Of the large number of diverse sets, numerical sets are especially interesting and important, i.e. those sets whose elements are numbers. Obviously, to work with numerical sets you need to have the skill of writing them down, as well as depicting them on a coordinate line.
Writing numerical sets
The generally accepted designation for any set is capital Latin letters. Number sets are no exception. For example, we can talk about number sets B, F or S, etc. However, there is also a generally accepted marking of numerical sets depending on the elements included in it:
N – set of all natural numbers; Z – set of integers; Q – set of rational numbers; J – set of irrational numbers; R – set of real numbers; C is the set of complex numbers.
It becomes clear that designating, for example, a set consisting of two numbers: - 3, 8 with the letter J can be misleading, since this letter marks a set of irrational numbers. Therefore, to designate the set - 3, 8, it would be more appropriate to use some kind of neutral letter: A or B, for example.
Let us also recall the following notation:
- ∅ is an empty set or a set without constituent elements;
- ∈ or ∉ is a sign of whether an element belongs or does not belong to a set. For example, the notation 5 ∈ N means that the number 5 is part of the set of all natural numbers. The notation - 7, 1 ∈ Z reflects the fact that the number - 7, 1 is not an element of the set Z, because Z – set of integers;
- signs that a set belongs to a set:
⊂ or ⊃ - “included” or “includes” signs, respectively. For example, the notation A ⊂ Z means that all elements of the set A are included in the set Z, i.e. the number set A is included in the set Z. Or vice versa, the notation Z ⊃ A will clarify that the set of all integers Z includes the set A.
⊆ or ⊇ are signs of the so-called non-strict inclusion. Mean "included or matches" and "includes or matches" respectively.
Let us now consider the scheme for describing numerical sets using the example of the main standard cases most often used in practice.
First we consider number sets containing finite and a small amount of elements. It is convenient to describe such a set by simply listing all its elements. Elements in the form of numbers are written, separated by a comma, and enclosed in curly braces (which corresponds to the general rules for describing sets). For example, we write the set of numbers 8, - 17, 0, 15 as (8, - 17, 0, 15).
It happens that the number of elements of a set is quite large, but they all obey a certain pattern: then an ellipsis is used in the description of the set. For example, we write the set of all even numbers from 2 to 88 as: (2, 4, 6, 8, …, 88).
Now let's talk about describing numerical sets in which the number of elements is infinite. Sometimes they are described using the same ellipsis. For example, we write the set of all natural numbers as follows: N = (1, 2, 3, ...).
It is also possible to write a numerical set with an infinite number of elements by specifying the properties of its elements. The notation (x | properties) is used. For example, (n | 8 n + 3, n ∈ N) defines the set of natural numbers that, when divided by 8, leave a remainder of 3. This same set can be written as: (11, 19, 27, …).
In special cases, numerical sets with an infinite number of elements are the well-known sets N, Z, R, etc., or numerical intervals. But basically, numerical sets are a union of their constituent numerical intervals and numerical sets with a finite number of elements (we talked about them at the very beginning of the article).
Let's look at an example. Suppose the components of a certain numerical set are the numbers - 15, - 8, - 7, 34, 0, as well as all the numbers of the segment [- 6, - 1, 2] and the numbers of the open number line (6, + ∞). In accordance with the definition of a union of sets, we write the given numerical set as: ( - 15 , - 8 , - 7 , 34 ) ∪ [ - 6 , - 1 , 2 ] ∪ ( 0 ) ∪ (6 , + ∞) . Such a notation actually means a set that includes all the elements of the sets (- 15, - 8, - 7, 34, 0), [- 6, - 1, 2] and (6, + ∞).
In the same way, by combining various numerical intervals and sets of individual numbers, it is possible to give a description of any numerical set consisting of real numbers. Based on the above, it becomes clear why they are introduced different kinds number intervals such as interval, half-interval, segment, open number ray and number ray. All these types of intervals, together with the designations of sets of individual numbers, make it possible to describe any numerical set through their combination.
