2 is an irrational number. Irrational numbers, definition, examples

The set of all natural numbers is denoted by the letter N. Natural numbers are the numbers that we use to count objects: 1,2,3,4, ... In some sources, the number 0 is also considered a natural number.

The set of all integers is denoted by the letter Z. Integers are all natural numbers, zero and negative numbers:

1,-2,-3, -4, …

Now let’s add to the set of all integers the set of all ordinary fractions: 2/3, 18/17, -4/5 and so on. Then we get the set of all rational numbers.

Set of rational numbers

The set of all rational numbers is denoted by the letter Q. The set of all rational numbers (Q) is a set consisting of numbers of the form m/n, -m/n and the number 0. Any natural number can act as n,m. It should be noted that all rational numbers can be represented as a finite or infinite PERIODIC decimal fraction. The converse is also true that any finite or infinite periodic decimal fraction can be written as a rational number.

But what about, for example, the number 2.0100100010...? It is an infinitely NON-PERIODIC decimal fraction. And it does not apply to rational numbers.

In the school algebra course, only real (or real) numbers are studied. The set of all real numbers is denoted by the letter R. The set R consists of all rational and all irrational numbers.

The concept of irrational numbers

Irrational numbers are all infinite decimal non-periodic fractions. Irrational numbers do not have a special designation.

For example, all numbers obtained by extracting the square root of natural numbers that are not squares of natural numbers will be irrational. (√2, √3, √5, √6, etc.).

But do not think that irrational numbers are obtained only by extracting square roots. For example, the number “pi” is also irrational, and it is obtained by division. And no matter how hard you try, you cannot get it by taking the square root of any natural number.

And they derived their roots from the Latin word “ratio”, which means “reason”. Based on the literal translation:

• A rational number is a “reasonable number.”
• An irrational number is, accordingly, an “unreasonable number.”

General concept of a rational number

A rational number is a number that can be written as:

1. An ordinary positive fraction.
2. Negative common fraction.
3. As a number zero (0).

In other words, the following definitions apply to a rational number:

• Any natural number is inherently rational, since any natural number can be represented as an ordinary fraction.
• Any integer, including the number zero, since any integer can be written either as a positive ordinary fraction, as a negative ordinary fraction, or as the number zero.
• Any ordinary fraction, and it does not matter whether it is positive or negative, also directly approaches the definition of a rational number.
• The definition can also include a mixed number, a finite decimal fraction or an infinite periodic fraction.

Rational number examples

Let's look at examples of rational numbers:

• Natural numbers - “4”, “202”, “200”.
• Integers - “-36”, “0”, “42”.
• Ordinary fractions.

From the above examples it is quite obvious that rational numbers can be both positive and negative. Naturally, the number 0 (zero), which in turn is also a rational number, at the same time does not belong to the category of a positive or negative number.

Hence, I would like to remind the general education program using the following definition: “Rational numbers” are those numbers that can be written as a fraction x/y, where x (numerator) is an integer, and y (denominator) is a natural number.

General concept and definition of an irrational number

In addition to “rational numbers,” we also know the so-called “irrational numbers.” Let's briefly try to define these numbers.

Even ancient mathematicians, wanting to calculate the diagonal of a square along its sides, learned about the existence of an irrational number.
Based on the definition of rational numbers, you can build a logical chain and give a definition of an irrational number.
So, in essence, those real numbers that are not rational are simply irrational numbers.
Decimal fractions, expressing irrational numbers, are not periodic and infinite.

Examples of an irrational number

For clarity, let's consider a small example of an irrational number. As we already understood, infinite decimal non-periodic fractions are called irrational, for example:

• The number “-5.020020002... (it is clearly visible that the twos are separated by a sequence of one, two, three, etc. zeros)
• The number “7.040044000444... (here it is clear that the number of fours and the number of zeros increases by one each time in a chain).
• Everyone knows the number Pi (3.1415...). Yes, yes - it is also irrational.

In general, all real numbers are both rational and irrational. In simple terms, an irrational number cannot be represented as a common fraction x/y.

General conclusion and brief comparison between numbers

We looked at each number separately, but the difference between a rational number and an irrational number remains:

1. An irrational number occurs when extracting the square root, when dividing a circle by its diameter, etc.
2. A rational number represents a common fraction.

Let's conclude our article with a few definitions:

• An arithmetic operation performed on a rational number, other than division by 0 (zero), will ultimately lead to a rational number.
• The final result, when performing an arithmetic operation on an irrational number, can lead to both a rational and an irrational value.
• If both numbers take part in an arithmetic operation (except for division or multiplication by zero), then the result will be an irrational number.

All rational numbers can be represented as a common fraction. This applies to whole numbers (for example, 12, –6, 0), and finite decimal fractions (for example, 0.5; –3.8921), and infinite periodic decimal fractions (for example, 0.11(23); –3 ,(87)).

