Rational numbers, definition, examples. What are rational numbers? What are the other

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's formulate more short definition 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 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 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 $8=2^3$.

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.

Integers

Natural numbers definition are positive integers. Natural numbers are used to count objects and for many other purposes. Here are the numbers:

This is natural series numbers.
Zero is a natural number? No, zero is not a natural number.
How many natural numbers are there? There is an infinite set of natural numbers.
What is the smallest natural number? One is the smallest natural number.
What is the largest natural number? It cannot be specified, because there is an infinite set of natural numbers.

The sum of natural numbers is a natural number. So, the addition of 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 natural numbers is a natural number, otherwise it is not.

The quotient of natural numbers There is not always a natural number. If for natural numbers a and b

where c is a natural number, it means that a is evenly 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 the natural number by which the first number is evenly divisible.

Every natural number is divisible by 1 and itself.

Simple integers are only divisible by 1 and itself. Here we mean divided completely. Example, numbers 2; 3; 5; 7 is only divisible by 1 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 is 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 natural numbers, zero and the opposite of natural numbers.

Numbers opposite to 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 integers and fractions.

Any rational number can be represented as a periodic fraction. Examples:

1,(0); 3,(6); 0,(0);...

It can be seen from the examples that any integer is a periodic fraction with a period of zero.

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

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" (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 belongs to the set of natural numbers.

For more examples of rational numbers, see the article.

Equal rational numbers

Different fractions can represent a single rational number.

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

Reduce the numerator and denominator by a common factor of 2:

6	=	2 * 3	=	3
10		2 * 5		5

We got the fraction 3/5, which means that

In this subsection we give several definitions of rational numbers. Despite the differences in wording, all these definitions have the same meaning: rational numbers combine integers and fractional numbers, just as integers combine natural numbers, their opposite numbers, 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 as the most natural.

Definition.

Rational numbers are numbers that can be written as positive common fraction, a negative common fraction, or the number zero.

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

any natural number n. Indeed, any natural number can be represented as an ordinary fraction, for example, 3=3/1 .

· Any integer, in particular, the number zero. Indeed, any integer can be written as either a positive common fraction, or as a negative common fraction, or as zero. For example, 26=26/1 , .

Any ordinary fraction (positive or negative). This is directly stated by the given definition of rational numbers.

· Any mixed number. Indeed, it is always possible to represent a mixed number as an improper common fraction. For example, and.

· Any finite decimal fraction or infinite periodic fraction. This is so because the specified decimal fractions are converted to ordinary fractions. For example, a 0,(3)=1/3 .

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

Now we can easily bring examples of rational numbers. Numbers 4 ,903 , 100 321 are rational numbers, since they are natural numbers. Whole numbers 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.

The above examples show 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 shorter form.

Definition.

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

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

Let us give examples of rational numbers, based on this definition. 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 also 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 any 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, numbers 5 , 0 , −13 , are examples of rational numbers, since they can be written as the following decimal fractions 5,0 , 0,0 ,−13,0 , 0,8 and −7,(18) .

We finish the theory of this section with the following statements:

integer and fractional numbers (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 is a rational number;

Every rational number can be represented as a finite or infinite periodic decimal fraction, and each such fraction represents some rational number.

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The addition of positive rational numbers is commutative and associative,

("a, b н Q +) a + b= b + a;

("a, b, c н Q +) (a + b)+ c = a + (b+ c)

Before formulating the definition of multiplication of positive rational numbers, consider the following problem: it is known that the length of the segment X is expressed as a fraction at unit length E, and the length of the unit segment is measured using the unit E 1 and is expressed as a fraction. How to find the number that will represent the length of the segment X, if you measure it using the unit of length E 1?

Since X=E, then nX=mE, and from the fact that E =E 1 it follows that qE=pE 1 . We multiply the first equality obtained by q, and the second by m. Then (nq)X \u003d (mq)E and (mq)E \u003d (mp)E 1, whence (nq)X \u003d (mp)E 1. This equality shows that the length of the segment x at unit length is expressed as a fraction, and hence , =, i.e. multiplication of fractions is associated with the transition from one unit of length to another when measuring the length of the same segment.

Definition. If a positive number a is represented by a fraction, and a positive rational number b is a fraction, then their product is the number a b, which is represented by a fraction.

Multiplication of positive rational numbers commutative, associative, and distributive with respect to addition and subtraction. The proof of these properties is based on the definition of multiplication and addition of positive rational numbers, as well as on the corresponding properties of addition and multiplication of natural numbers.

46. ​​As you know subtraction is the opposite of addition.

