Numbers – types, concepts and operations, natural and other types of numbers.
Number is a fundamental concept of mathematics, used to determine quantitative characteristics, numbering, comparison of objects and their parts. Various arithmetic operations are applicable to numbers: addition, subtraction, multiplication, division, exponentiation and others.
The numbers involved in the operation are called operands. Depending on the action performed, they receive different names. In general, the operation scheme can be represented as follows:<операнд1> <знак операции> <операнд2> = <результат>.
In a division operation, the first operand is called the dividend (this is the name of the number that is being divided). The second (by which they divide) is the divisor, and the result is the quotient (it shows how many times the dividend is greater than the divisor).
Types of numbers
Various numbers can be involved in the division operation. The result of division can be integer or fractional. In mathematics there are the following types of numbers:
- Natural numbers are numbers used in counting. Among them, a subset of prime numbers stands out, having only two divisors: one and itself. All others except 1 are called composite and have more than two divisors (examples of prime numbers: 2, 5, 7, 11, 13, 17, 19, etc.);
- Integers are a set consisting of negative, positive numbers and zero. When dividing one integer by another, the quotient can be an integer or a real (fractional). Among them we can distinguish a subset of perfect numbers - equal to the sum of all their divisors (including 1), except for themselves. The ancient Greeks knew only four perfect numbers. Sequence of perfect numbers: 6, 28, 496, 8128, 33550336... So far, not a single odd perfect number is known;
- Rational - representable as a fraction a/b, where a is the numerator and b is the denominator (the quotient of such numbers is usually not calculated);
- Real (real) – containing an integer and a fractional part. The set includes rational and irrational numbers (representable as a non-periodic infinite decimal fraction). The quotient of such numbers is usually a real value.
There are several features associated with performing the arithmetic operation - division. Understanding them is important to obtain the correct result:
- You cannot divide by zero (in mathematics this operation makes no sense);
- Integer division is an operation as a result of which only the integer part is calculated (the fractional part is discarded);
- Calculating the remainder of an integer division allows you to obtain as a result the integer remaining after the operation is completed (for example, when dividing 17 by 2, the integer part is 8, the remainder is 1).
The concept of a real number: real number- (real number), any non-negative or negative number or zero. Real numbers are used to express measurements of each physical quantity.
Real, or real number arose from the need to measure the geometric and physical quantities of the world. In addition, for performing root extraction operations, calculating logarithms, solving algebraic equations, etc.
Natural numbers were formed with the development of counting, and rational numbers with the need to manage parts of a whole, then real numbers (real) are used to measure continuous quantities. Thus, the expansion of the stock of numbers that are considered led to the set of real numbers, which, in addition to rational numbers, consists of other elements called irrational numbers.
Set of real numbers(denoted R) are sets of rational and irrational numbers collected together.
Real numbers divided byrational And irrational.
The set of real numbers is denoted and often called real or number line. Real numbers consist of simple objects: whole And rational numbers.
A number that can be written as a ratio, wherem is an integer, and n- natural number, isrational number.
Any rational number can easily be represented as a finite fraction or an infinite periodic decimal fraction.
Example,
Infinite decimal, is a decimal fraction that has an infinite number of digits after the decimal point.
Numbers that cannot be represented in the form are irrational numbers.
Example:
Any irrational number can easily be represented as an infinite non-periodic decimal fraction.
Example,
Rational and irrational numbers create set of real numbers. All real numbers correspond to one point on the coordinate line, which is called number line.
For numerical sets the following notation is used:
- N- set of natural numbers;
- Z- set of integers;
- Q- set of rational numbers;
- R- set of real numbers.
Theory of infinite decimal fractions.
A real number is defined as infinite decimal, i.e.:
±a 0 ,a 1 a 2 …a n …
where ± is one of the symbols + or −, a number sign,
a 0 is a positive integer,
a 1 ,a 2 ,…a n ,… is a sequence of decimal places, i.e. elements of a numerical set {0,1,…9}.
An infinite decimal fraction can be explained as a number that lies between rational points on the number line like:
±a 0 ,a 1 a 2 …a n And ±(a 0 ,a 1 a 2 …a n +10 −n) for all n=0,1,2,…
Comparison of real numbers as infinite decimal fractions occurs place-wise. For example, suppose we are given 2 positive numbers:
α =+a 0 ,a 1 a 2 …a n …
β =+b 0 ,b 1 b 2 …b n …
If a 0 0, That α<β ; If a 0 >b 0 That α>β . When a 0 =b 0 Let's move on to the comparison of the next category. Etc. When α≠β , which means that after a finite number of steps the first digit will be encountered n, such that a n ≠b n. If a n n, That α<β ; If a n > b n That α>β .
