What does a rational number mean? Minus before a rational number

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.

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
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: .
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: .
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.
Commutativity of addition. Changing the places of the rational terms does not change the sum.
Associativity of addition. The order in which three rational numbers are added does not affect the result.
Presence of zero. There is a rational number 0 that preserves every other rational number when added.
The presence of opposite numbers. Any rational number has an opposite rational number, which when added to gives 0.
Commutativity of multiplication. Changing the places of rational factors does not change the product.
Associativity of multiplication. The order in which three rational numbers are multiplied does not affect the result.
Availability of unit. There is a rational number 1 that preserves every other rational number when multiplied.
Presence of reciprocal numbers. Any non-zero rational number has an inverse rational number, which when multiplied by gives 1.
Distributivity of multiplication relative to addition. The multiplication operation is coordinated 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.
The connection between the order relation and the multiplication operation. The left and right sides of a rational inequality can be multiplied by the same positive rational number.
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

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

Older schoolchildren and mathematics students will probably answer this question with ease. But for those who are far from this by profession, it will be more difficult. What is it really?

Essence and designation

Rational numbers are those that can be represented as an ordinary fraction. Positive, negative, and zero are also included in this set. The numerator of the fraction must be an integer, and the denominator must be

This set in mathematics is denoted as Q and is called the “field of rational numbers”. It includes all integers and natural numbers, denoted respectively as Z and N. The set Q itself is included in the set R. It is this letter that denotes the so-called real or

Performance

As already mentioned, rational numbers are a set that includes all integer and fractional values. They can come in different forms. Firstly, in the form of an ordinary fraction: 5/7, 1/5, 11/15, etc. Of course, integers can also be written in a similar form: 6/2, 15/5, 0/1, - 10/2, etc. Secondly, another type of representation is a decimal fraction with a final fractional part: 0.01, -15.001006, etc. This is perhaps one of the most common forms.

But there is also a third one - a periodic fraction. This type is not very common, but is still used. For example, the fraction 10/3 can be written as 3.33333... or 3,(3). In this case, different representations will be considered similar numbers. Fractions that are equal to each other will also be called the same, for example 3/5 and 6/10. It seems that it has become clear what rational numbers are. But why is this term used to refer to them?

origin of name

The word “rational” in modern Russian generally has a slightly different meaning. It's more like "reasonable", "thought out". But the mathematical terms are close to the direct meaning of this. In Latin, "ratio" is a "ratio", "fraction" or "division". Thus, the name captures the essence of what rational numbers are. However, the second meaning

not far from the truth.

Actions with them

When solving mathematical problems, we constantly come across rational numbers without knowing it ourselves. And they have a number of interesting properties. All of them follow either from the definition of a set or from actions.

First, rational numbers have the order relation property. This means that there can only be one relationship between two numbers - they are either equal to each other, or one is greater or less than the other. That is:

or a = b ; or a > b, or a< b.

In addition, the transitivity of the relation also follows from this property. That is, if a more b, b more c, That a more c. In mathematical language it looks like this:

(a > b) ^ (b > c) => (a > c).

Secondly, there are arithmetic operations with rational numbers, that is, addition, subtraction, division and, of course, multiplication. At the same time, in the process of transformations, a number of properties can also be identified.

a + b = b + a (change of places of terms, commutativity);
0 + a = a + 0 ;
(a + b) + c = a + (b + c) (associativity);
a + (-a) = 0;
ab = ba;
(ab)c = a(bc) (distributivity);
a x 1 = 1 x a = a;
a x (1 / a) = 1 (in this case a is not equal to 0);
(a + b)c = ac + ab;
(a > b) ^ (c > 0) => (ac > bc).

When we are talking about ordinary numbers and not integers, working with them can cause certain difficulties. Thus, addition and subtraction are possible only if the denominators are equal. If they are initially different, you should find the common one by multiplying the entire fraction by certain numbers. Comparison is also most often possible only if this condition is met.

Division and multiplication of ordinary fractions are carried out in accordance with fairly simple rules. Reduction to a common denominator is not necessary. The numerators and denominators are multiplied separately, and in the process of performing the action, if possible, the fraction should be reduced and simplified as much as possible.

As for division, this action is similar to the first one with a slight difference. For the second fraction you should find the inverse, that is

"turn" it over. Thus, the numerator of the first fraction will need to be multiplied with the denominator of the second and vice versa.

