The following forms of complex numbers exist: algebraic(x+iy), trigonometric(r(cos+isin 
)),
indicative(re i 
).
Any complex number z=x+iy can be depicted on the XOU plane as a point A(x,y).
The plane on which complex numbers are depicted is called the plane of the complex variable z (we put the symbol z on the plane).
The OX axis is the real axis, i.e. it contains real numbers. OU is an imaginary axis with imaginary numbers.
x+iy- algebraic form of writing a complex number.
Let us derive the trigonometric form of writing a complex number.
We substitute the obtained values into the initial form: , i.e.
r(cos
+isin
)
- trigonometric form of writing a complex number.
The exponential form of writing a complex number follows from Euler’s formula: 
,Then 
z= re i 
- exponential form of writing a complex number.
Operations on complex numbers.
1. addition. z 1 +z 2 =(x1+iy1)+ (x2+iy2)=(x1+x2)+i(y1+y2);
2 . subtraction. z 1 -z 2 =(x1+iy1)- (x2+iy2)=(x1-x2)+i(y1-y2);
3. multiplication. z 1 z 2 =(x1+iy1)*(x2+iy2)=x1x2+i(x1y2+x2y1+iy1y2)=(x1x2-y1y2)+i(x1y2+x2y1);
4
. division. z 1 /z 2 =(x1+iy1)/(x2+iy2)=[(x1+iy1)*(x2-iy2)]/[ (x2+iy2)*(x2-iy2)]= 
Two complex numbers that differ only in the sign of the imaginary unit, i.e. z=x+iy (z=x-iy) are called conjugate.
Work.
z1=r(cos 
+isin 
); z2=r(cos 
+isin 
).
That product z1*z2 of complex numbers is found: , i.e. the modulus of the product is equal to the product of the moduli, and the argument of the product is equal to the sum of the arguments of the factors.
;
;
Private.
If complex numbers are given in trigonometric form.
If complex numbers are given in exponential form.
Exponentiation.
1. Complex number given in algebraic form.
z=x+iy, then z n is found by Newton's binomial formula:
- the number of combinations of n elements of m (the number of ways in which n elements from m can be taken).
; n!=1*2*…*n; 0!=1; 
.
Apply for complex numbers.
In the resulting expression, you need to replace the powers i with their values:
i 0 =1 Hence, in the general case we obtain: i 4k =1
i 1 =i i 4k+1 =i
i 2 =-1 i 4k+2 =-1
i 3 =-i i 4k+3 =-i
Example.
i 31 = i 28 i 3 =-i
i 1063 = i 1062 i=i
2. trigonometric form.
z=r(cos 
+isin 
), That
- Moivre's formula.
Here n can be either “+” or “-” (integer).
3. If a complex number is given in indicative form:
Root extraction.
Consider the equation: 
.
Its solution will be the nth root of the complex number z: 
.
The nth root of a complex number z has exactly n solutions (values). The nth root of a real number has only one solution. In complex ones there are n solutions.
If a complex number is given in trigonometric form:
z=r(cos 
+isin 
), then the nth root of z is found by the formula:
, where k=0.1…n-1.
Rows. Number series.
Let the variable a take sequentially the values a 1, a 2, a 3,…, a n. Such a renumbered set of numbers is called a sequence. It is endless.
A number series is the expression a 1 + a 2 + a 3 +…+a n +…= 
. The numbers a 1, a 2, a 3,..., and n are members of the series.
For example.
and 1 is the first term of the series.
and n is the nth or common term of the series.
A series is considered given if the nth (common term of the series) is known.
The number series has an infinite number of terms.
Numerators – arithmetic progression (1,3,5,7…).
The nth term is found by the formula a n =a 1 +d(n-1); d=a n -a n-1 .
Denominator – geometric progression. b n =b 1 q n-1 ; 
.
Consider the sum of the first n terms of the series and denote it Sn.
Sn=a1+a2+…+a n.
Sn is the nth partial sum of the series.
Consider the limit: 
S is the sum of the series.
Row convergent , if this limit is finite (a finite limit S exists).
Row divergent , if this limit is infinite.
In the future, our task is to establish which row.
One of the simplest but most common series is the geometric progression.
, C=const.
Geometric progression isconvergent
near, If 
, and divergent if 
.
Also found harmonic series(row 
). This row divergent
.
TICKET 1
Rational numbers – numbers written in the form p/q, where q is a natural number. number, and p is an integer.
