irrational number- this is real number, which is not rational, that is, cannot be represented as a fraction, where are integers, . An irrational number can be represented as an infinite non-repeating decimal.

Many ir rational numbers usually capitalized Latin letter in bold with no fill. Thus: , i.e. set of irrational numbers is difference of sets of real and rational numbers.

On the existence of irrational numbers, more precisely segments, incommensurable with a segment of unit length, were already known by ancient mathematicians: they knew, for example, the incommensurability of the diagonal and the side of the square, which is equivalent to the irrationality of the number.

Properties

• Any real number can be written as an infinite decimal fraction, while irrational numbers and only they are written as non-periodic infinite decimal fractions.
• Irrational numbers define Dedekind sections in the set of rational numbers that have no largest number in the lower class and no smallest number in the upper one.
• Every real transcendental number is irrational.
• Every irrational number is either algebraic or transcendental.
• The set of irrational numbers is everywhere dense on the real line: between any two numbers there is an irrational number.
• The order on the set of irrational numbers is isomorphic to the order on the set of real transcendental numbers.
• The set of irrational numbers is uncountable, is a set of the second category.

Examples

Irrational are:

Irrationality Proof Examples

Root of 2

Assume the opposite: rational , that is, it is represented in the form irreducible fraction, where is an integer and is a natural number. Let's square the supposed equality:

From this it follows that even, therefore, even and . Let where the whole. Then

Therefore, even, therefore, even and . We have obtained that and are even, which contradicts the irreducibility of the fraction . Hence, the original assumption was wrong, and is an irrational number.

Binary logarithm of the number 3

Assume the contrary: it is rational, that is, it is represented as a fraction, where and are integers. Since , and can be taken positive. Then

But it's clear, it's odd. We get a contradiction.

e

Story

The concept of irrational numbers was implicitly adopted by Indian mathematicians in the 7th century BC, when Manawa (c. 750 BC - c. 690 BC) found that square roots 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 lengths of the sides of a pentagram. At the time of the Pythagoreans, it was believed that there single unit length, sufficiently small and indivisible, which is an integer number of times in any segment. However, Hippasus argued 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.
• According to the Pythagorean theorem: a² = 2 b².
• Because a² even, a must be even (since the square of an odd number would be odd).
• Because the 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 alogos(inexpressible), but according to the legends, Hippasus was not paid due respect. There is a legend that Hippasus made the discovery while in sea ​​voyage, and was thrown overboard by other Pythagoreans "for creating an element of the universe which denies the doctrine that all entities in the universe can be reduced to whole numbers and their ratios." The discovery of Hippasus posed a serious problem for Pythagorean mathematics, destroying the assumption underlying the whole theory that numbers and geometric objects are one and inseparable.

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 to solve new emerging problems, the previously existing concepts of real or real, integer, natural and rational numbers were no longer enough. For example, in order to calculate what the square of 2 is, one must use non-periodic infinite decimals. In addition, many of the simplest 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 ​​\u200b\u200bcannot 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 encountered this phenomenon in the 7th century, when it was discovered that the square roots of some quantities cannot be explicitly indicated. 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. A serious contribution to the study of this set was made by some other scientists who lived before our era. The introduction of the concept of irrational numbers led to 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 opposite meaning. Thus, the name of the set of these numbers indicates that they cannot be correlated with an integer or fractional, they have a separate place. This follows from their nature.

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) apply to them.

a + b = b + a (commutativity);

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

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

ab = ba (displacement law);

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

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

a x 1/a = 1 (the existence of an inverse number);

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

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

Of course, all irrational numbers can be transformed using the basic arithmetic operations. None special rules while not.

In addition, the action of the axiom of Archimedes 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 enough times, you can beat b.

Usage

Despite the fact that in ordinary life not so often you have to deal with them, irrational numbers are not countable. Them great multitude but they are almost invisible. We are surrounded by irrational numbers everywhere. Examples familiar to all are pi, which is 3.1415926... or e, which is essentially the base natural logarithm, 2.718281828... In algebra, trigonometry and geometry, you have to use them all the time. By the way, famous meaning"golden section", that is, the ratio of both the larger part to the smaller, and vice versa, also

belongs to this set. Less known "silver" - too.

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

There are still many unresolved issues associated with this set. There are such criteria as the measure of irrationality and the normality of a number. Mathematicians continue to examine the most significant examples for their belonging to one group or another. For example, it is considered that e is a normal number, that is, the probability of different digits appearing in its entry is the same. As for pi, research is still underway regarding it. A measure of irrationality is a value that shows how well a particular number can be approximated by rational numbers.

Algebraic and transcendental

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

Hidden under this designation complex numbers, which include real or real.

So, an algebraic value 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 the solution to the equation x 2 - 2 = 0.

Yet the rest real numbers, which 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 given in 1882 and simplified in 1894, which put an end to the 2,500-year controversy about 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 approximate.

For e (the Euler or Napier number), a proof 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 non-zero values.

The set of irrational numbers is usually denoted by a capital Latin letter I (\displaystyle \mathbb (I) ) in bold with no fill. In this way: I = R ∖ Q (\displaystyle \mathbb (I) =\mathbb (R) \backslash \mathbb (Q) ), that is, the set of irrational numbers is the difference between the sets of real and rational numbers.

