Geometric representation of complex numbers. Display of real numbers on the number axis. Intervals Geometric representation of complex numbers
§1.1. Real numbers
The first acquaintance with real numbers occurs in the school mathematics course. Any real number is represented by a finite or infinite decimal fraction.
Real (real) numbers are divided into two classes: the class of rational and the class of irrational numbers. Rational are the numbers that have the form, where m and n - coprime integers, but 
... (The set of rational numbers is denoted by the letter Q). The rest of the 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.
For example, the number 
- rational, and 
, 
, 
etc. - irrational numbers.
Real numbers can also be divided into algebraic - roots of a polynomial with rational coefficients (these include, in particular, all rational numbers - roots of the equation 
) - and on transcendental - all others (for example, numbers 
other).
The sets of all natural, whole, real numbers are denoted as follows: NZ, R
(initial letters of the words Naturel, Zahl, Reel).
§1.2. Display of real numbers on the number axis. Intervals
Geometrically (for clarity), real numbers are represented by dots 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 scale unit is chosen, which is set aside indefinitely in both directions from point 0. This is how integers are displayed. To represent a number with one decimal place, each segment must be divided into ten parts, etc. Thus, each real number will be represented by a point on the number axis. Back to every point 
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. Thus, a one-to-one correspondence is established between the set of all real numbers and the set of all points of the numerical axis. The terms "real number" and "number axis point" are used as synonyms.
Symbol 
we will denote both the real number and the point corresponding to it. Positive numbers are located to the right of point 0, negative - to the left. If a 
, then on the numerical axis the point 
lies to the left of the point 
... Let the point 
corresponds to a number, then the number is called the coordinate of the point, they write 
; more often the point itself is denoted by the same letter as the number. Point 0 is the origin. The axis is also denoted by the letter 
(Figure 1.1).
Figure: 1.1. Number axis.
The collection 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 be 
... A collection of numbers that satisfy the condition 
, called an interval (in the narrow sense) or an open interval, denoted by the symbol 
(Figure 1.2).
Figure: 1.2. Interval
A collection of numbers such that 
is called a closed interval (segment, segment) and is denoted by 
; on the number axis is marked as follows:
Figure: 1.3. Closed interval
It differs from an open gap only by two points (ends) and. But this difference is fundamental, essential, as we will see later, for example, when studying the properties of functions.
Omitting the words "the set of all numbers (points) x such that ", etc., we note further:
and 
, denoted 
and 
half-open, or half-closed, intervals (sometimes: half-intervals);
or 
means: 
or 
and denoted 
or 
;
or 
means 
or 
and denoted 
or 
;
, denoted 
the set of all real numbers. Badges 
symbols of "infinity"; they are called improper or ideal numbers. 
§1.3. The absolute value (or modulus) of a real number
Definition. Absolute value (or modulus) numbers are called this number itself if 
or 
if a 
... The absolute value is indicated by the symbol 
... So, 
For instance, 
, 
, 
.
Geometrically means point distance a to the origin. If we have two points and, then the distance between them can be represented as 
(or 
). For instance, 
the distance 
.
Properties of absolute values.
1. It follows from the definition that 
, 
, i.e 
.
2. The absolute value of the sum and difference does not exceed the sum of the absolute values: 
.
1) If 
then 
... 2) If 
then. ▲
3. 
.
, then by property 2: 
, i.e. 
... Similarly, if we imagine 
, then we come to the inequality 
▲
4. 
- follows from the definition: consider cases 
and 
.
5. 
, provided that 
It also follows from the definition.
6. Inequality 
,
means 
... This inequality is satisfied by the points that lie between 
and 
.
7. Inequality 
tantamount to inequality 
, i.e. ... This is an interval centered at a point of length 
... It is called 
a neighborhood of a point (number). If a 
, then the neighborhood is called punctured: this or 
... (Fig. 1.4).
8. 
whence it follows that the inequality 
(
) is equivalent to the inequality 
or 
; and inequality 
defines the set of points for which 
, i.e. these are points lying outside the segment 
, exactly: 
and 
.
§1.4. Some concepts, designations
Here are some commonly used concepts, designations from set theory, mathematical logic and other branches of modern mathematics.
1 ... Concept multitudes is one of the basic in mathematics, the original, universal - and therefore defies definition. It can only be described (replaced with synonyms): it is a collection, a collection of some objects, things, united by some signs. These objects are called elements sets. Examples: many grains of sand on the shore, stars in the universe, students in the classroom, the roots of an equation, points of a line. Sets whose elements are numbers are called numerical sets... For some standard sets, special designations are introduced, for example, N, Z, R -see § 1.1.