It is also necessary to pay attention to the fact that individual numbers and numerical intervals when writing a set can be ordered in ascending order. In general, this is not a mandatory requirement, but such ordering allows you to represent a numerical set more simply, and also correctly display it on the coordinate line. It is also worth clarifying that such records do not use numerical intervals with common elements, since these records can be replaced by combining numerical intervals, eliminating common elements. For example, the union of numerical sets with common elements [- 15, 0] and (- 6, 4) will be the half-interval [- 15, 4). The same applies to the union of numerical intervals with the same boundary numbers. For example, the union (4, 7] ∪ (7, 9] is the set (4, 9]. This point will be discussed in detail in the topic of finding the intersection and union of numerical sets.
In practical examples, it is convenient to use the geometric interpretation of numerical sets - their image on a coordinate line. For example, this method will help in solving inequalities in which it is necessary to take into account ODZ - when you need to display numerical sets in order to determine their union and/or intersection.
We know that there is a one-to-one correspondence between the points of the coordinate line and the real numbers: the entire coordinate line is a geometric model of the set of all real numbers R. Therefore, to depict the set of all real numbers, we draw a coordinate line and apply shading along its entire length:
Often the origin and the unit segment are not indicated:
Consider an image of number sets consisting of a finite number of individual numbers. For example, let's display a number set (- 2, - 0, 5, 1, 2). The geometric model of a given set will be three points of the coordinate line with the corresponding coordinates:
In most cases, it is possible not to maintain the absolute accuracy of the drawing: a schematic image without respect to scale, but maintaining the relative position of the points relative to each other, is quite sufficient, i.e. any point with a larger coordinate must be to the right of a point with a smaller one. With that said, an existing drawing might look like this:
Separately from the possible numerical sets, numerical intervals are distinguished: intervals, half-intervals, rays, etc.)
Now let's consider the principle of depicting numerical sets, which are the union of several numerical intervals and sets consisting of individual numbers. There is no difficulty in this: according to the definition of a union, it is necessary to display on the coordinate line all the components of the set of a given numerical set. For example, let's create an illustration of the number set (- ∞ , - 15) ∪ ( - 10 ) ∪ [ - 3 , 1) ∪ ( log 2 5 , 5 ) ∪ (17 , + ∞) .
It is also quite common for the number set to be drawn to include the entire set of real numbers except one or more points. Such sets are often specified by conditions like x ≠ 5 or x ≠ - 1, etc. In such cases, the sets in their geometric model are the entire coordinate line with the exception of given points. It is generally accepted to say that these points need to be “plucked out” from the coordinate line. The punctured point is depicted as a circle with an empty center. To reinforce what was said practical example, display on the coordinate line a set with the given condition x ≠ - 2 and x ≠ 3:
The information provided in this article is intended to help you gain the skill of seeing the recording and representation of numerical sets as easily as individual numerical intervals. Ideally, the written numerical set should be immediately represented in the form of a geometric image on the coordinate line. And vice versa: from the image, a corresponding numerical set should be easily formed through the union of numerical intervals and sets that are separate numbers.
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From a huge variety of all kinds sets Of particular interest are the so-called number sets, that is, sets whose elements are numbers. It is clear that to work comfortably with them you need to be able to write them down. We will begin this article with the notation and principles of writing numerical sets. Next, let’s look at how numerical sets are depicted on a coordinate line.
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Writing numerical sets
Let's start with accepted notations. As you know, capital letters of the Latin alphabet are used to denote sets. Number sets like special case sets are also denoted. For example, we can talk about number sets A, H, W, etc. The sets of natural, integer, rational, real, complex numbers, etc. are of particular importance; their own notations have been adopted for them:
- N – set of all natural numbers;
- Z – set of integers;
- Q – set of rational numbers;
- J – set of irrational numbers;
- R – set of real numbers;
- C is the set of complex numbers.
From here it is clear that you should not 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 denote the specified numerical set, it is better to use some other “neutral” letter, for example, A.
Since we are talking about notation, let us also recall here about 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 whether an element belongs or does not belong to 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 - the decimal fraction 5.7 does not belong to the set of integers.
