Numbers that are rational. Infinite non-periodic fractions

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

Rational number- a number that is represented as a common 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. At the same time, 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. Presence of 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 operation of addition. The same rational number is added to the left and right sides of a rational inequality.

∀ 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 lesson we will learn about many rational numbers. Let's analyze the basic properties of rational numbers, learn how to convert decimal fractions to ordinary fractions and vice versa.

We have already talked about the sets of natural and integer numbers. The set of natural numbers is a subset of the integers.

Now we have learned what fractions are and learned how to work with them. A fraction, for example, is not a whole number. This means that we need to describe a new set of numbers, which will include all the fractions, and this set needs a name, a clear definition and designation.

Let's start with the name. The Latin word ratio is translated into Russian as ratio, fraction. The name of the new set “rational numbers” comes from this word. That is, “rational numbers” can be translated as “fractional numbers.”

Let's figure out what numbers this set consists of. We can assume that it consists of all fractions. For example, such - . But such a definition would not be entirely correct. A fraction is not a number itself, but a form of writing a number. In the example below, two different fractions represent the same number:

Then it would be more accurate to say that rational numbers are those numbers that can be represented as a fraction. And this, in fact, is almost the same definition that is used in mathematics.

This set is designated by the letter . How are the sets of natural and integer numbers related to the new set of rational numbers? A natural number can be written as a fraction in an infinite number of ways. And since it can be represented as a fraction, then it is also rational.

The situation is similar with negative integers. Any negative integer can be represented as a fraction . Is it possible to represent the number zero as a fraction? Of course you can, also in an infinite number of ways .

Thus, all natural numbers and all integers are also rational numbers. The sets of natural numbers and integers are subsets of the set of rational numbers ().

Closedness of sets with respect to arithmetic operations

The need to introduce new numbers - integers, then rationals - can be explained not only by problems from real life. The arithmetic operations themselves tell us this. Let's add two natural numbers: . We get a natural number again.

They say that the set of natural numbers is closed under the operation of addition (closed under addition). Think for yourself whether the set of natural numbers is closed under multiplication.

As soon as we try to subtract something equal or greater from a number, we are left short of natural numbers. Introducing zero and negative integers corrects the situation:

The set of integers is closed under subtraction. We can add and subtract any integer without fear of not having a number to write the result with (closed to addition and subtraction).

Is the set of integers closed under multiplication? Yes, the product of any two integers results in an integer (closed under addition, subtraction and multiplication).

There is one more action left - division. Is the set of integers closed under division? The answer is obvious: no. Let's divide by. Among the integers there is no such number to write down the answer: .

But using a fraction, we can almost always write down the result of dividing one integer by another. Why almost? Let us remember that, by definition, you cannot divide by zero.

Thus, the set of rational numbers (which arises when fractions are introduced) claims to be a set closed under all four arithmetic operations.

Let's check.

That is, the set of rational numbers is closed under addition, subtraction, multiplication and division, excluding division by zero. In this sense, we can say that the set of rational numbers is structured “better” than the previous sets of natural and integer numbers. Does this mean that rational numbers are the last number set we study? No. Subsequently, we will have other numbers that cannot be written as fractions, for example, irrational ones.

Numbers as a tool

Numbers are a tool that man created as needed.

Rice. 1. Using natural numbers

Later, when it was necessary to make monetary calculations, they began to put plus or minus signs in front of the number, indicating whether the original value should be increased or decreased. This is how negative and positive numbers appeared. The new set was called the set of integers ().

Rice. 2. Using fractions

Therefore, a new tool appears, new numbers - fractions. We write them in different equivalent ways: ordinary and decimal fractions ( ).

All numbers - “old” (integer) and “new” (fractional) - were combined into one set and called it the set of rational numbers ( - rational numbers)

So, a rational number is a number that can be represented as a common fraction. But this definition in mathematics is further refined. Any rational number can be represented as a fraction with a positive denominator, that is, the ratio of an integer to a natural number: .

Then we get the definition: a number is called rational if it can be represented as a fraction with an integer numerator and a natural denominator ( ).

In addition to ordinary fractions, we also use decimals. Let's see how they relate to the set of rational numbers.

There are three types of decimals: finite, periodic and non-periodic.

Infinite non-periodic fractions: such fractions also have an infinite number of decimal places, but there is no period. An example is the decimal notation of PI:

Any finite decimal fraction by definition is an ordinary fraction with a denominator, etc.

Let's read the decimal fraction out loud and write it in ordinary form: , .

When going back from writing as a fraction to a decimal, you can get finite decimal fractions or infinite periodic fractions.

Converting from a fraction to a decimal

The simplest case is when the denominator of a fraction is a power of ten: etc. Then we use the definition of a decimal fraction:

There are fractions whose denominator can easily be reduced to this form: . It is possible to go to such a notation if the expansion of the denominator includes only twos and fives.

