Geometrically real numbers, like rational numbers, are represented by points on a straight line.
Let l - an arbitrary straight line, and O - some of its points (Fig. 58). Every positive real number α put in correspondence the 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
2,1 < α
< 2,2
2,13 < α
< 2,14
etc. It is obvious that the point A in this case must be on the 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 line l the only point A, which we consider as the geometric image of a real number α = 2,1356... .
Likewise, every negative real number β put in correspondence the 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 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 - point B, lying 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 straightedge, one can find the points corresponding to the real numbers √2, √3, √4, √5, etc. To do this, we first of all show how to 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 restore a perpendicular to this segment and set aside on it the segment AC, equal to the segment AB. Then, applying the Pythagorean theorem to the right triangle ABC, we get; BC \u003d √AB 2 + AC 2 \u003d √1 + 1 \u003d √2
Therefore, the segment BC has length √2. Now let us restore the perpendicular to the segment BC at the point C and choose the point D on it so that the segment CD is equal to one AB length. Then from right triangle BCD find:
ВD \u003d √BC 2 + CD 2 \u003d √2 + 1 \u003d √3
Therefore, the segment BD has length √3. Continuing the described process further, we could get segments BE, BF, ..., whose lengths are expressed by the numbers √4, √5, etc.
Now on the line 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, the segment BC to the right of the point O (Fig. 61), we get the point C, which serves as a geometric representation 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, etc.
However, one should not think that with the help of a compass and a ruler on a number line l one can find a 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 ... . So on the number line l using such constructions, it is impossible to indicate a 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 of the line l . This point will be separated from the starting point O 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 . 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 a straight line be considered as a geometric image of some rational numbers? At that time 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 a straight line represents a rational number. But in this case, another question arises: can any point of the real 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 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 segment VO. Finally, the point O serves as a geometric representation of the number zero. It is clear that two distinct points of the line l cannot be the geometric image of the same real number.
For the reasons stated above, a straight line on which some point O is indicated as the "initial" point (for a given unit of length) is called number line.
Conclusion. Many of all real numbers and the set of all points of the real 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.
An expressive geometric representation of the system of rational numbers can be obtained as follows.
Rice. 8. Number axis
On some straight line, the "numerical axis", we mark the segment from 0 to 1 (Fig. 8). This sets 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 number axis, namely, positive numbers are marked to the right, and negative ones to the left of point 0. To depict numbers with a denominator, we divide each of the obtained segments of unit length by 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. This ratio is reflected on the real axis 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 indicated geometric ratio is established for any pair of rational points, it is natural to try to generalize the arithmetic inequality relation in such a way to preserve this geometric order for the points in question. This succeeds if we accept the following definition: one says 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 from this (for ) that the points (numbers) between are those that
simultaneously Each such pair of points, together with all points between them, is called a segment (or segment) and is denoted (and the set of intermediate points alone is called an interval (or interval), denoted by
The distance of an arbitrary point A from the origin 0, considered as 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 valid regardless of the signs
A fact of fundamental importance is expressed by the following proposition: rational points are everywhere dense on the number line. The meaning of this statement is that inside any interval, no matter how small it may be, 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 the given 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 can be imagined), within which there would be no rational points. This implies a further corollary: every interval contains an infinite number of rational points. Indeed, if some 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 proved.
From the vast variety of sets of particular interest are the so-called number sets, that is, sets whose elements are numbers. It is clear that for comfortable work with them you need to be able to write them down. With the notation and principles of writing numerical sets, we will begin this article. And then we will consider how numerical sets are depicted on the coordinate line.
Page navigation.
Writing Numeric Sets
Let's start with the accepted notation. As you know, to denote sets, we use capital letters Latin alphabet. Numerical sets, as a special case of sets, are also denoted. For example, we can talk about numerical sets A , H , W , etc. Of particular importance are the sets of natural, integer, rational, real, complex numbers etc., their designations were adopted for them:
- N is the set of all natural numbers;
- Z is the set of integers;
- Q is the set of rational numbers;
- J is the set of irrational numbers;
- R is the set of real numbers;
- C is the set of complex numbers.
From this it is clear that it is not necessary to denote a set consisting, for example, of two numbers 5 and −7 as Q, this designation will be misleading, since the letter Q usually denotes the set of all rational numbers. To designate the specified numerical set, it is better to use some other "neutral" letter, for example, A.