And let us also recall the notation adopted for including one set into another. It is clear that all elements of the set N are included in the set Z, thus 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. The relations not included and not included are indicated by ⊄ and , respectively. Non-strict inclusion signs of the form ⊆ and ⊇ are also used, meaning included or coincides and includes or coincides, respectively.
We've talked about 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. It is convenient to describe numerical sets consisting of a finite number of elements by listing all their elements. All number elements are written separated by commas and enclosed in , which is consistent with the general 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 a numerical set is quite large, but the elements obey a certain pattern, an ellipsis is used for description. 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 ellipses. 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) specifies the set of natural numbers that, when divided by 8, leave a remainder of 3. This same set can be described as (11,19, 27, ...).
In special cases, numerical sets with an infinite number of elements are the known sets N, Z, R, etc. or number intervals. Basically, numerical sets are represented as Union their constituent individual numerical intervals and numerical sets with a finite number of elements (which we talked about just above).
Let's show an example. Let the number set consist of the numbers −10, −9, −8.56, 0, all the numbers of the segment [−5, −1,3] and the numbers of the open number line (7, +∞). Due to the definition of a union of sets, the specified numerical set can be written as {−10, −9, −8,56}∪[−5, −1,3]∪{0}∪(7, +∞) . This notation actually means a set containing all the elements of the sets (−10, −9, −8.56, 0), [−5, −1.3] and (7, +∞).
Similarly, by combining different number intervals 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 interval, half-interval, segment, open numerical ray and numerical ray were introduced: all of them, coupled with notations for sets of individual numbers, make it possible to describe any numerical sets through their union.
Please note that when writing a number set, its constituent numbers and numerical intervals are ordered in ascending order. This is not a necessary but desirable condition, since an ordered numerical set is easier to imagine and depict on a coordinate line. Also note that such records do not use numeric intervals with common elements, since such records can be replaced by combining numeric intervals without common elements. For example, the union of numerical sets with common elements [−10, 0] and (−5, 3) is the 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
Representation of number sets on a coordinate line
In practice, it is convenient to use geometric images of numerical sets - their images on. For example, when solving inequalities, in which it is necessary to take into account ODZ, it is necessary to depict numerical sets in order to find their intersection and/or union. So it will be useful to have a good understanding of all the nuances of depicting numerical sets on a 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, to depict the set of all real numbers, you need to draw a coordinate line with shading along its entire length:
And often they don’t even indicate the origin and the unit segment: 
Now let's talk about the image of numerical sets, which represent a certain finite number of individual numbers. For example, let's depict 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 practical purposes there is no need to carry out the drawing exactly. Often a schematic drawing is sufficient, which implies that it is not necessary to maintain the scale; in this case, it is only important to maintain the relative position of the 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 kinds of numerical sets, numerical intervals (intervals, half-intervals, rays, etc.) are distinguished, which represent their geometric images; we examined them in detail in the section. We won't repeat ourselves here.
And it remains only to dwell on the image of numerical sets, which are a 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 it is necessary to depict all the components of the set of a given numerical set. As an example, let's show an image of a number set (−∞, −15)∪{−10}∪[−3,1)∪
(log 2 5, 5)∪(17, +∞) : 
And let us dwell on fairly common cases when the depicted numerical set represents the entire set of real numbers, with the exception of one or several 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 need to be “plucked out” from the coordinate line. They are depicted as circles with an empty center. For clarity, let us depict a numerical set corresponding to the conditions 
(this set essentially exists): 
Summarize. Ideally, the information from the previous paragraphs should form the same view of the recording and depiction 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 intervals and sets consisting of individual numbers.
Bibliography.
- Algebra: textbook for 8th grade. general education institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova]; edited by S. A. Telyakovsky. - 16th ed. - M.: Education, 2008. - 271 p. : ill. - ISBN 978-5-09-019243-9.
- Mordkovich A. G. Algebra. 9th grade. At 2 p.m. Part 1. Textbook for students educational institutions/ A. G. Mordkovich, P. V. Semenov. - 13th ed., erased. - M.: Mnemosyne, 2011. - 222 p.: ill. ISBN 978-5-346-01752-3.
