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 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.
Numbers are divided into classes. Positive integers - N = (1, 2, 3, ...) - make up the set of natural numbers. Often, 0 is considered a natural number.
The set of integers Z includes all natural numbers, the number 0 and all natural numbers taken with a minus sign: Z = (0, 1, -1, 2, -2, ...).
Each rational number x can be specified as a pair of integers (m, n), where m is the numerator, n is the denominator of the number: x = m/n. An equivalent representation of a rational number is to express it as a number written in positional decimal notation, where the fractional part of the number can be a finite or infinite periodic fraction. For example, the number x = 1/3 = 0,(3) is represented as an infinite periodic fraction.
Numbers defined by infinite non-periodic fractions are called irrational numbers. These are, for example, all numbers of the form vp, where p is a prime number. The numbers known to everyone and e are irrational.
The union of the sets of integers, rational and irrational numbers constitutes the set of real numbers. The geometric image of the set of real numbers is a straight line - the real axis, where each point on the axis corresponds to a certain real number, so that the real numbers densely and continuously fill the entire real axis.
The plane represents the geometric image of a set of complex numbers, where two axes are introduced - real and imaginary. Each complex number, defined by a pair of real numbers, is representable in the form: x = a+b*i, where a and b are real numbers, which can be considered as the Cartesian coordinates of the number on the plane.
Divisors and multipliers
Let us now consider a classification that divides the set of natural numbers into two subsets - prime and composite numbers. This classification is based on the concept of divisibility of natural numbers. If n is divisible by d, then we say that d “divides” n, and write it in the form: . Note that this definition may not correspond to the intuitive understanding: d "divides" n if n is divisible by d, and not vice versa. The number d is called the divisor of n. Every number n has two trivial factors - 1 and n. Divisors other than trivial ones are called factors of n. A number n is called prime if it has no divisors other than trivial ones. Prime numbers are divisible only by 1 and themselves. Numbers that have factors are called composite numbers. The number 1 is a special number because it is neither a prime nor a composite number. Negative numbers also do not belong to either prime or composite numbers, but you can always consider the modulus of a number and classify it as a prime or composite number.
Any composite number N can be represented as a product of its factors: . This representation is not unique, for example 96 = 8*12 = 2*3*16. However, for each composite number N there is a unique representation in the form of a product of powers of prime numbers: , where are prime numbers and . This representation is called the factorization of the number N into prime factors. For example 
.
If and , then d is a common divisor of the numbers m and n. Among all common divisors, we can distinguish the greatest common divisor, denoted as gcd(m,n). If gcd(m,n) = 1, then the numbers m and n are called coprime. Prime numbers are coprime, so gcd(q,p) =1 if q and p are prime numbers.
If and , then A is a common multiple of m and n. Among all common multiples, we can distinguish the least common multiple, denoted as LCM(m,n). If LCM(m,n) = m*n, then the numbers m and n are relatively prime. LCM(q, p) =q*p if q and p are prime numbers.
If we denote the sets of all prime factors of the numbers m and n by and, then
If the decomposition of the numbers m and n into prime factors is obtained, then, using the given relations, it is easy to calculate GCD(m,n) and LCM(m,n). There are also more efficient algorithms that do not require factoring a number.
Euclid's algorithm
An effective algorithm for calculating GCD(m,n) was proposed by Euclid. It is based on the following properties of GCD(m,n), the proof of which is left to the reader:
If , then according to the third property it can be reduced by the value n. If, then, according to the second property, the arguments can be swapped and again come to the previously considered case. When, as a result of these transformations, the values of the arguments become equal, the solution will be found. Therefore, we can propose the following scheme:
while(m != n) ( if(m< n) swap(m,n); m = m - n; } return(m);
Here the swap procedure exchanges the values of the arguments.
If you think a little, it becomes clear that it is not at all necessary to exchange values - it is enough to change the argument with the maximum value at each step of the loop. As a result, we come to the diagram:
while(m != n) ( if(m > n) m = m - n; else n = n - m; ) return(m);
If you think a little more, you can improve this scheme by moving to a loop with an identically true condition:
while(true) ( if(m > n) m = m - n; else if (n > m) n = n - m; else return(m); )
The last diagram is good because it clearly shows the need to prove the completeness of this cycle. It is not difficult to prove the completeness of a loop using the concept of a loop variant. For this loop, an option can be an integer function - max(m,n) , which decreases at each step, always remaining positive.
