A number is rational or irrational. What are rational and irrational numbers

Rational number- a number represented by an ordinary fraction m / n, where the numerator m is an integer and the denominator n is a natural number. Any rational number can be represented as a periodic infinite decimal... Lots of rational numbers is denoted by Q.

If the real number is not rational, then it irrational number... Decimal fractions expressing irrational numbers are infinite and not periodic. The set of irrational numbers is usually denoted by a capital letter I.

The real number is called algebraic if it is a root of some polynomial (nonzero degree) with rational coefficients. Any non-algebraic number is called transcendental.

Some properties:

The set of rational numbers is everywhere densely located on the number axis: between any two different rational numbers there is at least one rational number (and hence an infinite set of rational numbers). Nevertheless, it turns out that the set of rational numbers Q and the set natural numbers N are equivalent, that is, a one-to-one correspondence can be established between them (all elements of the set of rational numbers can be renumbered).

The set Q of rational numbers is closed with respect to addition, subtraction, multiplication and division, that is, the sum, difference, product and quotient of two rational numbers are also rational numbers.

All rational numbers are algebraic (the converse is not true).

Every real transcendental number is irrational.

Every irrational number is either algebraic or transcendental.

The set of irrational numbers is everywhere dense on the number line: between any two numbers there is an irrational number (and therefore, an infinite set of irrational numbers).

The set of irrational numbers is uncountable.

When solving problems, it is convenient, together with the irrational number a + b√ c (where a, b are rational numbers, c is an integer that is not a square of a natural number), to consider the “conjugate” number a - b√ c: its sum and product with the original one - rational numbers. So a + b√ c and a - b√ c are roots quadratic equation with integer coefficients.

Problems with solutions

1. Prove that

a) number √ 7;

b) the number lg 80;

c) the number √ 2 + 3 √ 3;

is irrational.

a) Suppose that the number √ 7 is rational. Then, there are coprime p and q such that √ 7 = p / q, whence we obtain p 2 = 7q 2. Since p and q are coprime, p is 2, and hence p is divisible by 7. Then p = 7k, where k is some natural number. Hence q 2 = 7k 2 = pk, which contradicts the fact that p and q are coprime.

So, the assumption is false, which means that the number √ 7 is irrational.

b) Suppose that the number lg 80 is rational. Then there exist natural numbers p and q such that lg 80 = p / q, or 10 p = 80 q, whence we obtain 2 p – 4q = 5 q – p. Taking into account that the numbers 2 and 5 are coprime, we see that the last equality is possible only for p – 4q = 0 and q – p = 0. Whence p = q = 0, which is impossible, since p and q are chosen natural.

So, the assumption is false, which means that the number lg 80 is irrational.

c) Let us denote this number by x.

Then (x - √ 2) 3 = 3, or x 3 + 6x - 3 = √ 2 · (3x 2 + 2). After squaring this equation, we find that x must satisfy the equation

x 6 - 6x 4 - 6x 3 + 12x 2 - 36x + 1 = 0.

Only numbers 1 and –1 can be its rational roots. Checking shows that 1 and –1 are not roots.

So, the given number √ 2 + 3 √ 3 ​​is irrational.

2. It is known that the numbers a, b, √ a –√ b,- rational. Prove that √ a and √ b Are also rational numbers.

Consider the product

(√ a - √ b) (√ a + √ b) = a - b.

Number √ a + √ b, which is equal to the ratio of the numbers a - b and √ a –√ b, is rational, since the quotient of dividing two rational numbers is a rational number. The sum of two rational numbers

½ (√ a + √ b) + ½ (√ a - √ b) = √ a

- rational number, their difference,

½ (√ a + √ b) - ½ (√ a - √ b) = √ b,

is also a rational number, as required.

3. Prove that there are positive irrational numbers a and b for which the number a b is natural.

4. Are there rational numbers a, b, c, d satisfying the equality

(a + b √ 2) 2n + (c + d√ 2) 2n = 5 + 4√ 2,

where n is a natural number?

If the equality given in the condition holds, and the numbers a, b, c, d are rational, then the equality holds:

(a - b √ 2) 2n + (c - d√ 2) 2n = 5 - 4√ 2.

But 5 - 4√2 (a - b√2) 2n + (c - d√2) 2n> 0. The resulting contradiction proves that the original equality is impossible.

Answer: do not exist.

5. If segments with lengths a, b, c form a triangle, then for all n = 2, 3, 4,. ... ... segments with lengths n √ a, n √ b, n √ c also form a triangle. Prove it.

