Fraction m/n we will consider it irreducible (after all, a reducible fraction can always be reduced to an irreducible form). By squaring both sides of the equality, we get m^2=2n^2. From here we conclude that m^2, and after this the number m- even. those. m = 2k. That's why m^2 = 4k^2 and therefore 4 k^2 =2n^2, or 2 k^2 = n^2. But then it turns out that n is also an even number, but this cannot be, since the fraction m/n irreducible. A contradiction arises. It remains to conclude: our assumption is incorrect and the rational number m/n, equal to √2, does not exist.”

That's all their proof.

A critical assessment of the evidence of the ancient Greeks

But…. Let's look at this proof of the ancient Greeks somewhat critically. And if you are more careful in simple mathematics, then you can see the following in it:

1) In the rational number adopted by the Greeks m/n numbers m And n- whole, but unknown(whether they even, whether they odd). And so it is! And in order to somehow establish any dependence between them, it is necessary to accurately determine their purpose;

2) When the ancients decided that the number m– even, then in the equality they accepted m = 2k they (intentionally or out of ignorance!) did not quite “correctly” characterize the number “ k " But here is the number k- This whole(WHOLE!) and quite famous a number that quite clearly defines what was found even number m. And don't be this way found numbers " k"the ancients could not in the future" use" and number m ;

3) And when from equality 2 k^2 = n^2 the ancients received the number n^2 is even, and at the same time n- even, then they would have to do not hurry with the conclusion about " the contradiction that has arisen", but it is better to make sure of the maximum accuracy accepted by them " choice» numbers « n ».

How could they do this? Yes, simple!
Look: from the equality they obtained 2 k^2 = n^2 one could easily obtain the following equality k√2 = n. And there is nothing reprehensible here - after all, they got from equality m/n=√2 is another equality adequate to it m^2=2n^2 ! And no one contradicted them!

But in the new equality k√2 = n for obvious INTEGERS k And n it is clear that from it Always get the number √2 - rational . Always! Because it contains numbers k And n- famous WHOLE ones!

But so that from their equality 2 k^2 = n^2 and, as a consequence, from k√2 = n get the number √2 – irrational (like that " wished"the ancient Greeks!), then it is necessary to have in them, least , number " k" as not whole (!!!) numbers. And this is exactly what the ancient Greeks did NOT have!

Hence the CONCLUSION: the above proof of the irrationality of the number √2, made by the ancient Greeks 2400 years ago, is frankly incorrect and mathematically incorrect, not to say rudely - it is simply fake .

In the small brochure F-6 shown above (see photo above), released in Krasnodar (Russia) in 2015 with a total circulation of 15,000 copies. (obviously with sponsorship investment) a new, extremely correct from the point of view of mathematics and extremely correct ] proof of the irrationality of the number √2 is given, which could have happened long ago if there were no hard " teacher n" to the study of the antiquities of History.

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 ir rational 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.

What are irrational numbers? Why are they called that? Where are they used and what are they? Few people can answer these questions without thinking. 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 represent 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 sufficient. For example, in order to calculate which quantity is the square of 2, you need to use non-periodic infinite decimals. In addition, many simple 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 ​​cannot be represented as a simple fraction, the numerator of which will be an integer, and the denominator will be

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 indicated explicitly. And the first proof of the existence of such numbers is attributed to the Pythagorean Hippasus, who did this in the process of studying the isosceles right triangle. Some other scientists who lived before our era made a serious contribution to the study of this set. The introduction of the concept of irrational numbers entailed a revision of the existing mathematical system, that's why they are so important.

origin of name

If ratio translated from Latin is “fraction”, “ratio”, then the prefix “ir”
gives this word the opposite meaning. Thus, the name of the set of these numbers indicates that they cannot be correlated with an integer or fraction and have a separate place. This follows from their essence.

