An irrational number can be represented as an infinite non-periodic fraction. The set of irrational numbers stands for $ I $ and it is equal to: $ I = R / Q $.

for instance... Irrational numbers are:

Operations on irrational numbers

On the set of irrational numbers, you can enter four basic arithmetic operations: addition, subtraction, multiplication and division; but for none of the listed operations the set of irrational numbers possesses the property of being closed. For example, the sum of two irrational numbers can be a rational number.

for instance... Find the sum of two irrational numbers $ 0.1010010001 \ ldots $ and $ 0.0101101110 \ ldots $. The first of these numbers is formed by a sequence of ones, separated, respectively, by one zero, two zeros, three zeros, etc., the second - by a sequence of zeros, between which one unit, two ones, three ones, etc. are placed:

$$ 0.1010010001 \ ldots + 0.0101101110 \ ldots = 0.1111111 = 0, (1) = \ frac (1) (9) $$

Thus, the sum of two given irrational numbers is the number $ \ frac (1) (9) $, which is rational.

Example

Exercise. Prove that the number $ \ sqrt (3) $ is irrational.

Proof. We will use the method of proving by contradiction. Suppose that $ \ sqrt (3) $ is a rational number, that is, it can be represented as a fraction $ \ sqrt (3) = \ frac (m) (n) $, where $ m $ and $ n $ are coprime naturals numbers.

We square both sides of the equality, we get

$$ 3 = \ frac (m ^ (2)) (n ^ (2)) \ Leftrightarrow 3 \ cdot n ^ (2) = m ^ (2) $$

The number 3 $ \ cdot n ^ (2) $ is divisible by 3. Therefore, $ m ^ (2) $ and, therefore, $ m $ is divisible by 3. Setting $ m = 3 \ cdot k $, the equality $ 3 \ cdot n ^ (2) = m ^ (2) $ can be written as

$$ 3 \ cdot n ^ (2) = (3 \ cdot k) ^ (2) \ Leftrightarrow 3 \ cdot n ^ (2) = 9 \ cdot k ^ (2) \ Leftrightarrow n ^ (2) = 3 \ cdot k ^ (2) $$

It follows from the last equality that $ n ^ (2) $ and $ n $ are divisible by 3, therefore, the fraction $ \ frac (m) (n) $ can be canceled by 3. But by assumption, the fraction $ \ frac (m) ( n) $ is irreducible. This contradiction proves that the number $ \ sqrt (3) $ is not representable as a fraction $ \ frac (m) (n) $ and, therefore, is irrational.

Q.E.D.

Example:
\ (4 \) is a rational number, because it can be written as \ (\ frac (4) (1) \);
\ (0,0157304 \) - also rational, because it can be written as \ (\ 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, as 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.

Impossible because it is endless fractions, and even non-periodic. Therefore, there are no integers that would divide by each other and give an irrational number.

Example:
\ (\ sqrt (2) ≈1,414213562 ... \) -irrational number;
\ (π≈3.1415926 ... \) -irrational number;
\ (\ log_ (2) (5) ≈2,321928 ... \) is an irrational number.

Example (Assignment from the OGE). The meaning of which expression 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, hence it is also impossible to represent a number as a fraction with whole numbers, 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, this \ (\ frac (-5) (1) \), so 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.

The ancient mathematicians already knew with a segment of unit length: they knew, for example, the incommensurability of the diagonal and the side of a square, which is tantamount to the irrationality of a number.

Irrational are:

Examples of proof of irrationality

Root of 2

Suppose the opposite: rational, that is, represented as an irreducible fraction, where and are integers. Let's square the assumed equality:

Hence it follows that even means even and. Let it be, where is the whole. Then

Therefore, even means even and. We got that and are even, which contradicts the irreducibility of the fraction. This means that the initial assumption was wrong, and - an irrational number.

Binary logarithm of 3

Suppose the opposite: rational, that is, represented as a fraction, where and are integers. Since, and can be chosen as positive. Then

But even and odd. We get a contradiction.

e

Story

The concept of irrational numbers was implicitly adopted by Indian mathematicians in the 7th century BC, when Manava (c. 750 BC - c. 690 BC) figured out that the square roots of some natural numbers such as 2 and 61 cannot be explicitly expressed.

The first proof of the existence of irrational numbers is usually attributed to Hippasus of Metapontus (c. 500 BC), a Pythagorean who found this proof by studying the side lengths of the pentagram. At the time of the Pythagoreans, it was believed that there is a single unit of length, sufficiently small and indivisible, which enters any segment an integer number of times. However, Hippasus proved that there is no single unit of length, since the assumption of its existence leads to a contradiction. He showed that if the hypotenuse of an isosceles right triangle contains an integer number of unit segments, then this number must be both even and odd at the same time. The proof looked like this:

• The ratio of the length of the hypotenuse to the length of the leg of an isosceles right triangle can be expressed as a:b, where a and b selected as the smallest possible.
• By the Pythagorean theorem: a² = 2 b².
• Because a² even, a must be even (since the square of an odd number would be odd).
• Insofar as a:b irreducible, b must be odd.
• Because a even, denote a = 2y.
• Then a² = 4 y² = 2 b².
• b² = 2 y², therefore b Is even, then b even.
• However, it has been proven that b odd. Contradiction.