Definition: Natural numbers are numbers that are used to count objects (1, 2, 3, 4, 5, ...) [The number 0 is not natural. It has its own separate history in the history of mathematics and appeared much later than natural numbers.]
The set of all natural numbers (1, 2, 3, 4, 5, ...) is denoted by the letter N.
Whole numbers
Having learned to count, the next thing we do is learn to perform arithmetic operations on numbers. Usually, addition and subtraction are taught first (using counting sticks).With addition, everything is clear: adding any two natural numbers, the result will always be the same natural number. But in subtraction we discover that we cannot subtract the larger from the smaller so that the result is a natural number. (3 − 5 = what?) This is where the idea of negative numbers comes into play. (Negative numbers are no longer natural numbers)
At the stage of occurrence of negative numbers (and they appeared later than fractional ones) there were also their opponents, who considered them nonsense. (Three objects can be shown on your fingers, ten can be shown, a thousand objects can be represented by analogy. And what is “minus three bags”? - At that time, numbers were already used on their own, in isolation from specific objects, the number of which they denote were still in the minds of people much closer to these specific subjects than today.) But, like the objections, the main argument in favor of negative numbers came from practice: negative numbers made it possible to conveniently count debts. 3 − 5 = −2 - I had 3 coins, I spent 5. This means that I not only ran out of coins, but I also owed someone 2 coins. If I return one, the debt will change −2+1=−1, but can also be represented by a negative number.
As a result, negative numbers appeared in mathematics, and now we have an infinite number of natural numbers (1, 2, 3, 4, ...) and there is the same number of their opposites (−1, −2, −3, −4 , ...). Let's add another 0 to them. And we will call the set of all these numbers integers.
Definition: The natural numbers, their opposites, and zero make up the set of integers. It is designated by the letter Z.
Any two integers can be subtracted from each other or added to form a whole number.
The idea of adding integers already presupposes the possibility of multiplication, as simply more fast way performing addition. If we have 7 bags of 6 kilograms each, we can add 6+6+6+6+6+6+6 (add 6 to the current total seven times), or we can simply remember that such an operation will always result in 42. Just like adding six sevens, 7+7+7+7+7+7 will also always give 42.
Results of the addition operation certain numbers with yourself certain the number of times for all pairs of numbers from 2 to 9 are written out and a multiplication table is made up. To multiply integers greater than 9, the column multiplication rule is invented. (Which also applies to decimal fractions, and which will be discussed in one of the following articles.) When multiplying any two integers by each other, the result will always be an integer.
Rational numbers
Now division. Just as subtraction is the inverse operation of addition, we come to the idea of division as the inverse operation of multiplication.When we had 7 bags of 6 kilograms, using multiplication we easily calculated that the total weight of the contents of the bags was 42 kilograms. Let's imagine that we poured the entire contents of all the bags into one common pile weighing 42 kilograms. And then they changed their minds and wanted to distribute the contents back into 7 bags. How many kilograms will end up in one bag if we distribute it equally? – Obviously, 6.
What if we want to distribute 42 kilograms into 6 bags? Here we will think that the same total 42 kilograms could be obtained if we poured 6 bags of 7 kilograms into a pile. And this means that when dividing 42 kilograms into 6 bags equally, we get 7 kilograms in one bag.
What if you divide 42 kilograms equally into 3 bags? And here, too, we begin to select a number that, when multiplied by 3, would give 42. For “tabular” values, as in the case of 6 · 7 = 42 => 42: 6 = 7, we perform the division operation simply by recalling the multiplication table. For more complex cases Column division is used, which will be discussed in one of the following articles. In the case of 3 and 42, you can “select” to remember that 3 · 14 = 42. This means 42:3 = 14. Each bag will contain 14 kilograms.
Now let's try to divide 42 kilograms equally into 5 bags. 42:5=?
We notice that 5 · 8 = 40 (few), and 5 · 9 = 45 (many). That is, we will not get 42 kilograms from 5 bags, neither 8 kilograms in a bag, nor 9 kilograms. At the same time, it is clear that in reality nothing prevents us from dividing any quantity (cereals, for example) into 5 equal parts.
