Prove that the number a is. Rational numbers, definition, examples. Definition and examples of rational numbers

Instructions

However, the search can be quite long if, say, you need to check a number like 136827658235479371. Therefore, it is worth paying attention to rules that can significantly reduce the calculation time.

If the number is composite, that is, it is a product of simple factors, then among these factors there must be at least one that is less than the square root of the given number. After all, the product of two, each of which is greater than the square root of some X, will certainly be greater than X, and these two numbers cannot in any way be its divisors.

Therefore, even with a simple search, you can limit yourself to checking only those integers that do not exceed the square root of a given number, rounded up. For example, when checking the number 157, you only try possible factors from 2 to 13.

If you don’t have a computer at hand, and you have to check the number for simplicity manually, then here too simple and obvious rules come to the rescue. What will help you most is knowing what you already know prime numbers. After all, there is no point in checking divisibility by composite numbers separately if you can check divisibility by their prime factors.

An even number, by definition, cannot be prime, since it is divisible by 2. Therefore, if the last digit of a number is even, then it is obviously composite.

Numbers divisible by 5 always end in a 5 or a zero. Looking at the last digit of the number will help weed them out.

If a number is divisible by 3, then the sum of its digits is also necessarily divisible by 3. For example, the sum of the digits of the number 136827658235479371 is 1 + 3 + 6 + 8 + 2 + 7 + 6 + 5 + 8 + 2 + 3 + 5 + 4 + 7 + 9 + 3 + 7 + 1 = 87. This number is divisible by 3 without a remainder: 87 = 29*3. Therefore, our number is also divisible by 3 and is composite.

The test of divisibility by 11 is also very simple. It is necessary to subtract the sum of all its even digits from the sum of all odd digits of a number. Evenness and oddness are determined by counting from the end, that is, from units. If the resulting difference is divisible by 11, then the entire given number is also divisible by it. For example, let the number 2576562845756365782383 be given. The sum of its even digits is 8 + 2 + 7 + 6 + 6 + 7 + 4 + 2 + 5 + 7 + 2 = 56. The sum of its odd digits is 3 + 3 + 8 + 5 + 3 + 5 + 5 + 8 + 6 + 6 + 5 = 57. The difference between them is 1. This number is not divisible by 11, and therefore 11 is not a divisor of the given number.

You can check the divisibility of a number by 7 and 13 in a similar way. Break the number into triplets of digits, starting from the end (this is done in typographic notation for ease of reading). The number 2576562845756365782383 becomes 2,576,562,845,756,365,782,383. Add the numbers in the odd places and subtract the sum of the numbers in the even places. In this case, you will get (383 + 365 + 845 + 576) - (782 + 756 + 562 + 2) = 67. This number is not divisible by either 7 or 13, which means they are not divisors of the given number.

Please note

The tests for divisibility by other prime numbers are much more complex, and in most cases it is easier to try to divide a given number by the intended divisor manually.

Sources:

Elementary mathematics - signs of divisibility

The most famous ways to find a list of prime numbers up to a certain value are the Sieve of Eratosthenes, the Sieve of Sundaram and the Sieve of Atkin. To check whether a given number is prime, there are primality tests

You will need

Calculator, sheet of paper and pencil (pen)

Instructions

Method 1. Sieve of Eratosthenes.
According to this method, in order to find all prime numbers not greater than a certain value X, you need to write down all the integers from one to X in a row. Let's take the number 2 as the first number. Let's cross out from the list all the numbers divisible by 2. Then take the next number after , not crossed out, and cross out from the list all the numbers divisible by the number we took. And then each time we will take the next number that has not been crossed out and cross out from the list all the numbers that are divisible by the number we took. And so on until the number we have chosen becomes greater than X/2. All remaining in the list not crossed out with simple

Method 2. Sundaram sieve.
From the series of natural numbers from 1 to N, all numbers of the form are excluded
x + y + 2xy,
where the indices x (not greater than y) run through all natural values, for which x+y+2xy is not greater than N, namely the values x=1, 2,...,((2N+1)1/2-1)/2 and x=y, x+1,.. .,(N-x)/(2x+1)yu. Then each of the remaining numbers is multiplied by 2 and 1. The resulting sequence is all odd prime numbers in the series from one to 2N+1.