Let be A - set and x is its element, then they write: 
; reads " x belongs A» ( 
inclusion sign for elements). If the object x not included in Athen write 
; reads: " x not belong A". For instance, 
N; 8,51
N; but 8.51 
R.
If a x is a general designation for the elements of the set Athen write 
... If it is possible to write down the designation of all elements, then write 
, 
etc. A set that does not contain any 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.
The set is called the finalif it consists of a finite number of elements. If, for whatever natural number N we take, in the set A there are more than N elements, then A called endless many: there are infinitely many elements in it.
If every element of the set ^ A belongs to the set Bthen 
called part or subset of the set B and write 
; reads " A contained in B» ( 
is the inclusion sign for sets). For instance, NZR.If and 
, then they say that the sets A and B equal and write 
... Otherwise, write 
... For example, if 
, and 
set of roots of the equation 
then.
The collection of elements of both sets A and B called unification sets and denoted 
(sometimes 
). A collection of elements belonging to and A and Bis called crossing sets and denoted 
... The collection of all elements of the set ^ Athat are not contained in Bis called difference sets and denoted 
... These operations can be schematically represented as follows:
If one can establish a one-to-one correspondence between the elements of the sets, then they say that these sets are equivalent and write 
... Every multitude A, equivalent to the set of natural numbers N\u003d called countable or countable. In other words, a set is called countable if its elements can be numbered, arranged in an infinite sequence 
, all members of which are different: 
at 
, and it can be written as. Other infinite sets are called uncountable... Countable, except for the set itself N, there will be, for example, the 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 cardinality of the continuum (cardinality is a generalization of the concept of the number (number) of elements for an infinite set).
2
... Let there be two statements, two facts: and 
... Symbol 
means: "if true, then true and" or "from follows", "implies is the root of the equation has a property from English Exist - exist.
Record: 
, or 
, means: there is (at least one) object with the property 
... And the record 
, or 
, means: all have the property. In particular, we can write: 
and.
TICKET 1
Rational numbers are numbers written in the form p / q, where q is a natural. a number, and p is an integer.
Two numbers a \u003d p1 / q1 and b \u003d p2 / q2 are called equal if p1q2 \u003d p2q1, and p2q1 and a\u003e b if p1q2
TICKET 2
Complex numbers.Complex numbers
An algebraic equation is an equation of the form: P n ( x) \u003d 0, where P n ( x) - polynomial n- oh degree. A couple of real numbers x and at will be called ordered if it is indicated which of them is considered the first and which is the second. Ordered pair notation: ( x, y). A complex number is an arbitrary ordered pair of real numbers. z = (x, y)-complex number.
x-material part z, yimaginary part z... If a x \u003d 0 and y \u003d 0, then z \u003d 0. Consider z 1 \u003d (x 1, y 1) and z 2 \u003d (x 2, y 2).
Definition 1. z 1 \u003d z 2 if x 1 \u003d x 2 and y 1 \u003d y 2.
Concepts\u003e and< для комплексных чисел не вводятся.
Geometric representation and trigonometric form of complex numbers.
M ( x, y) « z = x + iy.
½ OM½ \u003d r \u003d ½ z½ \u003d. (Picture)
r is called the modulus of a complex number z.
j is called the argument of the complex number z... It is determined to within ± 2p n.
x\u003d rcosj, y\u003d rsinj.
z= x+ iy \u003d r (cosj + isinj) is the trigonometric form of complex numbers.
Statement 3.
\u003d (cos + i sin),
\u003d (cos + i sin), then
\u003d (cos (+) + i sin (+)),
\u003d (cos (-) + i sin (-)) for ¹0.
Statement 4.
If a z \u003d r (cosj + i sinj), then "natural n:
\u003d (cos nj + i sin nj),
TICKET 3
Let be X-numeric set containing at least one number (non-empty set).
xÎ X- x contained in X. ; xÏ X- x not belong X.
Definition: Lots 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 is called the upper (lower) bound of the set X... Lots of X is called bounded from above if $ M, " x Î X: x £ M. Definition set unbounded from above. Lots of X is called unbounded from above if " M $ x Î X: x> M. Definition lots 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 of the upper bounds of a set bounded above X is called its exact upper bound, and is denoted by Sup X
(supremum). \u003d Sup X... Similarly, you can determine the exact
bottom edge. Equivalent definitionexact top edge:
The number is called the exact upper bound of the set X, if a: 1) " x Î X: x £ (this condition shows that is one of the upper bounds). 2) " < $ x Î X: x \u003e (this condition shows that -
the smallest of the top faces).
Sup X= :
1. " xÎ X: x £ .