The operation of dividing integers by each other does not necessarily result in an integer. This is how we came to the concept of fractions. 42:5 = 42/5 = 8 whole 2/5 (if counted in fractions) or 42:5 = 8.4 (if counted in decimals).
Common and decimal fractions
We can say that any ordinary fraction m/n (m is any integer, n is any natural number) is simply a special form of writing the result of dividing the number m by the number n. (m is called the numerator of the fraction, n is the denominator) The result of dividing, for example, the number 25 by the number 5 can also be written as an ordinary fraction 25/5. But this is not necessary, since the result of dividing 25 by 5 can simply be written as the integer 5. (And 25/5 = 5). But the result of dividing the number 25 by the number 3 can no longer be represented as an integer, so here the need arises to use a fraction, 25:3 = 25/3. (You can distinguish the whole part 25/3 = 8 whole 1/3. Ordinary fractions and operations with ordinary fractions will be discussed in more detail in the following articles.)The good thing about ordinary fractions is that in order to represent the result of dividing any two integers as such a fraction, you simply need to write the dividend in the numerator of the fraction and the divisor in the denominator. (123:11=123/11, 67:89=67/89, 127:53=127/53, ...) Then, if possible, reduce the fraction and/or isolate the whole part (these actions with ordinary fractions will be discussed in detail in the following articles ). The problem is that performing arithmetic operations (addition, subtraction) with ordinary fractions is no longer as convenient as with integers.
For the convenience of writing (in one line) and for the convenience of calculations (with the possibility of calculations in a column, as for ordinary integers), in addition to ordinary fractions, decimal fractions were also invented. A decimal fraction is a specially written ordinary fraction with a denominator of 10, 100, 1000, etc. For example, the common fraction 7/10 is the same as the decimal fraction 0.7. (8/100 = 0.08; 2 whole 3/10 = 2.3; 7 whole 1/1000 = 7, 001). A separate article will be devoted to converting ordinary fractions to decimals and vice versa. Operations with decimal fractions - other articles.
Any integer can be represented as a common fraction with a denominator of 1. (5=5/1; −765=−765/1).
Definition: All numbers that can be represented as a fraction are called rational numbers. The set of rational numbers is denoted by the letter Q.
When dividing any two integers by each other (except for the case of division by 0), the result will always be rational number. For ordinary fractions, there are rules for addition, subtraction, multiplication and division that allow you to perform the corresponding operation with any two fractions and also obtain a rational number (fraction or integer) as a result.
The set of rational numbers is the first of the sets we have considered in which you can add, subtract, multiply, and divide (except for division by 0), never going beyond the boundaries of this set (that is, always getting a rational number as a result) .
It would seem that there are no other numbers; all numbers are rational. But this is not true either.
Real numbers
There are numbers that cannot be represented as a fraction m/n (where m is an integer, n is a natural number).What are these numbers? We have not yet considered the operation of exponentiation. For example, 4 2 =4 ·4 = 16. 5 3 =5 ·5 ·5=125. Just as multiplication is a more convenient form of writing and calculating addition, so exponentiation is a form of writing the multiplication of the same number by itself a certain number of times.
But now let’s look at the inverse operation of exponentiation—root extraction. The square root of 16 is the number that squared will give 16, that is, the number 4. The square root of 9 is 3. But Square root of 5 or of 2, for example, cannot be represented by a rational number. (The proof of this statement, other examples of irrational numbers and their history can be found, for example, on Wikipedia)
In the GIA in grade 9 there is a task to determine whether a number containing a root in its notation is rational or irrational. The task is to try to convert this number to a form that does not contain a root (using the properties of roots). If you can’t get rid of the root, then the number is irrational.
Another example of an irrational number is the number π, familiar to everyone from geometry and trigonometry.
Definition: Rational and irrational numbers together are called real (or real) numbers. The set of all real numbers is denoted by the letter R.
In real numbers, as opposed to rational numbers, we can express the distance between any two points on a line or plane.
