The concepts of “set”, “element”, “belonging of an element to a set” are the primary concepts of mathematics. Many- any collection (set) of any objects .
A is a subset of set B, if every element of set A is an element of set B, i.e. AÌB Û (ХОА Þ ХОВ).
Two sets are equal, if they consist of the same elements. We are talking about set-theoretic equality (not to be confused with equality between numbers): A=B Û AÌB Ù VA.
Union of two sets consists of elements belonging to at least one of the sets, i.e. KHOAÈV Û KHOAÚ KHOV.
Intersection consists of all elements simultaneously belonging to both set A and set B: хОАХВ Û хОА Ù хОВ.
Difference consists of all elements of A that do not belong to B, i.e. xО A\B Û xОА ÙхПВ.
Cartesian product C=A´B of sets A and B is the set of all possible pairs ( x,y), where the first element X each pair contains A, and its second element at belongs to V.
A subset F of the Cartesian product A´B is called mapping set A to set B , if the condition is met: (" X OA)($! pair ( x.y)ÎF). At the same time they write: A V.
The terms “display” and “function” are synonymous. If ("хОА)($! уУВ): ( x,y)ОF, then the element atÎ IN called way X when displaying F and write it like this: at=F( X). Element X at the same time is prototype (one of the possible) element y.
Let's consider set of rational numbers Q - the set of all integers and the set of all fractions (positive and negative). Each rational number can be represented as a quotient, for example, 1 =4/3=8/6=12/9=…. There are many such representations, but only one of them is irreducible .
IN any rational number can be the only way represent as a fraction p/q, where pÎZ, qÎN, numbers p, q are coprime.
Properties of the set Q:
1. Closedness under arithmetic operations. The result of addition, subtraction, multiplication, raising to the natural power, division (except division by 0) of rational numbers is a rational number: ; ; 
.
2. Orderliness: (" x, yÎQ, x¹y)®( x
Moreover: 1) a>b, b>c Þ a>c; 2)a -b.
3. Density. Between any two rational numbers x, y there is a third rational number (for example, c= ):
("x, yÎQ, x<y)($cÎQ) : ( X
On the set Q you can perform 4 arithmetic operations, solve systems of linear equations, but quadratic equations of the form x 2 =a, aÎ N are not always solvable in the set Q.
Theorem. There is no number xÎQ, whose square is 2.
g Let there be such a fraction X=p/q, where the numbers p and q are coprime and X 2 =2. Then (p/q) 2 =2. Hence,
The right side of (1) is divisible by 2, which means p 2 is an even number. Thus p=2n (n-integer). Then q must be an odd number.
Returning to (1), we have 4n 2 =2q 2. Therefore q 2 =2n 2. Similarly, we make sure that q is divisible by 2, i.e. q is an even number. The theorem is proven by contradiction.n
geometric representation of rational numbers. Putting a unit segment from the origin of coordinates 1, 2, 3 ... times to the right, we obtain points of the coordinate line that correspond natural numbers. Shifting similarly to the left, we obtain points corresponding to negative integers. Let's take 1/q(q= 2,3,4 … ) part of a unit segment and we will place it on both sides of the origin r once. We obtain the points of the line corresponding to numbers of the form ±p/q (pОZ, qОN). If p, q run through all pairs of relatively prime numbers, then on the straight line we have all points corresponding to fractional numbers. Thus, According to the accepted method, each rational number corresponds to a single point on the coordinate line.
Is it possible to specify a single rational number for every point? Is the line filled entirely with rational numbers?
It turns out there are points on the coordinate line that do not correspond to any rational numbers. We construct an isosceles right triangle on a unit segment. Point N does not correspond to a rational number, because if ON=x- rationally, then x 2 = 2, which cannot be.
There are infinitely many points similar to point N on a straight line. Let us take the rational parts of the segment x=ON, those. X. If we put them to the right, then no rational number will correspond to each of the ends of any of these segments. Assuming that the length of the segment is expressed by a rational number x=, we get that x=- rational. This contradicts what was proven above.
