What letter represents a rational number? Integers and rational numbers
Number is an abstraction used to quantify objects. Numbers arose back in primitive society due to the need of people to count objects. Over time, as science developed, number turned into the most important mathematical concept.
To solve problems and prove various theorems, you need to understand what types of numbers there are. The main types of numbers include: integers, integers, rational numbers, real numbers.
Integers- these are numbers obtained by natural counting of objects, or rather by numbering them (“first”, “second”, “third”...). The set of natural numbers is denoted by a Latin letter N (you can remember based on English word natural). It can be said that N ={1,2,3,....}
Whole numbers- these are numbers from the set (0, 1, -1, 2, -2, ....). This set consists of three parts - natural numbers, negative integers (the opposite of natural numbers) and the number 0 (zero). Integers are denoted by a Latin letter Z . It can be said that Z ={1,2,3,....}.
Rational numbers are numbers represented as a fraction, where m is an integer and n is a natural number. The Latin letter is used to denote rational numbers Q . All natural numbers and integers are rational. Also, examples of rational numbers include: ,,.
Real numbers- these are numbers that are used to measure continuous quantities. A bunch of real numbers denoted by the Latin letter R. Real numbers include rational numbers and irrational numbers. Irrational numbers are numbers that result from doing various operations with rational numbers (for example, taking roots, calculating logarithms), but are not rational. Examples of irrational numbers are,,.
Any real number can be displayed on the number line:
For the sets of numbers listed above, the following statement is true:
That is, the set of natural numbers is included in the set of integers. The set of integers is included in the set of rational numbers. And the set of rational numbers is included in the set of real numbers. This statement can be illustrated using Euler circles.
State educational institution
average vocational education
"Aleksinsky Mechanical Engineering College"
Numerical
sets
Designed by
teacher
mathematicians
Khristoforova M.Yu.
Number - basic concept , used for characteristics, comparisons, and their parts. Written signs to denote numbers are , and mathematical .
The concept of number arose in ancient times from the practical needs of people and developed in the process of human development. Region human activity expanded and, accordingly, the need for quantitative description and research increased. At first, the concept of number was determined by the needs of counting and measurement that arose in human practical activity, becoming more and more complex. Later, number becomes the basic concept of mathematics, and the needs of this science determine the further development of this concept.
Sets whose elements are numbers are called numerical.
Examples of number sets are:
N=(1; 2; 3; ...; n; ... ) - set of natural numbers;
Zo=(0; 1; 2; ...; n; ... ) - set of non-negative integers;
Z=(0; ±1; ±2; ...; ±n; ...) - set of integers;
Q=(m/n: m Z,n N) is the set of rational numbers.
R-set of real numbers.
There is a relationship between these sets
N Zo Z Q R.
Numbers of the formN = (1, 2, 3, ....) are callednatural . Natural numbers appeared in connection with the need to count objects.
Any , greater than unity, can be represented as a product of powers prime numbers, and in a unique way up to the order of the factors. For example, 121968=2 4 ·3 2 ·7·11 2
Ifm, n, k - natural numbers, then whenm - n = k they say thatm - minuend, n - subtrahend, k - difference; atm: n = k they say thatm - dividend, n - divisor, k - quotient, numberm also calledmultiples numbersn, and the numbern - divisor numbersm, If the numberm- multiple of a numbern, then there is a natural numberk, such thatm = kn.
From numbers using arithmetic signs and parentheses, they are composednumeric expressions. If you perform the indicated actions in numerical expression, observing the accepted order, you will get a number calledthe value of the expression .
The order of arithmetic operations: the actions in brackets are performed first; Inside any parentheses, multiplication and division are performed first, and then addition and subtraction.
If a natural numberm not divisible by a natural numbern, those. there is no such thingnatural number k, Whatm =kn, then they considerdivision with remainder: m = np + r, Wherem - dividend, n - divisor (m>n), p - quotient, r - remainder .
If a number has only two divisors (the number itself and one), then it is calledsimple : if a number has more than two divisors, then it is calledcomposite.
Any composite natural number can befactorize , and only one way. When factoring numbers into prime factors, usesigns of divisibility .
a Andb can be foundgreatest common divisor. It is designatedD(a,b). If the numbersa Andb are such thatD(a,b) = 1, then the numbersa Andb are calledmutually simple.
For any given natural numbersa Andb can be foundleast common multiple. It is designatedK(a,b). Any common multiple of numbersa Andb divided byK(a,b).
If the numbersa Andb relatively prime , i.e.D(a,b) = 1, ThatK(a,b) = ab .
Numbers of the form:Z = (... -3, -2, -1, 0, 1, 2, 3, ....) are called integers , those. Integers are the natural numbers, the opposite of the natural numbers, and the number 0.
The natural numbers 1, 2, 3, 4, 5.... are also called positive integers. The numbers -1, -2, -3, -4, -5, ..., the opposite of the natural numbers, are called negative integers.
Significant numbers a number is all its digits except the leading zeros.
A sequentially repeating group of digits after the decimal point in the decimal notation of a number is calledperiod, and an infinite decimal fraction having such a period in its notation is calledperiodic . If the period begins immediately after the decimal point, then the fraction is calledpure periodic ; if there are other decimal places between the decimal point and the period, then the fraction is calledmixed periodic .
Numbers that are not integers or fractions are calledirrational .