However infinite non-periodic decimals cannot be represented as ordinary fractions. That's what they are irrational numbers(that is, irrational). An example of such a number is the number π, which is approximately equal to 3.14. However, what it exactly equals cannot be determined, since after the number 4 there is an endless series of other numbers in which repeating periods cannot be distinguished. Moreover, although the number π cannot be expressed precisely, it has a specific geometric meaning. The number π is the ratio of the length of any circle to the length of its diameter. Thus, irrational numbers actually exist in nature, just like rational numbers.

Another example of irrational numbers is the square roots of positive numbers. Extracting roots from some numbers gives rational values, from others - irrational. For example, √4 = 2, i.e. the root of 4 is a rational number. But √2, √5, √7 and many others result in irrational numbers, that is, they can only be extracted by approximation, rounding to a certain decimal place. In this case, the fraction becomes non-periodic. That is, it is impossible to say exactly and definitely what the root of these numbers is.

So √5 is a number lying between the numbers 2 and 3, since √4 = 2, and √9 = 3. We can also conclude that √5 is closer to 2 than to 3, since √4 is closer to √5 than √9 to √5. Indeed, √5 ≈ 2.23 or √5 ≈ 2.24.

Irrational numbers are also obtained in other calculations (and not just when extracting roots), and can be negative.

In relation to irrational numbers, we can say that no matter what unit segment we take to measure the length expressed by such a number, we will not be able to definitely measure it.

In arithmetic operations, irrational numbers can participate along with rational ones. At the same time, there are a number of regularities. For example, if only rational numbers are involved in an arithmetic operation, then the result is always a rational number. If only irrational ones participate in the operation, then it is impossible to say unambiguously whether the result will be a rational or irrational number.

For example, if you multiply two irrational numbers √2 * √2, you get 2 - this is a rational number. On the other hand, √2 * √3 = √6 is an irrational number.

If an arithmetic operation involves rational and irrational numbers, then the result will be irrational. For example, 1 + 3.14... = 4.14... ; √17 – 4.

Why is √17 – 4 an irrational number? Let's imagine that we get a rational number x. Then √17 = x + 4. But x + 4 is a rational number, because we assumed that x is rational. The number 4 is also rational, so x + 4 is rational. However, a rational number cannot be equal to the irrational number √17. Therefore, the assumption that √17 – 4 gives a rational result is incorrect. The result of an arithmetic operation will be irrational.

However, there is an exception to this rule. If we multiply an irrational number by 0, we get the rational number 0.

And π

Thus, the set of irrational numbers is the difference I = R ∖ Q (\displaystyle \mathbb (I) =\mathbb (R) \backslash \mathbb (Q) ) sets of real and rational numbers.

The existence of irrational numbers, more precisely, segments 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 a square, which is equivalent to the irrationality of the number 2 (\displaystyle (\sqrt (2))).

Properties

• The sum of two positive irrational numbers can be a rational number.
• Irrational numbers define Dedekind sections in the set of rational numbers that do not have a largest number in the lower class and do not have a smallest number in the upper class.
• The set of irrational numbers is dense everywhere on the number line: between any two distinct 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. [ ]

Algebraic and transcendental numbers

Every irrational number is either algebraic or transcendental. The set of algebraic numbers is a countable set. Since the set of real numbers is uncountable, the set of irrational numbers is uncountable.

The set of irrational numbers is a set of the second category.

Let's square the supposed equality:

Story

Antiquity

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

The first proof of the existence of irrational numbers, or more precisely the existence of incommensurable segments, is usually attributed to the Pythagorean Hippasus of Metapontum (approximately 470 BC). At the time of the Pythagoreans, it was believed that there was a single unit of length, sufficiently small and indivisible, which included an integer number of times in any segment [ ] .

There is no exact data on which number was proven irrational 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 since this is the ratio of the diagonal to the side in a regular pentagon.

Greek mathematicians called this ratio of incommensurable quantities alogos(unspeakable), but according to the legends they did not pay due respect to Hippasus. 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 that denies the doctrine that all entities in the universe can be reduced to integers and their ratios.” The discovery of Hippasus posed a serious problem for Pythagorean mathematics, destroying the underlying assumption that numbers and geometric objects were one and inseparable.

Later, Eudoxus of Cnidus (410 or 408 BC - 355 or 347 BC) developed a theory of proportions that took into account both rational and irrational relationships. This served as the basis for understanding the fundamental essence of irrational numbers. Quantity began to be considered not as a number, but as a designation of entities, such as line segments, angles, areas, volumes, time intervals - entities that can change continuously (in the modern sense of the word). Magnitudes were contrasted with numbers, which can only change by “jumps” from one number to the next, for example, from 4 to 5. Numbers are made up of the smallest indivisible quantity, while quantities can be reduced indefinitely.