Rational numbers are not enough to associate a certain rational number with each point on a coordinate line.
Let's build many real numbers R through endless decimals.
According to the “corner” division algorithm, any rational number can be represented as a finite or infinite periodic decimal fraction. When the denominator of the fraction p/q has no prime factors other than 2 and 5, i.e. q=2 m ×5 k, then the result will be the final decimal fraction p/q=a 0,a 1 a 2 …a n. Other fractions can only have infinite decimal expansions.
Knowing an infinite periodic decimal fraction, you can find the rational number of which it is a representation. But any finite decimal fraction can be represented as an infinite decimal fraction in one of the following ways:
a 0 ,a 1 a 2 …a n = a 0 ,a 1 a 2 …a n 000…=a 0 ,a 1 a 2 …(a n -1)999… (2)
For example, for an infinite decimal fraction X=0,(9) we have 10 X=9,(9). If we subtract the original number from 10x, we get 9 X=9 or 1=1,(0)=0,(9).
A one-to-one correspondence is established between the set of all rational numbers and the set of all infinite periodic decimal fractions if we identify the infinite decimal fraction with the number 9 in a period with the corresponding infinite decimal fraction with the number 0 in a period according to rule (2).
Let us agree to use such infinite periodic fractions that do not have the number 9 in the period. If an infinite periodic decimal fraction with the number 9 in the period arises in the process of reasoning, then we will replace it with an infinite decimal fraction with a zero in the period, i.e. instead of 1,999... we will take 2,000...
Definition of an irrational number. In addition to infinite decimal periodic fractions, there are non-periodic decimal fractions. For example, 0.1010010001... or 27.1234567891011... (natural numbers appear consecutively after the decimal point).
Consider an infinite decimal fraction of the form ±a 0, a 1 a 2 …a n … (3)
This fraction is determined by specifying a “+” or “–” sign, not a whole negative number a 0 and sequences of decimal places a 1 ,a 2 ,…,a n ,…(the set of decimal places consists of ten numbers: 0, 1, 2,…, 9).
Let us call any fraction of the form (3) real (real) number. If there is a “+” sign in front of the fraction (3), it is usually omitted and written a 0 , a 1 a 2 …a n … (4)
We will call a number of the form (4) non-negative real number, and in the case when at least one of the numbers a 0 , a 1 , a 2 , …, a n is different from zero, – positive real number. If the “-” sign is taken in expression (3), then this is a negative number.
Union of sets of rational and irrational numbers form the set of real numbers (QÈJ=R). If the infinite decimal fraction (3) is periodic, then it is a rational number, when the fraction is non-periodic, it is irrational.
Two non-negative real numbers a=a 0 ,a 1 a 2 …a n …, b=b 0 ,b 1 b 2 …b n …. called equal(they write a=b), If a n = b n at n=0,1,2… Number a is less than number b(they write a<b), if either a 0 or a 0 =b 0 and there is such a number m, What a k =b k (k=0,1,2,…m-1), A a m , i.e. a Û (a 0 Ú ($mÎN: a k =b k (k= ), a m ). The concept “ A>b».
To compare arbitrary real numbers, we introduce the concept “ modulus of number a» . Modulus of a real number a=±a 0 , a 1 a 2 …a n … such a non-negative real number is called, representable by the same infinite decimal fraction, but taken with the “+” sign, i.e. ½ A½= a 0 , a 1 a 2 …a n … and½ A½³0. If A - non-negative, b is a negative number, then consider a>b. If both numbers are negative ( a<0, b<0 ), then we will assume that: 1) a=b, if ½ A½ = ½ b½; 2) A , if ½ A½ > ½ b½.
Properties of the set R:
I. Properties of order:
1. For each pair of real numbers A And b there is one and only one relation: a=b, a
2. If a
3. If a , then there is a number c such that a< с .
II. Properties of addition and subtraction operations:
4. a+b=b+a(commutativity).
5. (a+b)+c=a+(b+c) (associativity).
6. a+0=a.
7. a+(-a)= 0.