Two numbers a=p1/q1 and b=p2/q2 are called equal if p1q2=p2q1, and p2q1 and a>b if p1q2
TICKET 2
Complex numbers. Complex numbers
An algebraic equation is an equation of the form: P n ( x) = 0, where P n ( x) - polynomial n- oh degree. A couple of real numbers x And at Let's call it ordered if it is indicated which of them is considered first and which is considered second. Ordered pair notation: ( x, y). A complex number is an arbitrary ordered pair of real numbers. z = (x, y)-complex number.
x-real part z, y-imaginary part z. If x= 0 and y= 0, then z= 0. Consider z 1 = (x 1 , y 1) and z 2 = (x 2 , y 2).
Definition 1. z 1 = z 2 if x 1 = x 2 and y 1 = y 2.
Concepts > and< для комплексных чисел не вводятся.
Geometric representation and trigonometric form of complex numbers.
M( x, y) « z = x + iy.
½ OM½ = r =½ z½ = .(picture)
r is called the modulus of a complex number z.
j is called the argument of a complex number z. It is determined with an accuracy of ± 2p n.
X= rcosj, y= rsinj.
z= x+ iy= r(cosj + i sinj) is the trigonometric form of complex numbers.
Statement 3.
= (cos + i sin),
= (cos + i sin), then
= (cos( + ) + i sin( + )),
= (cos( - )+ i sin( - )) at ¹0.
Statement 4.
If z=r(cosj+ i sinj), then "natural n:
= (cos nj + i sin nj),
TICKET 3
Let X- a numerical set containing at least one number (non-empty set).
xÎ X- x contained in X. ; xÏ X- x do not belong X.
Definition: A bunch of X is called bounded above (below) if there is a number M(m) such that for any x Î X inequality holds x £ M (x ³ m), while the number M called the upper (lower) bound of the set X. A bunch of X is said to be bounded above if $ M, " x Î X: x £ M. Definition unlimited set from above. A bunch of X is said to be unbounded from above if " M $ x Î X: x> M. Definition a bunch of X is called bounded if it is bounded above and below, that is $ M, m such that " x Î X: m £ x £ M. Equivalent definition of ogre mn-va: Set X is called bounded if $ A > 0, " x Î X: ½ x½£ A. Definition: The smallest upper bound of a set bounded above X is called its supremum, and is denoted Sup X
(supremum). =Sup X. Similarly, one can determine the exact
bottom edge. Equivalent definition exact upper bound:
The number is called the supremum of the set X, If: 1) " x Î X: X£ (this condition shows that is one of the upper bounds). 2) " < $ x Î X: X> (this condition shows that -
the smallest of the upper faces).
Sup X= :
1. " xÎ X: x £ .
2. " < $ xÎ X: x> .
inf X(infimum) is the exact lower bound. Let us pose the question: does every bounded set have exact edges?
Example: X= {x: x>0) does not have a smallest number.
Theorem on the existence of an exact top (bottom) face. Any non-empty upper (lower) limit xÎR has an exact upper (lower) face.
Theorem on the separability of numerical plurals:▀▀▄
TICKET 4
If each natural number n (n=1,2,3..) is assigned a corresponding number Xn, then they say that it is defined and given subsequence x1, x2..., write (Xn), (Xn). Example: Xn=(-1)^n: -1,1,-1,1,...The name of the limit. from above (from below) if the set of points x=x1,x2,…xn lying on the numerical axis is limited from above (from below), i.e. $C:Xn£C" Sequence limit: number a is called the limit of the sequence if for any ε>0 $ : N (N=N/(ε)). "n>N the inequality |Xn-a|<ε. Т.е. – ε
at n>N.
Uniqueness of the limit bounded and convergent sequence
Property1: A convergent sequence has only one limit.
Proof: by contradiction let A And b limits of a convergent sequence (x n), and a is not equal to b. consider infinitesimal sequences (α n )=(x n -a) and (β n )=(x n -b). Because all elements b.m. sequences (α n -β n ) have the same value b-a, then by the property of b.m. sequences b-a=0 i.e. b=a and we have arrived at a contradiction.
Property2: A convergent sequence is bounded.
Proof: Let a be the limit of a convergent sequence (x n), then α n =x n -a is an element of the b.m. sequences. Let's take any ε>0 and use it to find N ε: / x n -a/< ε при n>N ε . Let us denote by b the largest of the numbers ε+/а/, /х1/, /х2/,…,/х N ε-1 /,х N ε. It is obvious that / x n /
Note: the bounded sequence may not be convergent.
TICKET 6
The sequence a n is called infinitesimal, which means that the limit of this sequence after is 0.
a n – infinitesimal Û lim(n ® + ¥)a n =0 that is, for any ε>0 there exists N such that for any n>N |a n |<ε
Theorem. The sum of an infinitesimal is an infinitesimal.
a n b n ®infinitesimal Þ a n +b n – infinitesimal.
