Example:
\(4\) - rational number,because it can be written as \(\frac(4)(1)\) ;
\(0.0157304\) is also rational, because it can be written in the form \(\frac(157304)(10000000)\) ;
\(0.333(3)...\) - and this is a rational number: can be represented as \(\frac(1)(3)\) ;
\(\sqrt(\frac(3)(12))\) is rational, since 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.

It's impossible because it's endless fractions, and even non-periodic ones. Therefore, there are no integers that, when divided by each other, would give an irrational number.

Example:
\(\sqrt(2)≈1.414213562…\) is an irrational number;
\(π≈3.1415926… \) is an 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, which means It is also impossible to represent a number as a fraction with integers, 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 \(\frac(-5)(1)\), which means 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.

1.Proofs are examples of deductive reasoning and are different from inductive or empirical arguments. A proof must demonstrate that the statement being proven is always true, sometimes by listing all possible cases and showing that the statement holds in each of them. A proof may rely on obvious or generally accepted phenomena or cases known as axioms. Contrary to this, the irrationality of the “square root of two” is proven.
2. The intervention of topology here is explained by the very nature of things, which means that there is no purely algebraic way to prove irrationality, in particular based on rational numbers. Here is an example, the choice is yours: 1 + 1/2 + 1/4 + 1/8 ….= 2 or 1+1/2 + 1/4 + 1/8 …≠ 2 ???
If you accept 1+1/2 + 1/4 + 1/8 +…= 2, which is considered the “algebraic” approach, then it is not at all difficult to show that there exists n/m ∈ ℚ, which on an infinite sequence is irrational and finite number. This suggests that the irrational numbers are the closure of the field ℚ, but this refers to a topological singularity.
So for Fibonacci numbers, F(k): 1,1,2,3,5,8,13,21,34,55,89,144,233,377, … lim(F(k+1)/F(k)) = φ
This only shows that there is a continuous homomorphism ℚ → I, and it can be shown rigorously that the existence of such an isomorphism is not a logical consequence of the algebraic axioms.

This property plays an important role in solving differential equations. So, for example, the only solution to the differential equation

is a function

Where c- arbitrary constant.

• 1. Number e irrational and even transcendental. Its transcendence was proven only in 1873 by Charles Hermite. It is assumed that e - normal number, that is, the probability of different numbers appearing in its entry is the same.
• 2. Number e is a computable (and therefore arithmetic) number.

Euler's formula, in particular

5. so-called "Poisson integral" or "Gauss integral"

8. Catalan Representation:

9. Presentation through the work:

10. Through Bell numbers:

11. Measure of irrationality of a number e is equal to 2 (which is the smallest possible value for irrational numbers).

Proof of irrationality

Let's pretend that

where a and b are natural numbers. Considering this equality and considering the series expansion:

we get the following equality:

Let us imagine this sum as the sum of two terms, one of which is the sum of the terms of the series in terms of n from 0 to a, and the second is the sum of all other terms of the series:

Now let's move the first sum to the left side of the equality:

Let's multiply both sides of the resulting equality by. We get

Now let's simplify the resulting expression:

Let us consider the left side of the resulting equality. Obviously the number is an integer. A number is also an integer, since (it follows that all numbers of the form are integers). Thus, the left side of the resulting equality is an integer.

Let's move now to the right side. This amount has the form

According to Leibniz's criterion, this series converges, and its sum S is a real number enclosed between the first term and the sum of the first two terms (with signs), i.e.

Both of these numbers lie between 0 and 1. Therefore, i.e. - the right side of the equality - cannot be an integer. We get a contradiction: an integer cannot be equal to a number that is not an integer. This contradiction proves that the number e is not rational, and therefore is irrational.

Definition of an irrational number

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

So, 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 by trying to extract Square root from a natural number.

Properties of irrational numbers

Unlike numbers written as infinite decimals, only irrational numbers are written as non-periodic infinite decimals.
The sum of two non-negative irrational numbers can end up being a rational number.
Irrational numbers define Dedekind sections in the set of rational numbers, in the lower class which do not have the large number, and in the upper there is no less.
Any real transcendental number is irrational.
All irrational numbers are either algebraic or transcendental.
The set of irrational numbers on a line is densely located, and between any two of its numbers there is sure to be an irrational 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, 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 end up with a rational number.
The set of irrational numbers is not even.

Numbers are not irrational

Sometimes it is quite difficult to answer the question whether a number is irrational, especially in cases where the number has the form decimal or as a numeric 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;
Secondly, integers;
Third, ordinary fractions;
Fourthly, different mixed numbers;
Fifthly, these are infinite periodic decimal fractions.

In addition to all of the above, an irrational number cannot be any combination of rational numbers that is performed by the 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 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. King Frederick II dedicated the entire palace to this number.

It turns out that they tried to use the number Pi in the construction of the Tower of Babel. But unfortunately, 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…..

What numbers are irrational? Irrational number is not a rational real number, i.e. it cannot be represented as a fraction (as a ratio of two integers), where m- integer, n- natural number . Irrational number can be represented as an infinite non-periodic decimal fraction.

Irrational number may not have an exact meaning. Only in format 3.333333…. For example, the square root of two is an irrational number.

Which number is irrational? Irrational number(as opposed to rational) is called an infinite decimal non-periodic fraction.

Set of irrational numbers often denoted by a capital Latin letter in bold style without shading. That.:

Those. The set of irrational numbers is the difference between the sets of real and rational numbers.

Properties of irrational numbers.

• The sum of 2 non-negative irrational numbers can be a rational number.
• Irrational numbers define Dedekind cuts in the set of rational numbers, in the lower class of which there is no largest number, and in the upper class there is no smaller one.
• Every real transcendental number is an irrational number.
• All irrational numbers are either algebraic or transcendental.
• The set of irrational numbers is dense everywhere on the number line: between every pair of 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 infinite and is a set of the 2nd category.
• The result of every arithmetic operation with rational numbers (except division by 0) is a rational number. The result of arithmetic operations on irrational numbers can be either a rational or an irrational number.
• The sum of a rational and an irrational number will always be an irrational number.
• The sum of irrational numbers can be a rational number. For example, let x irrational then y=x*(-1) also irrational; x+y=0, and the number 0 rational (if, for example, we add the root of any degree of 7 and minus the root of the same degree of seven, we get the rational number 0).

Irrational numbers, examples.

γ — ζ (3) — ρ — √2 — √3 — √5 — φ — δs — α — e — π — δ