The advantage of this version of the Euclid algorithm is that at each step an elementary and fast operation on integers is used - subtraction. If you allow the operation of calculating the remainder when dividing by an integer, then the number of loop steps can be significantly reduced. The following property is true:
This results in the following diagram:
int temp; if(n>m) temp = m; m = n; n = temp; //swap(m,n) while(m != n) ( temp = m; m = n; n = temp%n; )
If you think about it a little, it becomes clear that it is not at all necessary to perform a check before starting the cycle. This leads to a simpler GCD calculation scheme, usually used in practice:
int temp; while(m != n) ( temp = m; m = n; n = temp%n; )
To calculate LCM(m, n) you can use the following relation:
Is it possible to calculate LCM(m, n) without using multiplication and division operations? It turns out that you can simultaneously calculate LCM(m,n) while calculating GCD(m,n). Here is the corresponding diagram:
int x = v = m, y = u = n,; while(x != y) ( if(x > y)( x = x - y; v = v + u;) else (y = y - x; u = u + v;) ) GCD = (x + y )/2; LCM = (u+v)/2;
The proof that this scheme correctly calculates the GCD follows from the previously given properties of the GCD. The correctness of the LCM calculation is less obvious. To prove this, notice that the loop invariant is the following expression:
This relationship is satisfied after the variables are initialized before the loop begins execution. At the end of the cycle, when x and y become equal to gcd, the correctness of the scheme follows from the truth of the invariant. It is easy to verify that the loop body statements leave the statement true. The details of the proof are left to the readers.
The concept of GCD and LCM can be expanded by defining them for all integers. The following relations are valid:
Extended Euclidean Algorithm
Sometimes it is useful to represent gcd(m,n) as a linear combination of m and n:
In particular, the calculation of coefficients a and b is necessary in the RSA algorithm - public key encryption. I will give an algorithm diagram that allows you to calculate the triple - d, a, b - the greatest common divisor and expansion coefficients. The algorithm can be conveniently implemented as a recursive procedure
ExtendedEuclid(int m, int n, ref int d, ref int a, ref int b),
which, given input arguments m and n, calculates the values of arguments d, a, b. The non-recursive branch of this procedure corresponds to the case n = 0, returning as a result the values: d = m, a = 1, b = 0. The recursive branch calls
ExtendedEuclid(n, m % n, ref d, ref a, ref b)
and then modifies the resulting values of a and b as follows:
It is not difficult to construct a proof of the correctness of this algorithm. For the non-recursive branch, the correctness is obvious, and for the recursive branch it is easy to show that from the truth of the result returned by the recursive call, it follows that it is true for the input arguments after recalculating the values of a and b.
How does this procedure work? First, a recursive descent occurs until n becomes zero.
At this point, the value of d and the values of the parameters a and b will be calculated for the first time. After this, the rise will begin and parameters a and b will be recalculated.
Tasks
- 49. Given m and n are natural numbers. Calculate gcd(m, n). When making calculations, do not use multiplication and division operations.
- 50. Given m and n are natural numbers. Calculate LCM(m, n).
- 51. Given m and n are natural numbers. Calculate LCM(m, n). When making calculations, do not use multiplication and division operations.
- 52. Given m and n are integers. Calculate gcd(m, n). When making calculations, do not use multiplication and division operations.
- 53. Given m and n are integers. Calculate LCM(m, n). When making calculations, do not use multiplication and division operations.
- 54. Given m and n are integers. Calculate gcd(m, n). When making calculations, use the operation of taking the remainder of division by an integer.
- 55. Given m and n are integers. Calculate LCM(m, n). When making calculations, use the operation of taking the remainder of division by an integer.
- 56. Given m and n are integers. Compute a triple of numbers - (d, a, b) using the extended Euclidean algorithm.
- 57. Given m and n are natural numbers. Think of GCD(m, n) as a linear combination of m and n.
- 58. Given m and n are integers. Think of GCD(m, n) as a linear combination of m and n.
- 59. Given m and n are integers. Check whether the numbers m and n are coprime.