Finally, another property inherent in rational numbers is called Archimedes’ axiom. Often in the literature the name “principle” is also found. It is valid for the entire set of real numbers, but not everywhere. Thus, this principle does not apply to some sets of rational functions. Essentially, this axiom means that given the existence of two quantities a and b, you can always take enough a to exceed b.

Application area

So, for those who have learned or remembered what rational numbers are, it becomes clear that they are used everywhere: in accounting, economics, statistics, physics, chemistry and other sciences. Naturally, they also have a place in mathematics. Not always knowing that we are dealing with them, we constantly use rational numbers. Even small children, learning to count objects, cutting an apple into pieces, or performing other simple actions, encounter them. They literally surround us. And yet, they are not enough to solve some problems; in particular, using the Pythagorean theorem as an example, one can understand the need to introduce the concept

The topic of rational numbers is quite extensive. You can talk about it endlessly and write entire works, each time being surprised by new features.

In order to avoid mistakes in the future, in this lesson we will delve a little deeper into the topic of rational numbers, glean the necessary information from it and move on.

Lesson content

What is a rational number

A rational number is a number that can be represented as a fraction, where a— this is the numerator of the fraction, b is the denominator of the fraction. Moreover b must not be zero because division by zero is not allowed.

Rational numbers include the following categories of numbers:

integers (for example −2, −1, 0 1, 2, etc.)

decimal fractions (for example 0.2, etc.)

infinite periodic fractions (for example 0, (3), etc.)

Each number in this category can be represented as a fraction.

Example 1. The integer 2 can be represented as a fraction. This means that the number 2 applies not only to integers, but also to rational ones.

Example 2. A mixed number can be represented as a fraction. This fraction is obtained by converting a mixed number to an improper fraction

This means that a mixed number is a rational number.

Example 3. The decimal 0.2 can be represented as a fraction. This fraction was obtained by converting the decimal fraction 0.2 into a common fraction. If you have difficulty at this point, repeat the topic.

Since the decimal fraction 0.2 can be represented as a fraction, it means that it also belongs to rational numbers.

Example 4. The infinite periodic fraction 0, (3) can be represented as a fraction. This fraction is obtained by converting a pure periodic fraction into an ordinary fraction. If you have difficulty at this point, repeat the topic.

Since the infinite periodic fraction 0, (3) can be represented as a fraction, it means that it also belongs to rational numbers.

In the future, we will increasingly call all numbers that can be represented as a fraction by one phrase - rational numbers.

Rational numbers on the coordinate line

We looked at the coordinate line when we studied negative numbers. Recall that this is a straight line on which many points lie. As follows:

This figure shows a small fragment of the coordinate line from −5 to 5.

Marking integers of the form 2, 0, −3 on the coordinate line is not difficult.

Things are much more interesting with other numbers: with ordinary fractions, mixed numbers, decimals, etc. These numbers lie between the integers and there are infinitely many of these numbers.

For example, let's mark a rational number on the coordinate line. This number is located exactly between zero and one

Let's try to understand why the fraction is suddenly located between zero and one.

As mentioned above, between the integers lie other numbers - ordinary fractions, decimals, mixed numbers, etc. For example, if you increase a section of the coordinate line from 0 to 1, you can see the following picture

It can be seen that between the integers 0 and 1 there are other rational numbers, which are familiar decimal fractions. Here you can see our fraction, which is located in the same place as the decimal fraction 0.5. A careful examination of this figure provides an answer to the question of why the fraction is located exactly there.

A fraction means dividing 1 by 2. And if we divide 1 by 2, we get 0.5

The decimal fraction 0.5 can be disguised as other fractions. From the basic property of a fraction, we know that if the numerator and denominator of a fraction are multiplied or divided by the same number, then the value of the fraction does not change.

If the numerator and denominator of a fraction are multiplied by any number, for example by the number 4, then we get a new fraction, and this fraction is also equal to 0.5

This means that on the coordinate line the fraction can be placed in the same place where the fraction was located

Example 2. Let's try to mark a rational number on the coordinate. This number is located exactly between numbers 1 and 2

Fraction value is 1.5

If we increase the section of the coordinate line from 1 to 2, we will see the following picture:

It can be seen that between the integers 1 and 2 there are other rational numbers, which are familiar decimal fractions. Here you can see our fraction, which is located in the same place as the decimal fraction 1.5.

We magnified certain segments on the coordinate line to see the remaining numbers lying on this segment. As a result, we discovered decimal fractions that had one digit after the decimal point.