8. from a Þ a+c
III. Properties of multiplication and division operations:
9. a×b=b×a .
10. (a×b)×c=a×(b×c).
11. a×1=a.
12. а×(1/а)=1 (а¹0).
13. (a+b)×c = ac + bc(distributivity).
14. if a and c>0, then а×с
IV. Archimedean property("cÎR)($nÎN) : (n>c).
Whatever the number cÎR, there is nÎN such that n>c.
V. Continuity property of real numbers. Let two non-empty sets AÌR and BÌR be such that any element A OA will be no more ( a£ b) of any element bОB. Then Dedekind's continuity principle asserts the existence of a number c such that for all AОА and bОB the following condition holds: a£c£ b:
("AÌR, BÌR):(" aÎA, bÎB ® a£b)($cÎR): (" aÎA, bÎB® a£c£b).
We will identify the set R with the set of points on the number line, and call the real numbers points.
Geometrically real numbers, like rational numbers, are represented by points on a line.
Let l is an arbitrary straight line, and O is some of its points (Fig. 58). Every positive real number α let us associate point A, lying to the right of O at a distance of α units of length.
If, for example, α = 2.1356..., then
2 < α
< 3
2,1 < α
< 2,2
2,13 < α
< 2,14
etc. Obviously, point A in this case must be on the straight line l to the right of the points corresponding to the numbers
2; 2,1; 2,13; ... ,
but to the left of the points corresponding to the numbers
3; 2,2; 2,14; ... .
It can be shown that these conditions define on the line l the only point A, which we consider as a geometric image of a real number α = 2,1356... .
Likewise, for every negative real number β let us associate point B lying to the left of O at a distance of | β | units of length. Finally, we associate the number “zero” with point O.
So, the number 1 will be depicted on a straight line l point A, located to the right of O at a distance of one unit of length (Fig. 59), the number - √2 - by point B, located to the left of O at a distance of √2 units of length, etc.
Let's show how on a straight line l using a compass and a ruler, you can find points corresponding to the real numbers √2, √3, √4, √5, etc. To do this, first of all, we will show how you can construct segments whose lengths are expressed by these numbers. Let AB be a segment taken as a unit of length (Fig. 60).
At point A, we construct a perpendicular to this segment and plot on it a segment AC equal to the segment AB. Then, applying the Pythagorean theorem to the rectangular triangle ABC, we get; BC = √AB 2 + AC 2 = √1+1 = √2
Therefore, the segment BC has length √2. Now let’s construct a perpendicular to the segment BC at point C and select point D on it so that the segment CD is equal to one unit of length AB. Then from right triangle Let's find BCD:
ВD = √ВC 2 + СD 2 = √2+1 = √3
Therefore, the segment BD has length √3. Continuing the described process further, we could obtain segments BE, BF, ..., the lengths of which are expressed by the numbers √4, √5, etc.
Now on a straight line l it is easy to find those points that serve as a geometric representation of the numbers √2, √3, √4, √5, etc.
By plotting, for example, the segment BC to the right of point O (Fig. 61), we obtain point C, which serves as a geometric image of the number √2. In the same way, putting the segment BD to the right of point O, we get point D", which is the geometric image of the number √3, etc.
However, one should not think that using a compass and ruler on the number line l one can find the point corresponding to any given real number. It has been proven, for example, that, having only a compass and a ruler at your disposal, it is impossible to construct a segment whose length is expressed by the number π = 3.14 ... . Therefore, on the number line l with the help of such constructions it is impossible to indicate the point corresponding to this number. Nevertheless, such a point exists.
So, for every real number α it is possible to associate some well-defined point with a straight line l . This point will be at a distance of | α | units of length and be to the right of O if α > 0, and to the left of O, if α < 0. Очевидно, что при этом двум неравным действительным числам будут соответствовать две various points direct l . In fact, let the number α point A corresponds, and the number β - point B. Then, if α > β , then A will be to the right of B (Fig. 62, a); if α < β , then A will lie to the left of B (Fig. 62, b).