Prime numbers
Among the even numbers there is only one prime number - this is 2. There are as many prime odd numbers as you like. It is not difficult to prove that the number , where are consecutive prime numbers, is prime. So, if prime numbers have been constructed, then we can construct another prime number greater than . It follows that the set of prime numbers is unlimited. Example: the number N = 2*3*5*7 + 1 = 211 is a prime number.
Sieve of Eratosthenes
How to determine that N is a prime number? If the operation N % m is valid, giving the remainder when dividing N by number m, then the simplest algorithm is to check that the remainder is not equal to zero when dividing N by all numbers m less than N. An obvious improvement of this algorithm is to reduce the test range - it is enough to consider the numbers m in the range .
Back in the 3rd century BC. Greek mathematician Eratosthenes proposed an algorithm for finding prime numbers in a range that does not require division operations. This algorithm is called the “Sieve of Eratosthenes”. In the computer version, the idea of this algorithm can be described as follows. Let's build an array Numbers, the elements of which contain consecutive odd numbers, starting with 3. Initially, all numbers in this array are considered uncrossed out. Let's put the first uncrossed number from this array into the SimpleNumbers array - and this will be the first odd prime number (3). Then we will perform sifting, going through the Numbers array with a step equal to the found prime number, crossing out all the numbers that come across during this pass. On the first pass, the number 3 and all numbers that are multiples of 3 will be crossed out. On the next pass, the next prime number 5 will be entered into the table of prime numbers, and numbers that are multiples of 5 will be crossed out from the Numbers array. The process is repeated until all numbers in the array are crossed out Numbers. As a result, the SimpleNumbers array will contain a table of the first prime numbers less than N.
This algorithm is good for finding relatively small prime numbers. But if you need to find a prime number with twenty significant digits, then the computer memory will no longer be enough to store the corresponding arrays. Note that modern encryption algorithms use prime numbers containing several hundred digits.
Prime Density
We have shown that the number of prime numbers is unlimited. It is clear that there are fewer of them than odd numbers, but how much less? What is the density of prime numbers? Let be a function that returns the number of prime numbers less than n. It is not possible to specify this function precisely, but there is a good estimate for it. The following theorem is true:
The function asymptotically approaches its limit from above, so the estimate gives slightly underestimated values. This estimate can be used in the Sieve of Eratosthenes algorithm to select the dimension of the SimpleNumbers array when the dimension of the Numbers array is given, and, conversely, given the dimension of the table of primes, one can select the appropriate dimension for the Numbers array.
Tabular algorithm for determining the primeness of numbers
If you keep a table of prime numbers, SimpleNumbers, in which the largest prime number is M, then you can simply determine whether the number N less than is prime. If N is less than M, then it is enough to check whether the number N is in the SimpleNumbers table. If N is greater than M, then it is enough to check whether the number N is divisible by numbers from the SimpleNumbers table that do not exceed the value of vN. It is clear that if the number N has no prime factors less than vN, then the number N is prime.
Using a prime number table requires adequate computer memory and therefore limits the algorithm's capabilities, preventing it from being used to find large prime numbers.
Trivial algorithm
If N is an odd number, then you can check that it is prime based on the definition of primeness of a number. In this case, no memory is required to store tables of numbers - but, as always, winning in memory, we lose in time. Indeed, it is enough to check whether the number N is divisible by consecutive odd numbers in the range . If the number N has at least one factor, then it is composite, otherwise it is prime.
All of the algorithms discussed stop working effectively when numbers go beyond the computer's bit grid for representing numbers, so if there is a need to work with integers outside the System.Int64 range, then the task of determining the primality of such a number becomes far from simple. There are some recipes to determine that a number is composite. Let us at least recall the algorithms known from school times. If the last digit of a number is divisible by 2, then the number is divisible by 2. If the last two digits of the number are divisible by 4, then the number is divisible by 4. If the sum of the digits is divisible by 3 (by 9), then the number is divisible by 3 (by 9). If the last digit is 0 or 5, then the number is divisible by 5. Mathematicians have spent a lot of effort proving that a number is (or is not) a prime number. Now there are special techniques that allow you to prove that numbers of a certain type are prime. The most suitable candidates for prime are numbers of the form , where p is a prime number. For example, a number with more than 6000 digits has been proven to be prime, but it cannot be said which prime numbers are the nearest neighbors of that number.