Place in the general classification

Irrational numbers, along with rational numbers, belong to the group of real or real numbers, which in turn belong to complex numbers. There are no subsets, but 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 (existence of the opposite number);

ab = ba (commutative law);

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

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

a x 1/a = 1 (existence of a reciprocal number);

The comparison is also carried out in accordance with 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 converted using basic arithmetic. There are no special rules for this.

In addition, the Archimedes axiom applies to irrational numbers. It states that for any two quantities a and b, it is true that if you take a as a term enough times, you can exceed b.

Usage

Despite the fact that in ordinary life It is not very often that one encounters them; irrational numbers cannot be counted. There are a huge number of them, but they are almost invisible. Irrational numbers are all around us. Examples that are familiar to everyone are pi, which is 3.1415926..., or e, which is essentially the base natural logarithm, 2.718281828... In algebra, trigonometry and geometry they have to be used constantly. By the way, the famous meaning of the “golden ratio”, that is, the ratio of both the larger part to the smaller part, and vice versa, also

belongs to this set. The lesser known “silver” one too.

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

There are still a lot of unsolved problems associated with this set. There are criteria such as the measure of irrationality and the normality of a number. Mathematicians continue to study the most significant examples to determine whether they belong to one group or another. For example, it is believed that e is a normal number, i.e., the probability of different digits appearing in its notation is the same. As for pi, research is still underway regarding it. The measure of irrationality is a value that shows how well a given number can be approximated by rational numbers.

Algebraic and transcendental

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

This designation hides complex numbers, which include real or real numbers.

So, algebraic is a value that is the root of a polynomial that is not identically equal to zero. For example, Square root of 2 would fall into this category because it is a 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 other were originally developed by mathematicians in this capacity; their irrationality and transcendence were proven many years after their discovery. For pi, the proof was given in 1882 and simplified in 1894, ending a 2,500-year debate about the problem of squaring the circle. It has still not been fully studied, so modern mathematicians have something to work on. By the way, the first fairly accurate calculation of this value was carried out by Archimedes. Before him, all calculations were too approximate.

For e (Euler's or Napier's number), proof of its transcendence was found in 1873. It is used in solving logarithmic equations.

Other examples include the values ​​of sine, cosine, and tangent for any algebraic non-zero value.

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 also includes 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.

From here, I would like to remind 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. Speaking in simple words, an irrational number cannot be represented as an ordinary 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.

Understanding numbers, especially natural numbers, is one of the oldest mathematical "skills". Many civilizations, even modern ones, have attributed certain mystical properties to numbers due to their enormous importance in describing nature. Although modern science and mathematics do not confirm these “magical” properties, the importance of number theory is undeniable.

Historically, a variety of natural numbers appeared first, then fairly quickly fractions and positive irrational numbers were added to them. Zero and negative numbers were introduced after these subsets of the set real numbers. The last set, the set of complex numbers, appeared only with the development of modern science.

In modern mathematics, numbers are not introduced in historical order, although quite close to it.

Natural numbers $\mathbb(N)$

The set of natural numbers is often denoted as $\mathbb(N)=\lbrace 1,2,3,4... \rbrace $, and is often padded with zero to denote $\mathbb(N)_0$.

$\mathbb(N)$ defines the operations of addition (+) and multiplication ($\cdot$) with the following properties for any $a,b,c\in \mathbb(N)$:

1. $a+b\in \mathbb(N)$, $a\cdot b \in \mathbb(N)$ the set $\mathbb(N)$ is closed under the operations of addition and multiplication
2. $a+b=b+a$, $a\cdot b=b\cdot a$ commutativity
3. $(a+b)+c=a+(b+c)$, $(a\cdot b)\cdot c=a\cdot (b\cdot c)$ associativity
4. $a\cdot (b+c)=a\cdot b+a\cdot c$ distributivity
5. $a\cdot 1=a$ is a neutral element for multiplication

Since the set $\mathbb(N)$ contains a neutral element for multiplication but not for addition, adding a zero to this set ensures that it includes a neutral element for addition.