Proof.
a n - infinitesimal Û "ε>0 $ N 1:" n >N 1 Þ |a n |<ε
b n - infinitesimal Û "ε>0 $ N 2:" n >N 2 Þ |b n |<ε
Let us set N=max(N 1 ,N 2 ), then for any n>N Þ both inequalities are simultaneously satisfied:
|a n |<ε |a n +b n |£|a n |+|b n |<ε+ε=2ε=ε 1 "n>N
Let us set "ε 1 >0, set ε=ε 1 /2. Then for any ε 1 >0 $N=maxN 1 N 2: " n>N Þ |a n +b n |<ε 1 Û lim(n ® ¥)(a n +b n)=0, то
is a n + b n – infinitesimal.
Theorem The product of an infinitesimal is an infinitesimal.
a n ,b n – infinitesimal Þ a n b n – infinitesimal.
Evidence:
Let's set "ε 1 >0, put ε=Öε 1, since a n and b n are infinitesimal for this ε>0, then there is N 1: " n>N Þ |a n |<ε
$N 2: " n>N 2 Þ |b n |<ε
Let's take N=max (N 1 ;N 2 ), then "n>N = |a n |<ε
|a n b n |=|a n ||b n |<ε 2 =ε 1
" ε 1 >0 $N:"n>N |a n b n |<ε 2 =ε 1
lim a n b n =0 Û a n b n – infinitesimal, which is what needed to be proved.
Theorem The product of a bounded sequence and an infinitesimal sequence is an infinitesimal sequence
and n is a bounded sequence
a n – infinitesimal sequence Þ a n a n – infinitesimal sequence.
Proof: Since a n is bounded Û $С>0: "nО NÞ |a n |£C
Let's set "ε 1 >0; put ε=ε 1 /C; since a n is infinitesimal, then ε>0 $N:"n>NÞ |a n |<εÞ |a n a n |=|a n ||a n | "ε 1 >0 $N: "n>N Þ |a n a n |=Cε=ε 1 Þ lim(n ® ¥) a n a n =0Û a n a n – infinitesimal The sequence is called BBP(in sequence) if they write. Obviously, the BBP is not limited. The opposite statement is generally false (example). If for big ones n members, then write this means that as soon as. The meaning of the entry is determined similarly Infinitely large sequences a n =2 n ;
b n =(-1) n 2 n ;c n =-2 n Definition(infinitely large sequences) 1) lim(n ® ¥)a n =+¥, if "ε>0$N:"n>N Þ a n >ε where ε is arbitrarily small. 2) lim(n ® ¥)a n =-¥, if "ε>0 $N:"n>N Þ a n<-ε 3) lim(n ® ¥)a n =¥ Û "ε>0 $N:"n>N Þ |a n |>ε TICKET 7 Theorem “On the convergence of monotone. last" Any monotonic sequence is convergent, i.e. has limits. Document Let the sequence (xn) be monotonically increasing. and is limited from above. X – the entire set of numbers that accepts the element of this sequence according to the convention. The theorems are limited in number, therefore, according to Theorem it has a finite exact upper limit. face supX xn®supX (we denote supX by x*). Because x* exact top. face, then xn£x* " n. " e >0 the nerve is out $ xm (let m be n with a lid): xm>x*-e with " n>m => from the indicated 2 inequalities we obtain the second inequality x*-e£xn£x*+e for n>m is equivalent to ½xn-x*1 TICKET 8 Exponent or number e R-Roman number sequence with a common term xn=(1+1/n)^n (to the power n)(1) . It turns out that the sequence (1) increases monotonically, is bounded from above and is convergent; the limit of this sequence is called an exponential and is denoted by the symbol e»2.7128... Number e TICKET 9 The principle of nested segments Let a number line be given a sequence of segments ,,...,,... Moreover, these segments satisfy the following. condition: 1) each subsequent one is nested in the previous one, i.e. М, "n=1,2,…; 2) The lengths of the segments ®0 as n increases, i.e. lim(n®¥)(bn-an)=0. Sequences with the specified strings are called nested. Theorem Any sequence of nested segments contains a single point c that belongs to all segments of the sequence simultaneously, with the common point of all segments to which they are contracted. Document(an) - sequence of left ends of segments of phenomena. monotonically non-decreasing and bounded above by the number b1. (bn) - the sequence of the right ends is not monotonically increasing, therefore these sequences of phenomena. convergent, i.e. there are numbers c1=lim(n®¥)an and c2=lim(n®¥)bn => c1=c2 => c - their common value. Indeed, it has the limit lim(n®¥)(bn-an)= lim(n®¥)(bn)- lim(n®¥)(an) due to condition 2) o= lim(n®¥)(bn- an)=с2-с1=> с1=с2=с It is clear that t.c is common for all segments, since "n an£c£bn. Now we will prove that it is one. Let us assume that $ is another c' to which all segments are contracted. If we take any non-intersecting segments c and c', then on one side the entire “tail” of the sequences (an), (bn) should be located in the vicinity of point c'' (since an and bn converge to c and c' simultaneously). The contradiction is true. TICKET 10 Bolzano-Weierstrass theorem
From any cut. Afterwards you can select the gathering. Subsyllabus 1. Since the sequence is limited, then $ m and M, such that " m£xn£M, " n. D1= – segment in which all t-ki sequences lie. Let's divide it in half. At least one of the halves will contain an infinite number of t-k sequences. D2 is the half where an infinite number of t-k sequences lie. We divide it in half. At least in one of the halves neg. D2 has an infinite number of sequences. This half is D3. Divide segment D3... etc. we obtain a sequence of nested segments, the lengths of which tend to 0. According to the rule about nested segments, $ units. t-ka S, cat. belonging all segments D1, any t-tu Dn1. In segment D2 I choose point xn2, so that n2>n1. In segment D3... etc. As a result, the last word is xnkÎDk. TICKET 11 TICKET 12 fundamental In conclusion, we consider the question of the criterion for the convergence of a numerical sequence. Let i.e.: We got the following statement: If the sequence converges, the condition is satisfied Cauchy: A number sequence that satisfies the Cauchy condition is called fundamental. It can be proven that the converse is also true. Thus, we have a criterion (necessary and sufficient condition) for the convergence of the sequence. Cauchy criterion. In order for a sequence to have a limit, it is necessary and sufficient that it be fundamental. The second meaning of the Cauchy criterion. Sequence members and where n And m– any approaching without limit at . TICKET 13 One-sided limits. Definition 13.11. Number A called the limit of the function y = f(x) at X, striving for x 0 left (right), if such that | f(x)-A|<ε при x 0 – x< δ