Method 3. Atkin's sieve.
Atkin's sieve is a complex modern algorithm for finding all prime numbers up to a given value X. The main essence of the algorithm is to represent prime numbers as integers with an odd number of representations in given quadratic forms. A separate stage of the algorithm eliminates numbers that are multiples of the squares of prime numbers in the range from 5 to X.

Simplicity tests.
Primality tests are algorithms to determine whether a particular number X is prime.
One of the simplest, but also time-consuming tests is enumeration of divisors. It consists of taking all integers from 2 to the square root of X and calculating the remainder when X is divided by each of these numbers. If the remainder of dividing the number X by a certain number (greater than 1 and less than X) is equal to zero, then the number X is composite. If it turns out that the number X cannot be reduced without a remainder by any of the numbers except one and itself, then the number X is prime.
In addition to this method, there are also a large number of other tests for testing the primality of a number. Most of these tests are probabilistic and are used in cryptography. The only test that guarantees an answer (the AKS test) is very difficult to calculate, which makes it difficult to use in practice.

In this article we will begin to explore rational numbers. Here we will give definitions of rational numbers, give the necessary explanations and give examples of rational numbers. After this, we will focus on how to determine whether a given number is rational or not.

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Definition and examples of rational numbers

In this section we will give several definitions of rational numbers. Despite differences in wording, all of these definitions have the same meaning: rational numbers unite integers and fractions, just as integers unite natural numbers, their opposites, 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 most naturally.

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

Any natural number n. Indeed, you can represent any natural number as an ordinary fraction, for example, 3=3/1.
Any integer, in particular the number zero. In fact, any integer can be written as either a positive fraction, a negative fraction, or zero. For example, 26=26/1, .
Any common fraction (positive or negative). This is directly confirmed by the given definition of rational numbers.
Any mixed number. Indeed, you can always represent a mixed number as an improper fraction. For example, and.
Any finite decimal fraction or infinite periodic fraction. This is so due to the fact that the indicated decimal fractions are converted into ordinary fractions. For example, , and 0,(3)=1/3.

It is also clear that any infinite non-periodic decimal NOT a rational number because it cannot be represented as a fraction.

Now we can easily give examples of rational numbers. The numbers 4, 903, 100,321 are rational numbers because they are natural numbers. The integers 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.

From the above examples it is clear 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 more concise form.

Definition.

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

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

Let us give examples of rational numbers based on this definition. The 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 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 every 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, the numbers 5, 0, −13, are examples of rational numbers because they can be written as the following decimal fractions 5.0, 0.0, −13.0, 0.8, and −7, (18).

Let’s finish the theory of this point with the following statements:

integers and fractions (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 represents a certain rational number;
every rational number can be represented as a finite or infinite periodic decimal fraction, and each such fraction represents a rational number.

Is this number rational?

In the previous paragraph, we found out that any natural number, any integer, any ordinary fraction, any mixed number, any finite decimal fraction, as well as any periodic decimal fraction is a rational number. This knowledge allows us to “recognize” rational numbers from a set of written numbers.

But what if the number is given in the form of some , or as , etc., how to answer the question whether this number is rational? In many cases it is very difficult to answer. Let us indicate some directions of thought.

If a number is given as a numeric expression that contains only rational numbers and arithmetic signs (+, −, · and:), then the value of this expression is a rational number. This follows from how operations with rational numbers are defined. For example, after performing all the operations in the expression, we get the rational number 18.

Sometimes, after simplifying the expressions and making them more complex, it becomes possible to determine whether a given number is rational.