Tasks
Projects
- 67. Construct a “Temperature” class that allows you to set temperature in different units of measurement. Build a Windows project that supports an interface for working with the class.
- 68. Construct a “Distance” class that allows you to use different systems of measures. Build a Windows project that supports an interface for working with the class.
- 69. Build a class "Prime numbers". Build a Windows project that supports an interface for working with the class.
- 70. Build a class “Number systems”. Build a Windows calculator that supports calculations in a given number system.
- 71. Construct a class "Rational numbers". Build a Windows calculator that supports calculations with these numbers.
- 72. Construct the class "Complex numbers". Build a Windows calculator that supports calculations with these numbers.
Find the points on the number circle with the given abscissa. Coordinates. Property of point coordinates. Center of the number circle. From circle to trigonometer. Find the points on the number circle. Dots with abscissa. Trigonometer. Mark a point on the number circle. Number circle on the coordinate plane. Number circle. Points with ordinate. Give the coordinate of the point. Name the line and coordinate of the point.
““Derivatives” 10th grade algebra” - Application of derivatives to study functions. The derivative is zero. Find the points. Let's summarize the information. The nature of the monotonicity of the function. Application of the derivative to the study of functions. Theoretical warm-up. Complete the statements. Choose the correct statement. Theorem. Compare. The derivative is positive. Compare the formulations of the theorems. The function increases. Sufficient conditions for an extremum.
““Trigonometric equations” grade 10” - Values from the interval. X= tan x. Provide roots. Is the equality true? Series of roots. Equation cot t = a. Definition. Cos 4x. Find the roots of the equation. Equation tg t = a. Sin x. Does the expression make sense? Sin x =1. Never do what you don't know. Continue the sentence. Let's take a sample of the roots. Solve the equation. Ctg x = 1. Trigonometric equations. The equation.
“Algebra “Derivatives”” - Tangent equation. Origin of terms. Solve a problem. Derivative. Material point. Differentiation formulas. Mechanical meaning of derivative. Evaluation criteria. Derivative function. Tangent to the graph of a function. Definition of derivative. Equation of a tangent to the graph of a function. Algorithm for finding the derivative. An example of finding the derivative. Structure of the topic study. The point moves in a straight line.
“Shortest path” - A path in a digraph. An example of two different graphs. Directed graphs. Examples of directed graphs. Reachability. The shortest path from vertex A to vertex D. Description of the algorithm. Advantages of a hierarchical list. Weighted graphs. Path in the graph. ProGraph program. Adjacent vertices and edges. Top degree. Adjacency matrix. Path length in a weighted graph. An example of an adjacency matrix. Finding the shortest path.
"The History of Trigonometry" - Jacob Bernoulli. Techniques for operating with trigonometric functions. The doctrine of measuring polyhedra. Leonard Euler. The development of trigonometry from the 16th century to the present day. The student has to meet trigonometry three times. Until now, trigonometry has been formed and developed. Construction of a general system of trigonometric and related knowledge. Time passes, and trigonometry returns to schoolchildren.
Figure 3 Organization chart
Adding an organizational chart is done using the Add diagram or organizational chart button, the original test is replaced in its blocks, after which the entire object is compressed vertically.
1.1 WordArt program
The program is designed for entering artistic inscriptions into a document, editing them, placing them in text, etc.
Inserting an object is done as follows:
left-click on a key Add objectWordArt, select the type of inscription, press the key OK;
in the window that appears Changing textWordArt set the font type, its size and style (bold, italic), enter the text and press the key OK.
a panel will appear WordArt, having the form (Fig. 4):
Figure 4 Toolbar WordArt
The panel contains buttons: Add objectWordArt,Change text…, CollectionWordArt,Object formatWordArt(colors and lines, size, position on the screen, wrapping, drawing, inscription), Menu Text-Shape(forms of inscriptions) , Vertical text and etc.
The text size can be changed using the white circles of the selection outline. Moving the text is done with the mouse, and you need to grab the text by its middle or the selection contour line. The rotation of the object is performed using green circles, the tilt of the inscription is
using yellow diamonds. The color and other parameters of the object are changed using the button Object FormatWordArt or from the main panel Drawing, with which you can additionally set shading and volumetric effects .