If a a and b - positive numbers, then subtracting the number b from the number a means finding a number c that, when added to the number b, gives the number a.
a - b = c or c + b = a
The definition of subtraction holds true for all rational numbers. That is, the subtraction of positive and negative numbers can be replaced by addition.
To subtract another from one number, you need to add the opposite number to the minuend.
Or, in another way, we can say that the subtraction of the number b is the same addition, but with the number opposite to the number b.
a - b = a + (- b)
Example.
6 - 8 = 6 + (- 8) = - 2
Example.
0 - 2 = 0 + (- 2) = - 2
It is worth remembering the expressions below.
0 - a = - a
a - 0 = a
a - a = 0

Rules for subtracting negative numbers
The subtraction of the number b is the addition with the number opposite to the number b.
This rule is preserved not only when subtracting a smaller number from a larger number, but also allows you to subtract a larger number from a smaller number, that is, you can always find the difference between two numbers.
The difference can be a positive number, a negative number, or zero.
Examples of subtracting negative and positive numbers.
- 3 - (+ 4) = - 3 + (- 4) = - 7
- 6 - (- 7) = - 6 + (+ 7) = 1
5 - (- 3) = 5 + (+ 3) = 8
It is convenient to remember the sign rule, which allows you to reduce the number of brackets.
The plus sign does not change the sign of the number, so if there is a plus in front of the bracket, the sign in the brackets does not change.
+ (+ a) = + a
+ (- a) = - a
The minus sign in front of the brackets reverses the sign of the number in the brackets.
- (+ a) = - a
- (- a) = + a
It can be seen from the equalities that if there are identical signs before and inside the brackets, then we get “+”, and if the signs are different, then we get “-”.
(- 6) + (+ 2) - (- 10) - (- 1) + (- 7) = - 6 + 2 + 10 + 1 - 7 = - 13 + 13 = 0
The rule of signs is also preserved if there is not one number in brackets, but an algebraic sum of numbers.
a - (- b + c) + (d - k + n) = a + b - c + d - k + n
Please note that if there are several numbers in brackets and there is a minus sign in front of the brackets, then the signs in front of all the numbers in these brackets must change.
To remember the rule of signs, you can make a table for determining the signs of a number.
Sign rule for numbers + (+) = + + (-) = -
- (-) = + - (+) = -
Or learn a simple rule.
Two negatives make an affirmative,
Plus times minus equals minus.

Rules for dividing negative numbers.
To find the modulus of the quotient, you need to divide the modulus of the dividend by the modulus of the divisor.
So, to divide two numbers with the same signs, you need:

Divide the modulus of the dividend by the modulus of the divisor;

Put a "+" sign in front of the result.

Examples of dividing numbers with different signs:

You can also use the following table to determine the quotient sign.
The rule of signs when dividing
+ : (+) = + + : (-) = -
- : (-) = + - : (+) = -

When calculating "long" expressions, in which only multiplication and division appear, it is very convenient to use the sign rule. For example, to calculate a fraction
You can pay attention that in the numerator there are 2 "minus" signs, which, when multiplied, will give a "plus". There are also three minus signs in the denominator, which, when multiplied, will give a minus. Therefore, in the end, the result will be with a minus sign.
Fraction reduction (further actions with modules of numbers) is performed in the same way as before:
The quotient of dividing zero by a non-zero number is zero.
0: a = 0, a ≠ 0
Do NOT divide by zero!
All previously known rules for dividing by one also apply to the set of rational numbers.
a: 1 = a
a: (- 1) = - a
a: a = 1, where a is any rational number.
The dependencies between the results of multiplication and division, which are known for positive numbers, are also preserved for all rational numbers (except for the number zero):
if a × b = c; a = c: b; b = c: a;
if a: b = c; a = c × b; b=a:c
These dependencies are used to find the unknown factor, dividend and divisor (when solving equations), as well as to check the results of multiplication and division.
An example of finding the unknown.
x × (-5) = 10
x=10: (-5)
x=-2

Similar information.

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

rational number - a number that is represented by a simple fraction m/n, where the numerator m are whole numbers, 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 the natural numbers).

Using rational numbers in real life.

AT 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 a rough estimate of 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 uniquely identify between them 1-but and only one of the 3 relations: “<», «>" 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 related by the same relationship as 2 integers m a⋅ nb and m b⋅ n a;
• 2 negative numbers a and b related by the same relation 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 puts them in correspondence with a certain rational number c. However, the number itself c- This sum numbers a and b and is referred to 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⋅ nb).

∀ 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⋅ nb.

∀a,b∈Q ∃(a⋅b)∈Q

4. Transitivity of the order relation. For any three 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.

∀ a,b,c∈ Q(a ∧ b ⇒ a ∧ (a=b∧ b=c⇒ a = c)

5. Commutativity of addition. From a change in the places of rational terms, the sum does not change.

∀ a,b∈ Qa+b=b+a

6. Associativity of addition. The order of addition of 3 rational numbers 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∈ Qa+0=a

8. Presence of opposite numbers. Every rational number has an opposite rational number, adding them together results in 0.

∀ a∈ Q∃ (−a)∈ Qa+(−a)=0

9. Commutativity of multiplication. By changing the places of rational factors, the product does not change.

∀ a,b∈ Qa⋅ b=b⋅ a

10. Associativity of multiplication. The order of multiplication of 3 rational numbers does not affect the result.

∀ a,b,c∈ Q(a⋅ b)⋅ c=a⋅ (b⋅ c)

11. Availability of a unit. 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 reciprocals. Any 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 with respect to addition. The multiplication operation is related to addition using the distribution law:

∀ a,b,c∈ Q(a+b)⋅ c=a⋅ c+b⋅ c

14. Connection of the order relation with the addition operation. The same rational number is added to the left and right sides of a rational inequality.

∀ a,b,c∈ Q a ⇒ a+c

15. Connection of the order relation with the operation of multiplication. The left and right sides of a rational inequality can be multiplied by the same non-negative rational number.

∀ a,b,c∈ Qc>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.