.