The denominator consists of three twos and one five. Each one forms a ten. This means we are missing two. Multiply by both the numerator and denominator:

It could have been done differently. Divide by a column (see Fig. 1).

Rice. 2. Column division

In the case of with, the denominator cannot be turned into or another digit number, since its expansion includes a triple. There is only one way left - to divide in a column (see Fig. 2).

Such a division at each step will give a remainder and a quotient. This process is endless. That is, we got an infinite periodic fraction with a period

Let's practice. Let's convert ordinary fractions to decimals.

In all of these examples, we ended up with a final decimal fraction because the denominator expansion included only twos and fives.

(let's check ourselves by dividing into a table - see Fig. 3).

Rice. 3. Long division

Rice. 4. Column division

(see Fig. 4)

The expansion of the denominator includes a triple, which means bringing the denominator to the form , etc. will not work. Divide by into a column. The situation will repeat itself. There will be an infinite number of triplets in the result record. Thus, .

(see Fig. 5)

Rice. 5. Column division

So, any rational number can be represented as an ordinary fraction. This is his definition.

And any ordinary fraction can be represented as a finite or infinite periodic decimal fraction.

Types of recording fractions:

writing a decimal fraction in the form of an ordinary fraction: ; ;

writing a common fraction as a decimal: (final fraction); (infinite periodic).

That is, any rational number can be written as a finite or periodic decimal fraction. In this case, the final fraction can also be considered periodic with a period of zero.

Sometimes a rational number is given exactly this definition: a rational number is a number that can be written as a periodic decimal fraction.

Periodic Fraction Conversion

Let's first consider a fraction whose period consists of one digit and has no pre-period. Let's denote this number with the letter . The method is to get another number with the same period:

This can be done by multiplying the original number by . So the number has the same period. Subtract from the number itself:

To make sure that we did everything correctly, let's now make the transition in the opposite direction, in a way already known to us - by dividing into a column by (see Fig. 1).

In fact, we obtain a number in its original form with a period.

Let's consider a number with a pre-period and a longer period: . The method remains exactly the same as in the previous example. We need to get a new number with the same period and a pre-period of the same length. To do this, it is necessary for the comma to move to the right by the length of the period, i.e. by two characters. Multiply the original number by:

Let us subtract the original expression from the resulting expression:

So, what is the translation algorithm? The periodic fraction must be multiplied by a number of the form, etc., which has as many zeros as there are digits in the period of the decimal fraction. We get a new periodic one. For example:

Subtracting another from one periodic fraction, we get the final decimal fraction:

It remains to express the original periodic fraction in the form of an ordinary fraction.

To practice, write down a few periodic fractions yourself. Using this algorithm, reduce them to the form of an ordinary fraction. To check on a calculator, divide the numerator by the denominator. If everything is correct, then you get the original periodic fraction

So, we can write any finite or infinite periodic fraction as an ordinary fraction, as the ratio of a natural number and an integer. Those. all such fractions are rational numbers.

What about non-periodic fractions? It turns out that non-periodic fractions cannot be represented as ordinary fractions (we will accept this fact without proof). This means they are not rational numbers. They are called irrational.

Infinite non-periodic fractions

As we have already said, a rational number in decimal notation is either a finite or a periodic fraction. This means that if we can construct an infinite non-periodic fraction, then we will get a non-rational, that is, an irrational number.

Here is one way to construct this: The fractional part of this number consists only of zeros and ones. The number of zeros between ones increases by . It is impossible to highlight the repeating part here. That is, the fraction is not periodic.

Practice constructing non-periodic decimal fractions, that is, irrational numbers, on your own

A familiar example of an irrational number is pi ( ). There is no period in this entry. But besides pi, there are infinitely many other irrational numbers. We'll talk more about irrational numbers later.

1. Mathematics 5th grade. Vilenkin N.Ya., Zhokhov V.I., Chesnokov A.S., Shvartsburd S.I., 31st ed., erased. - M: Mnemosyne, 2013.
2. Mathematics 5th grade. Erina T.M.. Workbook for the textbook Vilenkina N.Ya., M.: Exam, 2013.
3. Mathematics 5th grade. Merzlyak A.G., Polonsky V.B., Yakir M.S., M.: Ventana - Graf, 2013.

1. Math-prosto.ru ().
2. Cleverstudents.ru ().
3. Mathematics-repetition.com ().

Homework

Integers

Natural numbers definition are positive integers. 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 of composite numbers:

One is not considered a composite number.

The set of natural numbers consists of 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 is an integer and n is a natural number. Let's imagine the number 3,(6) from the previous example as such a fraction.

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

Page navigation.

Definition and examples of rational numbers

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

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

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

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

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

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

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

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

Definition.

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

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

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

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

Definition.

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

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

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

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

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

Is this number rational?

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

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

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

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

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

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

The method by contradiction allows 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]; 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 those entering technical schools): Proc. allowance.- M.; Higher school, 1984.-351 p., ill.