2. " < $ xÎ X: x> .
inf X (infimum) is the exact bottom edge. Let us pose the question: does every bounded set have sharp edges?
Example: X= {x: x\u003e 0) does not have the smallest number.
The theorem on the existence of the exact upper (lower) face... Any non-empty upper (lower) bound on the set xÎR has an upper (lower) bound.
Separability theorem for numeric mn:▀▀▄
TICKET 4
If each nature number n (n \u003d 1,2,3 ..) is assigned a certain number Xn, then they say that it is defined and given sequencex1, x2 ..., write (Xn), (Xn). Example: Xn \u003d (- 1) ^ n: -1,1, -1,1, ... Then the name is limited. from above (from below) if many points x \u003d x1, x2,… xn lie on the numerical axis bounded above (below), i.e. $ С: Xn £ C " Sequence limit:a number a is called the limit of the last, if for any ε\u003e 0 $: N (N \u003d N / (ε)). "n\u003e N the inequality | Xn-a |<ε. Т.е. – ε
at n\u003e N.
Uniqueness limit bounded and converging sequence
Property1: A converging sequence has only one limit.
Proof: by contradiction let and and b the limits of the converging sequence (x n), and a is not equal to b. consider infinitesimal sequences (α n) \u003d (x n -a) and (β n) \u003d (x n -b). Because all elements are b.m. sequences (α n -β n) have the same value b-a, then by the property of b.m. sequences b-a \u003d 0 i.e. b \u003d a and we come to a contradiction.
Property2: The converging sequence is limited.
Proof: Let a be the limit of a convergent sequence (x n), then α n \u003d x n -a is an element of infinitesimal place. sequence. Take some ε\u003e 0 and from it we find N ε: / x n -a /< ε при n> N ε. Let b denote the largest of the numbers ε + / a /, / x1 /, / x2 /, ..., / x N ε-1 /, x N ε. It is obvious that / x n /
Note: a bounded sequence may not be convergent.
SEASON 6
The sequence a n is called infinitesimal, which means that the limit of this sequence after is equal to 0.
a n is infinitesimal Û lim (n ® + ¥) a n \u003d 0 that is, for any ε\u003e 0 there exists N such that for any n\u003e N | a n |<ε
Theorem.The amount of infinitesimal is infinitesimal.
a n b n ®infinitesimal Þ a n + b n - infinitesimal.
Evidence.
a n - infinitesimal Û "ε\u003e 0 $ N 1:" n\u003e N 1 Þ | a n |<ε
b n - infinitesimal Û "ε\u003e 0 $ N 2:" n\u003e N 2 Þ | b n |<ε
We put N \u003d max (N 1, N 2), then for any n\u003e N Þ both inequalities hold simultaneously:
| a n |<ε |a n +b n |£|a n |+|b n |<ε+ε=2ε=ε 1 "n>N
We set "ε 1\u003e 0, we put ε \u003d ε 1/2. Then for any ε 1\u003e 0 $ N \u003d maxN 1 N 2:" n\u003e 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 the infinitely small is the infinitely small.
a n, b n - infinitesimal Þ a n b n - infinitesimal.
Proof:
Let us set "ε 1\u003e 0, set ε \u003d Öε 1, since a n and b n are infinitesimal for this ε\u003e 0, then there is N 1:" n\u003e N Þ | a n |<ε
$ N 2: "n\u003e N 2 Þ | b n |<ε
Take N \u003d max (N 1; N 2), then "n\u003e N \u003d | a n |<ε
| a n b n | \u003d | a n || b n |<ε 2 =ε 1
"ε 1\u003e 0 $ N:" n\u003e N | a n b n |<ε 2 =ε 1
lim a n b n \u003d 0 Û a n b n is infinitesimal, as required.
Theorem The product of a bounded sequence and an infinitesimal sequence is an infinitesimal sequence
and n is a limited sequence
a n is an infinitesimal sequence Þ a n a n is an infinitesimal sequence.