Since we are talking about notation, here we also recall the notation of an empty set, that is, a set that does not contain elements. It is denoted by the sign ∅.
Let us also recall the designation of membership and non-membership of an element in a set. To do this, use the signs ∈ - belongs and ∉ - does not belong. For example, the notation 5∈N means that the number 5 belongs to the set of natural numbers, and 5.7∉Z - decimal 5,7 does not belong to the set of integers.
Let us also recall the notation adopted to include one set in another. It is clear that all elements of the set N are included in the set Z , so the number set N is included in Z , this is denoted as N⊂Z . You can also use the notation Z⊃N , which means that the set of all integers Z includes the set N . Relations not included and not included are denoted by the signs ⊄ and , respectively. The non-strict inclusion signs of the form ⊆ and ⊇ are also used, meaning, respectively, included or matches and includes or matches.
We talked about the notation, let's move on to the description of numerical sets. In this case, we will only touch on the main cases that are most often used in practice.
Let's start with numerical sets containing a finite and small number of elements. Numerical sets consisting of a finite number of elements can be conveniently described by listing all their elements. All number elements are written separated by commas and enclosed in , which is consistent with common set description rules. For example, a set consisting of three numbers 0 , −0.25 and 4/7 can be described as (0, −0.25, 4/7) .
Sometimes, when the number of elements of a numerical set is large enough, but the elements obey some pattern, ellipsis is used to describe. For example, the set of all odd numbers from 3 to 99 inclusive can be written as (3, 5, 7, ..., 99) .
So we smoothly approached the description of numerical sets, the number of elements of which is infinite. Sometimes they can be described using all the same ellipsis. For example, let's describe the set of all natural numbers: N=(1, 2. 3, …) .
They also use the description of numerical sets by indicating the properties of its elements. In this case, the notation (x| properties) is used. For example, the notation (n| 8 n+3, n∈N) defines the set of such natural numbers that, when divided by 8, give a remainder of 3 . The same set can be described as (11,19, 27, ...) .
In special cases, numerical sets with an infinite number of elements are known sets N , Z , R , etc. or number gaps. And in general, numerical sets are represented as Union individual numerical intervals that make them up and numerical sets with a finite number of elements (which we talked about a little higher).
Let's show an example. Let the number set be the numbers −10 , −9 , −8.56 , 0 , all the numbers of the interval [−5, −1.3] and the numbers of the open number ray (7, +∞) . By virtue of the definition of the union of sets, the indicated numerical set can be written as {−10, −9, −8,56}∪[−5, −1,3]∪{0}∪(7, +∞) . Such a notation actually means a set containing all elements of the sets (−10, −9, −8.56, 0) , [−5, −1.3] and (7, +∞) .
Similarly, by combining various numerical ranges and sets of individual numbers, any number set (consisting of real numbers) can be described. Here it becomes clear why such types of numerical intervals as an interval, a half-interval, a segment, an open numerical ray and a numerical ray were introduced: all of them, coupled with the notation of sets of individual numbers, make it possible to describe any numerical sets through their union.
Please note that when writing a numerical set, its constituent numbers and numerical intervals are sorted in ascending order. This is not a mandatory, but desirable condition, since an ordered numerical set is easier to represent and depict on a coordinate line. Also note that such entries do not use numeric ranges with common elements, since such entries can be replaced by the union of numeric ranges without common elements. For example, the union of numerical sets with common elements [−10, 0] and (−5, 3) is a half-interval [−10, 3) . The same applies to the union of numerical intervals with the same boundary numbers, for example, the union (3, 5]∪(5, 7] is a set (3, 7] , we will dwell on this separately when we learn to find the intersection and union of numerical sets .
Image of number sets on the coordinate line
In practice, it is convenient to use the geometric images of numerical sets - their images on . For example, when solving inequalities, in which it is necessary to take into account the ODZ, it is necessary to depict numerical sets in order to find their intersection and / or union. So it will be useful to understand well all the nuances of the representation of numerical sets on the coordinate line.