If segments with lengths a, b, c form a triangle, then the triangle inequality gives

Therefore we have

(n √ a + n √ b) n> a + b> c = (n √ c) n,

N √ a + n √ b> n √ c.

The rest of the cases of checking the triangle inequality are considered in a similar way, from which the conclusion follows.

6. Prove that the infinite decimal fraction 0.1234567891011121314 ... (after the decimal point in a row are written out all natural numbers in order) is an irrational number.

As you know, rational numbers are expressed in decimal fractions, which have a period starting from a certain sign. Therefore, it suffices to prove that the given fraction is not periodic from any sign. Suppose that this is not the case, and some sequence T, consisting of n digits, is the period of the fraction, starting from the mth decimal place. It is clear that among the digits after the m-th character there are nonzero ones, therefore there is a nonzero digit in the sequence of digits T. This means that starting from the mth digit after the decimal point, there is a nonzero digit among any n consecutive digits. However, in the decimal notation of this fraction, there must be a decimal notation of the number 100 ... 0 = 10 k, where k> m and k> n. It is clear that this entry will occur to the right of the m-th digit and contains more than n zeros in a row. Thus, we obtain a contradiction that completes the proof.

7. You are given an infinite decimal fraction 0, a 1 a 2 .... Prove that the numbers in its decimal notation can be rearranged so that the resulting fraction expresses a rational number.

Recall that a fraction expresses a rational number if and only if it is periodic, starting with a certain sign. We divide the numbers from 0 to 9 into two classes: in the first class we include those numbers that occur in the original fraction a finite number of times, in the second class - those that occur in the original fraction an infinite number of times. Let's start writing out the periodic fraction, which can be obtained from the original permutation of the numbers. First, after zero and a comma, we write in random order all the numbers from the first class - each as many times as it occurs in the initial fraction. The first-class digits recorded will precede the period in the fractional part of the decimal fraction. Next, we write down in some order, one at a time, the numbers from the second class. We will declare this combination as a period and will repeat it an infinite number of times. Thus, we have written out the required periodic fraction, which expresses a certain rational number.

8. Prove that in each infinite decimal fraction there is a sequence of decimal places of arbitrary length, which occurs infinitely many times in the expansion of the fraction.

Let m be an arbitrarily given natural number. Let's split the given infinite decimal fraction into segments, with m digits in each. There will be infinitely many such segments. On the other hand, different systems consisting of m digits, there are only 10 m, that is, a finite number. Consequently, at least one of these systems must be repeated here infinitely many times.

Comment. For irrational numbers √ 2, π or e we do not even know which digit is repeated infinitely many times in the infinite decimal fractions representing them, although each of these numbers, as can easily be proved, contains at least two different such digits.

9. Prove in an elementary way that the positive root of the equation

is irrational.

For x> 0, the left side of the equation increases with increasing x, and it is easy to see that for x = 1.5 it is less than 10, and for x = 1.6 - more than 10. Therefore, the only positive root of the equation lies within the interval (1.5 ; 1.6).

We write the root as an irreducible fraction p / q, where p and q are some coprime natural numbers. Then, for x = p / q, the equation will take the following form:

p 5 + pq 4 = 10q 5,

whence it follows that p is a divisor of 10, therefore, p is equal to one of the numbers 1, 2, 5, 10. However, writing out fractions with numerators 1, 2, 5, 10, we immediately notice that none of them falls inside the interval (1.5; 1.6).

So, the positive root of the original equation cannot be represented as an ordinary fraction, which means it is an irrational number.

10. a) Are there three points A, B and C on the plane such that for any point X the length of at least one of the segments XA, XB and XC is irrational?

b) The coordinates of the vertices of the triangle are rational. Prove that the coordinates of the center of its circumcircle are also rational.

c) Is there such a sphere on which there is exactly one rational point? (A rational point is a point at which all three Cartesian coordinates are rational numbers.)

a) Yes, there are. Let C be the midpoint of the segment AB. Then XC 2 = (2XA 2 + 2XB 2 - AB 2) / 2. If the number AB 2 is irrational, then the numbers XA, XB and XC cannot be rational at the same time.

b) Let (a 1; b 1), (a 2; b 2) and (a 3; b 3) be the coordinates of the vertices of the triangle. The coordinates of the center of its circumscribed circle are given by a system of equations:

(x - a 1) 2 + (y - b 1) 2 = (x - a 2) 2 + (y - b 2) 2,

(x - a 1) 2 + (y - b 1) 2 = (x - a 3) 2 + (y - b 3) 2.