Proof: Since a n is bounded Û $ С\u003e 0: "nÎ N Þ | a n | £ C
We set "ε 1\u003e 0; put ε \u003d ε 1 / C; since a n is infinitesimal, then ε\u003e 0 $ N:" n\u003e NÞ | a n |<εÞ |a n a n |=|a n ||a n | "ε 1\u003e 0 $ N:" n\u003e N Þ | a n a n | \u003d Cε \u003d ε 1 Þ lim (n ® ¥) a n a n \u003d 0Û a n a n - infinitesimal The sequence is called BBP (in sequence) if Write. Obviously, BBP is not limited. The converse statement is generally not true (example). If for large nmembers, then write this means that as soon as. The meaning of the entry Infinitely large sequencesa n \u003d 2 n ;
b n \u003d (- 1) n 2 n; c n \u003d -2 n Definition (infinitely large sequences) 1) lim (n ® ¥) a n \u003d + ¥ if "ε\u003e 0 $ N:" n\u003e N Þ a n\u003e ε where ε is arbitrarily small. 2) lim (n ® ¥) a n \u003d - ¥ if "ε\u003e 0 $ N:" n\u003e N Þ a n<-ε 3) lim (n ® ¥) a n \u003d ¥ Û "ε\u003e 0 $ N:" n\u003e N Þ | a n |\u003e ε SEASON 7 Theorem “On convergence monotone. last " Any monotonous message is converging, i.e. has limits. DocLet the last (xn) be monotonically ascending. and bounded from above. X - all the plurality of numbers that receive the email of this message according to conv. Theorems are many limited., Therefore, acc. In the theorem, it has a finite exact upper. supX xn®supX (we denote supX by x *). Because x * exact top. face, then xn £ x * "n." e\u003e 0 the output is $ xm (let m be n with a cover): xm\u003e x * -e for "n\u003e m \u003d\u003e from the indicated 2 inequalities, we obtain the second inequality x * -e £ xn £ x * + e for n\u003e m is equivalent to 1/2 xn-x * 1/2 SEASON 8 Exponent or number e R-rim number. send with a common term xn \u003d (1 + 1 / n) ^ n (to the power n) (1). It turns out that the last (1) rises monotonously, is bounded from above and slowly converges, the limit of this post is called an exponent and is denoted by the symbol e "2.7128 ... Number e SEASON 9 The nested line principle Let the number of line segments be given on the number line ,, ... ,, ... Moreover, these segments satisfy sl. conv .: 1) each last one is nested in the previous one, i.e. Ì, "n \u003d 1,2, ...; 2) The lengths of the segments ®0 with increasing n, that is, lim (n® ¥) (bn-an) \u003d 0. Send with the specified sv-you will be called nested. TheoremAny last of nested segments contains a single t-ku with belonging to all segments last at the same time, with a common point of all segments to which they contract. Doc(an) -send left endpoints of segments yavl. monotonically non-decreasing and bounded from above by the number b1. (bn) - the sequence of the right ends is monotonically non-increasing, therefore these sequences are. converging, i.e. there are numbers с1 \u003d lim (n® ¥) an and c2 \u003d lim (n® ¥) bn \u003d\u003e c1 \u003d c2 \u003d\u003e c - their common meaning. Indeed, lim (n® ¥) (bn-an) \u003d lim (n® ¥) (bn) - lim (n® ¥) (an) by condition 2) o \u003d lim (n® ¥) (bn- an) \u003d c2-c1 \u003d\u003e c1 \u003d c2 \u003d c It is clear that m. C is common for all segments, since "n an £ c £ bn. Now we will prove that it is one. Let us assume that $ is different with 'to which all segments are drawn. If we take any disjoint segments with and c ', then on one side the entire “tail” of the sequences (an), (bn) must be located in the vicinity of t-ki c' '(since an and bn converge to c and c 'simultaneously). The contradiction proves t-mu. SEASON 10 Bolzano-Weierstrass theorem
From any facet. afterwards you can select the exit. submissive. 1. Since the last is bounded, then $ m and M, such that "m £ xn £ M," n. D1 \u003d - the segment in which all the last points lie. Let's split it in half. At least in one of the halves there will be an infinite number of t-k last. D2 - that half, where there is an infinite number of t-to-last. We divide it in half. At least in one of the halves of neg. D2 nah-Xia infinite number of t-to the last. This half is D3. We divide the segment D3 ... and so on. we obtain the last nested segments, the lengths of which tend to 0. According to the n-th nested segments, $ unities. t-ka C, cat. accessories to all segments D1, any m-ku Dn1. In the segment D2 I choose t-ku xn2 so that n2\u003e n1. In the segment D3 ... etc. As a result, I'll send it to xnkÎDk. SEASON 11 SEASON 12 fundamental In conclusion, consider the question of the criterion for the convergence of a numerical sequence. Let that is: We got the following statement: If the sequence converges, the condition Cauchy: A numerical sequence satisfying the Cauchy condition is called fundamental... It can be proved 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 the sequence to have a limit it is necessary and sufficient that it be fundamental. Second meaning of the Cauchy criterion. Sequence members and where nand m- any unlimited approaching at. SEASON 13 One-sided limits. Definition 13.11.Number AND called the limit of the function y \u003d f (x) at xaiming for x 0 left (right), if such that | f (x) -A|<ε при x 0 - x< δ
(x - x 0< δ
). Legend: Theorem 13.1 (second definition of the limit). Function y \u003d f (x) has at x, aiming for x 0, limit equal to AND, if and only if both of its one-sided limits at this point exist and are equal AND. Evidence. 1) If, then 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<
δ 2. Choosing the lesser of 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 the limit 13.7 and the conditions for the existence and equality of unilateral limits has been proven, this condition can be considered the second definition of the limit. Definition 4 (according to Heine) Number AND is called the limit of the function if any BBP of the values \u200b\u200bof the argument, the sequence of the corresponding values \u200b\u200bof the function converges to AND. Definition 4 (by Cauchy). Number AND called if. It is proved that these definitions are equivalent. TICKET 14 and 15 Properties of the limit f-tion at a point 1) If the limit exists in t-ke, then it is the only one 2) If at x0 the limit of the function f (x) lim (x®x0) f (x) \u003d A lim (x®x0) g (x) £ B \u003d\u003e then in this m-k $ the limit of the sum, difference, product and quotient. Branch of these 2 f-tions. a) lim (x®x0) (f (x) ± g (x)) \u003d A ± B b) lim (x®x0) (f (x) * g (x)) \u003d A * B c) lim (x®x0) (f (x): g (x)) \u003d A / B d) lim (x®x0) C \u003d C e) lim (x®x0) C * f (x) \u003d C * A Theorem 3. If a ( resp A ) then the $ neighborhood in which the inequality \u003e B (resp Setting in Theorem 3 B \u003d 0, we get: if ( resp ), then $, at all points, which will be \u003e 0 (resp<0),