But it is tedious to pay attention to the fact that the number a 0 ,a 1 a 2 …a n (9)=a 0 ,a 1 a 2 …a n +10 −n . Therefore, if the record of one of the numbers being compared, starting from a certain digit, is a periodic decimal fraction with 9 in the period, then it must be replaced with an equivalent record with a zero in the period.
Arithmetic operations with infinite decimal fractions are a continuous continuation of the corresponding operations with rational numbers. For example, the sum of real numbers α And β is a real number α+β , which satisfies the following conditions:
∀ a′,a′′,b′,b′′∈ Q(a′⩽ α ⩽ a′′)∧ (b′⩽ β ⩽ b′′)⇒ (a′+b′⩽ α + β ⩽ a′′+b′′)
The operation of multiplying infinite decimal fractions is defined similarly.
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
The concept of a real number: real number- (real number), any non-negative or negative number or zero. Real numbers are used to express measurements of each physical quantity.
Real, or real number arose from the need to measure the geometric and physical quantities of the world. In addition, for performing root extraction operations, calculating logarithms, solving algebraic equations, etc.
Natural numbers were formed with the development of counting, and rational numbers with the need to manage parts of a whole, then real numbers (real) are used to measure continuous quantities. Thus, the expansion of the stock of numbers that are considered led to the set of real numbers, which, in addition to rational numbers, consists of other elements called irrational numbers.
Set of real numbers(denoted R) are sets of rational and irrational numbers collected together.
Real numbers divided byrational And irrational.
The set of real numbers is denoted and often called real or number line. Real numbers consist of simple objects: whole And rational numbers.
A number that can be written as a ratio, wherem is an integer, and n- natural number, isrational number.
Any rational number can easily be represented as a finite fraction or an infinite periodic decimal fraction.
Example,
Infinite decimal, is a decimal fraction that has an infinite number of digits after the decimal point.
Numbers that cannot be represented in the form are irrational numbers.
Example:
Any irrational number can easily be represented as an infinite non-periodic decimal fraction.
Example,
Rational and irrational numbers create set of real numbers. All real numbers correspond to one point on the coordinate line, which is called number line.
For numerical sets the following notation is used:
- N- set of natural numbers;
- Z- set of integers;
- Q- set of rational numbers;
- R- set of real numbers.
Theory of infinite decimal fractions.
A real number is defined as infinite decimal, i.e.:
±a 0 ,a 1 a 2 …a n …
where ± is one of the symbols + or −, a number sign,
a 0 is a positive integer,
a 1 ,a 2 ,…a n ,… is a sequence of decimal places, i.e. elements of a numerical set {0,1,…9}.
An infinite decimal fraction can be explained as a number that lies between rational points on the number line like:
±a 0 ,a 1 a 2 …a n And ±(a 0 ,a 1 a 2 …a n +10 −n) for all n=0,1,2,…
Comparison of real numbers as infinite decimal fractions occurs place-wise. For example, suppose we are given 2 positive numbers:
α =+a 0 ,a 1 a 2 …a n …
β =+b 0 ,b 1 b 2 …b n …
If a 0 0, That α<β ; If a 0 >b 0 That α>β . When a 0 =b 0 Let's move on to the comparison of the next category. Etc. When α≠β , which means that after a finite number of steps the first digit will be encountered n, such that a n ≠b n. If a n n, That α<β ; If a n > b n That α>β .
But it is tedious to pay attention to the fact that the number a 0 ,a 1 a 2 …a n (9)=a 0 ,a 1 a 2 …a n +10 −n . Therefore, if the record of one of the numbers being compared, starting from a certain digit, is a periodic decimal fraction with 9 in the period, then it must be replaced with an equivalent record with a zero in the period.
Arithmetic operations with infinite decimal fractions are a continuous continuation of the corresponding operations with rational numbers. For example, the sum of real numbers α And β is a real number α+β , which satisfies the following conditions:
∀ a′,a′′,b′,b′′∈ Q(a′⩽ α ⩽ a′′)∧ (b′⩽ β ⩽ b′′)⇒ (a′+b′⩽ α + β ⩽ a′′+b′′)
The operation of multiplying infinite decimal fractions is defined similarly.
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.