Speaking in § 37 about the geometric image of rational numbers, we posed the question: can any point on a line be considered as a geometric image of some rational numbers? We could not answer this question then; Now we can answer it quite definitely. There are points on the line that serve as a geometric representation of irrational numbers (for example, √2). Therefore, not every point on a line represents a rational number. But in this case, another question arises: can any point on the number line be considered as a geometric image of some valid numbers? This issue has already been resolved positively.
Indeed, let A be an arbitrary point on the line l , lying to the right of O (Fig. 63).
The length of the segment OA is expressed by some positive real number α (see § 41). Therefore, point A is a geometric image of the number α . It is similarly established that each point B lying to the left of O can be considered as a geometric image of a negative real number - β , Where β - length of segment VO. Finally, point O serves as a geometric representation of the number zero. It is clear that two different points on a line l cannot be a geometric image of the same real number.
For the reasons stated above, a straight line on which a certain point O is indicated as the “initial” point (for a given unit of length) is called number line.
Conclusion. The set of all real numbers and the set of all points on the number line are in a one-to-one correspondence.
This means that each real number corresponds to one, well-defined point on the number line, and, conversely, to each point on the number line, with such a correspondence, there corresponds one, well-defined real number.
Complex numbers
Basic Concepts
The initial data on the number dates back to the Stone Age - Paleomelithic era. These are “one”, “few” and “many”. They were recorded in the form of notches, knots, etc. The development of labor processes and the emergence of property forced man to invent numbers and their names. Natural numbers appeared first N, obtained by counting objects. Then, along with the need to count, people had a need to measure lengths, areas, volumes, time and other quantities, where they had to take into account parts of the measure used. This is how fractions came into being. The formal substantiation of the concepts of fractional and negative numbers was carried out in the 19th century. Set of integers Z– these are natural numbers, natural numbers with a minus sign and zero. Whole and fractional numbers formed a set of rational numbers Q, but it also turned out to be insufficient for the study of continuously changing variables. Genesis again showed the imperfection of mathematics: the impossibility of solving an equation of the form X 2 = 3, which is why irrational numbers appeared I. Union of the set of rational numbers Q and irrational numbers I– set of real (or real) numbers R. As a result, the number line was filled: each real number corresponded to a point on it. But on many R there is no way to solve an equation of the form X 2 = – A 2. Consequently, the need arose again to expand the concept of number. This is how complex numbers appeared in 1545. Their creator J. Cardano called them “purely negative.” The name “imaginary” was introduced in 1637 by the Frenchman R. Descartes, in 1777 Euler proposed using the first letter French number i to denote the imaginary unit. This symbol came into general use thanks to K. Gauss.
During the 17th and 18th centuries, the discussion of the arithmetic nature of imaginaries and their geometric interpretation continued. The Dane G. Wessel, the Frenchman J. Argan and the German K. Gauss independently proposed to represent a complex number as a point on coordinate plane. Later it turned out that it is even more convenient to represent a number not by the point itself, but by a vector going to this point from the origin.
Only towards the end of the 18th and beginning of the 19th centuries did complex numbers take their rightful place in mathematical analysis. Their first use is in theory differential equations and in the theory of hydrodynamics.
Definition 1.Complex number is called an expression of the form , where x And y are real numbers, and i– imaginary unit, .
Two complex numbers and equal if and only if , .
If , then the number is called purely imaginary; if , then the number is a real number, this means that the set R WITH, Where WITH– many complex numbers.
Conjugate to a complex number is called a complex number.
Geometric image complex numbers.
Any complex number can be represented by a point M(x, y) plane Oxy. A pair of real numbers also denotes the coordinates of the radius vector 
, i.e. between the set of vectors on the plane and the set of complex numbers, one can establish a one-to-one correspondence: .
Definition 2.Real part X.
Designation: x=Re z(from Latin Realis).
Definition 3.Imaginary part complex number is a real number y.
Designation: y= Im z(from Latin Imaginarius).