those. the function retains the sign of its limit. Theorem 4 (on the passage to the limit in the inequality). If in some neighborhood of the 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 preservation of a function, we have the value of its limit, but Theorem 5.(about the limit of the intermediate function). (1) If SEASON 16 Definition 14.1.Function y \u003d α (x) is called infinitesimal for x → x 0, if a Properties of infinitesimal. 1. The sum of two infinitesimal is infinitesimal. Evidence. If a α (x) and β (x) Are infinitesimal for 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. Hence it follows that the sum of any finite number of infinitesimal is infinitesimal. 2. If α ( x) Is infinitely small for x → x 0, and f (x) Is a function bounded in some neighborhood x 0then α (x) f (x) Is infinitely small for x → x 0. Evidence. Let's choose a number M such that | f (x) | Corollary 1. The product of an infinitely small by a finite number is an infinitesimal. Corollary 2. The product of two or more infinitesimal is infinitesimal. Corollary 3. A linear combination of infinitesimal 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 as f (x) \u003d A + α (x), where α (x) Is infinitely small for x → x 0. Evidence. 1)
Let Then | f (x) -A|<ε при x → x 0, i.e α (x) \u003d f (x) -A - infinitely small at x → x 0. Hence , f (x) \u003d A + α (x). 2) Let f (x) \u003d A + α (x). Then Comment. Thus, we have obtained one more definition of the limit, equivalent to the two previous ones. Infinitely large functions. Definition 15.1. The function f (x) is called infinitely large for x x 0 if For infinitely large, you can introduce the same classification system as for infinitesimal, namely: 1. Infinitely large f (x) and g (x) are considered quantities of the same order if 2. If, then f (x) is considered to be infinitely large of a higher order than g (x). 3. An infinitely large f (x) is called a k-th order relative to an infinitely large g (x) if. Comment. Note that ax is infinitely large (for a\u003e 1 and x) of higher order than x k for any k, and log ax is infinitely large of 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. Evidence. 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) |\u003e M. Hence, that is, 1 / α (x) is infinitely large as x → x 0. SEASON 17 Theorem 14.7 (first remarkable limit). ... Evidence. Consider a circle of unit radius centered at the origin and assume that the AOB angle is x (radian). Let us compare the areas of the AOB triangle, the AOB sector and the AOC triangle, where the line OS is the 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 get from here that An expressive geometric representation of the system of rational numbers can be obtained as follows. Figure: 8. Number axis On some straight line, "numerical axis", we mark the segment from 0 to 1 (Fig. 8). This establishes the length of the unit segment, which, generally speaking, can be chosen arbitrarily. Positive and negative integers are then depicted as a set of equally spaced points on the numerical axis, namely, positive numbers are marked to the right, and negative ones are marked to the left of point 0. To represent numbers with a denominator, divide each of the obtained unit length segments into equal parts; division points will represent fractions with a denominator. If we do this for the values \u200b\u200bcorresponding to all natural numbers, then each rational number will be represented by some point on the numerical axis. We will agree to call these points "rational"; in general, the terms "rational number" and "rational point" will be used as synonyms. In Chapter I, § 1, the inequality relation for natural numbers was defined. On the number axis, this ratio is reflected as follows: if a natural number A is less than a natural number B, then point A lies to the left of point B. Since the specified geometric relationship is established for any pair of rational points, it is natural to try to generalize the arithmetic relation of inequality in this way, to preserve this geometric order for the points in question. This is possible if we accept the following definition: they say that the rational number A is less than the Rational number or that the number B is greater than the number if the difference is positive. It follows (for) 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 only intermediate points is denoted by an interval (or an interval) denoted by 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 equality is true, if they have different signs, then. Combining 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 densely located on the number line. The meaning of this statement is that within any interval, no matter how small, there are 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 this interval; then at least one of the points of the form will be inside this interval. So, there is no such interval on the number axis (even the smallest, which one can imagine), inside which there would be no rational points. Hence, a further consequence follows: 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 adjacent such points, rational points would no longer exist, and this contradicts what has just been proved. The concepts of "set", "element", "belonging of an element to a set" are the primary concepts of mathematics. Lots of-
any collection (collection) of any items .