It is known that there is a one-to-one correspondence between the points of the coordinate line and the real numbers, which means that the coordinate line itself is a geometric model of the set of all real numbers R. Thus, in order to depict the set of all real numbers, it is necessary to draw a coordinate line with hatching along its entire length:
And often they don’t even indicate the origin and a single segment: 
Now let's talk about the image of numerical sets, which are some finite number of individual numbers. For example, let's draw the number set (−2, −0.5, 1.2) . The geometric image of this set, consisting of three numbers -2, -0.5 and 1.2 will be three points of the coordinate line with the corresponding coordinates: 
Note that usually for the needs of practice there is no need to perform the drawing accurately. Often a schematic drawing is sufficient, which means that it is not necessary to maintain scale, while it is only important to maintain mutual arrangement points relative to each other: any point with a smaller coordinate must be to the left of a point with a larger coordinate. The previous drawing will schematically look like this: 
Separately, from all possible numerical sets, numerical intervals (intervals, half-intervals, rays, etc.) are distinguished, which represent their geometric images, we examined in detail in the section. We will not repeat ourselves here.
And it remains only to dwell on the image of numerical sets, which are the union of several numerical intervals and sets consisting of individual numbers. There is nothing tricky here: according to the meaning of the union, in these cases, on the coordinate line, you need to depict all the components of the set of a given numerical set. As an example, let's show the image of a number set (−∞, −15)∪{−10}∪[−3,1)∪
(log 2 5, 5)∪(17, +∞) : 
And let's dwell on quite common cases when the depicted numerical set is the entire set of real numbers, with the exception of one or more points. Such sets are often specified by conditions like x≠5 or x≠−1 , x≠2 , x≠3,7 etc. In these cases, geometrically, they represent the entire coordinate line, with the exception of the corresponding points. In other words, these points must be “punched out” from the coordinate line. They are depicted as circles with an empty center. For clarity, we depict a numerical set corresponding to the conditions 
(this set is essentially ): 
Summarize. Ideally, the information of the previous paragraphs should form the same view of the recording and representation of numerical sets as the view of individual numerical intervals: the recording of a numerical set should immediately give its image on the coordinate line, and from the image on the coordinate line, we should be ready to easily describe the corresponding numerical set through the union of individual gaps and sets consisting of individual numbers.
Bibliography.
- Algebra: textbook for 8 cells. general education institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova]; ed. S. A. Telyakovsky. - 16th ed. - M. : Education, 2008. - 271 p. : ill. - ISBN 978-5-09-019243-9.
- Mordkovich A. G. Algebra. Grade 9 At 2 p.m. Part 1. Student's textbook educational institutions/ A. G. Mordkovich, P. V. Semenov. - 13th ed., Sr. - M.: Mnemosyne, 2011. - 222 p.: ill. ISBN 978-5-346-01752-3.
§1.1. Real numbers
The first acquaintance with real numbers occurs in school course mathematics. 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 numbers are called that look like , where m And n- integer mutually prime numbers, 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 (same as common fractions), then those and only those real numbers that can be represented by infinite non-periodic fractions will be irrational.
For example, number 
- rational and 
, 
, 
and so on. are irrational numbers.
Real numbers can also be divided into algebraic ones - the roots of a polynomial with rational coefficients (these include, in particular, all rational numbers - the roots of the equation 
) - and transcendental - all the rest (for example, numbers 
and others).
The sets of all natural, integer, real numbers are denoted respectively as follows: NZ, R
(the initial letters of the words Naturel, Zahl, Reel).
§1.2. Image of real numbers on the number line. Intervals
Geometrically (for clarity), real numbers are represented by points on an infinite (in both directions) straight line, called numerical
axis. For this purpose, a point is taken on the line under consideration (the reference point 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 on both sides of the point 0. This is how integers are displayed. To depict a number with one decimal place, each segment must be divided into ten parts, and so on. Thus, each real number is represented by a point on the number line. Conversely, every point 
corresponds to a real number equal to the length of the segment 
and taken with the sign "+" or "-", 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 "point of the numerical axis" are used as synonyms.
Symbol 
we will denote both a real number and a point corresponding to it. Positive numbers are located to the right of the point 0, negative - to the left. If 
, then on the real 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 of coordinates. The axis is also denoted by the letter 
(fig.1.1).
Rice. 1.1. Numeric axis.
The set of all numbers lying between given numbers and is called an interval or gap; the ends may or may not belong to him. Let's clarify this. Let 
. The set of numbers that satisfy the condition 
, is called an interval (in the narrow sense) or an open interval, denoted by the symbol 
(fig.1.2).