Let's move on. The number 2 is a rational number, since any natural number is rational. What about the number? Is it rational? It turns out that no, it is not a rational number, it is an irrational number (the proof of this fact by contradiction is given in the algebra textbook for grade 8, listed below in the list of references). It has also been proven that the square root of a natural number is a rational number only in those cases when under the root there is a number that is the perfect square of some natural number. For example, and are rational numbers, since 81 = 9 2 and 1 024 = 32 2, and the numbers and are not rational, since the numbers 7 and 199 are not perfect squares of natural numbers.

Is the number rational or not? In this case, it is easy to notice that, therefore, this number is rational. Is the number rational? It has been proven that the kth root of an integer is a rational number only if the number under the root sign is the kth power of some integer. Therefore, it is not a rational number, since there is no integer whose fifth power is 121.

The method by contradiction allows you to prove that the logarithms of some numbers are not rational numbers for some reason. For example, let us prove that - is not a rational number.

Let's assume the opposite, that is, let's say that is a rational number and can be written as an ordinary fraction m/n. Then we give the following equalities: . The last equality is impossible, since on the left side there is odd number 5 n, and on the right side is the even number 2 m. Therefore, our assumption is incorrect, thus not a rational number.

In conclusion, it is worth especially noting that when determining the rationality or irrationality of numbers, one should refrain from making sudden conclusions.

For example, you should not immediately assert that the product of the irrational numbers π and e is an irrational number; this is “seemingly obvious”, but not proven. This raises the question: “Why would a product be a rational number?” And why not, because you can give an example of irrational numbers, the product of which gives a rational number: .

It is also unknown whether numbers and many other numbers are rational or not. For example, there are irrational numbers whose irrational power is a rational number. For illustration, we present a degree of the form , the base of this degree and the exponent are not rational numbers, but , and 3 is a rational number.

References.

Mathematics. 6th grade: educational. for general education institutions / [N. Ya. Vilenkin and others]. - 22nd ed., rev. - M.: Mnemosyne, 2008. - 288 p.: ill. ISBN 978-5-346-00897-2.
Algebra: textbook for 8th grade. general education institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova]; edited by S. A. Telyakovsky. - 16th ed. - M.: Education, 2008. - 271 p. : ill. - ISBN 978-5-09-019243-9.
Gusev V. A., Mordkovich A. G. Mathematics (a manual for those entering technical schools): Proc. allowance.- M.; Higher school, 1984.-351 p., ill.

Here we will look at the definition final limit sequences. The case of a sequence converging to infinity is considered on the page "Definition of an infinitely large sequence".

Definition

(xn), if for any positive number ε > 0 there is a natural number N ε depending on ε such that for all natural numbers n > N ε the inequality
| x n - a|< ε .
The sequence limit is denoted as follows:
.
Or at .

Let's transform the inequality:
;
;
.

An open interval (a - ε, a + ε) is called ε - neighborhood of point a.

A sequence that has a limit is called convergent sequence. It is also said that the sequence converges to a. A sequence that has no limit is called divergent.

From the definition it follows that if a sequence has a limit a, no matter what ε-neighborhood of point a we choose, outside it there can be only a finite number of elements of the sequence, or none at all (the empty set). And any ε-neighborhood contains an infinite number of elements. In fact, having given a certain number ε, we thereby have the number . So all elements of the sequence with numbers , by definition, are located in the ε - neighborhood of point a . The first elements can be located anywhere. That is, outside the ε-neighborhood there can be no more than elements - that is, a finite number.

We also note that the difference does not have to monotonically tend to zero, that is, decrease all the time. It can tend to zero non-monotonically: it can either increase or decrease, having local maxima. However, these maxima, as n increases, should tend to zero (possibly also not monotonically).

Using the logical symbols of existence and universality, the definition of a limit can be written as follows:
(1) .

Determining that a is not a limit

Now consider the converse statement that the number a is not the limit of the sequence.

Number a is not the limit of the sequence, if there is such that for any natural number n there is such a natural m > n, What
.

Let's write this statement using logical symbols.
(2) .

Statement that number a is not the limit of the sequence, means that
you can choose such an ε - neighborhood of point a, outside of which there will be an infinite number of elements of the sequence.