A is a subset of the set B,if each element of the set A is an element of the set B, i.e. AÌB Û (xÎA Þ xÎB). Two sets are equalif they consist of the same elements. It's about set-theoretic equality (not to be confused with equality between numbers): A \u003d B Û AÌB Ù BÌA. Union of two sets consists of elements belonging to at least one of the sets, i.e. хÎАÈВ Û хÎАÚ хÎВ. Crossing consists of all elements simultaneously belonging to both set A and set B: xÎAÇB Û xÎA Ù xÎB. Difference consists of all elements of A that do not belong to B, i.e. xÎ A \\ B Û xÎA ÙxÏB. Cartesian product C \u003d A´B of sets A and B is called the set of all possible pairs ( x, y), where the first element x each pair belongs to A, and its second element at belongs to V. The subset F of the Cartesian product A´B is called mapping the set A into the set B
if the condition is met: (" xОА) ($! Pair ( xy) ÎF). At the same time they write: A. The terms "display" and "function" are synonymous. If ("хÎА) ($! УÎВ): ( x, y) ÎF, then the element atÎ IN called way
x when displaying F and write it like this: at\u003d F ( x). Element x while is prototype
(one of the possible) element y. Consider the set of rational numbers Q
- the set of all integers and the set of all fractions (positive and negative). Each rational number can be represented as a quotient, for example, 1 \u003d 4/3 \u003d 8/6 \u003d 12/9 \u003d…. There are many such ideas, but only one of them is irreducible. .
IN any rational number can be uniquely represented as a fraction p / q, where pÎZ, qÎN, the numbers p, q are coprime. Properties of the set Q:
1. Closure with respect to arithmetic operations.The result of addition, subtraction, multiplication, raising to a natural power, division (except division by 0) of rational numbers is a rational number:; ; 2. Ordering: (" x, yÎQ, huy)®( x Moreover: 1) a\u003e b, b\u003e c Þ a\u003e c;2) a -b. 3. Density... Between any two rational numbers x, y there is a third rational number (for example, c \u003d ):
("x, y ÎQ, x<y) ($ 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 \u003d a, aÎN are not always decidable in the set Q. Theorem. There is no number xÎQwhose square is 2. g Let there exist a fraction x\u003d p / q, where the numbers p and q are coprime and x 2 \u003d 2. Then (p / q) 2 \u003d 2. Hence, The right side of (1) is divisible by 2, so p 2 is an even number. Thus p \u003d 2n (n-integer). Then q must be odd. Returning to (1), we have 4n 2 \u003d 2q 2. Therefore, q 2 \u003d 2n 2. Similarly, we make sure that q is divisible by 2, i.e. q is an even number. The theorem is proved by contradiction. N geometric image of rational numbers.Putting off the unit segment from the origin of coordinates 1, 2, 3… times to the right, we get points of the coordinate line that correspond to natural numbers. Putting aside similarly to the left, we get points corresponding to negative integers. Let's take 1 / q(q \u003d2,3,4 …
) a part of a unit segment and we will postpone it on both sides of the origin rtime. We get points of a straight line corresponding to numbers of the form ± p / q (pÎZ, qÎN). If p, q run through all pairs of coprime numbers, then on the line we have all points corresponding to fractional numbers. In this way, each rational number corresponds, according to the accepted method, to a single point of the coordinate line. Can a single rational number be specified for any point? Is the straight line filled entirely with rational numbers? It turns out that there are points on the coordinate line that do not correspond to any rational numbers. We build an isosceles right-angled triangle on a unit segment. Point N does not correspond to a rational number, since if ON \u003d x - rationally, then x 2 \u003d2, which cannot be. There are infinitely many points similar to the point N on the straight line. Take the rational parts of the segment x \u003d ON, those. x... If we postpone them to the right, then no rational number will correspond to each of the ends of any of such segments. Assuming that the length of the segment is expressed by a rational number x \u003d, we get that x \u003d - rational. This contradicts what was proved above. Rational numbers are not enough to associate some rational number with each point of the coordinate line. Let's build the set of real numbers R