If you draw a straight line and select two arbitrary points on it or select two arbitrary points on a plane, it may turn out that the exact distance between these points cannot be expressed as a rational number. (Example - the hypotenuse of a right triangle with legs 1 and 1, according to the Pythagorean theorem, will be equal to the root of two - that is, an irrational number. This also includes the exact length of the diagonal of a tetrad cell (the length of the diagonal of any ideal square with integral sides).)
And in the set of real numbers, any distances on a line, in a plane or in space can be expressed by the corresponding real number.
Integers
Natural numbers definition are positive integers. Natural numbers are used to count objects and for many other purposes. These are the numbers:
This is a natural series of numbers.
Is zero a natural number? No, zero is not a natural number.
How many natural numbers are there? There is an infinite number of natural numbers.
What is the smallest natural number? One is the smallest natural number.
What is the largest natural number? It is impossible to specify it, because there is an infinite number of natural numbers.
The sum of natural numbers is a natural number. So, adding natural numbers a and b:
The product of natural numbers is a natural number. So, the product of natural numbers a and b:
c is always a natural number.
Difference of natural numbers There is not always a natural number. If the minuend is greater than the subtrahend, then the difference of the natural numbers is a natural number, otherwise it is not.
The quotient of natural numbers is not always a natural number. If for natural numbers a and b
where c is a natural number, this means that a is divisible by b. In this example, a is the dividend, b is the divisor, c is the quotient.
The divisor of a natural number is a natural number by which the first number is divisible by a whole.
Every natural number is divisible by one and itself.
Prime natural numbers are divisible only by one and themselves. Here we mean divided entirely. Example, numbers 2; 3; 5; 7 is only divisible by one and itself. These are simple natural numbers.
One is not considered a prime number.
Numbers that are greater than one and that are not prime are called composite numbers. Examples of composite numbers:
One is not considered a composite number.
The set of natural numbers consists of one, prime numbers and composite numbers.
The set of natural numbers is denoted Latin letter N.
Properties of addition and multiplication of natural numbers:
commutative property of addition
associative property of addition
(a + b) + c = a + (b + c);
commutative property of multiplication
associative property of multiplication
(ab) c = a (bc);
distributive property of multiplication
A (b + c) = ab + ac;
Whole numbers
Integers are the natural numbers, zero, and the opposites of the natural numbers.
The opposite of natural numbers are negative integers, for example:
1; -2; -3; -4;...
The set of integers is denoted by the Latin letter Z.
Rational numbers
Rational numbers are whole numbers and fractions.
Any rational number can be represented as a periodic fraction. Examples:
1,(0); 3,(6); 0,(0);...
From the examples it is clear that any integer is a periodic fraction with period zero.
Any rational number can be represented as a fraction m/n, where m is an integer number,n natural number. Let's imagine the number 3,(6) from the previous example as such a fraction.
Number- an important mathematical concept that has changed over the centuries.
The first ideas about number arose from counting people, animals, fruits, various products, etc. The result is natural numbers: 1, 2, 3, 4, ...
Historically, the first extension of the concept of number is the addition of fractional numbers to the natural number.
Fraction a part (share) of a unit or several equal parts is called.
Designated by: , where m, n- whole numbers;
Fractions with denominator 10 n, Where n- an integer, called decimal: .
Among decimal fractions, a special place is occupied by periodic fractions: 
- pure periodic fraction, 
- mixed periodic fraction.
Further expansion of the concept of number is caused by the development of mathematics itself (algebra). Descartes in the 17th century. introduces the concept negative number.
The numbers integers (positive and negative), fractions (positive and negative), and zero are called rational numbers. Any rational number can be written as a finite and periodic fraction.
To study continuously changing variable quantities, it turned out that a new expansion of the concept of number was necessary - the introduction of real (real) numbers - by adding irrational numbers to rational numbers: irrational numbers are infinite decimal non-periodic fractions.
Irrational numbers appeared when measuring incommensurable segments (the side and diagonal of a square), in algebra - when extracting roots, an example of a transcendental, irrational number is π, e .
Numbers natural(1, 2, 3,...), whole(..., –3, –2, –1, 0, 1, 2, 3,...), rational(representable as a fraction) and irrational(not representable as a fraction ) form a set real (real) numbers.
Complex numbers are distinguished separately in mathematics.