But these were not the only numbers lying on these segments. There are infinitely many numbers lying on the coordinate line.

It is not difficult to guess that between decimal fractions that have one digit after the decimal point, there are other decimal fractions that have two digits after the decimal point. In other words, hundredths of a segment.

For example, let's try to see the numbers that lie between the decimal fractions 0.1 and 0.2

Another example. Decimal fractions that have two digits after the decimal point and lie between zero and the rational number 0.1 look like this:

Example 3. Let us mark a rational number on the coordinate line. This rational number will be very close to zero

The value of the fraction is 0.02

If we increase the segment from 0 to 0.1, we will see exactly where the rational number is located

It can be seen that our rational number is located in the same place as the decimal fraction 0.02.

Example 4. Let us mark the rational number 0 on the coordinate line, (3)

The rational number 0, (3) is an infinite periodic fraction. Its fractional part never ends, it is infinite

And since the number 0,(3) has an infinite fractional part, this means that we will not be able to find the exact place on the coordinate line where this number is located. We can only indicate this place approximately.

The rational number 0.33333... will be located very close to the common decimal fraction 0.3

This figure does not show the exact location of the number 0,(3). This is just an illustration to show how close the periodic fraction 0.(3) can be to the regular decimal fraction 0.3.

Example 5. Let us mark a rational number on the coordinate line. This rational number will be located in the middle between the numbers 2 and 3

This is 2 (two integers) and (one second). A fraction is also called “half”. Therefore, we marked two whole segments and another half segment on the coordinate line.

If we convert a mixed number to an improper fraction, we get an ordinary fraction. This fraction on the coordinate line will be located in the same place as the fraction

The value of the fraction is 2.5

If we increase the section of the coordinate line from 2 to 3, we will see the following picture:

It can be seen that our rational number is located in the same place as the decimal fraction 2.5

Minus before a rational number

In the previous lesson, which was called, we learned how to divide integers. Both positive and negative numbers could act as dividend and divisor.

Let's consider the simplest expression

(−6) : 2 = −3

In this expression, the dividend (−6) is a negative number.

Now consider the second expression

6: (−2) = −3

Here the divisor (−2) is already a negative number. But in both cases we get the same answer -3.

Considering that any division can be written as a fraction, we can also write the examples discussed above as a fraction:

And since in both cases the value of the fraction is the same, the minus in either the numerator or the denominator can be made common by placing it in front of the fraction

Therefore, you can put an equal sign between the expressions and and because they carry the same meaning

In the future, when working with fractions, if we encounter a minus in the numerator or denominator, we will make this minus common by placing it in front of the fraction.

Opposite rational numbers

Like an integer, a rational number has its opposite number.

For example, for a rational number, the opposite number is . It is located on the coordinate line symmetrically to the location relative to the origin of coordinates. In other words, both of these numbers are equidistant from the origin

Converting mixed numbers to improper fractions

We know that in order to convert a mixed number into an improper fraction, we need to multiply the whole part by the denominator of the fractional part and add it to the numerator of the fractional part. The resulting number will be the numerator of the new fraction, but the denominator remains the same.

For example, let's convert a mixed number to an improper fraction

Multiply the whole part by the denominator of the fractional part and add the numerator of the fractional part:

Let's calculate this expression:

(2 × 2) + 1 = 4 + 1 = 5

The resulting number 5 will be the numerator of the new fraction, but the denominator will remain the same:

This procedure is written in full as follows:

To return the original mixed number, it is enough to select the whole part in the fraction

But this method of converting a mixed number into an improper fraction is only applicable if the mixed number is positive. This method will not work for a negative number.

Let's consider the fraction. Let's select the whole part of this fraction. We get

To return the original fraction, you need to convert the mixed number to an improper fraction. But if we use the old rule, namely, multiply the whole part by the denominator of the fractional part and add the numerator of the fractional part to the resulting number, we get the following contradiction:

We received a fraction, but we should have received a fraction.

We conclude that the mixed number was converted to an improper fraction incorrectly

To correctly convert a negative mixed number into an improper fraction, you need to multiply the whole part by the denominator of the fractional part, and from the resulting number subtract numerator of the fractional part. In this case, everything will fall into place for us

A negative mixed number is the opposite of a mixed number. If a positive mixed number is located on the right side and looks like this

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.

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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 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'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 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 you 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.

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.