For example, the name of the newspaper "Znamya" after entering and customizing using the WordArt program may look like (Fig. 5):
Example 3
Figure 5 The inscription "Banner"
2 Development of a wall advertisement
When developing it we use text fields, which are created using a button Inscription. An inscription is a frame, a “patch” that is superimposed on a document and can contain any data - text, tables, pictures and other objects. Such an advertisement usually consists of a picture, the text of the advertisement, the name of the organization and sheets of “tear-off telephone numbers”. All ad elements are entered into their text fields No. 1-No. 5:
Example 4: Sequence of actions (possible) when creating a wall ad using text fields:
Using a button Inscription toolbars Drawing create a text field #1 that matches the size of the ad.
On the menu Format select item Borders and Shading and create a frame around text field No. 1 - these are the dimensional boundaries of the ad. The frame can be double, bold, dotted, etc.
In the upper left corner of field No. 1, create field No. 2 (without border), in
which will contain the name of the organization.
In the Draw panel, select Add WordArt.
A WordArt window will appear on the screen, select the raised text, click OK. In the Text entry field, enter the name of the organization "student". Set the font type to Arial, size 18, style - bold, italic, click OK. The name of the organization will appear in text field No. 2, curved in an arc; stretch it vertically.
Create a text field number 3, the size of which fits into the arc of the word “student”. Place the drawing inside the arched text. To do this in the menu Insert select item Drawing\Pictures, in the dialog box that opens, select the appropriate image in the list of files and click the button OK. The inserted picture is surrounded by a frame with white squares. If the picture does not match the size of field No. 3, then it can be reduced by moving these squares with the mouse, and the picture is cropped. To make it smaller proportionally, you need to click on the picture with the mouse, a frame with black squares will appear, with which you can adjust the size of the picture without cropping.
Create a text field No. 4 and type in the ad text “Abstracts, coursework, dissertations: PRINTING, DESIGN”. Select and format the text according to field size No. 4, Arial Narrow font, font size 16, bold, positioned in width, colors dark red, dark blue and autoflower (black).
Create a text field #5 in the line where the first tear-off phone on the left will be located. Add a WordArt object with a vertical text effect and enter a phone number.
Copy text field No. 5 with the phone number using the mouse while pressing the Ctrl key as many times as it will fit in width in text field No. 1. You can use the clipboard, i.e. select an object, copy it to the clipboard with the command Edit\Copy or button Copy on the panel Standard, then place the cursor at the insertion point and execute the command Edit\Paste or button Insert, but when pasting, the copies will overlap each other and will have to be additionally moved into a row manually.
Grouping all objects in order to later use them as a single object, for example, when copying. If this is not done, then each object (picture, phone shortcut, name...) will be copied separately. Grouping objects can be done in two ways:
While holding down the key Shift, click on each of the objects, so they will all be selected at the same time. Then
expand the toolbar Drawing and press the G button group. A common frame will appear around the objects (they will become a single object);
Press the button Selecting objects on the panel Drawing and stretch the grid around all ad objects, they will all be highlighted at the same time and press the button Group. If necessary, objects can be ungrouped using the button Ungroup.
Mouse with key Ctrl or via the clipboard, as indicated in paragraph 9.
Now the advertisement page can be printed and cut into
A sheet of A4 format can accommodate 8 advertisements of this size.
Save the resulting wall announcement (Fig. 6) on a floppy disk with the command File\Save As... .
It should be noted that pictures and text fields can be superimposed on each other in several layers in different sequences, and also placed on top of or behind the main level - the text. For this purpose, 6 toolbar commands are used Drawing\Order.
ABOUT 
Objects created in WordArt can be edited later. To do this, just click on the object, the WordArt menu will open, and change the text effect, font, etc. in it.
To insert an object into text, you need to select the object and in the menu Format, team Borders and Shading, in the window Object Format
in the tab Position choose
required text wrapping.
Figure 6 Wall notice
f Format the object and fill around the frame? 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 For Fig. 6 the flow “along the contour” is performed.
The considered sequence of actions when creating a wall ad is not the only and optimal one. However, it allows you to gain experience using the WordArt program
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.