Rational numbers and measurements. Definition of rational numbers

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 exists? Exists infinite set 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 composite numbers:

One is not considered a composite number.

The set of natural numbers is one, prime numbers and composite numbers.

The set of natural numbers is denoted Latin letter N.

Properties of addition and multiplication of natural numbers:

commutative property of addition

associative property addition

(a + b) + c = a + (b + c);

commutative property of multiplication

associative property of multiplication

(ab) c = a (bc);

distributive property 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 These 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 periodic fraction with period zero.

Any rational number can be represented as a fraction m/n, where m is an integer number,n natural number. Let's imagine the number 3,(6) from the previous example as such a fraction.

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

Rational number- the number that is represented ordinary fraction m/n, where the numerator m are integers, and the denominator n- integers, for example 2/3.

Endless non-periodic fractions 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⋅ nb +mb⋅ 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∈ Qa⋅ 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∈ Qa⋅ 1=a

12. Availability reciprocal numbers . Every rational number other than zero has an inverse rational number, multiplying by which we get 1 .

∀ a∈ Q∃ a−1∈ Qa⋅ 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. To the left and right side rational inequality add the same rational number.

∀ a,b,c∈ Qa ⇒ 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.

) 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.

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)

Summation Rule looks like that:

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

∀ a,b∈ Q∃ !(a+b)∈ Q

Multiplication rule looks like that: m a n a⋅ m b n b =m a⋅ m b n a⋅ 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∈ Qa⋅ 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∈ Qa⋅ 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∈ Qa⋅ 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∈ Qa ⇒ 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.

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.

To avoid mistakes in the future, this lesson we'll go a little deeper into the topic of rational numbers and draw from it necessary information and let's 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 obtained by converting a mixed number to improper fraction

Means mixed number refers to rational numbers.

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.

Because the decimal 0.2 can be represented as a fraction, which means 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 lies 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 they weren't singular 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. His fraction never ends, it's endless

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 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. For a negative number this method won't work.

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

Rational numbers

Quarters

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 relationship 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
Addition operation. For any rational numbers a And b there is a so-called summation rule c. At the same time, 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: .
Multiplication operation. For any rational numbers a And b there is a so-called multiplication rule, which assigns them some rational number c. At the same time, 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: .
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.
Associativity of addition. Order adding three rational numbers 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 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. /pictures/wiki/files/51/358b88fcdff63378040f8d9ab9ba5048.png" border="0">
Axiom of Archimedes. Whatever the rational number a, you can take so many units that their sum 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.

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

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 is created ordinary fractions, 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 fractions 1/1 are assigned to the number 1, the fractions 2/1 to the number 2, etc. It should be noted that only irreducible fractions. 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 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

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