Table of Contents

## Hexadecimal SS

This is an integer system with base 16 (characters of hexadecimal number system 0-9 and letters A. F). Used in the implementation of computer programming and documentation at a low level, as an 8-bit byte, for recording which it is convenient to use 2 digits of the hexadecimal system.

The Unicode standard uses 4 or more characters of the 16th SS.

This system is also used to record the colors red, green and blue (R, G and B).

### Algorithm for converting numbers to 16CC

The method of transformation is similar to the previous ones. writing down the number as a polynomial, taking into account the degrees 16. To do this, the number is divided by 16, resulting in a list of the remainder of the division written in reverse.

There are calculators on the web, capable of converting numbers to various CC and vice versa (some even with a detailed description of the process).

## Chapter 4.What number systems do professionals use to **work** with a computer?

In the 1950s, mathematicians and designers faced the problem of finding such systems of notation, which would meet the requirements of both computer developers and designers of software. One of the outcomes of this research was a major change in thinking about number systems and computing methods. It turned out that arithmetic, which mankind has been using since ancient times, can be improved, sometimes quite unexpectedly and surprisingly efficiently.

Specialists distinguished the so called “machine” group of number systems and developed ways of converting numbers of this group.

The “machine” number systems include:

Octal (using the digits 0, 1. 7);

Hexadecimal (for the first whole numbers from zero to nine, the digits 0, 1 are used). 9, and for the following numbers, from ten to fifteen, the symbols A, B, C, D, E, F are used as digits).

The official birth of binary arithmetic is associated with the name of G.В. Leibniz’s 1703. an article in which he discussed the rules for performing arithmetic operations on binary numbers.

Why do people use the decimal number system and computers. Binary, octal, hexadecimal?

People prefer the decimal system, probably because since ancient times people have been counting by the fingers, and people have ten fingers on their hands and toes. People don’t always and everywhere use the decimal number system. In China, for example, they used the pentatonic notation system for a long time.

And computers use the binary system because it has several advantages over other systems:

for its realization we need technical devices with two stable states (current. no current, magnetized. not magnetized, etc.).п.) and not, for example, with ten. as in decimal;

the representation of information by means of only two states is reliable and noise-resistant;

it is possible to use the apparatus of Boolean algebra to perform logical transformations of information;

Binary arithmetic is much simpler than decimal arithmetic.

The disadvantage of the binary system is the rapid growth of the number of bits needed to write the numbers.

The binary system convenient for computers is inconvenient for a human because of its unwieldiness and unusual notation.

Conversion of numbers from decimal to binary and vice versa is performed by a machine. However, to use a computer professionally, one must learn to understand the machine’s word. This is what the octal and hexadecimal systems are for.

The numbers in these systems are almost as easy to read as decimal numbers and require three (octal) and four (hexadecimal) times as few digits as in the binary system (because 8 and 16 are, respectively, the third and fourth powers of 2).

The translation of octal and hexadecimal numbers into binary is very simple: simply replace each digit with an equivalent binary triad (three digits) or tetrad (four digits).

To convert a number from binary to octal or hexadecimal, it must be divided left and right of the comma into triads (for octal) or tetrodes (for hexadecimal) and each such group is replaced by the corresponding octal.

In the main part, the 10 ancient numbering systems were discussed, the history of the development of numbering systems. Out of all considered systems of notation the most interesting for me was ancient Chinese numbering. For it is the closest to our “Arab” numbering system. The most beautiful numbers in the ancient Egyptian numbering system. During my research I found out how counting was done orally in ancient times (addition, subtraction, multiplication, and division), how counting boards were used (such as the Greek abacus), how ancient numbers were used to represent fractions, and what numbering systems were used by different peoples.

I also discovered that binary numbering is much older than electronic machines. People have been interested in the binary number system for a long time. This interest was especially strong from the late 16th century to the 19th century. The famous Leibniz found the binary notation system simple, convenient, and beautiful. Even at his request, a medal was struck in honor of this “dyadic” system (as the binary number system was then called).Binary notation is the simplest and most convenient for automation. The presence of only two symbols in a system makes it easier to convert them into electrical signals. It is possible to convert from any number system to a binary code. Almost all computers use either the binary number system directly, or the binary coding of some other number system.

But the binary system also has disadvantages:

Gashkov S.Б. Number Systems and their Applications. ICNMO, 2004.

N. Ugrinovich.Т. Computer Science and Information Technology. Textbook for Grades 10-11 М.: Laboratory of Basic Knowledge. 2003.