Complex numbers arise in connection with the problem of solving squares for the case D< 0 (здесь D– discriminant of a quadratic equation). For a long time, these numbers did not find physical application, which is why they were called “imaginary” numbers. However, now they are very widely used in various fields of physics and technology: electrical engineering, hydro- and aerodynamics, elasticity theory, etc.
Complex numbers are written in the form: z= a+ bi. Here a And b – real numbers, A i – imaginary unit, i.e.e. i 2 = -1. Number a called abscissa, a b –ordinate complex number a+ bi. Two complex numbers a+ bi And a–bi are called conjugate complex numbers.
Properties:
1. Real number A can also be written in complex number form: a+ 0i or a – 0i. For example 5 + 0 i and 5 – 0 i mean the same number 5.
2. Complex number 0 + bi called purely imaginary number. Record bi means the same as 0 + bi.
3. Two complex numbers a+ bi And c+ di are considered equal if a= c And b= d. Otherwise complex numbers not equal.
Actions:
Addition. Sum of complex numbers a+ bi And c+ di is called a complex number ( a+ c) + (b+ d)i. Thus, When adding complex numbers, their abscissas and ordinates are added separately.
Subtraction. The difference of two complex numbers a+ bi(diminished) and c+ di(subtrahend) is called a complex number ( a–c) + (b–d)i. Thus, When subtracting two complex numbers, their abscissas and ordinates are subtracted separately.
Multiplication. Product of complex numbers a+ bi And c+ di is called a complex number:
(ac–bd) + (ad+ bc)i. This definition follows from two requirements:
1) numbers a+ bi And c+ di must be multiplied like algebraic binomials,
2) number i has the main property: i 2 = –1.
EXAMPLE ( a+ bi)(a–bi)= a 2 +b 2 . Hence, workof two conjugate complex numbers is equal to a positive real number.
Division. Divide a complex number a+ bi(divisible) by another c+ di (divider) - means to find the third number e+ f i(chat), which when multiplied by a divisor c+ di, results in the dividend a+ bi. If the divisor is not zero, division is always possible.
EXAMPLE Find (8 + i) : (2 – 3i) .
Solution. Let's rewrite this ratio as a fraction:
Multiplying its numerator and denominator by 2 + 3 i and after performing all the transformations, we get:
Task 1: Add, subtract, multiply and divide z 1 on z 2
Extracting the square root: Solve the equation x 2 = -a. To solve this equation we are forced to use numbers of a new type - imaginary numbers . Thus, imaginary the number is called the second power of which is a negative number. According to this definition of imaginary numbers we can define and imaginary unit:
Then for the equation x 2 = – 25 we get two imaginary root:
Task 2: Solve the equation:
1)x 2 = – 36; 2) x 2 = – 49; 3) x 2 = – 121
Geometric representation of complex numbers. Real numbers are represented by points on the number line:
Here is the point A means the number –3, dot B–number 2, and O-zero. In contrast, complex numbers are represented by points on the coordinate plane. For this purpose, we choose rectangular (Cartesian) coordinates with the same scales on both axes. Then the complex number a+ bi will be represented by a dot P with abscissaA and ordinateb. This coordinate system is called complex plane .
Module complex number is the length of the vector OP, representing a complex number on the coordinate ( comprehensive) plane. Modulus of a complex number a+ bi denoted | a+ bi| or) letter r and is equal to:
Conjugate complex numbers have the same modulus.
The rules for drawing up a drawing are almost the same as for a drawing in a Cartesian coordinate system. Along the axes you need to set the dimension, note:
e 
unit along the real axis; Re z
imaginary unit along the imaginary axis. Im z
Task 3. Construct the following complex numbers on the complex plane: , , , , , , ,
1. Numbers are exact and approximate. The numbers we encounter in practice are of two kinds. Some give the true value of the quantity, others only approximate. The first are called exact, the second - approximate. Most often it is convenient to use an approximate number instead of an exact one, especially since in many cases it is impossible to find an exact number at all.
