IV. Archimedovo Property("Cîr) ($ Nîn): (n\u003e C).

What would be the number of Cîr, there is Nîn that n\u003e c.

V. The property of continuity of valid numbers. Let two non-empty sets of AR and BìR are such that any element butÎa will not more ( a.£ b.) Any element BîB. Then the principle of continuity of Dedekinduapprove of such a number with what for all butÎA and BîB holds a condition a.£ C £. b.:

("AìR, BìR) :(" a.Îa, Bîb ® a.£ b) ($ Cîr): (" a.Îa, Bîb® a.£ C £ B).

We will identify the set r with a plurality of numerical straight points, and real numbers Call dots.

Complex numbers

Basic concepts

The initial data on the number belongs to the era of the Stone Age - Paleomelitis. This is "one", "little" and "a lot." They were recorded in the form of scubons, nodules, etc. The development of labor processes and the appearance of ownership forced a person to invent numbers and their names. Natural numbers appeared first N.Received with the score of items. Then, along with the need for an account, people have the need to measure lengths, squares, volumes, time and other values, where we had to take into account parts of the used measure. Thus arose fractions. The formal substantiation of the concepts of fractional and negative number was carried out in the 19th century. Many integers Z. - These are natural numbers, natural with a minus and zero sign. Whole I. fractional numbers Formed a combination of rational numbers Q,but it was insufficient to study continuously changing variables. Being again showed the imperfection of mathematics: the inability to solve the equation of the form h. 2 \u003d 3, in connection with which the irrational numbers appeared I.Combining a set of rational numbers Q.and irrational numbers I.- Many valid (or real) numbers R.. As a result, the numerical straight line was filled: each actual number corresponded to it. But on the set R. There is no possibility to solve the equation of the form h. 2 = – but 2. Consequently, the need to expand the concept of the number again. So in 1545 comprehensive numbers appeared. Their creator of J. Kardano called them "purely negative." The name "Mimic" introduced the Frenchman R. Descarten in 1637, in 1777, Euler offered to use the first letter of the French number i. To indicate an imaginary unit. This symbol entered into universal use thanks to K. Gauss.

During the 17th - 18th centuries, the discussion of the arithmetic nature of the differences, their geometric interpretation continued. Danchanin G. Vessel, Frenchman J. Argan and German K. Gauss independently of each other offered to portray a complex number of point on coordinate plane. Later it turned out that it is even more convenient to depict the number not the point itself, and the vector that goes to this point from the start of the coordinates.

Only by the end of the 18th - the beginning of the 19th century, complex numbers occupied a worthy place in mathematical analysis. Their first use - in theory differential equations and in the theory of hydrodynamics.

Definition 1.Integrated number called the expression of the view where x. and y. - Actual numbers, and I. - Imaginary unit,.

Two complex numbers and equal Then and only when,.

If, the number is called purely imaginary; If, the number is a valid number, it means that the set R. FROMwhere FROM - lots of complex numbers.

Conjugatean integrated number is called a complex number.

Geometric image of complex numbers.

Any integrated number can be depicted by a point. M.(x., y.) Plane Oxy.A pair of valid numbers are indicated by the coordinates of the radius-vector . A multiple correspondence can be installed between the set of vectors on the plane and many complex numbers :.

Definition 2.The actual part h..

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

Definition 3.Imaginary part integrated number is called a valid number y..

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

Re. z. postponed on the axis ( Oh)IM. z. postponed on the axis ( OY.), Then the vector corresponding to the integrated number is the radius-vector point M.(x., y.), (or M. (Re. z.IM. z.)) (Fig. 1).

Definition 4.The plane whose points are put in compliance with many complex numbers, called complex plane. The abscissa axis is called valid axisSince it is active numbers. The ordinate axis is called imaginary axisIt is purely imaginary complex numbers. Many complex numbers are indicated FROM.

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

Definition 6.Argument The integrated number is called the angle between the positive axis direction ( Oh) and vector: .

§1.1. Actual numbers
The first acquaintance with valid numbers occurs in school course mathematics. Any valid number is represented by a finite or infinite decimal fraction.

Valid (real) numbers are divided into two classes: the class of rational and class of irrational numbers. Rational called numbers that have a view where m. and n. - whole mutually simple numbers, but
. (Many rational numbers are denoted by the letter Q.). The remaining valid numbers are called irrational. Rational numbers are represented by a finite or infinite periodic fraction (the same as ordinary fractions), then the irrational will be those and only those actual numbers that can be represented by endless non-periodic fractions.

For example, the number
- rational, and
,
,
etc. - irrational numbers.

Actual numbers can also be divided into algebraic - roots of a polynomial with rational coefficients (they include, in particular, all rational numbers - the roots of the equation
) - and on transcendent - all others (for example, numbers
other).