through infinite decimal fractions. According to the “corner” division algorithm, any rational number can be represented as a finite or infinite periodic decimal fraction. When the fraction p / q has no prime factors other than 2 and 5, i.e. q \u003d 2 m × 5 k, then the result will be the final decimal fraction p / q \u003d a 0, a 1 a 2… a n. The rest of the fractions can only have infinite decimal expansions. Knowing the infinite periodic decimal fraction, you can find the rational number that it represents. But any final decimal fraction can be represented as an infinite decimal fraction in one of the following ways: a 0, a 1 a 2… a n \u003d a 0, a 1 a 2… a n 000… \u003d a 0, a 1 a 2… (a n -1) 999… (2) For example, for infinite decimal x\u003d 0, (9) we have 10 x\u003d 9, (9). If we subtract the original number from 10x, we get 9 x\u003d 9 or 1 \u003d 1, (0) \u003d 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 the infinite decimal fraction is identified with the digit 9 in the period with the corresponding infinite decimal fraction with the digit 0 in the period according to the 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 a 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 are sequentially 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 the sign "+" or "-", a non-negative integer a 0 and a sequence of decimal places a 1, a 2, ..., an, ... (the set of decimal places consists of ten numbers: 0, 1, 2, ..., nine). Any fraction of the form (3) is called 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) A number of the form (4) will be called 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 in expression (3) the sign "-" is taken, then this is a negative number. The union of the sets of rational and irrational numbers form the set of real numbers (QÈJ \u003d R). If the infinite decimal fraction (3) is periodic, then this is a rational number, when the fraction is non-periodic, it is irrational. Two non-negative real numbers a \u003d a 0, a 1 a 2… a n…, b \u003d b 0, b 1 b 2… b n…. called equal (write a \u003d b),
if a a n \u003d b nat n \u003d 0,1,2 ... Number a is less than number b (write a<b),
if either a 0 or a 0 \u003d b 0 and there is such a number m,what a k \u003d b k (k \u003d 0,1,2, ... m-1),and a m , i.e. a Û (a 0 Ú ($mÎN: a k \u003d b k (k \u003d), a m ). The concept of “ and> b». To compare arbitrary real numbers, we introduce the concept “ modulus of the number a» .
By the modulus of a real number a \u003d ± a 0, a 1 a 2 ... a n ... is called such a non-negative real number represented by the same infinite decimal fraction, but taken with the sign "+", i.e. ½ and½= a 0, a 1 a 2 ... a n ... and 1/2 and½³0. If a and -non-negative, b Is a negative number, then a\u003e b... If both numbers are negative ( a<0, b<0
), then we will assume that: 1) a \u003d bif ½ and½ =
½ b½; 2) and if ½ and½ >
½ b½. Properties of the set R:
I. Order properties: 1. For each pair of real numbers and and b there is one and only one relation: a \u003d b, a 2. If a 3. If a , then there is a number c such that a< с . II. Properties of addition and subtraction actions: 4. a + b \u003d b + a (commutability). 5. (a + b) + c \u003d a + (b + c) (associativity). 6. a + 0 \u003d a. 7. a + (- a) \u003d0. 8.from a Þ a + c III. Properties of multiplication and division actions: 9. a × b \u003d b × a . 10. (a × b) × c \u003d a × (b × c). 11. a × 1 \u003d a. 12. a × (1 / a) \u003d 1 (a¹0). 13. (a + b) × c \u003d ac + bc(distribution). 14.if a and c\u003e 0, then a × c IV. Archimedean property("cÎR) ($ nÎN): (n\u003e c). Whatever the number cÎR, there is nÎN such that n\u003e c. V. Continuity property of real numbers. Let two non-empty sets АÌR and BÌR be such that any element andÎA will be no more ( a£ b) of any element bÎB. Then dedekind continuity principleasserts the existence of such a number with that for all andОА and bÎB the condition 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 of the number line, and call the real numbers points. REAL NUMBERS II Section 44 Geometric representation of real numbers Geometrically, real numbers, as well as rational numbers, are represented by points on a straight line. Let be l
- an arbitrary straight line, and O - some of its point (Fig. 58). To every positive real number α
we put in correspondence point A, lying to the right of O at a distance of α
units of length. If, for example, α
\u003d 2.1356 ..., then 2 < α
< 3 and so on. 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 determine 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
we put in correspondence a point B lying to the left of O at a distance of | β |
units of length. Finally, we assign the point O to the number "zero". So, the number 1 will be displayed on the line l
point A, located to the right of O at a distance of one unit of length (Fig. 59), the number - √2 - point B, lying to the left of O at a distance of √2 units of length, and so on. Let's show how on a straight line l
