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Rational and irrational numbers. Irrational numbers: what are they and what are they used for? How to prove that an expression is irrational

Definition of an irrational number

Irrational numbers are those numbers that in decimal notation represent endless non-periodic decimal fractions.



So, for example, numbers obtained by taking the square root of natural numbers are irrational and are not squares of natural numbers. But not all irrational numbers are obtained by taking square roots, because the number pi obtained by division is also irrational, and you are unlikely to get it by trying to extract the square root of a natural number.

Properties of irrational numbers

Unlike numbers written as infinite decimals, only irrational numbers are written as non-periodic infinite decimals.
The sum of two non-negative irrational numbers can end up being a rational number.
Irrational numbers define Dedekind cuts in the set of rational numbers, in the lower class of which there is no largest number, and in the upper class there is no smaller one.
Any real transcendental number is irrational.
All irrational numbers are either algebraic or transcendental.
The set of irrational numbers on a line is densely located, and between any two of its numbers there is sure to be an irrational number.
The set of irrational numbers is infinite, uncountable and is a set of the 2nd category.
When performing any arithmetic operation on rational numbers, except division by 0, the result will be a rational number.
When adding a rational number to an irrational number, the result is always an irrational number.
When adding irrational numbers, we can end up with a rational number.
The set of irrational numbers is not even.

Numbers are not irrational

Sometimes it is quite difficult to answer the question of whether a number is irrational, especially in cases where the number is in the form of a decimal fraction or in the form of a numerical expression, root or logarithm.

Therefore, it will not be superfluous to know which numbers are not irrational. If we follow the definition of irrational numbers, then we already know that rational numbers cannot be irrational.

Irrational numbers are not:

First, all natural numbers;
Secondly, integers;
Third, ordinary fractions;
Fourthly, various mixed numbers;
Fifthly, these are infinite periodic decimal fractions.

In addition to all of the above, an irrational number cannot be any combination of rational numbers that is performed by the signs of arithmetic operations, such as +, -, , :, since in this case the result of two rational numbers will also be a rational number.

Now let's see which numbers are irrational:



Do you know about the existence of a fan club, where fans of this mysterious mathematical phenomenon are looking for more and more information about Pi, trying to unravel its mystery? Any person who knows by heart a certain number of Pi numbers after the decimal point can become a member of this club;

Did you know that in Germany, under the protection of UNESCO, there is the Castadel Monte palace, thanks to the proportions of which you can calculate Pi. King Frederick II dedicated the entire palace to this number.

It turns out that they tried to use the number Pi in the construction of the Tower of Babel. But unfortunately, this led to the collapse of the project, since at that time the exact calculation of the value of Pi was not sufficiently studied.

Singer Kate Bush in her new disc recorded a song called “Pi”, in which one hundred and twenty-four numbers from the famous number series 3, 141… were heard.


The material in this article provides initial information about irrational numbers. First we will give the definition of irrational numbers and explain it. Below we give examples of irrational numbers. Finally, let's look at some approaches to figuring out whether a given number is irrational or not.

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Definition and examples of irrational numbers

When studying decimals, we separately considered infinite non-periodic decimals. Such fractions arise when measuring decimal lengths of segments that are incommensurable with a unit segment. We also noted that infinite non-periodic decimal fractions cannot be converted to ordinary fractions (see converting ordinary fractions to decimals and vice versa), therefore, these numbers are not rational numbers, they represent the so-called irrational numbers.

So we come to definition of irrational numbers.

Definition.

Numbers that represent infinite non-periodic decimal fractions in decimal notation are called irrational numbers.

The voiced definition allows us to give examples of irrational numbers. For example, the infinite non-periodic decimal fraction 4.10110011100011110000... (the number of ones and zeros increases by one each time) is an irrational number. Let's give another example of an irrational number: −22.353335333335... (the number of threes separating eights increases by two each time).

It should be noted that irrational numbers are quite rarely found in the form of endless non-periodic decimal fractions. They are usually found in the form , etc., as well as in the form of specially entered letters. The most famous examples of irrational numbers in this notation are the arithmetic square root of two, the number “pi” π=3.141592..., the number e=2.718281... and the golden number.

Irrational numbers can also be defined in terms of real numbers, which combine rational and irrational numbers.

Definition.

Irrational numbers are real numbers that are not rational numbers.

Is this number irrational?

When a number is given not in the form of a decimal fraction, but in the form of some root, logarithm, etc., then answering the question of whether it is irrational is in many cases quite difficult.

Undoubtedly, when answering the question posed, it is very useful to know which numbers are not irrational. From the definition of irrational numbers it follows that irrational numbers are not rational numbers. Thus, irrational numbers are NOT:

  • finite and infinite periodic decimal fractions.

