What numbers are called rational examples. Definition of rational numbers

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

rational number- a number that is represented by a simple fraction m/n, where the numerator m are whole numbers, 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 the natural numbers).

Using rational numbers in real life.

AT 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 a rough estimate of 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 uniquely identify between them 1-but and only one of the 3 relations: “<», «>" 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 related by the same relationship as 2 integers m a⋅ nb and m b⋅ n a;
2 negative numbers a and b related by the same relation 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 puts them in correspondence with a certain rational number c. However, the number itself c- this is sum numbers a and b and is referred to as (a+b) summation.

Summation rule looks like that:

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

∀ 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⋅ nb.

∀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, then a less c, what if a equals b and b equals c, then a equals c.

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

5. Commutativity of addition. From a change in the places of rational terms, the sum does not change.

∀ a,b∈ Qa+b=b+a

6. Associativity of addition. The order of addition of 3 rational numbers 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∈ Qa+0=a

8. Presence of opposite numbers. Every rational number has an opposite rational number, adding them together results in 0.

∀ a∈ Q∃ (−a)∈ Qa+(−a)=0

9. Commutativity of multiplication. By changing the places of rational factors, the product does not change.

∀ a,b∈ Q a⋅ b=b⋅ a

10. Associativity of multiplication. The order of multiplication of 3 rational numbers does not affect the result.

∀ a,b,c∈ Q(a⋅ b)⋅ c=a⋅ (b⋅ c)

11. Availability of a unit. 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. Availability reciprocal numbers . Any 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 with respect to addition. The multiplication operation is related to addition using the distribution law:

∀ a,b,c∈ Q(a+b)⋅ c=a⋅ c+b⋅ c

14. Connection of the order relation with the addition operation. The same rational number is added to the left and right sides of a rational inequality.

∀ a,b,c∈ Qa ⇒ a+c

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

∀ a,b,c∈ Qc>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.

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 next view: .
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, what 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. Such additional properties lots of. It makes sense here to cite just a few of them.

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

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, i.e. establishes a bijection between the sets of rational and natural numbers.

The simplest of these algorithms is as follows. An infinite table of ordinary fractions is compiled, 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 .

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, what 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.

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

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, i.e., establishes a bijection between the sets of rational and natural numbers.

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.

Insufficiency of rational numbers

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

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 .

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

rational number- a number that is represented by a simple fraction m/n, where the numerator m are whole numbers, 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 the 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 a rough estimate of the spatial relationships of extended objects.

Properties of rational numbers.

Basic properties of rational numbers.

2 positive numbers a=m a /n a and b=m b /n b related by the same relationship as 2 integers m a⋅ nb and m b⋅ n a;
2 negative numbers a and b related by the same relation as 2 positive numbers |b| and |a|;
when a positive, and b- negative, then a>b.

∀ a,b∈ Q(a ∨ a>b∨ a=b)

Summation rule looks like that:

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

∀ a,b∈ Q∃ !(a+b)∈ Q

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

∀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, then a less c, what if a equals b and b equals c, then a equals c.

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

5. Commutativity of addition. From a change in the places of rational terms, the sum does not change.

∀ a,b∈ Qa+b=b+a

6. Associativity of addition. The order of addition of 3 rational numbers 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∈ Qa+0=a

8. Presence of opposite numbers. Every rational number has an opposite rational number, adding them together results in 0.

∀ a∈ Q∃ (−a)∈ Qa+(−a)=0

9. Commutativity of multiplication. By changing the places of rational factors, the product does not change.

∀ a,b∈ Q a⋅ b=b⋅ a

10. Associativity of multiplication. The order of multiplication of 3 rational numbers does not affect the result.

∀ a,b,c∈ Q(a⋅ b)⋅ c=a⋅ (b⋅ c)

11. Availability of a unit. 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 reciprocals. Any 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 with respect to addition. The multiplication operation is related to addition using the distribution law:

∀ a,b,c∈ Q(a+b)⋅ c=a⋅ c+b⋅ c

14. Connection of the order relation with the addition operation. The same rational number is added to the left and right sides of a rational inequality.

∀ a,b,c∈ Qa ⇒ a+c

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

∀ a,b,c∈ Qc>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 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 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 either as a positive common fraction, 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.

It can be seen from the above examples 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 by a fraction, then their product is called 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 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 subtracting from a smaller number more, that is, you can always find the difference of two numbers.
The difference can be a positive number, negative number or the number 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, 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 unknown multiplier, 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