The sets of all natural, integer, valid numbers are indicated accordingly: N.Z., R.
(initial letters of the words Naturel, Zahl, Reel).

§1.2. An image of real numbers on a numeric axis. Intervals

Geometrically (for clarity) Actual numbers are depicted by points on the endless (in both directions) of the straight line, called numerical axis. To this end, the point is taken on the direct line (the beginning of the reference - point 0), the positive direction depicted by the arrow (usually right) is indicated and the unit of scale is elected, which is deficked unlimited in both directions from point 0. So the integers are depicted. To portray the number with one decimal sign, it is necessary to divide each segment for ten parts, etc. Thus, each actual number is represented by a point on the numeric axis. Back, every point
corresponds to a valid number equal to the length of the segment
And taken with the "+" or "-" sign, depending on whether the point is to the right or the left of the beginning of the reference. Thus, a mutually valuable correspondence is established between the set of all valid numbers and the set of all points of the numerical axis. Terms "Valid" and "Point of Numerical Axis" are used as synonyms.

Symbol We will denote the actual number, and the point corresponding to it. Positive numbers There are to the right point 0, negative - to the left. If a
then on the numeric axis point lies to the left of the point . Let the point
corresponds to the number, then the number is called the coordinate point, write
; More often, the point itself is indicated by the same letter as the number. Point 0 - the beginning of the coordinates. The axis designate the letter too (Fig.1.1).

Fig. 1.1. Number axis.
A combination of all numbers lying between data numbers and is called interval or gap; Ends and can belong to him, and may not belong. Claim it. Let be
. A combination of numbers satisfying the condition
, is called interval (in a narrow sense) or an open interval, indicated by the symbol
(Fig.1.2).

Fig. 1.2. Interval
The totality of numbers such that
called a closed interval (segment, segment) and denoted through
; The numeric axis is said like this:

Fig. 1.3. Closed interval
From the open gap, it differs only in two points (ends) and. But this difference is fundamental, substantial, as we will see in the future, for example, when studying the properties of functions.

Omitting the words "Many of all numbers (points) x. Such that "etc., we note further:

and
, denotes
and
semi-open, or semi-jammed, intervals (sometimes: semi-intervals);

or
Means:
or
And denotes
or
;

or
means
or
And denotes
or
;

, denotes
many of all valid numbers. Icons
Symbols of "Infinity"; They are called incomprehensible or ideal numbers.

§1.3. Absolute value (or module) of a valid number
Definition. Absolute value (or module) numbers are called this number itself if
or
if a
. Designated absolute value symbol . So,

For example,
,
,
.

Geometrically means distance point a. before the start of coordinates. If we have two points and, then the distance between them can be represented as
(or
). For example,
That distance
.

Properties absolute values.

1. From the definition it follows that

,
, i.e
.

2. The absolute amount of the amount and difference does not exceed the amount of absolute values:
.

1) if
T.
. 2) if
then. ▲.

3.
.

, then by property 2:
.
. Similarly, if you submit
, then come to inequality
▲

4.
- It follows from the definition: consider cases
and
.

5.
, provided that
Also follows from the definition.

6. Inequality
,
means
. This inequality satisfies the points that lie between
and
.

7. Inequality
equivalent to inequality
. . This is the interval with the center at the length of the length.
. It is called
neighborhood points (numbers). If a
, the neighborhood is called puncture: this or
. (Fig.1.4).

8.
From where it follows that inequality
(
) It is equivalent to inequality
or
; And inequality
determines the set of points for which
. These are points lying outside the segment
, exactly:
and
.

§1.4. Some concepts, notation
We present some widespread concepts, designations from the theory of sets, mathematical logic and other sections of modern mathematics.

1 . Concept Set It is one of the main in mathematics, the initial, universal - and therefore cannot be determined. It can only be described (replace synonyms): This is a collection, a set of some objects, things combined by any signs. These objects are called elements sets. Examples: a plurality of sands on the shore, stars in the universe, students in the audience, the roots of the equation, the dots of the segment. Sets whose elements are the essence of the number are called numerical sets. For some standard sets, special designations are introduced, for example, N., Z., R -see § 1.1.

Let be A. - Most I. x. It is his element, then they write:
; read " x. belongs A.» (
inclusion sign for items). If the object x. Not included in A., then write
; Reading: " x. not belong A." For example,
N.; 8,51N.; But 8,51 R..

If a x. is the general designation of the elements of the set A., then write
. If it is possible to write down the designation of all elements, then write
,
and so on. A set that does not contain a single element is called an empty set and denotes the symbol ; For example, the set of roots (valid) equations
There is empty.

