Geometric representation of complex numbers. Geometric representation of real numbers Real numbers geometric representation of real numbers
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 infimum. 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. $С: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+/a/, /x1/, /x2/,…,/x N ε-1 /,x 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 proven.
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 the 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 saints 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 general meaning. 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. subseq. 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 infinite number t-k after. 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 exists 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 REAL NUMBERS II § 37 Geometric representation of rational numbers Let Δ
is a segment taken as a unit of length, and l
- arbitrary straight line (Fig. 51). Let's take some point on it and designate it with the letter O. Every positive rational number m /
n
let's match the point to a straight line l
, lying to the right of C at a distance of m /
n
units of length. For example, the number 2 will correspond to point A, lying to the right of O at a distance of 2 units of length, and the number 5/4 will correspond to point B, lying to the right of O at a distance of 5/4 units of length. Every negative rational number k /
l
let us associate a point with a straight line lying to the left of O at a distance of | k /
l
| units of length. So, the number - 3 will correspond to point C, lying to the left of O at a distance of 3 units of length, and the number - 3/2 to point D, lying to the left of O at a distance of 3/2 units of length. Finally, we associate the rational number “zero” with point O. Obviously, with the chosen correspondence, equal rational numbers (for example, 1/2 and 2/4) will correspond to the same point, and different points of the line will not correspond to equal numbers. Let's assume that the number m /
n
point P corresponds, and the number k /
l
point Q. Then if m /
n
> k /
l
, then point P will lie to the right of point Q (Fig. 52, a); if m /
n
< k /
l
, then point P will be located to the left of point Q (Fig. 52, b). So, any rational number can be represented geometrically as some well-defined point on a line. Is the opposite statement true? Can every point on a line be considered as a geometric image of some rational number? We will postpone the decision of this issue until § 44. Exercises
296. Draw the following rational numbers as points on a line: 3; - 7 / 2 ; 0 ; 2,6. 297. It is known that point A (Fig. 53) serves as a geometric image of the rational number 1/3. What numbers represent points B, C and D? 298. Two points are given on a line, which serve as a geometric representation of rational numbers A
And b
a + b
And a - b
. 299. Two points are given on a line, which serve as a geometric representation of rational numbers a + b
And a - b
. Find the points representing numbers on this line A
And b
. 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 sets 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, we divide each of the resulting segments of unit length into 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 the rational number B (A is greater than the number A (B>A), if the difference B-A is positive. It follows (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 The concept of “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 densely located 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 a finite number of 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. 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 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 straight 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 P
> 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? The following forms of complex numbers exist: algebraic(x+iy), trigonometric(r(cos+isin Any complex number z=x+iy can be represented 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 The exponential form of writing a complex number follows from Euler’s formula: z= re i 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 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: 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 2.
i 1063 = i 1062 i=i
form. trigonometric -
), That. Moivre's formula 3.
Here n can be either “+” or “-” (integer). If a complex number is given in
indicative form: 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 i 1063 = i 1062 i=i
form: trigonometric 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 +…= 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. A 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. Geometric progression isconvergent
near, If Also found harmonic series(row 
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 
2,1 < α
< 2,2
2,13 < α
< 2,14
)),
indicative(re i 
).
+isin
)
- trigonometric form of writing a complex number.
,Then 
- exponential form of writing a complex number.Operations on complex numbers.
+isin 
); z2=r(cos 
+isin 
).
;
;
- the number of combinations of n elements of m (the number of ways in which n elements from m can be taken).
; 
.
+isin 
z=r(cos
.
.
+isin 
), then the nth root of z is found by the formula:
, where k=0.1…n-1.Rows. Number series.
. The numbers a 1, a 2, a 3,..., and n are members of the series.
.
, C=const.
, and divergent if 
.
). This series divergent
.