Let's look at an example. Let a sequence with a common element be given
(3)
Any neighborhood of a point contains an infinite number of elements. However, this point is not the limit of the sequence, since any neighborhood of the point also contains an infinite number of elements. Let's take ε - a neighborhood of a point with ε = 1 . This will be the interval (-1, +1) . All elements except the first one with even n belong to this interval. But all elements with odd n are outside this interval, since they satisfy the inequality x n > 2 . Since the number of odd elements is infinite, there will be an infinite number of elements outside the chosen neighborhood. Therefore, the point is not the limit of the sequence.

Now we will show this, strictly adhering to statement (2). The point is not a limit of sequence (3), since there exists such that, for any natural n, there is an odd one for which the inequality holds
.

It can also be shown that any point a cannot be a limit of this sequence. We can always choose an ε - neighborhood of point a that does not contain either point 0 or point 2. And then outside the chosen neighborhood there will be an infinite number of elements of the sequence.

Equivalent definition

We can give an equivalent definition of the limit of a sequence if we expand the concept of ε - neighborhood. We will obtain an equivalent definition if, instead of an ε-neighborhood, it contains any neighborhood of the point a.

Determining the neighborhood of a point
Neighborhood of point a any open interval containing this point is called. Mathematically, the neighborhood is defined as follows: , where ε 1 and ε 2 - arbitrary positive numbers.

Then the definition of the limit will be as follows.

Equivalent definition of sequence limit
The number a is called the limit of the sequence, if for any neighborhood of it there is a natural number N such that all elements of the sequence with numbers belong to this neighborhood.

This definition can also be presented in expanded form.

The number a is called the limit of the sequence, if for any positive numbers and there exists a natural number N depending on and such that the inequalities hold for all natural numbers
.

Proof of equivalence of definitions

Let us prove that the two definitions of the limit of a sequence presented above are equivalent.

Let the number a be the limit of the sequence according to the first definition. This means that there is a function, so that for any positive number ε the following inequalities hold:
(4) at .

Let us show that the number a is the limit of the sequence by the second definition. That is, we need to show that there is such a function such that for any positive numbers ε 1 and ε 2 the following inequalities are satisfied:
(5) at .

Let us have two positive numbers: ε 1 and ε 2 . And let ε be the smallest of them: . Then ; ; . Let's use this in (5):
.
But the inequalities are satisfied for . Then inequalities (5) are also satisfied for .

That is, we have found a function for which inequalities (5) are satisfied for any positive numbers ε 1 and ε 2 .
The first part has been proven.

Now let the number a be the limit of the sequence according to the second definition. This means that there is a function such that for any positive numbers ε 1 and ε 2 the following inequalities are satisfied:
(5) at .

Let us show that the number a is the limit of the sequence by the first definition. To do this you need to put . Then when the following inequalities hold:
.
This corresponds to the first definition with .
The equivalence of the definitions has been proven.

Examples

Here we will look at several examples in which we need to prove that a given number a is the limit of a sequence. In this case, you need to specify an arbitrary positive number ε and define a function N of ε such that the inequality is satisfied for all.

Example 1

Prove that .

(1) .
In our case;
.

.
Let's take advantage properties of inequalities. Then if and , then
.

.
Then
at .
This means that the number is the limit of the given sequence:
.

Example 2

Using the definition of the limit of a sequence, prove that
.

Let us write down the definition of the limit of a sequence:
(1) .
In our case,;
.

Enter positive numbers and :
.
Let's take advantage properties of inequalities. Then if and , then
.

That is, for any positive, we can take any natural number greater than or equal to:
.
Then
at .
.

Example 3

We introduce the notation , .
Let's transform the difference:
.
For natural n = 1, 2, 3, ... we have:
.

Let us write down the definition of the limit of a sequence:
(1) .
Enter positive numbers and :
.
Then if and , then
.

That is, for any positive, we can take any natural number greater than or equal to:
.
At the same time
at .
This means that the number is the limit of the sequence:
.

Example 4