In addition to these two operations, the “less than” relations ($

1. $a b$ trichotomy
2. if $a\leq b$ and $b\leq a$, then $a=b$ antisymmetry
3. if $a\leq b$ and $b\leq c$, then $a\leq c$ is transitive
4. if $a\leq b$ then $a+c\leq b+c$
5. if $a\leq b$ then $a\cdot c\leq b\cdot c$

Integers $\mathbb(Z)$

Examples of integers:
$1, -20, -100, 30, -40, 120...$

Solving the equation $a+x=b$, where $a$ and $b$ are known natural numbers, and $x$ is an unknown natural number, requires the introduction of a new operation - subtraction(-). If there is a natural number $x$ satisfying this equation, then $x=b-a$. However, this particular equation does not necessarily have a solution on the set $\mathbb(N)$, so practical considerations require expanding the set of natural numbers to include solutions to such an equation. This leads to the introduction of a set of integers: $\mathbb(Z)=\lbrace 0,1,-1,2,-2,3,-3...\rbrace$.

Since $\mathbb(N)\subset \mathbb(Z)$, it is logical to assume that the previously introduced operations $+$ and $\cdot$ and the relations $ 1. $0+a=a+0=a$ there is a neutral element for addition
2. $a+(-a)=(-a)+a=0$ there is an opposite number $-a$ for $a$

Property 5.:
5. if $0\leq a$ and $0\leq b$, then $0\leq a\cdot b$

The set $\mathbb(Z)$ is also closed under the subtraction operation, that is, $(\forall a,b\in \mathbb(Z))(a-b\in \mathbb(Z))$.

Rational numbers $\mathbb(Q)$

Examples of rational numbers:
$\frac(1)(2), \frac(4)(7), -\frac(5)(8), \frac(10)(20)...$

Now consider equations of the form $a\cdot x=b$, where $a$ and $b$ are known integers, and $x$ is an unknown. For the solution to be possible, it is necessary to introduce the division operation ($:$), and the solution takes the form $x=b:a$, that is, $x=\frac(b)(a)$. Again the problem arises that $x$ does not always belong to $\mathbb(Z)$, so the set of integers needs to be expanded. This introduces the set of rational numbers $\mathbb(Q)$ with elements $\frac(p)(q)$, where $p\in \mathbb(Z)$ and $q\in \mathbb(N)$. The set $\mathbb(Z)$ is a subset in which each element $q=1$, therefore $\mathbb(Z)\subset \mathbb(Q)$ and the operations of addition and multiplication extend to this set according to the following rules, which preserve all the above properties on the set $\mathbb(Q)$:
$\frac(p_1)(q_1)+\frac(p_2)(q_2)=\frac(p_1\cdot q_2+p_2\cdot q_1)(q_1\cdot q_2)$
$\frac(p-1)(q_1)\cdot \frac(p_2)(q_2)=\frac(p_1\cdot p_2)(q_1\cdot q_2)$

The division is introduced as follows:
$\frac(p_1)(q_1):\frac(p_2)(q_2)=\frac(p_1)(q_1)\cdot \frac(q_2)(p_2)$

On the set $\mathbb(Q)$, the equation $a\cdot x=b$ has a unique solution for each $a\neq 0$ (division by zero is undefined). This means that there is an inverse element $\frac(1)(a)$ or $a^(-1)$:
$(\forall a\in \mathbb(Q)\setminus\lbrace 0\rbrace)(\exists \frac(1)(a))(a\cdot \frac(1)(a)=\frac(1) (a)\cdot a=a)$

The order of the set $\mathbb(Q)$ can be expanded as follows:
$\frac(p_1)(q_1)

The set $\mathbb(Q)$ has one important property: between any two rational numbers there are infinitely many other rational numbers, therefore, there are no two adjacent rational numbers, unlike the sets of natural numbers and integers.

