What numbers are irrational. Rational and irrational numbers

Definition of an irrational number

Irrational numbers are those numbers that, in decimal notation, are infinite non-periodic decimal fractions.

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 extracting square roots, because the number "pi" obtained by division is also irrational, and you are unlikely to get it when trying to extract the square root of a natural number.

Properties of irrational numbers

Unlike numbers written in infinite decimal fractions, only irrational numbers are written in non-periodic infinite decimal fractions.
The sum of two non-negative irrational numbers can eventually be a rational number.
Irrational numbers define Dedekind sections in the set of rational numbers, in the lower class which do not have the a large number, and there is no smaller one in the upper one.
Any real transcendental number is irrational.
All irrational numbers are either algebraic or transcendental.
The set of irrational numbers on the line are densely packed, and between any two of its numbers there is necessarily an ir rational 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, its 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 get a rational number as a result.
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;
Second, integers;
Thirdly, ordinary fractions;
Fourth, different mixed numbers;
Fifth, these are infinite periodic decimal fractions.

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

Now let's see which of the 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. A whole palace was dedicated to this number by King Frederick II.

It turns out that they tried to use the number Pi in the construction of the Tower of Babel. But to our great regret, 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 sounded ... ..

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 referred to natural numbers.

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 we 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 the set consisting of numbers of the form m/n, -m/n and the number 0. In as n,m can be any natural number. 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. And it does not apply to rational numbers.

In the school course of algebra, only real (or real) numbers are studied. Many of all real numbers 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 have no special notation.

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 can't get it by taking the square root of any natural number.

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

History

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).
• 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 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.

see also

Notes

Numerical systems
Counting sets	Natural numbers () Integers ()

A rational number is a number that can be represented as a fraction, where . Q is the set of all rational numbers.

Rational numbers are divided into: positive, negative and zero.

Each rational number can be associated with a single point on the coordinate line. The relation "to the left" for points corresponds to the relation "less than" for the coordinates of these points. It can be seen that every negative number is less than zero and every positive number; of two negative numbers, the one whose modulus is greater is less. So, -5.3<-4.1, т.к. |5.3|>|4.1|.

Any rational number can be represented as a decimal periodic fraction. For example, .

Algorithms for operations on rational numbers follow from the rules of signs for the corresponding operations on zero and positive fractions. Q performs division other than division by zero.

Any linear equation, i.e. equation of the form ax+b=0, where , is solvable on the set Q, but not any quadratic equation kind , is solvable in rational numbers. Not every point on a coordinate line has a rational point. Even at the end of the 6th century BC. n. e in the school of Pythagoras, it was proved that the diagonal of a square is not commensurate with its height, which is tantamount to the statement: "The equation has no rational roots." All of the above led to the need to expand the set Q, the concept of an irrational number was introduced. Denote the set of irrational numbers by the letter J .

On a coordinate line, all points that do not have rational coordinates have irrational coordinates. , where r are sets of real numbers. in a universal way assignments of real numbers are decimals. Periodic decimals define rational numbers, and non-periodic decimals define irrational numbers. So, 2.03 (52) is a rational number, 2.03003000300003 ... (the period of each following digit “3” is written one zero more) is an irrational number.

The sets Q and R have the properties of positivity: between any two rational numbers there is a rational number, for example, ecoi a

For every irrational number α one can specify a rational approximation both with a deficiency and with an excess with any accuracy: a< α

The operation of extracting a root from some rational numbers leads to irrational numbers. Extracting the root of a natural degree is an algebraic operation, i.e. its introduction is connected with the solution of an algebraic equation of the form . If n is odd, i.e. n=2k+1, where , then the equation has a single root. If n is even, n=2k, where , then for a=0 the equation has a single root x=0, for a<0 корней нет, при a>0 has two roots that are opposite to each other. Extracting a root is the reverse operation of raising to a natural power.

The arithmetic root (for brevity, the root) of the nth degree of a non-negative number a is a non-negative number b, which is the root of the equation. The root of the nth degree from the number a is denoted by the symbol. For n=2, the degree of the root 2 is not indicated: .

For example, , because 2 2 =4 and 2>0; , because 3 3 =27 and 3>0; does not exist because -4<0.

For n=2k and a>0, the roots of equation (1) are written as and . For example, the roots of the equation x 2 \u003d 4 are 2 and -2.

For n odd, equation (1) has a single root for any . If a≥0, then - the root of this equation. If a<0, то –а>0 and - the root of the equation. So, the equation x 3 \u003d 27 has a root.

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, you need to use non-recurring 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 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 the word the 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 ones, 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 the 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 transformed using the basic arithmetic operations. There are no special rules for this.

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, it is possible to surpass b.

Usage

Despite the fact that in ordinary life not so often you have to deal with them, irrational numbers are not countable. There are a lot of them, 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, the famous meaning of the "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 problems 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.

Under this designation, complex numbers are hidden, which include real or real numbers.

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.

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 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.