(x - x 0< δ
). Designations: Theorem 13.1 (second definition of limit). Function y=f(x) has at X, striving for X 0, limit equal to A, if and only if both of its one-sided limits at this point exist and are equal A. Proof. 1) If , then and for x 0 – x< δ, и для x - x 0< δ
|f(x) - A|<ε, то есть 1) If , then there is δ 1: | f(x) - A| < ε при x 0 – x< δ 1 и δ 2: |f(x) - A| < ε при x - x 0<
δ2. Choosing the smaller one from the numbers δ 1 and δ 2 and taking it as δ, we obtain that for | x - x 0| < δ |f(x) - A| < ε, то есть . Теорема доказана. Comment. Since the equivalence of the requirements contained in the definition of limit 13.7 and the conditions for the existence and equality of one-sided limits has been proven, this condition can be considered the second definition of the limit. Definition 4 (according to Heine) Number A is called the limit of a function if any BBP of argument values, the sequence of corresponding function values converges to A. Definition 4 (according to Cauchy). Number A called if . It is proved that these definitions are equivalent. TICKET 14 and 15 Properties of the function limit at a point 1) If there is a limit, then it is the only one 2) If in tka x0 the limit of the function f(x) lim(x®x0)f(x)=A lim(x®x0)g(x)£B=> then in this case $ is the limit of the sum, difference, product and quotient. Separation of these 2 functions. a) lim(x®x0)(f(x)±g(x))=A±B b) lim(x®x0)(f(x)*g(x))=A*B c) lim(x®x0)(f(x):g(x))=A/B d) lim(x®x0)C=C e) lim(x®x0)C*f(x)=C*A Theorem 3. If ( resp A ) then $ the neighborhood in which the inequality holds >B (resp Assuming in Theorem 3 B=0, we get: if ( resp), then $ , at all points, which will be >0 (resp<0),
those. the function preserves the sign of its limit. Theorem 4(on passage to the limit in inequality). If in some neighborhood of a point (except perhaps this point itself) the condition is satisfied and these functions have limits at the point, then . In the language and. Let's introduce the function. It is clear that in the vicinity of t. . Then, by the theorem on the conservation of a function, we have the value of its limit, but Theorem 5.(on the limit of an intermediate function). (1) If TICKET 16 Definition 14.1. Function y=α(x) is called infinitesimal at x→x 0, If Properties of infinitesimals. 1. The sum of two infinitesimals is infinitesimal. Proof. If α(x) And β(x) – infinitesimal at x→x 0, then there exist δ 1 and δ 2 such that | α(x)|<ε/2 и |β(x)|<ε/2 для выбранного значения ε. Тогда |α(x)+β(x)|≤|α(x)|+|β(x)|<ε, то есть |(α(x)+β(x))-0|<ε. Следовательно, Comment. It follows that the sum of any finite number of infinitesimals is infinitesimal. 2. If α( X) – infinitesimal at x→x 0, A f(x) – a function bounded in a certain neighborhood x 0, That α(x)f(x) – infinitesimal at x→x 0. Proof. Let's choose a number M such that | f(x)| Corollary 1. The product of an infinitesimal by a finite number is an infinitesimal. Corollary 2. The product of two or more infinitesimals is an infinitesimal. Corollary 3. A linear combination of infinitesimals is infinitesimal. 3. (Third definition of limit). If , then a necessary and sufficient condition for this is that the function f(x) can be represented in the form f(x)=A+α(x), Where α(x) – infinitesimal at x→x 0. Proof. 1)
Let Then | f(x)-A|<ε при x→x 0, that is α(x)=f(x)-A– infinitesimal at x→x 0 . Hence , f(x)=A+α(x). 2) Let f(x)=A+α(x). Then Comment. Thus, another definition of the limit is obtained, equivalent to the previous two. Infinitely large functions. Definition 15.1. The function f(x) is said to be infinitely large for x x 0 if For infinitely large, you can introduce the same classification system as for infinitely small, namely: 1. Infinitely large f(x) and g(x) are considered quantities of the same order if 2. If , then f(x) is considered infinitely large of a higher order than g(x). 