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

In order to avoid mistakes in the future, in this lesson we will delve a little into the topic of rational numbers, draw 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 - is the numerator of a fraction b is the denominator of the fraction. And b must not be zero, since 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. So 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 the mixed number to an improper fraction.

So 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 an ordinary fraction. If you are having difficulty at this point, repeat the topic.

Since the decimal fraction 0.2 can be represented as a fraction, it means that it also applies 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 are having 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, all numbers that can be represented as a fraction, we will increasingly call one phrase - rational numbers.

Rational numbers on the coordinate line

We considered 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.

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

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

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

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

As mentioned above, between integers lie other numbers - ordinary fractions, decimal fractions, mixed numbers, etc. For example, if you increase the 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 already other rational numbers, which are decimal fractions familiar to us. Our fraction is also visible here, which is located in the same place as the decimal fraction 0.5. A careful examination of this figure gives an answer to the question why the fraction is located exactly there.

A fraction means to divide 1 by 2. And if we divide 1 by 2, then 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 will not change.

If the numerator and denominator of a fraction are multiplied by any number, for example by the number 4, then we will 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 the numbers 1 and 2

The value of the fraction is 1.5

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

It can be seen that between the integers 1 and 2 there are already other rational numbers, which are decimal fractions familiar to us. Our fraction is also visible here, which is located in the same place as the decimal fraction 1.5.

We increased certain segments on the coordinate line in order to see the rest of the numbers lying on this segment. As a result, we found 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 easy to guess that between decimal fractions that have one digit after the decimal point, there are already 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 decimal fractions 0.1 and 0.2

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

Example 3 We 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 where exactly 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 a 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 very close to the usual decimal 0.3

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

Example 5 We 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 a "half". Therefore, we marked two whole segments and another half of the segment on the coordinate line.

If we translate a mixed number into 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, then 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. The dividend and divisor could be both positive and negative numbers.

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.

Given 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 standing either in the numerator or in the denominator can be made common by putting it in front of the fraction

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

In the future, working with fractions, if we encounter a minus in the numerator or in the denominator, we will make this minus common, putting 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. In other words, both of these numbers are equidistant from the origin

Convert mixed numbers to improper fractions

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

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

Multiply the integer 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, and the denominator will remain the same:

The entire process is written as follows:

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

But this way of converting a mixed number to an improper fraction is applicable only if the mixed number is positive. For a negative number, this method will not work.

Let's consider a fraction. Let's take the integer part of this fraction. 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, we multiply the integer part by the denominator of the fractional part and add the numerator of the fractional part to the resulting number, then we get the following contradiction:

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

We conclude that the mixed number was translated incorrectly into an improper fraction

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

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

As we have seen, the set of natural numbers

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

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

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

Case 1. .

Case 2. .

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

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

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

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

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

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

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

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

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

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

High school students and students of mathematical specialties are likely to easily answer this question. 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 a fraction. Positive, negative, as well as zero are also included in this set. The numerator of a fraction must be an integer, and the denominator must be

This set is denoted in mathematics 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 be presented in different forms. First, 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 - a periodic fraction. This type is not very common, but 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. Equal fractions will also be called, 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 is rather "reasonable", "considered". But mathematical terms are close to the direct meaning of this. In Latin, "ratio" is "ratio", "fraction" or "division". Thus, the name reflects 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 encounter 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 only one ratio can exist between two numbers - they are either equal to each other, or one is greater or less than the other. i.e.:

or a = b or a > b or a< b.

Moreover, this property also implies the transitivity of the relation. That is, if a more b, b more c, then a more c. In the language of mathematics, 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, a number of properties can also be distinguished in the process of transformations.

• a + b = b + a (replacement 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 it comes to ordinary, and not or integers, operations with them can cause certain difficulties. So, addition and subtraction are possible only if the denominators are equal. If they are initially different, you should find a common one, using the multiplication of 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 performed in accordance with fairly simple rules. Reduction to a common denominator is not necessary. The numerators and denominators are multiplied separately, while 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 with a slight difference. For the second fraction, you should find the reciprocal, that is,

"flip" it. 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. The term "principle" is also often found in the literature. It is valid for the entire set of real numbers, but not everywhere. Thus, this principle does not work for some collections of rational functions. In essence, this axiom means that given the existence of two quantities a and b, you can always take enough a to surpass 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 young 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

Definition of rational numbers

Rational numbers are:

• Natural numbers that can be represented as a fraction. For example, $7=\frac(7)(1)$.
• Integers, including the number zero, that can be expressed as positive or negative fractions, or as zero. For example, $19=\frac(19)(1)$, $-23=-\frac(23)(1)$.
• Ordinary fractions (positive or negative).
• Mixed numbers that can be represented as an improper common fraction. For example, $3 \frac(11)(13)=\frac(33)(13)$ and $-2 \frac(4)(5)=-\frac(14)(5)$.
• A finite decimal and an infinite periodic fraction, which can be represented as a common fraction. For example, $-7,73=-\frac(773)(100)$, $7,(3)=-7 \frac(1)(3)=-\frac(22)(3)$.

Remark 1

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

Example 1

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

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

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

Thus, rational numbers are divided into positive and negative. Zero is a rational number, but it is not a positive or negative rational number.

Let us formulate a shorter definition of rational numbers.

Definition 3

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

The following conclusions can be drawn:

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

How to determine if a number is rational

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

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

Rational numbers

quarters

1. Orderliness. a and b there is a rule that allows you to uniquely identify between them one and only one of the three relationships: “< », « >' 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

2. 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: .
3. 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: .
4. Transitivity of the order relation. For any triple of rational numbers a , b and c if a smaller b and b smaller c, then a smaller c, and if a equals b and b equals c, then a equals c. 6435">Commutativity of addition. The sum does not change from changing the places of rational terms.
5. Associativity of addition. The order in which three rational numbers are added does not affect the result.
6. The presence of zero. There is a rational number 0 that preserves every other rational number when summed.
7. The presence of opposite numbers. Any rational number has an opposite rational number, which, when summed, gives 0.
8. Commutativity of multiplication. By changing the places of rational factors, the product does not change.
9. Associativity of multiplication. The order in which three rational numbers are multiplied does not affect the result.
10. The presence of a unit. There is a rational number 1 that preserves every other rational number when multiplied.
11. The presence of reciprocals. Any rational number has an inverse rational number, which, when multiplied, gives 1.
12. Distributivity of multiplication with respect to addition. The multiplication operation is consistent with the addition operation through the distribution law:
13. Connection of the order relation with the operation of addition. The same rational number can be added to the left and right sides of a rational inequality. /pictures/wiki/files/51/358b88fcdff63378040f8d9ab9ba5048.png" border="0">
14. Axiom of Archimedes. Whatever the rational number a, you can take so many units that their sum will exceed a. src="/pictures/wiki/files/55/70c78823302483b6901ad39f68949086.png" border="0">

Additional properties

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

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Set countability

Numbering of rational numbers

To estimate the number of rational numbers, you need to find the cardinality of their set. It is easy to prove that the set of rational numbers is countable. To do this, it suffices to give an algorithm that enumerates rational numbers, 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. A formal sign of irreducibility is the equality to one of the greatest common divisor of the numerator and denominator of a 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 misleading 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

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