Irrational numbers $\mathbb(I)$

Examples of irrational numbers:
$\sqrt(2) \approx 1.41422135...$
$\pi\approx 3.1415926535...$

Since between any two rational numbers there are infinitely many other rational numbers, it is easy to erroneously conclude that the set of rational numbers is so dense that there is no need to expand it further. Even Pythagoras made such a mistake in his time. However, his contemporaries already refuted this conclusion when studying solutions to the equation $x\cdot x=2$ ($x^2=2$) on the set of rational numbers. To solve such an equation, it is necessary to introduce the concept of a square root, and then the solution to this equation has the form $x=\sqrt(2)$. An equation like $x^2=a$, where $a$ is a known rational number and $x$ is an unknown one, does not always have a solution on the set of rational numbers, and again the need arises to expand the set. A set of irrational numbers arises, and numbers such as $\sqrt(2)$, $\sqrt(3)$, $\pi$... belong to this set.

Real numbers $\mathbb(R)$

The union of the sets of rational and irrational numbers is the set of real numbers. Since $\mathbb(Q)\subset \mathbb(R)$, it is again logical to assume that the introduced arithmetic operations and relations retain their properties on the new set. The formal proof of this is very difficult, so the above-mentioned properties of arithmetic operations and relations on the set of real numbers are introduced as axioms. In algebra, such an object is called a field, so the set of real numbers is said to be an ordered field.

In order for the definition of the set of real numbers to be complete, it is necessary to introduce an additional axiom that distinguishes the sets $\mathbb(Q)$ and $\mathbb(R)$. Suppose that $S$ is a non-empty subset of the set of real numbers. An element $b\in \mathbb(R)$ is called the upper bound of a set $S$ if $\forall x\in S$ holds $x\leq b$. Then we say that the set $S$ is bounded above. The smallest upper bound of the set $S$ is called the supremum and is denoted $\sup S$. The concepts are introduced similarly lower limit, a set bounded below, and an infinum $\inf S$ . Now the missing axiom is formulated as follows:

Any non-empty and upper-bounded subset of the set of real numbers has a supremum.
It can also be proven that the field of real numbers defined in the above way is unique.

Complex numbers$\mathbb(C)$

Examples of complex numbers:
$(1, 2), (4, 5), (-9, 7), (-3, -20), (5, 19),...$
$1 + 5i, 2 - 4i, -7 + 6i...$ where $i = \sqrt(-1)$ or $i^2 = -1$

The set of complex numbers represents all ordered pairs of real numbers, that is, $\mathbb(C)=\mathbb(R)^2=\mathbb(R)\times \mathbb(R)$, on which the operations of addition and multiplication are defined as follows way:
$(a,b)+(c,d)=(a+b,c+d)$
$(a,b)\cdot (c,d)=(ac-bd,ad+bc)$

There are several forms of writing complex numbers, of which the most common is $z=a+ib$, where $(a,b)$ is a pair of real numbers, and the number $i=(0,1)$ is called the imaginary unit.

It is easy to show that $i^2=-1$. Extending the set $\mathbb(R)$ to the set $\mathbb(C)$ allows us to determine the square root of negative numbers, which was the reason for introducing the set of complex numbers. It is also easy to show that a subset of the set $\mathbb(C)$, given by $\mathbb(C)_0=\lbrace (a,0)|a\in \mathbb(R)\rbrace$, satisfies all the axioms for real numbers, therefore $\mathbb(C)_0=\mathbb(R)$, or $R\subset\mathbb(C)$.

The algebraic structure of the set $\mathbb(C)$ with respect to the operations of addition and multiplication has the following properties:
1. commutativity of addition and multiplication
2. associativity of addition and multiplication
3. $0+i0$ - neutral element for addition
4. $1+i0$ - neutral element for multiplication
5. Multiplication is distributive with respect to addition
6. There is a single inverse for both addition and multiplication.