Numbers. Real numbers. Numbers: natural, integer, rational, irrational, real Any rational number is real

This article contains basic information about real numbers. First, the definition of real numbers is given and examples are given. The position of the real numbers on the coordinate line is shown next. And in conclusion, it is analyzed how real numbers are given in the form of numerical expressions.

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Definition and examples of real numbers

Real numbers as expressions

From the definition of real numbers, it is clear that real numbers are:

any natural number;
any integer ;
any ordinary fraction (both positive and negative);
any mixed number;
any decimal(positive, negative, finite, infinite periodic, infinite non-periodic).

But very often real numbers can be seen in the form , etc. Moreover, the sum, difference, product, and quotient of real numbers are also real numbers (see operations with real numbers). For example, these are real numbers.

And if we go further, then from real numbers with the help of arithmetic signs, root signs, degrees, logarithmic, trigonometric functions etc. you can compose all kinds of numerical expressions, the values of which will also be real numbers. For example, expression values and are real numbers.

In conclusion of this article, we note that the next step in expanding the concept of number is the transition from real numbers to complex numbers.

Bibliography.

Vilenkin N.Ya. etc. Mathematics. Grade 6: textbook for educational institutions.
Makarychev Yu.N., Mindyuk N.G., Neshkov K.I., Suvorova S.B. Algebra: textbook for 8 cells. educational institutions.
Gusev V.A., Mordkovich A.G. Mathematics (a manual for applicants to technical schools).

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This article is devoted to the study of the topic "Rational numbers". The following are definitions of rational numbers, examples are given, and how to determine whether a number is rational or not.

Rational numbers. Definitions

Before giving a definition of rational numbers, let's remember what other sets of numbers are and how they are related to each other.

Natural numbers, together with their opposites and the number zero, form a set of integers. In turn, the set of integer fractional numbers forms the set of rational numbers.

Definition 1. Rational numbers

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

Thus, we can leave a number of properties of rational numbers:

Any natural number is a rational number. Obviously, every natural number n can be represented as a fraction 1 n .
Any integer, including the number 0 , is a rational number. Indeed, any positive integer and negative integer can be easily represented as a positive or negative common fraction, respectively. For example, 15 = 15 1 , - 352 = - 352 1 .
Any positive or negative common fraction a b is a rational number. This follows directly from the above definition.
Any mixed number is rational. Indeed, after all, a mixed number can be represented as an ordinary improper fraction.
Any finite or periodic decimal fraction can be represented as a common fraction. Therefore, every periodic or final decimal is a rational number.
Infinite and non-recurring decimals are not rational numbers. They cannot be represented in the form of ordinary fractions.

Let us give examples of rational numbers. The numbers 5 , 105 , 358 , 1100055 are natural, positive and integer. After all, these are rational numbers. The numbers - 2 , - 358 , - 936 are negative integers, and they are also rational by definition. The common fractions 3 5 , 8 7 , - 35 8 are also examples of rational numbers.

The above definition of rational numbers can be formulated more concisely. Let's answer the question again, what is a rational number.

Definition 2. Rational numbers

Rational numbers are those numbers that can be represented as a fraction ± z n, where z is an integer, n is a natural number.

It can be shown that this definition is equivalent to the previous definition of rational numbers. To do this, remember that the bar of a fraction is the same as the division sign. Taking into account the rules and properties of the division of integers, we can write the following fair inequalities:

0 n = 0 ÷ n = 0 ; - m n = (- m) ÷ n = - m n .

Thus, one can write:

z n = z n , p p and z > 0 0 , p p and z = 0 - z n , p p and z< 0

Actually, this record is proof. We give examples of rational numbers based on the second definition. Consider the numbers - 3 , 0 , 5 , - 7 55 , 0 , 0125 and - 1 3 5 . All these numbers are rational, since they can be written as a fraction with an integer numerator and natural denominator: - 3 1 , 0 1 , - 7 55 , 125 10000 , 8 5 .

We present one more equivalent form of the definition of rational numbers.

Definition 3. Rational numbers

A rational number is a number that can be written as a finite or infinite periodic decimal fraction.

This definition follows directly from the very first definition of this paragraph.

To summarize and formulate a summary on this item:

Positive and negative fractional and integer numbers make up the set of rational numbers.
Every rational number can be represented as a fraction, the numerator of which is an integer and the denominator a natural number.
Every rational number can also be represented as a decimal fraction: finite or infinite periodic.

Which number is rational?

As we have already found out, any natural number, integer, regular and improper ordinary fraction, periodic and final decimal fraction are rational numbers. Armed with this knowledge, you can easily determine whether a number is rational.

However, in practice, one often has to deal not with numbers, but with numerical expressions that contain roots, powers, and logarithms. In some cases, the answer to the question "Is a number rational?" is far from obvious. Let's take a look at how to answer this question.

If a number is given as an expression containing only rational numbers and arithmetic operations between them, then the result of the expression is a rational number.

For example, the value of the expression 2 · 3 1 8 - 0 , 25 0 , (3) is a rational number and equals 18 .

Thus, simplifying a complex numerical expression allows you to determine whether the number given by it is rational.

Now let's deal with the sign of the root.

It turns out that the number m n given as the root of the degree n of the number m is rational only when m is the nth power of some natural number.

Let's look at an example. The number 2 is not rational. Whereas 9, 81 are rational numbers. 9 and 81 are the perfect squares of the numbers 3 and 9, respectively. The numbers 199 , 28 , 15 1 are not rational numbers, since the numbers under the root sign are not perfect squares of any natural numbers.

Now let's take a more complicated case. Is the number 243 5 rational? If you raise 3 to the fifth power, you get 243 , so the original expression can be rewritten like this: 243 5 = 3 5 5 = 3 . Therefore, this number is rational. Now let's take the number 121 5 . This number is not rational, since there is no natural number whose raising to the fifth power will give 121.

In order to find out whether the logarithm of some number a to the base b is a rational number, it is necessary to apply the contradiction method. For example, let's find out if the number log 2 5 is rational. Let's assume that this number is rational. If so, then it can be written as an ordinary fraction log 2 5 = m n. By the properties of the logarithm and the properties of the degree, the following equalities are true:

5 = 2 log 2 5 = 2 m n 5 n = 2 m

Obviously, the last equality is impossible, since the left and right sides contain odd and even numbers, respectively. Therefore, the assumption made is wrong, and the number log 2 5 is not a rational number.

It is worth noting that when determining the rationality and irrationality of numbers, one should not make sudden decisions. For example, the result of a product of irrational numbers is not always an irrational number. illustrative example: 2 2 = 2 .

There are also irrational numbers whose raising to an irrational power gives a rational number. In a power of the form 2 log 2 3, the base and exponent are irrational numbers. However, the number itself is rational: 2 log 2 3 = 3 .

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

This is a natural series of 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 natural numbers are only divisible by 1 and themselves. 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 whole numbers 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 integer,n natural number. Let's represent the number 3,(6) from the previous example as such a fraction:

Another example: the rational number 9 can be represented as a simple fraction as 18/2 or as 36/4.

Another example: the rational number -9 can be represented as a simple fraction as -18/2 or as -72/8.