So, if they say that there are 29 students in a class, then the number 29 is accurate. If they say that the distance from Moscow to Kyiv is 960 km, then here the number 960 is approximate, since, on the one hand, our measuring instruments are not absolutely accurate, on the other hand, the cities themselves have a certain extent.
The result of actions with approximate numbers is also an approximate number. By performing some operations on exact numbers (division, root extraction), you can also obtain approximate numbers.
The theory of approximate calculations allows:
1) knowing the degree of accuracy of the data, evaluate the degree of accuracy of the results;
2) take data with an appropriate degree of accuracy sufficient to ensure the required accuracy of the result;
3) rationalize the calculation process, freeing it from those calculations that will not affect the accuracy of the result.
2. Rounding. One source of obtaining approximate numbers is rounding. Both approximate and exact numbers are rounded.
Rounding a given number to a certain digit is called replacing it with a new number, which is obtained from the given one by discarding all its digits written to the right of the digit of this digit, or by replacing them with zeros. These zeros are usually underlined or written smaller. To ensure that the rounded number is as close as possible to the one being rounded, you should use the following rules: to round a number to one of a certain digit, you must discard all the digits after the digit of this digit, and replace them with zeros in the whole number. The following are taken into account:
1) if the first (on the left) of the discarded digits is less than 5, then the last remaining digit is not changed (rounding down);
2) if the first digit to be discarded is greater than 5 or equal to 5, then the last digit left is increased by one (rounding with excess).
Let's show this with examples. Round:
a) up to tenths 12.34;
b) to hundredths 3.2465; 1038.785;
c) up to thousandths 3.4335.
d) up to thousand 12375; 320729.
a) 12.34 ≈ 12.3;
b) 3.2465 ≈ 3.25; 1038.785 ≈ 1038.79;
c) 3.4335 ≈ 3.434.
d) 12375 ≈ 12,000; 320729 ≈ 321000.
3. Absolute and relative errors. The difference between the exact number and its approximate value is called the absolute error of the approximate number. For example, if the exact number 1.214 is rounded to the nearest tenth, we get an approximate number of 1.2. IN in this case the absolute error of the approximate number 1.2 is 1.214 - 1.2, i.e. 0.014.
But in most cases, the exact value of the value under consideration is unknown, but only an approximate one. Then the absolute error is unknown. In these cases, indicate the limit that it does not exceed. This number is called the limiting absolute error. They say that the exact value of a number is equal to its approximate value with an error less than the marginal error. For example, the number 23.71 is an approximate value of the number 23.7125 with an accuracy of 0.01, since the absolute error of the approximation is 0.0025 and less than 0.01. Here the limiting absolute error is 0.01 *.
Boundary absolute error of the approximate number A denoted by the symbol Δ a. Record
x≈a(±Δ a)
should be understood as follows: the exact value of the quantity x is between the numbers A– Δ a And A+ Δ A, which are called the lower and upper bounds, respectively X and denote NG x VG X.
For example, if x≈ 2.3 (±0.1), then 2.2<x< 2,4.
Vice versa, if 7.3< X< 7,4, тоX≈ 7.35 (±0.05). The absolute or marginal absolute error does not characterize the quality of the measurement performed. The same absolute error can be considered significant and insignificant depending on the number with which the measured value is expressed. For example, if we measure the distance between two cities with an accuracy of one kilometer, then such accuracy is quite sufficient for this change, but at the same time, when measuring the distance between two houses on the same street, such accuracy will be unacceptable. Consequently, the accuracy of the approximate value of a quantity depends not only on the magnitude of the absolute error, but also on the value of the measured quantity. Therefore, the relative error is a measure of accuracy.
Relative error is the ratio of the absolute error to the value of the approximate number. The ratio of the limiting absolute error to the approximate number is called the limiting relative error; they designate it like this: . Relative and marginal relative errors are usually expressed as percentages. For example, if measurements showed that the distance X between two points is more than 12.3 km, but less than 12.7 km, then the arithmetic mean of these two numbers is taken as its approximate value, i.e. their half-sum, then the marginal absolute error is equal to the half-difference of these numbers. In this case X≈ 12.5 (±0.2). Here the limiting absolute error is 0.2 km, and the limiting relative