Re z is deposited on the axis ( Oh), Im z is deposited on the axis ( Oh), then the vector corresponding to the complex number is the radius vector of the point M(x, y), (or M(Re z, Im z)) (Fig. 1).
Definition 4. A plane whose points are associated with a set of complex numbers is called complex plane. The abscissa axis is called real axis, since it contains real numbers. The ordinate axis is called imaginary axis, it contains purely imaginary complex numbers. The set of complex numbers is denoted WITH.
Definition 5.Module complex number z = (x, y) is called the length of the vector: , i.e. 
.
Definition 6.Argument complex number is the angle between the positive direction of the axis ( Oh) and vector: 
.
REAL NUMBERS II
§ 44 Geometric representation of real numbers
Geometrically real numbers, like rational numbers, are represented by points on a line.
Let l is an arbitrary straight line, and O is some of its points (Fig. 58). Every positive real number α let us associate point A, lying to the right of O at a distance of α units of length.
If, for example, α = 2.1356..., then
2 < α
< 3
2,1 < α
< 2,2
2,13 < α
< 2,14
etc. Obviously, point A in this case must be on the straight line l to the right of the points corresponding to the numbers
2; 2,1; 2,13; ... ,
but to the left of the points corresponding to the numbers
3; 2,2; 2,14; ... .
It can be shown that these conditions define on the line l the only point A, which we consider as a geometric image of a real number α = 2,1356... .
Likewise, for every negative real number β let us associate point B lying to the left of O at a distance of | β | units of length. Finally, we associate the number “zero” with point O.
So, the number 1 will be depicted on a straight line l point A, located to the right of O at a distance of one unit of length (Fig. 59), the number - √2 - by point B, located to the left of O at a distance of √2 units of length, etc.
Let's show how on a straight line l using a compass and a ruler, you can find points corresponding to the real numbers √2, √3, √4, √5, etc. To do this, first of all, we will show how you can construct segments whose lengths are expressed by these numbers. Let AB be a segment taken as a unit of length (Fig. 60).
At point A, we construct a perpendicular to this segment and plot on it a segment AC equal to the segment AB. Then, applying the Pythagorean theorem to right triangle ABC, we get; BC = √AB 2 + AC 2 = √1+1 = √2
Therefore, the segment BC has length √2. Now let’s construct a perpendicular to the segment BC at point C and select point D on it so that the segment CD is equal to one unit of length AB. Then from the right triangle BCD we find:
ВD = √ВC 2 + СD 2 = √2+1 = √3
Therefore, the segment BD has length √3. Continuing the described process further, we could obtain segments BE, BF, ..., the lengths of which are expressed by the numbers √4, √5, etc.
Now on a straight line l it is easy to find those points that serve as a geometric representation of the numbers √2, √3, √4, √5, etc.
By laying off, for example, the segment BC to the right of point O (Fig. 61), we obtain point C, which serves as a geometric image of the number √2. In the same way, putting the segment BD to the right of point O, we get point D", which is the geometric image of the number √3, etc.
However, one should not think that using a compass and ruler on the number line l one can find the point corresponding to any given real number. It has been proven, for example, that, having only a compass and a ruler at your disposal, it is impossible to construct a segment whose length is expressed by the number π = 3.14 ... . Therefore, on the number line l with the help of such constructions it is impossible to indicate the point corresponding to this number. Nevertheless, such a point exists.
So, for every real number α it is possible to associate some well-defined point with a straight line l . This point will be at a distance of | α | units of length and be to the right of O if α > 0, and to the left of O, if α < 0. Очевидно, что при этом двум неравным действительным числам будут соответствовать две различные точки прямой l . In fact, let the number α point A corresponds, and the number β - point B. Then, if α > β , then A will be to the right of B (Fig. 62, a); if α < β , then A will lie to the left of B (Fig. 62, b).
Speaking in § 37 about the geometric image of rational numbers, we posed the question: can any point on a line be considered as a geometric image of some rational numbers? We could not answer this question then; Now we can answer it quite definitely. There are points on the line that serve as a geometric representation of irrational numbers (for example, √2). Therefore, not every point on a line represents a rational number. But in this case, another question arises: can any point on the number line be considered as a geometric image of some valid numbers? This issue has already been resolved positively.