Also, any composition of rational numbers connected by the signs of arithmetic operations (+, −, ·, :) is not an irrational number. This is because the sum, difference, product and quotient of two rational numbers is a rational number. For example, the values ​​of expressions and are rational numbers. Here we note that if such expressions contain one single irrational number among the rational numbers, then the value of the entire expression will be an irrational number. For example, in the expression the number is irrational, and the remaining numbers are rational, therefore it is an irrational number. If it were a rational number, then the rationality of the number would follow, but it is not rational.

If the expression that specifies the number contains several irrational numbers, root signs, logarithms, trigonometric functions, numbers π, e, etc., then it is necessary to prove the irrationality or rationality of the given number in each specific case. However, there are a number of results already obtained that can be used. Let's list the main ones.

It has been proven that a kth root of an integer is a rational number only if the number under the root is the kth power of another integer; in other cases, such a root specifies an irrational number. For example, the numbers and are irrational, since there is no integer whose square is 7, and there is no integer whose raising to the fifth power gives the number 15. And the numbers are not irrational, since and .

As for logarithms, it is sometimes possible to prove their irrationality using the method of contradiction. As an example, let's prove that log 2 3 is an irrational number.

Let's assume that log 2 3 is a rational number, not an irrational one, that is, it can be represented as an ordinary fraction m/n. and allow us to write the following chain of equalities: . The last equality is impossible, since on its left side odd number, and on the right side – even. So we came to a contradiction, which means that our assumption turned out to be incorrect, and this proved that log 2 3 is an irrational number.

Note that lna for any positive and non-one rational a is an irrational number. For example, and are irrational numbers.

It is also proven that the number e a for any non-zero rational a is irrational, and that the number π z for any non-zero integer z is irrational. For example, numbers are irrational.

Irrational numbers are also the trigonometric functions sin, cos, tg and ctg for any rational and non-zero value of the argument. For example, sin1 , tan(−4) , cos5,7 are irrational numbers.

There are other proven results, but we will limit ourselves to those already listed. It should also be said that when proving the above results, the theory associated with algebraic numbers And transcendental numbers.

In conclusion, we note that we should not make hasty conclusions regarding the irrationality of the given numbers. For example, it seems obvious that an irrational number to an irrational degree is an irrational number. However, this is not always the case. To confirm the stated fact, we present the degree. It is known that - is an irrational number, and it has also been proven that - is an irrational number, but is a rational number. You can also give examples of irrational numbers, the sum, difference, product and quotient of which are rational numbers. Moreover, the rationality or irrationality of the numbers π+e, π−e, π·e, π π, π e and many others have not yet been proven.

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.

Example:
\(4\) is a rational number, because it can be written as \(\frac(4)(1)\) ;
\(0.0157304\) is also rational, because it can be written in the form \(\frac(157304)(10000000)\) ;
\(0.333(3)...\) - and this is a rational number: can be represented as \(\frac(1)(3)\) ;
\(\sqrt(\frac(3)(12))\) is rational, since it can be represented as \(\frac(1)(2)\) . Indeed, we can carry out a chain of transformations \(\sqrt(\frac(3)(12))\) \(=\)\(\sqrt(\frac(1)(4))\) \(=\) \ (\frac(1)(2)\)


Irrational number is a number that cannot be written as a fraction with an integer numerator and denominator.

It's impossible because it's endless fractions, and even non-periodic ones. Therefore, there are no integers that, when divided by each other, would give an irrational number.

Example:
\(\sqrt(2)≈1.414213562…\) is an irrational number;
\(π≈3.1415926… \) is an irrational number;
\(\log_(2)(5)≈2.321928…\) is an irrational number.


Example (Assignment from the OGE). The meaning of which of the expressions is a rational number?
1) \(\sqrt(18)\cdot\sqrt(7)\);
2)\((\sqrt(9)-\sqrt(14))(\sqrt(9)+\sqrt(14))\);
3) \(\frac(\sqrt(22))(\sqrt(2))\);
4) \(\sqrt(54)+3\sqrt(6)\).

Solution:

1) \(\sqrt(18)\cdot \sqrt(7)=\sqrt(9\cdot 2\cdot 7)=3\sqrt(14)\) – the root of \(14\) cannot be taken, which means It is also impossible to represent a number as a fraction with integers, therefore the number is irrational.

2) \((\sqrt(9)-\sqrt(14))(\sqrt(9)+\sqrt(14))= (\sqrt(9)^2-\sqrt(14)^2)=9 -14=-5\) – there are no roots left, the number can be easily represented as a fraction, for example \(\frac(-5)(1)\), which means it is rational.

3) \(\frac(\sqrt(22))(\sqrt(2))=\sqrt(\frac(22)(2))=\sqrt(\frac(11)(1))=\sqrt( 11)\) – the root cannot be extracted - the number is irrational.

