What letter represents prime numbers? A manual for students "numeric sets"

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Numerical

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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 further development 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 - integers, 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 signs arithmetic operations and brackets are composednumeric expressions. If in numerically perform the indicated actions, observing the accepted order, then you 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.

Consistently repeating group of digits after the decimal point in decimal notation numbers are 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 is 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 different numbers a and b contain 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).

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:

1. When multiplying and dividing positive numbers, the resulting number will be positive
2. When multiplying and dividing negative numbers, the resulting number will be positive
3. When multiplying and dividing negative and positive numbers, the resulting number will be negative

Real numbers can also be compared with each other.

The phrase " number sets" is quite common in mathematics textbooks. There you can often find phrases like this:

“Blah blah blah, where belongs to the set of natural numbers.”

Often, instead of the end of a phrase, you can see something like this. It means the same as the text a little above - a number belongs to the set of natural numbers. Many people quite often do not pay attention to which set this or that variable is defined in. As a result, completely incorrect methods are used when solving a problem or proving a theorem. This occurs because the properties of numbers belonging to different sets may differ.

There are not so many numerical sets. Below you can see the definitions of various number sets.

The set of natural numbers includes all integers greater than zero—positive integers.

For example: 1, 3, 20, 3057. The set does not include the number 0.

This number set includes all integers greater and less than zero, and also zero.

For example: -15, 0, 139.

Rational numbers, generally speaking, are a set of fractions that cannot be canceled (if a fraction is canceled, then it will already be an integer, and for this case there is no need to introduce another number set).

An example of numbers included in the rational set: 3/5, 9/7, 1/2.

,

where is a finite sequence of digits of the integer part of a number belonging to the set of real numbers. This sequence is finite, that is, the number of digits in the integer part of a real number is finite.

– an infinite sequence of numbers that are in the fractional part of a real number. It turns out that the fractional part contains an infinite number of numbers.

Such numbers cannot be represented as a fraction. Otherwise, such a number could be classified as a set of rational numbers.

Examples of real numbers:

Let's take a closer look at the meaning of the root of two. The integer part contains only one digit - 1, so we can write:

In the fractional part (after the dot), the numbers 4, 1, 4, 2 and so on appear sequentially. Therefore, for the first four digits we can write:

I dare to hope that now the definition of the set of real numbers has become clearer.

Conclusion

It should be remembered that the same function can exhibit completely different properties depending on which set the variable belongs to. So remember the basics - they will come in handy.

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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.

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.

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 by the 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 and n is a natural number. Let's imagine the number 3,(6) from the previous example as such a fraction.