Rational fractional numbers. Numbers

Definition of rational numbers:

A rational number is a number that can be represented as a fraction. The numerator of such a fraction belongs to the set of integers, and the denominator belongs to the set of natural numbers.

Why are numbers called rational?

In Latin, ratio means ratio. Rational numbers can be represented as a ratio, i.e. in other words, as a fraction.

Rational number example

The number 2/3 is a rational number. Why? This number is represented as a fraction, the numerator of which belongs to the set of integers, and the denominator to the set of natural numbers.

For more examples of rational numbers, see the article.

Equal rational numbers

Different fractions can represent the same rational number.

Consider the rational number 3/5. This rational number is equal to

Let's reduce the numerator and denominator by a common factor of 2:

We got the fraction 3/5, which means that

Rational numbers

Quarters

1. Orderliness. a And b there is a rule that allows one to uniquely identify one and only one of three relationships between them: “< », « >" or " = ". This rule is called ordering rule and is formulated as follows: two non-negative numbers and are related by the same relation as two integers and ; two non-positive numbers a And b are related by the same relationship as two non-negative numbers and ; if suddenly a non-negative, but b- negative, then a > b. src="/pictures/wiki/files/57/94586b8b651318d46a00db5413cf6c15.png" border="0">

   

   Adding Fractions

2. Addition operation. For any rational numbers a And b there is a so-called summation rule c. Moreover, the number itself c called amount numbers a And b and is denoted by , and the process of finding such a number is called summation. The summation rule has the following form: .
3. Multiplication operation. For any rational numbers a And b there is a so-called multiplication rule, which assigns them some rational number c. Moreover, the number itself c called work numbers a And b and is denoted by , and the process of finding such a number is also called multiplication. The multiplication rule looks like this: .
4. Transitivity of the order relation. For any triple of rational numbers a , b And c If a less b And b less c, That a less c, and if a equals b And b equals c, That a equals c. 6435">Commutativity of addition. Changing the places of rational terms does not change the sum.
5. Associativity of addition. The order in which three rational numbers are added does not affect the result.
6. Presence of zero. There is a rational number 0 that preserves every other rational number when added.
7. The presence of opposite numbers. Any rational number has an opposite rational number, which when added to gives 0.
8. Commutativity of multiplication. Changing the places of rational factors does not change the product.
9. Associativity of multiplication. The order in which three rational numbers are multiplied does not affect the result.
10. Availability of unit. There is a rational number 1 that preserves every other rational number when multiplied.
11. Presence of reciprocal numbers. Any rational number has an inverse rational number, which when multiplied by gives 1.
12. Distributivity of multiplication relative to addition. The multiplication operation is coordinated with the addition operation through the distribution law:
13. Connection of the order relation with the operation of addition. The same rational number can be added to the left and right sides of a rational inequality. /pictures/wiki/files/51/358b88fcdff63378040f8d9ab9ba5048.png" border="0">
14. Axiom of Archimedes. Whatever the rational number a, you can take so many units that their sum exceeds a. src="/pictures/wiki/files/55/70c78823302483b6901ad39f68949086.png" border="0">

Additional properties

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

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Countability of a set

Numbering of rational numbers

To estimate the number of rational numbers, you need to find the cardinality of their set. It is easy to prove that the set of rational numbers is countable. To do this, it is enough to give an algorithm that enumerates rational numbers, i.e., establishes a bijection between the sets of rational and natural numbers.

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

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

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

In the process of such a traversal, each new rational number is associated with another natural number. That is, the fraction 1/1 is assigned to the number 1, the fraction 2/1 to the number 2, etc. It should be noted that only irreducible fractions are numbered. 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.

The statement about the countability of the set of rational numbers may cause some confusion, since at first glance it seems that it is much more extensive than the set of natural numbers. In fact, this is not so and there are enough natural numbers to enumerate all rational ones.

Lack of rational numbers

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

Rational numbers of the form 1 / n at large n arbitrarily small quantities can be measured. This fact creates the misleading impression that rational numbers can be used to measure any geometric distances. It is easy to show that this is not true.

Notes

Literature

• I. Kushnir. Handbook of mathematics for schoolchildren. - Kyiv: ASTARTA, 1998. - 520 p.
• P. S. Alexandrov. Introduction to set theory and general topology. - M.: chapter. ed. physics and mathematics lit. ed. "Science", 1977
• I. L. Khmelnitsky. Introduction to the theory of algebraic systems

Links

Wikimedia Foundation. 2010.

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.

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 incorrectly converted into an improper fraction:

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

Definition of rational numbers

Rational numbers include:

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

Note 1

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

Example 1

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

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

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

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

Let us formulate a more concise definition of rational numbers.

Definition 3

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

The following conclusions can be drawn:

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

How to Determine If a Number is Rational

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

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

As we have already seen, the set of natural numbers

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

closed under addition, multiplication and subtraction. However, neither of these sets is closed under division, since division of integers can result in fractions, as in the cases of 4/3, 7/6, -2/5, etc. 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 in the form , where a and d are integers, and d is not equal to zero. Let us make a few comments about this definition.

1) We required that d be non-zero. This requirement (mathematically written as 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 the dividend and 7 the divisor, we get the quotient of 3 and the remainder of 4. So the word divisor is used here in a more general sense and applies to a greater number of cases than in Chap. I. However, in cases like Case 1, the concept of a divisor introduced in Chap. I; therefore it is necessary, as in ch. I, exclude the possibility of d = 0.

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

3) The definition of a rational number includes the expression “a number that can be represented in the form , 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 the 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 entry we have chosen are of the form , where a and d are integers.

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

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 case of fraction conversion

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

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