Many called endIf it consists of a finite number of items. If whatever natural number n neither take, in a variety A. There are elements more than N, then A. called infinite A variety: in it elements infinitely a lot.

If every element of the set ^ A. Belongs and set B.T. called part or a subset of the set B. and write
; read " A. contained in B.» (
There is a sign on set). For example, N.Z.R.If
, then they say that many A. and B. equal and write
. Otherwise write
. For example, if
, but
many roots equation
then.

A combination of elements of both sets A. and B. called Association Sets and designated
(sometimes
). The combination of elements belonging and A. and B., called intersection Sets and designated
. A combination of all elements of the set ^ A.not contained in B., called difference Sets and designated
. Schematically, these operations can be depicted as:

If there is a multiple matching between sets of sets, it is said that these sets are equivalent and written
. Any many A.Equivalent to set natural numbers N.\u003d called accounting or calculated. In other words, the set is called accountable if its elements can be numbered, to locate in the infinite sequence
, all members of which are different:
for
And it can be written in the form. Other endless sets are called unsecured. Accidents other than the most N, There will be, for example, sets
, Z. It turns out that many rational and algebraic numbers - Accidents, and equivalent among themselves many of all irrational, transcendental, real numbers and points of any interval - unpleasant. It is said that the latter have the power of the continuum (power - generalization of the concept of quantity (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 it follows", "implicit eating the root of the equation has a property from English EXIST - exist.

Recording:

, or
means: exists (at least one) item . And record
, or
means: everyone has a property. In particular, we can write:
and.

From the huge variety of all kinds set Of particular interest are the so-called numeric sets, that is, the sets whose elements are numbers. It is clear that for a comfortable work with them you need to be able to record them. With the designations and principles of recording numerical sets, we will begin this article. And then consider how numeric sets are depicted on the coordinate direct.

Navigating page.

Recording numerical sets

Let's start with the adopted designations. As you know, to designate sets are used capital letters Latin alphabet. Numerical sets like private case Sets are indicated also. For example, you can talk about numeric sets a, h, w, etc. Of particular importance are of many natural, integer, rational, real, integrated numbers, etc., for them they were adopted:

• N - the set of all natural numbers;
• Z - Many integers;
• Q - many rational numbers;
• J - many irrational numbers;
• R is a lot of valid numbers;
• C - many complex numbers.

It is clear that it is not necessary to designate a set, consisting, for example, from two numbers 5 and -7 as Q, this designation will be misled, since the letter Q is usually denoted by many rational numbers. To denote the specified numeric set, it is better to use some other "neutral" letter, for example, a.

Since we started talking about the designations, here we will remind the designation of the empty set, that is, the sets that do not contain elements. It is indicated by the sign ∅.

We also remind the designation of the belonging and non-delicacy of the element set. For this, signs ∈ - belongs and ∉ does not belong. For example, recording 5∈N means that the number 5 belongs to a set of natural numbers, and 5,7 ∉z - the decimal fraction 5.7 does not belong to the set of integers.

And we will recall the designations adopted to include one set to another. It is clear that all elements of the set n are included in the set z, thus, the numerical set n is included in Z, this is indicated as N⊂z. You can also use Z⊃n record, which means that the set of all integers Z includes the set n. The relationship is not included and does not include signs according to signs and. The signs of non-strict inclusion of the form ⊆ and ⊇ are also used, meaning it is turned on or coincided and includes or coincides.

We talked about the designations, go to the description of numerical sets. At the same time, only the main cases that are most often used in practice are affected.

Let's start with numeric sets containing a finite and small number of items. Numeric sets consisting of a finite number of elements are conveniently described by lifying all their elements. All elements numbers are recorded through the comma and are concluded, which is consistent with the common rules for describing sets. For example, a set consisting of three numbers 0, -0.25 and 4/7 can be described as (0, -0.25, 4/7).

Sometimes, when the number of elements of the numerical set is sufficiently large, but the elements are subject to some patterns, a dot is used to describe. For example, a set of all odd numbers from 3 to 99 inclusive can be written as (3, 5, 7, ..., 99).

So we smoothly approached the description of numerical sets, the number of elements of which is infinite. Sometimes they can be described using everything too much. For example, we describe the set of all natural numbers: n \u003d (1, 2. 3, ...).

Also use the description of numerical sets by specifying the properties of its elements. At the same time apply the designation (x | properties). For example, the recording (n | 8 · n + 3, n∈N) sets many of these natural numbers, which the residue 3 gives the residue at 8. This set can be described as (11.19, 27, ...).

In particular cases, numerical sets with an infinite number of elements are known sets N, Z, R, and the like. or numerical gaps. And mostly numerical sets are represented as an association The components of their individual numerical intervals and numeric sets with a finite number of elements (which we talked slightly above).