It is easy to check that these equations are linear, which means that the solution of the considered system of equations is rational.

c) Such a sphere exists. For example, a sphere with the equation

(x - √ 2) 2 + y 2 + z 2 = 2.

Point O with coordinates (0; 0; 0) is a rational point lying on this sphere. The rest of the points of the sphere are irrational. Let's prove it.

Suppose the opposite: let (x; y; z) be a rational point of the sphere, different from the point O. It is clear that x is different from 0, since at x = 0 there is a unique solution (0; 0; 0), which we are not now interested. Let's expand the brackets and express √ 2:

x 2 - 2√ 2 x + 2 + y 2 + z 2 = 2

√ 2 = (x 2 + y 2 + z 2) / (2x),

which cannot be for rational x, y, z and irrational √ 2. So, O (0; 0; 0) is the only rational point on the considered sphere.

Tasks without solutions

1. Prove that the number

\ [\ sqrt (10+ \ sqrt (24) + \ sqrt (40) + \ sqrt (60)) \]

is irrational.

2. For which integers m and n does the equality (5 + 3√ 2) m = (3 + 5√ 2) n hold?

3. Is there a number a such that the numbers a - √ 3 and 1 / a + √ 3 are integers?

4. Can the numbers 1, √ 2, 4 be members (not necessarily adjacent) of an arithmetic progression?

5. Prove that for any natural number n the equation (x + y√ 3) 2n = 1 + √ 3 has no solutions in rational numbers (x; y).

A lot of irrational numbers are usually indicated by a capital Latin letter I (\ displaystyle \ mathbb (I)) in bold, no fill. 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 ancient mathematicians already knew about the existence of irrational numbers, more precisely, segments incommensurable with a segment of unit length: they knew, for example, the incommensurability of the diagonal and the side of a square, which is equivalent to the irrationality of a number.

Collegiate YouTube

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  Irrational are:

  Examples of proof of irrationality

  Root of 2

  Suppose 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 assumed equality:

  2 = mn ⇒ 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)).

  History

  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 explicitly expressed [ ] .

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

  There is no exact data about the irrationality of which number was proved 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 it was the golden ratio [ ] .

  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 entire theory that numbers and geometric objects are one and indivisible.

  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 Square root from 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 its numbers there will necessarily 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 with rational numbers, other than 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 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 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, common 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 that is performed by signs of arithmetic operations, like +, -,,:, 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 construction. 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 and twenty-four numbers from the famous number series 3, 141 ... ..

  We have shown earlier that $ 1 \ frac25 $ is close to $ \ sqrt2 $. If it were exactly $ \ sqrt2 $,. Then the ratio - $ \ frac (1 \ frac25) (1) $, which can be turned into the ratio of integers $ \ frac75 $, multiplying the upper and lower parts of the fraction by 5, and would be the desired value.

  Unfortunately, $ 1 \ frac25 $ is not an exact value for $ \ sqrt2 $. A more accurate answer $ 1 \ frac (41) (100) $, gives us the relation $ \ frac (141) (100) $. We achieve even greater precision when we equate $ \ sqrt2 $ with $ 1 \ frac (207) (500) $. In this case, the ratio in integers will be $ \ frac (707) (500) $. But also $ 1 \ frac (207) (500) $ is not the exact value of the square root of 2. Greek mathematicians spent a lot of time and effort to calculate the exact value of $ \ sqrt2 $, but they never succeeded. They could not represent the ratio $ \ frac (\ sqrt2) (1) $ as a ratio of integers.

  Finally, the great Greek mathematician Euclid proved that no matter how the accuracy of the calculations increases, it is impossible to get the exact value of $ \ sqrt2 $. There is no fraction that, when squared, will result in 2. They say that Pythagoras was the first to come to this conclusion, but this inexplicable fact so amazed the scientist that he himself swore and took an oath from his students to keep this discovery a secret. ... However, it is possible that this information does not correspond to reality.