3. An infinitely large f(x) is called a quantity of kth order relative to an infinitely large g(x) if . Comment. Note that a x is infinitely large (for a>1 and x) of a higher order than x k for any k, and log a x is infinitely large of a lower order than any power of x k. Theorem 15.1. If α(x) is infinitely small as x→x 0, then 1/α(x) is infinitely large as x→x 0. Proof. Let us prove that for |x - x 0 |< δ. Для этого достаточно выбрать в качестве ε 1/M. Тогда при |x - x 0 | < δ |α(x)|<1/M, следовательно, |1/α(x)|>M. This means, that is, 1/α(x) is infinitely large as x→x 0. TICKET 17 Theorem 14.7 (first remarkable limit). . Proof. Consider a circle of unit radius with a center at the origin and assume that angle AOB is equal to x (radians). Let's compare the areas of triangle AOB, sector AOB and triangle AOC, where straight line OS is tangent to the circle passing through the point (1;0). It's obvious that . Using the corresponding geometric formulas for the areas of figures, we obtain from this that From a huge variety of all kinds sets Of particular interest are the so-called number sets, that is, sets whose elements are numbers. It is clear that to work comfortably with them you need to be able to write them down. We will begin this article with the notation and principles of writing numerical sets. Next, let’s look at how numerical sets are depicted on a coordinate line. Page navigation. Let's start with the accepted notation. As you know, capital letters of the Latin alphabet are used to denote sets. Numerical sets, as a special case of sets, are also designated. For example, we can talk about number sets A, H, W, etc. The sets of natural, integer, rational, real, complex numbers, etc. are of particular importance; their own notations have been adopted for them: From here it is clear that you should not denote a set consisting, for example, of two numbers 5 and −7 as Q, this designation will be misleading, since the letter Q usually denotes the set of all rational numbers. To denote the specified numerical set, it is better to use some other “neutral” letter, for example, A. Since we are talking about notation, let us also recall here about the notation of an empty set, that is, a set that does not contain elements. It is denoted by the sign ∅. Let us also recall the designation of whether an element belongs or does not belong to a set. To do this, use the signs ∈ - belongs and ∉ - does not belong. For example, the notation 5∈N means that the number 5 belongs to the set of natural numbers, and 5.7∉Z - the decimal fraction 5.7 does not belong to the set of integers. And let us also recall the notation adopted for including one set into another. It is clear that all elements of the set N are included in the set Z, thus the number set N is included in Z, this is denoted as N⊂Z. You can also use the notation Z⊃N, which means that the set of all integers Z includes the set N. The relations not included and not included are indicated by ⊄ and , respectively. Non-strict inclusion signs of the form ⊆ and ⊇ are also used, meaning included or coincides and includes or coincides, respectively. We've talked about notation, let's move on to the description of numerical sets. In this case, we will only touch on the main cases that are most often used in practice. Let's start with numerical sets containing a finite and small number of elements. It is convenient to describe numerical sets consisting of a finite number of elements by listing all their elements. All number elements are written separated by commas and enclosed in , which is consistent with the general rules for describing sets. For example, a set consisting of three numbers 0, −0.25 and 4/7 can be described as (0, −0.25, 4/7). Sometimes, when the number of elements of a numerical set is quite large, but the elements obey a certain pattern, an ellipsis is used for description. For example, the set of all odd numbers from 3 to 99 inclusive can be written as (3, 5, 7, ..., 99). So we smoothly approached the description of numerical sets, the number of elements of which is infinite. Sometimes they can be described using all the same ellipses. For example, let’s describe the set of all natural numbers: N=(1, 2. 3, …) . They also use the description of numerical sets by indicating the properties of its elements. In this case, the notation (x| properties) is used. For example, the notation (n| 8·n+3, n∈N) specifies the set of natural numbers that, when divided by 8, leave a remainder of 3. This same set can be described as (11,19, 27, ...). In special cases, numerical sets with an infinite number of elements are the known sets N, Z, R, etc. or number intervals. Basically, numerical sets are represented as Union their constituent individual numerical intervals and numerical sets with a finite number of elements (which we talked about just above). Let's show an example. Let the number set consist of the numbers −10, −9, −8.56, 0, all the numbers of the segment [−5, −1,3] and the numbers of the open number line (7, +∞). Due to the definition of a union of sets, the specified numerical set can be written as {−10, −9, −8,56}∪[−5, −1,3]∪{0}∪(7, +∞)