using a compass and a ruler, you can find the points corresponding to the real numbers √2, √3, √4, √5, etc. For 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 raise a perpendicular to this segment and put on it the segment AC, equal to the segment AB. Then, applying the Pythagorean theorem to the right-angled triangle ABC, we obtain; ВС \u003d √АВ 2 + АС 2 \u003d √1 + 1 \u003d √2 Therefore, the segment BC has length √2. Now let us restore the perpendicular to the segment BC at point C and choose point D on it so that the segment CD is equal to the unit of length AB. Then from the rectangular triangle BCD we find: ВD \u003d √ВC 2 + СD 2 \u003d √2 + 1 \u003d √3 Consequently, the segment BD has length √3. Continuing the described process further, we could get segments BE, BF, ..., the lengths of which are expressed by numbers √4, √5, etc. Now on the straight l
it is easy to find those points that serve as a geometric representation of the numbers √2, √3, √4, √5, etc. Putting, for example, to the right of the point O segment BC (Fig. 61), we get point C, which serves as a geometric image of the number √2. In the same way, putting the segment BD to the right of the point O, we get the point D ", which is the geometric image of the number √3, and so on. However, one should not think that with the help of a compass and a ruler on the number line l
you can find a point that matches any given real number. It has been proved, for example, that having only a compass and a ruler at our disposal, it is impossible to construct a segment whose length is expressed by the number π
\u003d 3.14 .... Therefore, on the number line l
with the help of such constructions it is impossible to indicate a point corresponding to this number. Nevertheless, such a point exists. So, to each real number α
can be associated with some well-defined point of the straight line l
... This point will be spaced from the starting point O at a distance of | α
| units of length and be to the right of O if α
\u003e 0, and to the left of 0, if α
< 0. Очевидно, что при этом двум неравным действительным числам будут соответствовать две различные точки прямой l
... Indeed, let the number α
corresponds to point A, 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 representation of rational numbers, we posed the question: can any point of the line be regarded as a geometric image of some rational numbers? Then we could not give an answer to this question; 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 the line represents a rational number. But in this case, another question arises: can any point of the number line be considered as a geometric image of some actual numbers? This issue is already being resolved positively. Indeed, let A be an arbitrary point of the straight 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 the geometric image of the number α
... Similarly, it is 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
- the length of the VO segment. Finally, point O serves as a geometric representation of the number zero. It is clear that two different points of the line l
cannot be the geometric image of the same real number. For the reasons stated above, the straight line on which some point O is indicated as "initial" (for a given unit of length) is called number line. Conclusion. The set of all real numbers and the set of all points of the number line are in one-to-one correspondence. This means that each real number corresponds to one, well-defined point of the number line and, conversely, to each point of the number line, with such a correspondence, there corresponds one, well-defined real number. Exercises
320. Find out which of the two points is on the number line to the left and which to the right, 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 on the number line further from the starting point O, if these points correspond to 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 for constructing a segment of length √ n
using a compass and a ruler, you can do the following: first construct a segment of length √2, then a segment of length √3, and so on, until we reach a segment of length √ n
... But for every fixed p
\u003e 3 this process can be accelerated. How would you, for example, start building a segment of length √10? 323 *. How to use a compass and a ruler to find on the number line the point corresponding to the number 1 / α
if the position of the point corresponding to the number α
, you know?
along with a natural number, another natural number can be substituted into the last inequality 
,then
and condition (2) is satisfied in some neighborhood of m (except perhaps m itself), then the function has a limit at m and this limit is AND. by condition (1) $ for (here is the smallest neighborhood of the point). But then, by virtue of condition (2), the value will also be in the - neighborhood of the point AND, those. ...
, i.e α (x) + β (x) Is infinitely small.
means, | f (x) -A|<ε при |x - x 0| < δ(ε). Cледовательно, .
, or sinx 
.
2,1 < α
< 2,2
2,13 < α
< 2,14