3.Wikipedia Encyclopedia [Electronic resource]: Access mode: http://ru.wikipedia.org, free

4.Urnov V.А. etc. Teaching Computer Science in the Computer Classroom, M.: Prosveshcheniye, 1990, pp. 17

5.Zavarykin V.М. Fundamentals of computer science and computer engineering, M.: Prosveshcheniye, 1989, p.19

6.Gein A.Г. Fundamentals of Computer Science and Engineering, M.: Prosveshcheniye, 1992, pp.231

In the 1950s, mathematicians and designers faced the problem of finding such systems of arithmetic that would satisfy the requirements of both **computer** developers and designers of software. One of the results of this research was a considerable change in ideas about numeral systems and methods of computation. It turned out that arithmetic, which mankind has been using since ancient times, could be improved, sometimes quite unexpectedly and surprisingly efficiently.

Specialists have distinguished the so-called “machine” group of number systems and have developed ways of converting numbers in this group.

The “machine” number systems were:

Octal (using the digits 0, 1. 7);

Hexadecimal (the numbers 0, 1 are used for the first integers from zero to nine. 9, and for the following numbers, from ten to fifteen, the symbols A, B, C, D, E, F are used as digits).

The official birth of binary arithmetic is associated with the name of G.В. Leibniz, who published in 1703. an article in which he reviewed the rules for performing arithmetic operations on binary numbers.

Why do people use the decimal numeration system and computers use the. Binary, octal, hexadecimal?

People prefer the decimal system, probably because since ancient times they have counted by the fingers, and people have ten fingers on their hands and feet. Not always and not everywhere people use decimal notation. In China, for example, they used the pentatonic notation for a long time.

And computers use the binary system because it has some advantages over other systems:

for its realization we need technical devices with two stable states (current. no current, magnetized. not magnetized, etc.).п.) rather than, for example, with ten. as in decimal;

The representation of information by means of only two states is reliable and noise-resistant;

Boolean algebra apparatus may be used to perform logical transformations of information;

binary arithmetic is much simpler than decimal arithmetic.

A disadvantage of the binary system is the rapid increase in the number of digits needed to write numbers.

The binary system, convenient for computers, is inconvenient for humans because of its unwieldiness and unusual record.

Conversion of numbers from decimal to binary and vice versa is done by a machine. However, in order to use a computer professionally, one must learn to understand the machine’s word. This is what the octal and hexadecimal systems are for.

Numbers in these systems are almost as easy to read as decimal numbers and require three (octal) and four (hexadecimal) times less digits than in the binary system (because 8 and 16 are, respectively, the third and fourth powers of 2).

The conversion of octal and hexadecimal numbers into binary is very simple: just replace each digit with an equivalent binary triad (three digits) or tetrad (four digits).

To translate a number from binary into octal or hexadecimal, it must be divided left and right of the comma into triads (for octal) or tetrodes (for hexadecimal) and each such group must be replaced by the corresponding octal.

The main part covered 10 ancient numeration systems, the history of development of numeration systems. Of all examined systems of notation the most interesting to me is ancient Chinese numeration. Since it is closest to our “Arabic” numbering system. The most beautiful figures and numbers in the ancient Egyptian numbering system. In the course of the research I found out how in ancient times they used oral arithmetic (addition, subtraction, multiplication, and division), how counting boards (such as the Greek abacus) were used, how fractions were represented with the help of ancient numbers, and what numbering systems were used by different peoples.

I also found out that binary number systems are much older than electronic machines. People have been interested in the binary notation system for a long time. This fascination was particularly strong from the late 16th to the 19th century. The famous Leibniz thought the binary number system was simple, convenient and beautiful. Even at his request was a medal in honor of this “dyadic” system (as they called the binary number **system** at the time).Binary notation is the easiest and most convenient for automation. Having only two symbols in the system makes it easy to convert them into electrical signals. Any number system can be converted to binary. Almost all computers use either directly a binary number system or a binary coding of some other number system.

But the binary system also has disadvantages:

Gashkov S.Б. Number systems and their applications. ICNMO, 2004.

Ugrinovich N.Т. Computer Science and Information Technology. Textbook for Grades 10-11 М.: Laboratory of Basic Knowledge. 2003.

3.Wikipedia Encyclopedia [Electronic resource]: Access mode: http://ru.wikipedia.org, free

4.Urnov V.А. etc. Teaching Computer Science in the Computer Classroom, M.: Prosveshcheniye, 1990, pp. 17

5.Zavarykin V.М. Fundamentals of Informatics and Computer Science, M.: Prosveshcheniye, 1989, pp.19

6.Gein A.Г. Fundamentals of Computer Science and Engineering, M.: Prosveshcheniye, 1992, p.231

## Value of numbers

But this is not always the case. This way of representing values is relatively modern and requires the use of zeros as placeholders. This is why older number systems such as Roman numerals or Egyptian hieroglyphics do not use this place value, but add all the digits to get the total value.

Using this basic concept of place value, we have created different numbering systems or ways of writing numbers. They are named by the number of increments per place, that is, how many times you can increase the value of one place before you have to “move” it to the next. For example, on a decimal basis, we can increase the number of units by a factor of nine with 10 different digits (counting zero) before going from units to tens.