Indeed, let A be an arbitrary point on the line l , lying to the right of O (Fig. 63).
The length of the segment OA is expressed by some positive real number α (see § 41). Therefore, point A is a geometric image of the number α . It is similarly established that each point B lying to the left of O can be considered as a geometric image of a negative real number - β , Where β - length of segment VO. Finally, point O serves as a geometric representation of the number zero. It is clear that two different points on a line l cannot be a geometric image of the same real number.
For the reasons stated above, a straight line on which a certain point O is indicated as the “initial” point (for a given unit of length) is called number line.
Conclusion. The set of all real numbers and the set of all points on the number line are in a one-to-one correspondence.
This means that each real number corresponds to one, well-defined point on the number line, and, conversely, to each point on the number line, with such a correspondence, there corresponds one, well-defined real number.
Exercises
320. Find out which of the two points is to the left and which is to the right on the number line, if these points correspond to numbers:
a) 1.454545... and 1.455454...; c) 0 and - 1.56673...;
b) - 12.0003... and - 12.0002...; d) 13.24... and 13.00....
321. Find out which of the two points is located on the number line further from the starting point O, if these points correspond to the numbers:
a) 5.2397... and 4.4996...; .. c) -0.3567... and 0.3557... .
d) - 15.0001 and - 15.1000...;
322. In this section it was shown that to construct a segment of length √ n using a compass and a ruler, you can proceed as follows: first construct a segment of length √2, then a segment of length √3, etc., until we reach a segment of length √ n . But for every fixed n > 3 this process can be accelerated. How, for example, would you begin to construct a segment of length √10?
323*. How to use a compass and ruler to find the point on the number line corresponding to the number 1 / α , if the position of the point corresponding to the number α , is it known?
An expressive geometric representation of the system of rational numbers can be obtained as follows.
On a certain straight line, the “numerical axis,” we mark the segment from O to 1 (Fig. 8). This establishes the length of a unit segment, which, generally speaking, can be chosen arbitrarily. Positive and negative integers are then represented by a set of equally spaced points on the number axis, namely positive numbers are marked to the right, and negative numbers to the left of point 0. To depict numbers with a denominator n, divide each of the resulting segments of unit length by n equal parts; The division points will represent fractions with denominator n. If we do this for the values of n corresponding to all natural numbers, then each rational number will be depicted by some point on the number axis. We will agree to call these points “rational”; In general, we will use the terms “rational number” and “rational point” as synonyms.
In Chapter I, § 1, the inequality relation A was defined for any pair of rational points, then it is natural to try to generalize the arithmetic inequality relation in such a way as to preserve this geometric order for the points under consideration. This is possible if we accept the following definition: they say that a rational number A less than a rational number B (A is greater than the number A (B>A), if difference VA positive. This implies (for A between A and B are those that are both >A and a segment (or segment) and is denoted by [A, B] (and the set of intermediate points alone is interval(or in between), denoted (A, B)).
The distance of an arbitrary point A from the origin 0, considered as a positive number, is called absolute value A and is indicated by the symbol
Concept " absolute value" is defined as follows: if A≥0, then |A| = A; if A
|A + B|≤|A| + |B|,
which is true regardless of the signs of A and B.
A fact of fundamental importance is expressed by the following sentence: rational points are located densely everywhere on the number line. The meaning of this statement is that every interval, no matter how small, contains rational points. To verify the validity of the stated statement, it is enough to take the number n so large that the interval will be less than the given interval (A, B); then at least one of the view points will be inside this interval. So, there is no such interval on the number line (even the smallest one imaginable) within which there would be no rational points. This leads to a further corollary: every interval contains an infinite set of rational points. Indeed, if a certain interval contained only final number rational points, then inside the interval formed by two neighboring such points, there would no longer be rational points, and this contradicts what has just been proven.