Set of rational numbers

The set of rational numbers is denoted and can be written as follows:

It turns out that different notations can represent the same fraction, for example, and , (all fractions that can be obtained from each other by multiplying or dividing by the same natural number represent the same rational number). Since by dividing the numerator and denominator of a fraction by their greatest common divisor we can obtain a single irreducible representation of a rational number, we can speak of their set as the set irreducible fractions with mutually prime integer numerator and natural denominator:

Here is the greatest common divisor of the numbers and .

The set of rational numbers is a natural generalization of the set of integers. It is easy to see that if a rational number has a denominator , then it is an integer. The set of rational numbers is located everywhere densely on the number axis: between any two different rational numbers there is at least one rational number (and therefore an infinite set of rational numbers). However, it turns out that the set of rational numbers has countable cardinality (that is, all its elements can be renumbered). Let us note, by the way, that the ancient Greeks were convinced of the existence of numbers that cannot be represented as a fraction (for example, they proved that there is no rational number whose square is 2).

Terminology

Formal definition

Formally, rational numbers are defined as the set of equivalence classes of pairs with respect to the equivalence relation if. In this case, the operations of addition and multiplication are defined as follows:

Related definitions

Proper, improper and mixed fractions

Correct A fraction whose numerator is less than its denominator is called a fraction. Proper fractions represent rational numbers modulo less than one. A fraction that is not proper is called wrong and represents a rational number greater than or equal to one in modulus.

An improper fraction can be represented as the sum of a whole number and a proper fraction, called mixed fraction . For example, . A similar notation (with the addition sign missing), although used in elementary arithmetic, is avoided in strict mathematical literature due to the similarity of the notation for a mixed fraction with the notation for the product of an integer and a fraction.

Shot height

Height of a common shot is the sum of the modulus of the numerator and denominator of this fraction. Height of a rational number is the sum of the modulus of the numerator and the denominator of the irreducible ordinary fraction corresponding to this number.

For example, the height of a fraction is . The height of the corresponding rational number is equal to , since the fraction can be reduced by .

A comment

Term fraction (fraction) Sometimes [ specify] is used as a synonym for the term rational number, and sometimes a synonym for any non-integer number. In the latter case, fractional and rational numbers are different things, since then non-integer rational numbers are just a special case of fractional numbers.

Properties

Basic properties

The set of rational numbers satisfy sixteen basic properties, which can easily be derived from the properties of integers.

1. Orderliness. For any rational numbers, there is a rule that allows you to uniquely identify one and only one of three relations between them: “”, “” or “”. This rule is called ordering rule and is formulated as follows: two positive numbers and are related by the same relation as two integers and ; two non-positive numbers and are related by the same relation as two non-negative numbers and ; if suddenly it is not negative, but - negative, then .

   

   Adding Fractions

2. Addition operation. summation rule amount numbers and 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 there is a so-called multiplication rule, which puts them in correspondence with some rational number. In this case, the number itself is called work numbers and and is denoted by , and the process of finding such a number is also called multiplication. The multiplication rule has the following form: .
4. Transitivity of the order relation. For any triple of rational numbers, and if less and less, then less, and if equal and equal, then equal.
5. Commutativity of addition. Changing the places of the rational terms does not change the sum.
6. Associativity of addition. The order in which three rational numbers are added does not affect the result.
7. Presence of zero. There is a rational number 0 that preserves every other rational number when added.
8. The presence of opposite numbers. Any rational number has an opposite rational number, which when added to gives 0.
9. Commutativity of multiplication. Changing the places of rational factors does not change the product.
10. Associativity of multiplication. The order in which three rational numbers are multiplied does not affect the result.
11. Availability of unit. There is a rational number 1 that preserves every other rational number when multiplied.
12. Presence of reciprocal numbers. Any non-zero rational number has an inverse rational number, which when multiplied by gives 1.
13. Distributivity of multiplication relative to addition. The multiplication operation is coordinated with the addition operation through the distribution law:
14. 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.
15. Connection of the order relation with the multiplication operation. The left and right sides of a rational inequality can be multiplied by the same positive rational number.
16. Axiom of Archimedes. Whatever the rational number , you can take so many units that their sum exceeds .

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.

Countability of a set

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. An example of such a construction is the following simple algorithm. An endless table of ordinary fractions is compiled, on each row in each column of which a fraction is located. For definiteness, it is assumed that the rows and columns of this table are numbered starting from one. Table cells are designated , where is the number of the table row in which the cell is located, and is the 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, fractions are assigned the number 1, fractions are assigned the number 2, etc. It should be noted that only irreducible fractions are numbered. A formal sign of irreducibility is that the greatest common divisor of the numerator and denominator of the fraction is equal 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.

Of course, there are other ways to enumerate rational numbers. For example, for this you can use structures such as the Kalkin-Wilf tree, the Stern-Broko tree or the Farey series.

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

see also

Notes

Literature

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