Show an example. Let the numeric set make up the number -10, -9, -8.56, 0, all numbers of the segment [-5, -1.3] and the number of the open numeric beam (7, + ∞). Due to the definition of unification of sets, the specified numeric set can be written as {−10, −9, −8,56}∪[−5, −1,3]∪{0}∪(7, +∞) . Such a record actually means a set containing all elements of sets (-10, -9, -8.56, 0), [-5, -1.3] and (7, + ∞).

Similarly, combining different numeric gaps and sets of individual numbers, you can describe any numeric set (consisting of valid numbers). It becomes clear here why such types of numerical gaps were introduced as an interval, half-interval, segment, an open numeric beam and a numeric beam: all of them in a compartment with the symbols of sets of individual numbers allow you to describe any numerical sets through their association.

Please note that when recording a numerical set, the components of its numbers and numeric gaps are ordered ascending. This is not a mandatory, but desirable condition, since the ordered numeric set is easier to imagine and depict on the coordinate direct. Also note that these records are not used numerical intervals with common elementsSince such entries can be replaced by combining numerical intervals without common elements. For example, the combination of numerical sets with common elements [-10, 0] and (-5, 3) is semi-interval [-10, 3). The same applies to the combination of numerical gaps with the same boundary numbers, for example, the union (3, 5] ∪ (5, 7] is a set (3, 7], we will separately stop on this when we learn to find the intersection and combining numeric Sets.

An image of numerical sets on the coordinate direct

In practice, it is convenient to use geometric images of numerical sets - their images on. For example, for decision inequalitiesIn which it is necessary to consider OTZ, you have to portray numerical sets to find their intersection and / or union. So it will be useful will be well able to deal with all the nuances of the image of numerical sets on the coordinate direct.

It is known that between points in the coordinate direct and valid numbers, there is a mutually unambiguous compliance, which means that the coordinate direct is the geometric model of a plurality of all valid numbers R. Thus, to portray many valid numbers, it is necessary to draw the coordinate straight with the hatching throughout it:

And often do not even indicate the beginning of the reference and a single segment:

Now let's talk about the image of numerical sets, which are some finite number of individual numbers. For example, you will depict a numeric set (-2, -0,5, 1,2). The geometric manner of this set consisting of three numbers -2, -0.5 and 1.2 will be three points of the coordinate direct with the corresponding coordinates:

Note that usually for the needs of practice there is no need to perform the drawing for sure. It is often quite a schematic drawing, which implies an optional maintenance of scale, while it is important only to maintain the mutual location of the points relative to each other: any point with a smaller coordinate should be the left of the point with the greater coordinate. The previous drawing will schematically look like this:

Separately, out of all sorts of numeric sets, numerical intervals (intervals, semi-intervals, rays, etc.) are isolated, which represent their geometric images, we figured out in detail in the section. Here we will not repeat.

And it remains to be stopped only on the image of numerical sets, which are combining several numerical intervals and sets consisting of individual numbers. There is nothing cunning here: in terms of the meaning of combining in these cases, on the coordinate direct, you need to portray all the components of the set of this numerical set. As an example, we show the image of a numerical set (−∞, −15)∪{−10}∪[−3,1)∪ (Log 2 5, 5) ∪ (17, + ∞):

And we will focus on fairly common cases when the numeric set represents the entire set of valid numbers, with the exception of one or more points. Such sets are often set by the conditions of type x ≠ 5 or x ≠ -1, x ≠ 2, x ≠ 3.7, and the like. In these cases, they are geometrically represent the entire coordinate direct, with the exception of the corresponding points. In other words, from the coordinate direct you need to "buy" these points. They are depicted with circles with an empty center. For clarity, you will show a numerical set corresponding to the conditions (This is a lot of essentially):

Summarize. Ideally, the information of the previous items should form the same look at the record and image of numerical sets, as well as a look at the individual numerical gaps: the recording of a numerical set should immediately give its image on the coordinate direct, and on the image on the coordinate direct, we must be ready to describe The corresponding numerical set through the combination of individual intervals and sets consisting of individual numbers.

Bibliography.

• Algebra: studies. For 8 cl. general education. institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorov]; Ed. S. A. Telikovsky. - 16th ed. - M.: Enlightenment, 2008. - 271 p. : IL. - ISBN 978-5-09-019243-9.
• Mordkovich A. G. Algebra. Grade 9. In 2 tsp. 1. Tutorial for students of general educational institutions / A. Mordkovich, P. V. Semenov. - 13th ed., Even. - M.: Mnemozina, 2011. - 222 C.: Il. ISBN 978-5-346-01752-3.