Since no quantitative value was correlated with magnitude, Eudoxus was able to cover both commensurate and incommensurable quantities when defining a fraction as the ratio of two quantities, and proportion as the equality of two fractions. By removing quantitative values ​​(numbers) from the equations, he avoided the trap of having to call an irrational quantity a number. Eudoxus's theory allowed Greek mathematicians to make incredible progress in geometry, providing them with the necessary logical basis for working with incommensurable quantities. The tenth book of Euclid's Elements is devoted to the classification of irrational quantities.

Middle Ages

The Middle Ages were marked by the adoption of concepts such as zero, negative numbers, integers and fractions, first by Indian and then by Chinese mathematicians. Later, Arab mathematicians joined in and were the first to consider negative numbers as algebraic objects (along with positive numbers), which made it possible to develop the discipline now called algebra.

Arab mathematicians combined the ancient Greek concepts of “number” and “magnitude” into a single, more general idea of ​​real numbers. They were critical of Euclid's ideas about relations; in contrast, they developed a theory of relations of arbitrary quantities and expanded the concept of number to relations of continuous quantities. In his commentary on Euclid's Book 10 Elements, the Persian mathematician Al Makhani (c. 800 CE) explored and classified quadratic irrational numbers (numbers of the form) and the more general cubic irrational numbers. He defined rational and irrational quantities, which he called irrational numbers. He easily operated with these objects, but talked about them as separate objects, for example:

In contrast to Euclid's concept that quantities are primarily line segments, Al Makhani considered integers and fractions to be rational quantities, and square and cube roots to be irrational. He also introduced the arithmetic approach to the set of irrational numbers, since it was he who showed the irrationality of the following quantities:

The Egyptian mathematician Abu Kamil (c. 850 CE - c. 930 CE) was the first to consider it acceptable to recognize irrational numbers as solutions to quadratic equations or as coefficients in equations - generally in quadratic or cubic form roots, as well as roots of the fourth degree. In the 10th century, the Iraqi mathematician Al Hashimi produced general proofs (rather than visual geometric demonstrations) of the irrationality of the product, quotient, and results of other mathematical transformations over irrational and rational numbers. Al Khazin (900 AD - 971 AD) gives the following definition of rational and irrational quantity:

Many of these ideas were later adopted by European mathematicians after the translation of Arabic texts into Latin in the 12th century. Al Hassar, an Arab mathematician from the Maghreb who specialized in Islamic inheritance laws, introduced modern symbolic mathematical notation for fractions in the 12th century, dividing the numerator and denominator by a horizontal bar. The same notation then appeared in the works of Fibonacci in the 13th century. During the XIV-XVI centuries. Madhava of Sangamagrama and representatives of the Kerala School of Astronomy and Mathematics investigated infinite series converging to certain irrational numbers, such as π, and also showed the irrationality of certain trigonometric functions. Jestadeva presented these results in the book Yuktibhaza. (proving at the same time the existence of transcendental numbers), thereby rethinking the work of Euclid on the classification of irrational numbers. Works on this topic were published in 1872

Continued fractions, closely related to irrational numbers (a continued fraction representing a given number is infinite if and only if the number is irrational), were first explored by Cataldi in 1613, then came to attention again in the work of Euler, and in the early 19th century century - in the works of Lagrange. Dirichlet also made significant contributions to the development of the theory of continued fractions. In 1761, Lambert used continued fractions to show that π (\displaystyle \pi ) is not a rational number, and also that e x (\displaystyle e^(x)) And tg ⁡ x (\displaystyle \operatorname (tg) x) are irrational for any non-zero rational x (\displaystyle x). Although Lambert's proof can be called incomplete, it is generally considered to be quite rigorous, especially considering the time it was written. Legendre in 1794, after introducing the Bessel-Clifford function, showed that π 2 (\displaystyle \pi ^(2)) irrational, where does irrationality come from? π (\displaystyle \pi ) follows trivially (a rational number squared would give a rational).

The existence of transcendental numbers was proven by Liouville in 1844-1851. Later, Georg Cantor (1873) showed their existence using a different method, and argued that any interval of the real series contains an infinite number of transcendental numbers. Charles Hermite proved in 1873 that e transcendental, and Ferdinand Lindemann in 1882, based on this result, showed transcendence π (\displaystyle \pi ) Literature

The set of irrational numbers is usually denoted by a capital letter I (\displaystyle \mathbb (I) ) in bold style without shading. Thus: 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 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 a square, which is equivalent to the irrationality of the number.

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  Irrational are:

  Examples of proof of irrationality

  Root of 2

  Let's assume 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 Manava (c. 750 BC - c. 690 BC) figured out that the square roots of some natural numbers, such as 2 and 61 cannot be expressed explicitly [ ] .

  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 was a single unit of length, sufficiently small and indivisible, which included an integer number of times in any segment [ ] .

  There is no exact data on which number was proven irrational 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(unspeakable), but according to the legends they did not pay due respect to Hippasus. 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 that denies the doctrine that all entities in the universe can be reduced to integers and their ratios.” The discovery of Hippasus posed a serious problem for Pythagorean mathematics, destroying the underlying assumption that numbers and geometric objects were one and inseparable.