Greek mathematicians called this ratio of incommensurable quantities aalogos(ineffable), however, according to the legends, they did not give Hippas the respect he deserved. Legend has it that Hippasus made a discovery while on a sea voyage and was thrown overboard by other Pythagoreans "for creating an element of the universe that denies the doctrine that all entities in the universe can be reduced to whole numbers and their relationships." The discovery of Hippasus posed a serious problem for Pythagorean mathematics, destroying the assumption underlying the whole theory that numbers and geometrical objects are one and indivisible.

see also

Notes (edit)

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

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Definition and examples of irrational numbers

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

So we came to definition of irrational numbers.

Definition.

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

The sounded definition allows you to bring 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 triples separating the eights is increased by two each time).

It should be noted that irrational numbers are rarely found precisely in the form of infinite non-periodic decimal fractions. Usually they are found in the form, etc., as well as in the form of specially introduced letters. The most famous examples of irrational numbers in this notation are the arithmetic square root of two, pi = 3.141592 ..., e = 2.718281 ... and the golden number.

Irrational numbers can also be defined through real numbers, which combine rational and irrational numbers.

Definition.

Irrational numbers- it real numbers that are not rational.

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 it is quite difficult to answer the question of whether it is irrational in many cases.

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

• finite and infinite periodic decimal fractions.

Also, any composition of rational numbers connected by 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 the expressions and are rational numbers. Here we note that if in such expressions among the rational numbers there is one single irrational number, then the value of the whole expression will be an irrational number. For example, in the expression, the number is irrational, and the rest of the numbers are rational, therefore, an irrational number. If it were a rational number, then the rationality of the number would follow from this, 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 required to prove the irrationality or rationality of a 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 is proved that a root of degree k from 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 defines 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 numbers and are not irrational, as well as.

As for logarithms, it is sometimes possible to prove their irrationality by contradiction. As an example, let us prove that log 2 3 is an irrational number.

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

Note that lna is an irrational number for any rational a that is positive and different from unity. For example, and are irrational numbers.

It was also proved that the number e a for any nonzero rational a is irrational, and that the number π z for any nonzero integer z is irrational. For example, numbers are irrational.

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

There are other proven results, but we will restrict ourselves to those already listed. It should also be said that in the proof of the results sounded above, the theory associated with algebraic numbers and transcendental numbers.

In conclusion, we note that one should not make hasty conclusions about the irrationality of the given numbers. For example, it seems clear that an irrational number to an irrational degree is an irrational number. However, this is not always the case. As a confirmation of the voiced fact, we give the degree. It is known that is an irrational number, and it is also proved 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 has not yet been proven.

Bibliography.

• Mathematics. Grade 6: textbook. for general education. institutions / [N. Ya. Vilenkin and others]. - 22nd ed., Rev. - M .: Mnemosina, 2008 .-- 288 p.: Ill. ISBN 978-5-346-00897-2.
• Algebra: study. for 8 cl. general education. institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova]; ed. S. A. Telyakovsky. - 16th ed. - M.: Education, 2008 .-- 271 p. : ill. - ISBN 978-5-09-019243-9.
• Gusev V.A., Mordkovich A.G. Mathematics (manual for applicants to technical schools): Textbook. manual. - M .; Higher. shk., 1984.-351 p., ill.

What are irrational numbers? Why are they called that? Where are they used and what are they? Few can answer these questions without hesitation. 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 The need to introduce this concept is due to the fact that the previously existing concepts of real or real, integer, natural and rational numbers were not enough to solve new emerging problems. For example, in order to figure out how square 2 is, you need to use non-periodic infinite decimal fractions. In addition, many of the simplest equations also do not have a solution without introducing the concept of an irrational number.

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

For the first time, one way or another, Indian mathematicians faced this phenomenon in the 7th century when it was discovered that the square roots of some quantities could not be indicated explicitly. And the first proof of the existence of such numbers is attributed to the Pythagorean Hippasus, who did this in the process of studying an isosceles right triangle. Some 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 name

If ratio in Latin is "fraction", "ratio", then the prefix "ir"
gives this word the opposite meaning. Thus, the name of the set of these numbers means that they cannot be correlated with whole or fractional numbers, they 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 are complex. There are no subsets, however, 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) are applicable to them.

a + b = b + a (commutativity);

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

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

ab = ba (displacement law);

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

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

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

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

If a> b and b> c, then a> c (the 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 action of the Archimedes axiom extends to irrational numbers. It says that for any two quantities a and b, the statement is true that by taking a as a term a sufficient number of times, you can exceed b.

Usage

Despite the fact that in everyday life you do not have to deal with them very often, irrational numbers cannot be counted. There are a lot of them, but they are almost invisible. We are surrounded by irrational numbers everywhere. Examples familiar to everyone are pi, equal to 3.1415926 ..., or e, which is essentially the base 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 greater part to the lesser, and vice versa, is also

refers to this set. The less well-known "silver" - too.

On the number line, they are located very densely, so that between any two quantities referred to the set of rational ones, an irrational one is necessarily encountered.

Until now, there are 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 examine the most significant examples for belonging to one group or another. For example, it is considered that e - normal number, that is, the probability of different numbers appearing in its record is the same. As for pi, research is underway on it. The measure of irrationality is a quantity that shows how well a particular 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.

Under this designation are hidden complex numbers that include real or tangible.

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

All the rest 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 second was originally deduced by mathematicians in this capacity, their irrationality and transcendence were proved many years after their discovery. For pi, the proof was presented in 1882 and simplified in 1894, ending the 2,500 year controversy over the problem of squaring the circle. It is still not fully understood, so modern mathematicians have something to work on. By the way, the first sufficiently accurate calculation of this value was carried out by Archimedes. Before him, all calculations were too rough.

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

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