. This notation actually means a set containing all the elements of the sets (−10, −9, −8.56, 0), [−5, −1.3] and (7, +∞). Similarly, by combining different numerical intervals and sets of individual numbers, any numerical set (consisting of real numbers) can be described. Here it becomes clear why such types of numerical intervals as interval, half-interval, segment, open numerical ray and numerical ray were introduced: all of them, coupled with notations for sets of individual numbers, make it possible to describe any numerical sets through their union. Please note that when writing a number set, its constituent numbers and numerical intervals are ordered in ascending order. This is not a necessary but desirable condition, since an ordered numerical set is easier to imagine and depict on a coordinate line. Also note that such records do not use numeric intervals with common elements, since such records can be replaced by combining numeric intervals without common elements. For example, the union of numerical sets with common elements [−10, 0] and (−5, 3) is the half-interval [−10, 3) . The same applies to the union of numerical intervals with the same boundary numbers, for example, the union (3, 5]∪(5, 7] is a set (3, 7] , we will dwell on this separately when we learn to find the intersection and union of numerical sets In practice, it is convenient to use geometric images of numerical sets - their images on. For example, when solving inequalities, in which it is necessary to take into account ODZ, it is necessary to depict numerical sets in order to find their intersection and/or union. So it will be useful to have a good understanding of all the nuances of depicting numerical sets on a coordinate line. It is known that there is a one-to-one correspondence between the points of the coordinate line and the real numbers, which means that the coordinate line itself is a geometric model of the set of all real numbers R. Thus, to depict the set of all real numbers, you need to draw a coordinate line with shading along its entire length: And often they don’t even indicate the origin and the unit segment: Now let's talk about the image of numerical sets, which represent a certain finite number of individual numbers. For example, let's depict the number set (−2, −0.5, 1.2) . The geometric image of this set, consisting of three numbers −2, −0.5 and 1.2, will be three points of the coordinate line with the corresponding coordinates: Note that usually for practical purposes there is no need to carry out the drawing exactly. Often a schematic drawing is sufficient, which implies that it is not necessary to maintain the scale; in this case, it is only important to maintain the relative position of the points relative to each other: any point with a smaller coordinate must be to the left of a point with a larger coordinate. The previous drawing will schematically look like this: Separately, from all kinds of numerical sets, numerical intervals (intervals, half-intervals, rays, etc.) are distinguished, which represent their geometric images; we examined them in detail in the section. We won't repeat ourselves here. And it remains only to dwell on the image of numerical sets, which are a union of several numerical intervals and sets consisting of individual numbers. There is nothing tricky here: according to the meaning of the union in these cases, on the coordinate line it is necessary to depict all the components of the set of a given numerical set. As an example, let's show an image of a number set (−∞, −15)∪{−10}∪[−3,1)∪
(log 2 5, 5)∪(17, +∞) : And let us dwell on fairly common cases when the depicted numerical set represents the entire set of real numbers, with the exception of one or several points. Such sets are often specified by conditions like x≠5 or x≠−1, x≠2, x≠3.7, etc. In these cases, geometrically they represent the entire coordinate line, with the exception of the corresponding points. In other words, these points need to be “plucked out” from the coordinate line. They are depicted as circles with an empty center. For clarity, let us depict a numerical set corresponding to the conditions Summarize. Ideally, the information from the previous paragraphs should form the same view of the recording and depiction of numerical sets as the view of individual numerical intervals: the recording of a numerical set should immediately give its image on the coordinate line, and from the image on the coordinate line we should be ready to easily describe the corresponding numerical set through the union of individual intervals and sets consisting of individual numbers. Bibliography. An expressive geometric representation of the system of rational numbers can be obtained as follows. Rice. 8. Number axis On a certain straight line, the “number axis,” we mark the segment from 0 to 1 (Fig. 8). This sets the length of a unit segment, which, generally speaking, can be chosen arbitrarily. Positive and negative integers are then depicted 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, we divide each of the resulting segments of unit length into equal parts; division points will represent fractions with a denominator. If we do this for the values 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 for natural numbers was defined. On the number axis this relationship is reflected as follows: if the natural number A is less than the natural number B, then point A lies to the left of point B. Since the indicated geometric relationship is established for any pair of rational points, it is natural to try to generalize the arithmetic inequality relation in this way, to maintain this geometric order for the points in question. This is possible if we accept the following definition: we say that a rational number A is less than a Rational number or that a number B is greater than a number if the difference is positive. It follows (at ) that the points (numbers) between are