The existence of irrational numbers, more precisely segments that are incommensurable with a segment of unit length, was already known to ancient mathematicians: they knew, for example, the incommensurability of the diagonal and the side of the square, which is equivalent to the irrationality of the number.

Encyclopedic YouTube

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  Irrational are:

  Irrationality Proof Examples

  Root of 2

  Let's say the opposite: 2 (\displaystyle (\sqrt (2))) rational, that is, represented as a fraction m n (\displaystyle (\frac (m)(n))), where m (\displaystyle m) is an integer, and n (\displaystyle n)- natural number .

  Let's square the supposed equality:

  2 = m n ⇒ 2 = m 2 n 2 ⇒ m 2 = 2 n 2 (\displaystyle (\sqrt (2))=(\frac (m)(n))\Rightarrow 2=(\frac (m^(2 ))(n^(2)))\Rightarrow m^(2)=2n^(2)).

  Story

  Antiquity

  The concept of irrational numbers was implicitly adopted by Indian mathematicians in the 7th century BC, when Manawa (c. 750 BC - c. 690 BC) found 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. At the time of the Pythagoreans, it was believed that there is a single unit of length, sufficiently small and indivisible, which is an integer number of times included in any segment [ ] .

  There is no exact data on the irrationality of which number was proved by Hippasus. According to legend, he found it by studying the lengths of the sides of the pentagram. Therefore, it is reasonable to assume that this was the golden ratio [ ] .

  Greek mathematicians called this ratio of incommensurable quantities alogos(inexpressible), but according to the legends, Hippasus was not paid due respect. There is a legend that Hippasus made the discovery while on a sea voyage and was thrown overboard by other Pythagoreans "for creating an element of the universe, which denies the doctrine that all entities in the universe can be reduced to whole numbers and their ratios." The discovery of Hippas put before Pythagorean mathematics serious problem, destroying the assumption underlying the whole theory that numbers and geometric objects are one and inseparable.

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

  Irrational are:

  Irrationality Proof Examples

  Root of 2

  Assume the contrary: it is rational, that is, it is represented as an irreducible fraction, where and are integers. Let's square the supposed equality:

  .

  From this it follows that even, therefore, even and . Let where the whole. Then

  Therefore, even, therefore, even and . We have obtained that and are even, which contradicts the irreducibility of the fraction . Hence, the original assumption was wrong, and is an irrational number.

  Binary logarithm of the number 3

  Assume the contrary: it is rational, that is, it is represented as a fraction, where and are integers. Since , and can be taken positive. Then

  But it's clear, it's odd. We get a contradiction.

  e

  Story

  The concept of irrational numbers was implicitly adopted by Indian mathematicians in the 7th century BC, when Manawa (c. 750 BC - c. 690 BC) found 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 lengths of the sides of a pentagram. In the time of the Pythagoreans, it was believed that there is a single unit of length, sufficiently small and indivisible, which is an integer number of times included in any segment. However, Hippasus argued 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.
  • According to the Pythagorean theorem: a² = 2 b².
  • Because a² even, a must be even (since the square of an odd number would be odd).
  • Because the 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 alogos(inexpressible), but according to the legends, Hippasus was not paid due respect. There is a legend that Hippasus made the discovery while on a sea voyage and was thrown overboard by other Pythagoreans "for creating an element of the universe, which denies the doctrine that all entities in the universe can be reduced to whole numbers and their ratios." The discovery of Hippasus posed a serious problem for Pythagorean mathematics, destroying the assumption underlying the whole theory that numbers and geometric objects are one and inseparable.

  see also

  Notes

  Numerical systems
  Counting sets	Natural numbers () Integers ()

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

  Irrational are:

  Irrationality Proof Examples

  Root of 2

  Assume the contrary: it is rational, that is, it is represented as an irreducible fraction, where and are integers. Let's square the supposed equality:

  .

  From this it follows that even, therefore, even and . Let where the whole. Then

  Therefore, even, therefore, even and . We have obtained that and are even, which contradicts the irreducibility of the fraction . Hence, the original assumption was wrong, and is an irrational number.

  Binary logarithm of the number 3

  Assume the contrary: it is rational, that is, it is represented as a fraction, where and are integers. Since , and can be taken positive. Then

  But it's clear, it's odd. We get a contradiction.

  e

  Story

  The concept of irrational numbers was implicitly adopted by Indian mathematicians in the 7th century BC, when Manawa (c. 750 BC - c. 690 BC) found 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 lengths of the sides of a pentagram. In the time of the Pythagoreans, it was believed that there is a single unit of length, sufficiently small and indivisible, which is an integer number of times included in any segment. However, Hippasus argued 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.
  • According to the Pythagorean theorem: a² = 2 b².
  • Because a² even, a must be even (since the square of an odd number would be odd).
  • Because the 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 alogos(inexpressible), but according to the legends, Hippasus was not paid due respect. There is a legend that Hippasus made the discovery while on a sea voyage and was thrown overboard by other Pythagoreans "for creating an element of the universe, which denies the doctrine that all entities in the universe can be reduced to whole numbers and their ratios." The discovery of Hippasus posed a serious problem for Pythagorean mathematics, destroying the assumption underlying the whole theory that numbers and geometric objects are one and inseparable.

  see also

  Notes

  Numerical systems
  Counting sets	Natural numbers () Integers ()