The difference between an irrational and a rational number is an example. Irrational numbers

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

A lot of irrational numbers are usually indicated by an uppercase Latin letter in bold, no fill. Thus: i.e. the set of irrational numbers is the difference between the 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 to the ancient mathematicians: they knew, for example, the incommensurability of the diagonal and the side of a square, which is tantamount to the irrationality of a number.

Properties

• Any real number can be written in the form of an infinite decimal fraction, while irrational numbers and only they are written in non-periodic infinite decimal fractions.
• Irrational numbers define Dedekind sections in the set of rational numbers, which do not have the largest number in the lower class and do not have the smallest number in the upper class.
• Every real transcendental number is irrational.
• Each irrational number is either algebraic or transcendental.
• The set of irrational numbers is everywhere dense on the number line: there is an irrational number between any two numbers.
• 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, it is a set of the second category.

Examples of

Irrational numbers - ζ (3) - √2 - √3 - √5 - - - - -

Irrational are:

Examples of proof of irrationality

Root of 2

Suppose the opposite: rational, that is, it is represented as an irreducible fraction, where is an integer and is a natural number. 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.

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 categorized as positive, negative and zero.

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

Any number rationally can be represented by a decimal periodic fraction. For instance, .

Algorithms for actions on rational numbers follow from the sign rules of the corresponding actions on zero and positive fractions. Division is performed in Q, except 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 of the kind , is decidable in rational numbers. Not every point on the coordinate line has a rational point. Even at the end of the 6th century before. n. BC in the school of Pythagoras it was proved that the diagonal of a square is not commensurate with its height, which is equivalent 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. Let us denote the set of irrational numbers by the letter J .

On the 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 are rational numbers, and non-periodic decimals are irrational numbers. So, 2.03 (52) is a rational number, 2.03003000300003 ... (the period of each next digit "3" is written one more zero) is an irrational number.

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

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

The operation of extracting a root from some rational numbers leads to irrational numbers. Extraction of a root of 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 inverse operation of raising to a natural power.

An arithmetic root (for brevity, a 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 n-th root of the number a is denoted by the symbol. For n = 2, the degree of the root 2 is not indicated:.

For example, since 2 2 = 4 and 2> 0; since 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 follows. For example, the roots of the equation x 2 = 4 are 2 and -2.

When n is odd, equation (1) has a unique root for any. If a≥0, then is the root of this equation. If a<0, то –а>0 and is the root of the equation. So, the equation x 3 = 27 has a root.

Definition of an irrational number

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

So, for example, numbers obtained by extracting 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 by 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 end up as a rational number.
Irrational numbers define Dedekind sections in the set of rational numbers, in the lower class which do not have the 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 straight line are densely packed, and between any two of it there is always an irrational number.
The set of irrational numbers is infinite, uncountable and is a set of the 2nd category.
When performing any arithmetic operation with 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 get a rational number as a result.
The set of irrational numbers is not even.

Numbers are not irrational

Sometimes it is 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;
Third, ordinary fractions;
Fourth, different mixed numbers;
Fifth, these are infinite periodic decimal fractions.

In addition to all of the above, an irrational number cannot be any combination of rational numbers, which is performed by 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 of the numbers are irrational:

And 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 his secret. Any person who knows by heart a certain number of pi 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 pi can be calculated. An entire palace was dedicated to this number by King Frederick II.

It turns out that Pi was tried to be used 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 had not been sufficiently studied.

Singer Keith Bush in her new disc recorded a song called "Pi", which sounded one hundred twenty-four numbers from the famous number series 3, 141 ... ..

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

The 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. Any natural number can be used 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-TRANSITION 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.

Irrational numbers

Irrational numbers are all infinite decimal non-periodic fractions. Irrational numbers have no 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 extraction of square roots. For example, the number "pi" is also irrational, and it is obtained by division. No matter how hard you try, you cannot get it by taking the square root of any natural number.