those that simultaneously Each such pair of points, together with all the points between them, is called a segment (or segment) and is denoted (and the set of intermediate points alone is called an interval (or interval), denoted The distance of an arbitrary point A from the origin 0, considered as a positive number, is called the absolute value of A and is denoted by the symbol The concept of “absolute value” is defined as follows: if , then if then It is clear that if the numbers have the same sign, then the equality is true; if they have different signs, then . Putting these two results together we arrive at the general inequality which is true regardless of the signs 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 a number so large that the interval ( will be less than the given interval; then at least one of the points of the form will be inside the given interval. So, there is no such interval on the number axis (even the smallest one, imaginable), inside which there would be no rational points. A further corollary follows: every interval contains an infinite number of rational points. Indeed, if a certain interval contained only a finite number of rational points, then within the interval formed by two adjacent such points. points, there would no longer be rational points, and this contradicts what has just been proven. §1.1. Real numbers Real numbers are divided into two classes: the class of rational numbers and the class of irrational numbers. Rational are numbers that have the form , where m And n are coprime integers, but For example, number Real numbers can also be divided into algebraic ones - the roots of a polynomial with rational coefficients (these include, in particular, all rational numbers - the roots of the equation The sets of all natural, integer, and real numbers are denoted accordingly as follows: NZ, R §1.2. Image of real numbers on the number line. Intervals Geometrically (for clarity), real numbers are represented by points on an infinite (in both directions) straight line called numerical
axis. For this purpose, a point is taken on the line under consideration (the origin is point 0), a positive direction is indicated, depicted by an arrow (usually to the right) and a unit of scale is selected, which is set aside indefinitely on both sides of point 0. This is how integers are depicted. To represent a number with one decimal place, you need to divide each segment into ten parts, etc. Thus, each real number is represented by a point on the number line. Back to each point Symbol Rice. 1.1. Number axis. Rice. 1.2. Interval Rice. 1.3. Closed interval Omitting the words “the set of all numbers (points) x such that”, etc., we note further: §1.3. Absolute value (or modulus) of a real number For example, Geometrically means point distance a to the origin. If we have two points and , then the distance between them can be represented as Properties of absolute quantities. 1. From the definition it follows that 2. The absolute value of the sum and difference does not exceed the sum of the absolute values: 3. 4. 5. 6. Inequality 7. Inequality 8. §1.4. Some concepts and notations 1
. Concept sets is one of the fundamental ones in mathematics, initial, universal - and therefore cannot be defined. It can only be described (replaced with synonyms): it is a collection, a collection of some objects, things, united by some characteristics. These objects are called elements multitudes. Examples: many grains of sand on the shore, stars in the Universe, students in the classroom, roots of an equation, points of a segment. Sets whose elements are numbers are called numerical sets. For some standard sets, special notation is introduced, for example, N,Z,R- see § 1.1. Let A– many and x is its element, then they write: If x is a general designation for elements of a set A, then they write The set is called final, if it consists of a finite number of elements. If, no matter what natural number N is taken, in the set A there are more elements than N, then A called endless set: there are infinitely many elements in it. If every element of the set ^A belongs to many B, That The set of elements of both sets A And B called unification sets and is denoted If a one-to-one correspondence can be established between the elements of sets, then they say that these sets are equivalent and write 2
. Let there be two statements, two facts: and Entry: 
Along with a natural number, you can substitute another natural number into the last inequality 
,Then
and in some neighborhood of the point (except perhaps the point itself) condition (2) is satisfied, then the function has a limit in the point and this limit is equal to A. by condition (1) $ for (here is the smallest neighborhood of the point ). But then, due to condition (2), the value will also be located in the vicinity of the point A, those. .
, that is α(x)+β(x) – infinitesimal.
means | f(x)-A|<ε при |x - x 0| < δ(ε). Cледовательно, .
, or sinx Writing numerical sets
Representation of number sets on a coordinate line
(this set essentially exists): 
The first acquaintance with real numbers occurs in a school mathematics course. Every real number is represented by a finite or infinite decimal fraction.
