Display of real numbers on the number axis. Intervals

Complex numbers

Basic concepts

The initial data on the number date back to the Stone Age - paleomelite. These are "one", "little" and "many". They were recorded in the form of notches, nodules, etc. The development of labor processes and the emergence of property forced man to invent numbers and their names. Natural numbers appeared first Nreceived when counting items. Then, along with the need for counting, people had a need to measure lengths, areas, volumes, time and other quantities, where it was necessary to take into account parts of the measure used. This is how fractions arose. The formal substantiation of the concepts of fractional and negative numbers was carried out in the 19th century. Lots of integers Z - these are natural numbers, natural numbers with minus and zero signs. Integers and fractional numbers formed a set of rational numbers Q,but it also turned out to be insufficient for studying continuously changing variables. Genesis again showed the imperfection of mathematics: the impossibility of solving an equation of the form x 2 \u003d 3, in connection with which irrational numbers appeared I.Union of the set of rational numbers Qand irrational numbers I- a set of real (or real) numbers R... As a result, the number line was filled: a point on it corresponded to each real number. But on the set R there is no way to solve an equation of the form x 2 = – and 2. Consequently, the need arose again to expand the concept of number. This is how complex numbers appeared in 1545. Their creator J. Cardano called them "purely negative." The name "imaginary" was introduced in 1637 by the Frenchman R. Descartes, in 1777 Euler proposed to use the first letter of the French number i to denote an imaginary unit. This symbol came into general use thanks to K. Gauss.

During the 17th - 18th centuries, the discussion of the arithmetic nature of imaginations, their geometric interpretation continued. The Dane G. Wessel, the Frenchman J. Argan and the German K. Gauss independently of each other proposed to represent a complex number by a point on the coordinate plane. Later it turned out that it is even more convenient to represent the number not by the point itself, but by the vector going to this point from the origin.

Only by the end of the 18th and the beginning of the 19th century did complex numbers take their rightful place in mathematical analysis. Their first use was in the theory of differential equations and in the theory of hydrodynamics.

Definition 1.Complex number is called an expression of the form, where x and y Are real numbers, and i Is an imaginary unit,.

Two complex numbers and are equal if and only if,.

If, then the number is called purely imaginary; if, then the number is a real number, this means that the set R FROMwhere FROM - a set of complex numbers.

Conjugatedto a complex number is called a complex number.

Geometric representation of complex numbers.

Any complex number can be represented by a dot M(x, y) plane Oxy.A pair of real numbers denotes the coordinates of the radius vector , i.e. between the set of vectors on the plane and the set of complex numbers, you can establish a one-to-one correspondence:.

Definition 2.The real part x.

Designation: x \u003d Re z(from the Latin Realis).

Definition 3.The imaginary part complex number is called a real number y.

Designation: y \u003d Im z(from Latin Imaginarius).

Re z is plotted on the axis ( Oh), Im z is plotted on the axis ( Oy), then the vector corresponding to the complex number is the radius vector of the point M(x, y), (or M (Re z, Im z)) (Fig. 1).

Definition 4.The plane, the points of which are associated with the set of complex numbers, is called complex plane... The abscissa axis is called real axissince it contains real numbers. The ordinate axis is called imaginary axis, it contains purely imaginary complex numbers. The set of complex numbers is denoted FROM.

Definition 5.Modulecomplex number z = (x, y) is the length of the vector:, i.e. .

Definition 6.The argument complex number is the angle between the positive direction of the axis ( Oh) and vector: .

REAL NUMBERS II

§ 37 Geometric representation of rational numbers

Let be Δ is a segment taken as a unit of length, and l - an arbitrary straight line (fig. 51). Take some point on it and designate it with the letter O.

To every positive rational number m / n match the point of the 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 to the number 5/4, point B, lying to the right of O at a distance of 5/4 units of length. To every negative rational number k / l we put in correspondence a point of 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 the point C, lying to the left of O at a distance of 3 units of length, and the number - 3/2 point D, lying to the left of O at a distance of 3/2 units of length. Finally, to the rational number "zero" we associate the point O.

Obviously, with the chosen correspondence, equal rational numbers (for example, 1/2 and 2/4) will correspond to the same point, and not to equal numbers between different points of the line. Suppose that the number m / n corresponds to point P, 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 to the left of point Q (Fig. 52, b).

So, any rational number can be geometrically depicted as a certain, well-defined point of a straight line. Is the converse true? Can any point on a straight line be regarded as a geometric image of some rational number? We will postpone the decision of this question until § 44.

Exercises

296. Draw the following rational numbers by points of a straight 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 straight line, which serve as a geometric representation of rational numbers and and b a + b and a - b .

299. Two points are given on a straight line, which serve as a geometric representation of rational numbers a + b and a - b ... Find on this line the points representing numbers and and b .

CHAPTER 1. Variables and Functions

§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
and others).

The sets of all natural, integer, real numbers are denoted as follows: NZ, R
(initial letters of the words Naturel, Zahl, Reel).

§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 depicted. 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
, 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
... 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
, 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, notation 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 by 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 do not belong A". For instance,
N; 8,51N; but 8.51 R.

If 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
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.

An expressive geometric representation of the system of rational numbers can be obtained as follows.

On some straight line, "numerical axis", 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, it is the positive numbers that are marked to the right, and the negative ones, to the left of point 0. To represent the numbers with the denominator n, we divide each of the obtained segments of unit length into n equal parts; division points will represent fractions with denominator n. If we do this for the values \u200b\u200bof n corresponding to all natural numbers, then each rational number will be represented by some point of 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 was defined for any pair of rational points, 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 succeeds if we accept the following definition: they say that a rational number A lessthan a rational number B (is greater than the number A (B\u003e A) if the difference B-A is positive. Hence it follows (for A between A and B are those that are simultaneously\u003e A and a segment (or segment) and is denoted by [A, B] (and the set of only intermediate points is interval (or in between), denoted by (A, B)).

The distance of an arbitrary point A from the origin 0, considered as a positive number, is called absolute value A and denoted by the symbol

The concept of "absolute value" is defined as follows: if A≥0, then | A | \u003d A; if A

| A + B | ≤ | A | + | B |,

which is true regardless of the signs of A and B.

The 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 statement made, it is enough to take a number n so large that the interval will be less than the given interval (A, B); then at least one of the points of the view will be within this interval. So, there is no such interval on the number axis (not even the smallest one that can be imagined), within which there would be no rational points. This implies a further corollary: every interval contains an infinite set of rational points. Indeed, if some interval contained only a finite number of rational points, then there would be no rational points inside the interval formed by two adjacent such points, and this contradicts what has just been proved.

Geometrically, real numbers, as well as rational numbers, are represented by points on a straight line.

Let be l - an arbitrary 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
2,1 < α < 2,2
2,13 < α < 2,14

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, the number "zero" is associated with the point O.

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 we will restore the perpendicular to the segment BC at point C and select 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

Therefore, the segment BD has length √3. Continuing the described process further, we could get segments BE, BF, ..., the lengths of which are expressed by the 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 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 corresponding to 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 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 ... 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 - 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.

Output. 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.