Rice. 1.2. Interval
A collection of numbers such that 
is called a closed interval (segment, segment) and is denoted by 
; on the numerical axis is marked as follows:
Rice. 1.3. closed interval
It differs from an open gap only in 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. Absolute value (or modulus) of a real number
Definition. Absolute value (or module) number is called the number itself, if 
or 
If 
. The absolute value is denoted by the symbol 
. So, 
For example, 
, 
, 
.
Geometrically means point distance a to the origin of coordinates. If we have two points and , then the distance between them can be represented as 
(or 
). For example, 
that distance 
.
Properties of absolute values.
1. It follows from the definition that 
, 
, that is 
.
2. The absolute value of the sum and difference does not exceed the sum of the absolute values: 
.
1) If 
, That 
. 2) If 
, That . ▲
3. 
.
, then by property 2: 
, i.e. 
. Similarly, if we imagine 
, then we arrive at the inequality 
▲
4. 
– follows from the definition: consider cases 
And 
.
5. 
, provided that 
The same follows from the definition.
6. Inequality 
,
, means 
. This inequality is satisfied by the points that lie between 
And 
.
7. Inequality 
is equivalent to the inequality 
, i.e. . It is an interval centered at the point of length 
. It is called 
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 
determines the set of points for which 
, i.e. are points outside the segment 
, exactly: 
And 
.
§1.4. Some concepts, designations
Let us give some widely used concepts, notations from set theory, mathematical logic and other branches of modern mathematics.
1 . concept sets is one of the basic in mathematics, initial, universal - and therefore cannot be defined. 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 the equation, points of the segment. Sets whose elements are numbers are called numerical sets. For some standard sets, special notation is introduced, for example, N,Z,R- see § 1.1.
Let A- set and x is its element, then we write: 
; reads " x belongs A» ( 
inclusion sign for elements). If the object x not included in A, then they write 
; reads: " x do not belong A". For example, 
N; 8,51
N; but 8.51 
R.
If x is a general designation for elements of a set A, then they write 
. If it is possible to write out the designation of all elements, then write 
, 
etc. A set that does not contain a single element is called an empty set and is denoted by the symbol ; for example, the set of (real) roots of the equation 
there is an empty one.
The set is called final if it consists of a finite number of elements. If, however, no matter what natural number N is taken, in the set A there are more elements than N, then A called endless set: there are infinitely many elements in it.
If every element of the set ^A belongs to the set B, That 
called a part or subset of a set B and write 
; reads " A contained in B» ( 
there is an inclusion sign for sets). For example, NZR. If 
, then we say that the sets A And B equal and write 
. Otherwise, write 
. For example, if 
, A 
set of roots of the equation 
, That .
The set of elements of both sets A And B called association sets and is denoted 
(Sometimes 
). The set of elements belonging to and A And B, is called intersection sets and is denoted 
. The set of all elements of the set ^A, which are not included in B, is called difference sets and is denoted 
. Schematically, these operations can be depicted as follows:
If a one-to-one correspondence can be established between the elements of the sets, then they say that these sets are equivalent and write 
. Any set A, equivalent to the set of natural numbers N= called countable or countable. In other words, a set is called countable if its elements can be numbered, placed in an infinite subsequence 
, 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, sets 
, Z. It turns out that the sets of all rational and algebraic numbers are countable, and the equivalent sets of all irrational, transcendental, real numbers and points of any interval are uncountable. They say that the latter have the power of the continuum (the power 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 "follows", "implies there is a root of the equation has a property from English Exist- exist.
Recording: 
, or 
, means: there is (at least one) object that has the property 
. A 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, the "numerical axis", we mark the segment from O to 1 (Fig. 8). This sets 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 number axis, it is positive numbers that are marked to the right, and negative ones to the left of point 0. To depict numbers with a 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 of n corresponding to all natural numbers, then each rational number will be depicted by some point on the numerical axis. We shall 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 A 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 is possible if we accept the following definition: we say that the rational number A less than the rational number B (A is greater than the number A (B>A), if difference B-A positive. This implies (for A between A and B are those that are both > A and segment (or segment) and is denoted by [A, B] (and the set of only intermediate points - interval(or gap), 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 is denoted 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 A and B.
A fact of fundamental importance is expressed by the following proposition: rational points are everywhere dense on the number line. The meaning of this statement is that inside any interval, no matter how small it may be, there are 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 the given interval. So, there is no such interval on the number line (even the smallest imaginable) 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 no longer be rational points inside the interval formed by two neighboring such points, and this contradicts what has just been proved.