  But if the number $ \ frac (\ sqrt2) (1) $ cannot be represented as a ratio of integers, then none containing $ \ sqrt2 $, for example $ \ frac (\ sqrt2) (2) $ or $ \ frac (4) (\ sqrt2) $ also cannot be represented as a ratio of integers, since all such fractions can be converted to $ \ frac (\ sqrt2) (1) $ multiplied by some number. So $ \ frac (\ sqrt2) (2) = \ frac (\ sqrt2) (1) \ times \ frac12 $. Or $ \ frac (\ sqrt2) (1) \ times 2 = 2 \ frac (\ sqrt2) (1) $, which can be transformed by multiplying the top and bottom by $ \ sqrt2 $ to get $ \ frac (4) (\ sqrt2) $. (Remember that no matter what the number $ \ sqrt2 $ is, if we multiply it by $ \ sqrt2 $, we get 2.)
  
  Since the number $ \ sqrt2 $ cannot be represented as a ratio of integers, it is called irrational number... On the other hand, all numbers that can be represented as a ratio of integers are called rational.

  All integers and fractional numbers, both positive and negative, are rational.

  As it turns out, most square roots are irrational numbers. Only numbers in a series of square numbers have rational square roots. These numbers are also called perfect squares. Rational numbers are also fractions made up of these perfect squares. For example, $ \ sqrt (1 \ frac79) $ is a rational number because $ \ sqrt (1 \ frac79) = \ frac (\ sqrt16) (\ sqrt9) = \ frac43 $ or $ 1 \ frac13 $ (4 is the root square of 16, and 3 is the square root of 9).

  Understanding numbers, especially natural numbers, is one of the oldest mathematical "skills". Many civilizations, even modern ones, have attributed some mystical properties to numbers because of their great importance in describing nature. Although modern science and mathematics does not confirm these "magic" properties, the significance of number theory is undeniable.

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

  In modern mathematics, numbers are not entered in historical order, although in a rather 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 zero-padded to denote $ \ mathbb (N) _0 $.

  In $ \ mathbb (N) $, the operations of addition (+) and multiplication ($ \ cdot $) with 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 $ distributive
  5. $ a \ cdot 1 = a $ is the neutral element for multiplication

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

  In addition to these two operations, on the set $ \ mathbb (N) $, the relations "less than" ($

  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 transitivity
  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:
  $1, -20, -100, 30, -40, 120...$

  The solution of 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 $ that satisfies 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 extending the set of natural numbers to include solutions to such an equation. This results in 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 additions
  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 unknown. For the solution to be possible, it is necessary to enter 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 must be expanded. Thus, we introduce 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 is $ q = 1 $, therefore $ \ mathbb (Z) \ subset \ mathbb (Q) $ and the operations of addition and multiplication are extended to this set according to the following rules, which preserve all of 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) $

  Division is introduced in this way:
  $ \ 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 not defined). This means that there is an inverse $ \ 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 extended 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, in contrast to the sets of naturals and integers.

  Irrational numbers $ \ mathbb (I) $

  Examples of irrational numbers:
  $0.333333...$
  $ \ sqrt (2) \ approx 1.41422135 ... $
  $ \ pi \ approx 3.1415926535 ... $

  In view of the fact that between any two rational numbers there are infinitely many other rational numbers, it is easy to make an erroneous conclusion that the set of rational numbers is so dense that there is no need for its further expansion. Even Pythagoras made such a mistake in his time. However, already his contemporaries refuted this conclusion when studying the solutions of 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 of the type $ x ^ 2 = a $, where $ a $ is a known rational number, and $ x $ is an unknown, does not always have a solution on the set of rational numbers, and again there is a need to expand the set. A set of irrational numbers arises, and such numbers 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, therefore 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 they say that the set of real numbers is 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 distinguishing the sets $ \ mathbb (Q) $ and $ \ mathbb (R) $. Suppose that $ S $ is a non-empty subset of the set of real numbers. The element $ b \ in \ mathbb (R) $ is called the upper bound of the set $ S $ if $ \ forall x \ in S $ is true $ x \ leq b $. Then the set $ S $ is said to be bounded above. The smallest upper bound of the set $ S $ is called the supremum and is denoted by $ \ sup S $. The concepts of a lower bound, a set bounded from below, and an infinum $ \ inf S $ are introduced similarly. The missing axiom is now formulated as follows:

  Any non-empty and upper-bounded subset of the set of real numbers has a supremum.
  You can also prove 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 notation for complex numbers, the most common of which is $ z = a + ib $, where $ (a, b) $ is a pair of real numbers, and the number $ i = (0,1) $ is called an 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 the introduction of a set of complex numbers. It is also easy to show that a subset of the set $ \ mathbb (C) $, defined as $ \ 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.commutability 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 element for both addition and multiplication.