. (The set of rational numbers is denoted by the letter Q). The remaining real numbers are called irrational. Rational numbers are represented by a finite or infinite periodic fraction (the same as ordinary fractions), then those and only those real numbers that can be represented by infinite non-periodic fractions will be irrational.
- rational, and 
, 
, 
and so on. – irrational numbers.
) – and to transcendental ones – all the rest (for example, numbers 
and others).
(initial letters of the words Naturel, Zahl, Reel).
corresponds to a real number equal to the length of the segment 
and taken with a “+” or “–” sign, depending on whether the point lies to the right or to the left of the origin. In this way, a one-to-one correspondence is established between the set of all real numbers and the set of all points on the number axis. The terms “real number” and “number axis point” are used as synonyms. 
We will denote both a real number and the point corresponding to it. Positive numbers are located to the right of point 0, negative numbers are located to the left. If 
, then on the number axis the point 
lies to the left of the point 
. Let the point 
corresponds to the number, then the number is called the coordinate of the point, write 
; More often the point itself is denoted by the same letter as the number. Point 0 is the origin of coordinates. The axis is also designated by the letter 
(Fig. 1.1).
The set of all numbers lying between given numbers and is called an interval or interval; the ends may or may not belong to him. Let's clarify this. Let 
. A set of numbers that satisfy the condition 
, called an interval (in the narrow sense) or an open interval, denoted by the symbol 
(Fig. 1.2).
A set of numbers such that 
is called a closed interval (segment, segment) and is denoted by 
; on the number axis it is marked as follows:
It differs from the open gap only by two points (ends) and . But this difference is fundamental, significant, as we will see later, for example, when studying the properties of functions.
And 
, denoted 
And 
half-open or half-closed intervals (sometimes: half-intervals);
or 
means: 
or 
and is designated 
or 
;
or 
means 
or 
and is designated 
or 
;
, denoted 
the set of all real numbers. Badges 
"infinity" symbols; they are called improper or ideal numbers. 
Definition. Absolute value (or module) number is called the number itself if 
or 
If 
. The absolute value is indicated by the symbol 
. So, 
, 
, 
.
(or 
). For example, 
then the distance 
.
, 
, that is 
.
. 
1) If 
, That 
. 2) If 
, That . ▲
.
, then by property 2: 
, i.e. 
. Likewise, if you imagine 
, then we arrive at the inequality 
▲ 
– follows from the definition: consider cases 
And 
. 
, provided that 
The same follows from the definition.
,
, means 
. This inequality is satisfied by points that lie between 
And 
.
tantamount to inequality 
, i.e. . This is an interval centered at a point of length 
. It is called 
neighborhood of a point (number). If 
, then the neighborhood is called punctured: this is or 
. (Fig.1.4).
whence it follows that the inequality 
(
) is equivalent to the inequality 
or 
; and inequality 
defines a set of points for which 
, i.e. these are points lying outside the segment 
, exactly: 
And 
.
Let us present some widely used concepts and notations from set theory, mathematical logic and other branches of modern mathematics.
; reads " x belongs A» ( 
inclusion sign for elements). If the object x not included in A, then they write 
; reads: " x do not belong A" For example, 
N; 8,51
N; but 8.51 
R.
. If it is possible to write down the designation of all elements, then write 
, 
etc. A set that does not contain a single element is called an empty set and is denoted by the symbol ; for example, the set of roots (real) of the equation 
there is empty.
called a part or subset of a set B and write 
; reads " A contained in B» ( 
there is an inclusion sign for sets). For example, NZR. If 
, then they say that the sets A And B are equal and write 
. Otherwise they write 
. For example, if 
, A 
set of roots of the equation 
, That .
(Sometimes 
). A set of elements belonging to and A And B, called intersection sets and is denoted 
. The set of all elements of a set ^A, which are not contained in B, called difference sets and is denoted 
. These operations can be represented schematically as follows:
. Any set A, equivalent to the set of natural numbers N= called countable or countable. In other words, a set is called countable if its elements can be numbered and arranged in an infinite subsequence 
, all members of which are different: 
at 
, and it can be written in the form . Other infinite sets are called countless. Countable, except for the set itself N, there will be, for example, sets 
, Z. It turns out that the sets of all rational and algebraic numbers are countable, and the equivalent sets of all irrational, transcendental, real numbers and points of any interval are uncountable. They say that the latter have the power of continuum (power is a generalization of the concept of the number (number) of elements for an infinite set).
. Symbol 
means: “if true, then true and” or “it follows”, “implies that the root of the equation has the property from English Exist- exist.
, or 
, means: there is (at least one) object having the property 
. And the recording 
, or 
, means: everyone has the property. In particular, we can write: 
And .