4) \(\sqrt(54)+3\sqrt(6)=\sqrt(9\cdot 6)+3\sqrt(6)=3\sqrt(6)+3\sqrt(6)=6\sqrt (6)\) is also irrational.

What are irrational numbers? Why are they called that? Where are they used and what are they? Few people can answer these questions without thinking. But in fact, the answers to them are quite simple, although not everyone needs them and in very rare situations

Essence and designation

Irrational numbers are infinite non-periodic numbers. The need to introduce this concept is due to the fact that to solve new problems that arise, the previously existing concepts of real or real, integer, natural and rational numbers were no longer sufficient. For example, in order to calculate which quantity is the square of 2, you need to use non-periodic infinite decimals. In addition, many simple equations also have no solution without introducing the concept of an irrational number.

This set is denoted as I. And, as is already clear, these values ​​cannot be represented as a simple fraction, the numerator of which will be an integer, and the denominator will be

For the first time, one way or another, Indian mathematicians encountered this phenomenon in the 7th century when it was discovered that the square roots of some quantities cannot be indicated explicitly. And the first proof of the existence of such numbers is attributed to the Pythagorean Hippasus, who did this while studying an isosceles right triangle. Some other scientists who lived before our era made a serious contribution to the study of this set. The introduction of the concept of irrational numbers entailed a revision of the existing mathematical system, which is why they are so important.

Origin of the name

If ratio translated from Latin is “fraction”, “ratio”, then the prefix “ir”
gives this word the opposite meaning. Thus, the name of the set of these numbers indicates that they cannot be correlated with an integer or fraction and have a separate place. This follows from their essence.

Place in the general classification

Irrational numbers, along with rational numbers, belong to the group of real or real numbers, which in turn belong to complex numbers. There are no subsets, but there are algebraic and transcendental varieties, which will be discussed below.

Properties

Since irrational numbers are part of the set of real numbers, all their properties that are studied in arithmetic (they are also called basic algebraic laws) apply to them.

a + b = b + a (commutativity);

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

a + (-a) = 0 (existence of the opposite number);

ab = ba (commutative law);

(ab)c = a(bc) (distributivity);

a(b+c) = ab + ac (distribution law);

a x 1/a = 1 (existence of a reciprocal number);

The comparison is also carried out in accordance with general laws and principles:

If a > b and b > c, then a > c (transitivity of the relation) and. etc.

Of course, all irrational numbers can be converted using basic arithmetic. There are no special rules for this.

In addition, the Archimedes axiom applies to irrational numbers. It states that for any two quantities a and b, it is true that if you take a as a term enough times, you can beat b.

Usage

Despite the fact that you don’t encounter them very often in everyday life, irrational numbers cannot be counted. There are a huge number of them, but they are almost invisible. Irrational numbers are all around us. Examples that are familiar to everyone are the number pi, equal to 3.1415926..., or e, which is essentially the base of the natural logarithm, 2.718281828... In algebra, trigonometry and geometry, they have to be used constantly. By the way, the famous meaning of the “golden ratio”, that is, the ratio of both the larger part to the smaller part, and vice versa, also

belongs to this set. The lesser known “silver” one too.

On the number line they are located very densely, so that between any two quantities classified as rational, an irrational one is sure to occur.

There are still a lot of unsolved problems associated with this set. There are criteria such as the measure of irrationality and the normality of a number. Mathematicians continue to study the most significant examples to determine whether they belong to one group or another. For example, it is believed that e is a normal number, i.e., the probability of different digits appearing in its notation is the same. As for pi, research is still underway regarding it. The measure of irrationality is a value that shows how well a given number can be approximated by rational numbers.

Algebraic and transcendental

As already mentioned, irrational numbers are conventionally divided into algebraic and transcendental. Conditionally, since, strictly speaking, this classification is used to divide the set C.

This designation hides complex numbers, which include real or real numbers.

So, algebraic is a value that is the root of a polynomial that is not identically equal to zero. For example, the square root of 2 would be in this category because it is a solution to the equation x 2 - 2 = 0.

All other real numbers that do not satisfy this condition are called transcendental. This variety includes the most famous and already mentioned examples - the number pi and the base of the natural logarithm e.

Interestingly, neither one nor the other were originally developed by mathematicians in this capacity; their irrationality and transcendence were proven many years after their discovery. For pi, the proof was given in 1882 and simplified in 1894, ending a 2,500-year debate about the problem of squaring the circle. It has still not been fully studied, so modern mathematicians have something to work on. By the way, the first fairly accurate calculation of this value was carried out by Archimedes. Before him, all calculations were too approximate.

For e (Euler's or Napier's number), proof of its transcendence was found in 1873. It is used in solving logarithmic equations.

Other examples include the values ​​of sine, cosine, and tangent for any algebraic nonzero value.