Each irrational number represented as a non-periodic infinite decimal fraction.
The set of all finite and infinite decimals calledmany real numbers : rational and irrational.
The set R of real numbers has the following properties.
1. It is ordered: for any two different numbers α and b, one of two relations holds: a
2. The set R is dense: between any two distinct numbers a and b there is an infinite set of real numbers x, i.e. numbers satisfying the inequality a<х
So, if a
(a 2a< A+bA+b<2b 2 A A<(a+b)/2
Real numbers can be represented as points on a number line. To define a number line, you need to mark a point on the line, which will correspond to the number 0 - the origin, and then select a unit segment and indicate the positive direction.
Each point on the coordinate line corresponds to a number, which is defined as the length of the segment from the origin to the point in question, with a unit segment taken as the unit of measurement. This number is the coordinate of the point. If a point is taken to the right of the origin, then its coordinate is positive, and if to the left, it is negative. For example, points O and A have coordinates 0 and 2, respectively, which can be written as follows: 0(0), A(2).
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:
- ∅ – an empty set or a set that has no 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.
We will first consider numerical sets containing a finite and small number 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 various types of numerical intervals are introduced, such as interval, half-interval, segment, open numerical ray and numerical 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, excluding 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 support what has been said with a practical example, let us 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.
If you notice an error in the text, please highlight it and press Ctrl+Enter
Integers
The numbers used in counting are called natural numbers. For example, $1,2,3$, etc. The natural numbers form the set of natural numbers, which is denoted by $N$. This designation comes from the Latin word naturalis- natural.
Opposite numbers
Definition 1
If two numbers differ only in signs, they are called in mathematics opposite numbers.
For example, the numbers $5$ and $-5$ are opposite numbers, because They differ only in signs.
Note 1
For any number there is an opposite number, and only one.
Note 2
The number zero is the opposite of itself.
Whole numbers
Definition 2
Whole numbers are the natural numbers, their opposites, and zero.
The set of integers includes the set of natural numbers and their opposites.
Denote integers $Z.$
Fractional numbers
Numbers of the form $\frac(m)(n)$ are called fractions or fractional numbers. Fractional numbers can also be written in decimal form, i.e. in the form of decimal fractions.
For example: $\ \frac(3)(5)$ , $0.08$ etc.
Just like whole numbers, fractional numbers can be either positive or negative.
Rational numbers
Definition 3
Rational numbers is a set of numbers containing a set of integers and fractions.
Any rational number, both integer and fractional, can be represented as a fraction $\frac(a)(b)$, where $a$ is an integer and $b$ is a natural number.
Thus, the same rational number can be written in different ways.
For example,
This shows that any rational number can be represented as a finite decimal fraction or an infinite decimal periodic fraction.
The set of rational numbers is denoted by $Q$.
As a result of performing any arithmetic operation on rational numbers, the resulting answer will be a rational number. This is easily provable, due to the fact that when adding, subtracting, multiplying and dividing ordinary fractions, you get an ordinary fraction
Irrational numbers
While studying a mathematics course, you often have to deal with numbers that are not rational.
For example, to verify the existence of a set of numbers other than rational ones, let’s solve the equation $x^2=6$. The roots of this equation will be the numbers $\surd 6$ and -$\surd 6$. These numbers will not be rational.
Also, when finding the diagonal of a square with side $3$, we apply the Pythagorean theorem and find that the diagonal will be equal to $\surd 18$. This number is also not rational.
Such numbers are called irrational.
So, an irrational number is an infinite non-periodic decimal fraction.
One of the frequently encountered irrational numbers is the number $\pi $
When performing arithmetic operations with irrational numbers, the resulting result can be either a rational or an irrational number.
Let's prove this using the example of finding the product of irrational numbers. Let's find:
$\ \sqrt(6)\cdot \sqrt(6)$
$\ \sqrt(2)\cdot \sqrt(3)$
By decision
$\ \sqrt(6)\cdot \sqrt(6) = 6$
$\sqrt(2)\cdot \sqrt(3)=\sqrt(6)$
This example shows that the result can be either a rational or an irrational number.
If rational and irrational numbers are involved in arithmetic operations at the same time, then the result will be an irrational number (except, of course, multiplication by $0$).
Real numbers
The set of real numbers is a set containing the set of rational and irrational numbers.
The set of real numbers is denoted by $R$. Symbolically, the set of real numbers can be denoted by $(-?;+?).$
We said earlier that an irrational number is an infinite decimal non-periodic fraction, and any rational number can be represented as a finite decimal fraction or an infinite decimal periodic fraction, so any finite and infinite decimal fraction will be a real number.
When performing algebraic operations the following rules will be followed:
- When multiplying and dividing positive numbers, the resulting number will be positive
- When multiplying and dividing negative numbers, the resulting number will be positive
- When multiplying and dividing negative and positive numbers, the resulting number will be negative
Real numbers can also be compared with each other.
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.
Page navigation.
Writing numerical sets
Let's start with the accepted notation. As you know, capital letters of the Latin alphabet are used to denote sets. Numerical sets, as a special case of sets, are also designated. 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. In 2 hours. Part 1. Textbook for students of general education institutions / A. G. Mordkovich, P. V. Semenov. - 13th ed., erased. - M.: Mnemosyne, 2011. - 222 p.: ill. ISBN 978-5-346-01752-3.