## “Representation of numerical information in a computer. Numbering Systems”.

Please note that in accordance with Federal Law N 273-FZ “On Education in the Russian Federation” in organizations carrying out educational activities, training and education of students with disabilities is organized both together with other students, and in separate classes or groups.

Topic: “Representation of Numerical Information in the **Computer**. Number Systems”.

*Educational:* To reveal the concepts of “number”, “number”, “numbering system”, “positional numbering system”, “non-positional numbering system”, “basis for positional numbering **system**“; to introduce students to the history of the emergence and development of numbering systems, with ways of representing numbers in positional and nonpositional numbering systems; to indicate the main drawbacks and advantages of nonpositional and positional numbering systems; to improve skills of using information technology (skills of working with the testing program easyQuizzy).

*developing:* promote the development of cognitive interest, logical thinking (methods of analysis, synthesis), speech, attention, activity, independence of students.

*educational:* develop neatness, love for the subject, help students develop skills of self-organization and initiative, and the ability to evaluate their own and their classmates’ **work**.

Equipment: multimedia projector, computer, didactic handouts.

Type of the lesson: combined lesson.

**Work** on comprehension and assimilation of new material (22 min.)

Testing. Reconfirmation of what has been learned (10 min).)

*Greeting. Checking students’ readiness for the lesson.*

(Classroom duty officer announces the absentees)

*Creation of friendly atmosphere. Psychological mood for success.*

Guys, let’s have a look at **your** classmate. Smile at each other! I think you are in a good mood, so let’s work with that cheerful mood in the lesson. Be active and positive.

*(Processor) frontal questioning. Repetition of Background Knowledge.*

Before we begin our lesson, let’s have an intellectual warm-up and guess the topic of today’s lesson. You will have to guess the crossword puzzle and the rebus.

Please look at the crossword puzzle (Presentation 1, slide 2).

3.A computer device that processes information and controls other devices. The most important chip, the “brain” of a computer (processor).

It comes in matrix, inkjet, and laser. Output device (printer).

A special device for controlling the cursor, a manipulator (mouse).

What word came up vertically? (Answer: system.).

Systems come in many forms. Let’s find out which systems we’re going to study with you today by guessing the rebus.

What do the commas at the beginning or end of the diagram mean?? (Answer: as many letters must be discarded first or at the end.). What would the notation “e=and” mean?? (Answer: replace the letter e in the word with i).

So now that our mental warm-up has finished we have understood that today we are going to study number systems.

**Work** on comprehension and assimilation of the new material (25 min.)

Open **your** notebooks and sign the number and the topic of the lesson: “Representation of numerical information in a computer. Number systems”. (Presentation 1, slide 3)

Today in this lesson we will take a journey into the history of numbers, learn the concepts of numbers, numbers and number systems and find out which number systems are called positive and which are called non-positive. You will learn how to write numbers in different number systems, determine the basis of a number system, give examples of numbers in different number systems, and write numbers in expanded form.

*Initial perception of the new material.*

The epigraph of our lesson is “Everything is a number. This is what the Pythagoreans said. What did the ancient sages have in mind? (Answer: the important role of numbers in human practice).

“Everything is a number.”. the sages used to say, stressing the extraordinary importance of numbers in people’s lives. Every day modern man remembers the numbers of cars and telephones, counts the cost of purchases in a store, keeps a family budget, etc.д. etc.п. Numbers, numbers they are with us everywhere.

People have always written numbers, even five thousand years ago. But they wrote them completely differently, according to different rules. But in any case numbers were represented by symbols which are called numerals.

Who can tell me the definition of “number”?? Numbers are symbols involved in recording numbers and make up some alphabet. (Presentation 1, slide 4)

What then is a number?? The concept of number has evolved over many years, and today is considered a fundamental concept not only in mathematics, but also in computer science. A number is a quantity.

Numbers are put together by special rules. At different stages of human development, different peoples had different rules and today we call them number systems.

What are number systems?? A number system is a numbering system in which numbers are written according to certain rules using signs of some alphabet, called digits. (Presentation 1, slide 5)

Let’s write this definition down in our notebook.

__Writing in a Notebook__*: A numbering system is a system of signs where numbers are recorded according to certain rules by means of symbols of some alphabet called digits.*

All number systems are divided into positional and non-positional. Non-positional number systems are those in which the numerical value of a number **does** not depend on its position in the number. Positional number systems are systems in which the numerical value of a digit depends on its position in a number. (Presentation 1, slide 6)

We are now going to get acquainted with nonpositional number systems and find out what their peculiarities are. (Presentation 1, Slide 7)

Non-positional systems began before positional ones. The latter are in turn the result of the long historical development of non-positional numbering systems.

Unit numeral system (Presentation 1, Slide 8)

In ancient times, when people began to count, there was a need to record numbers. The number of items, such as bags, was depicted by drawing dashes or notches on some hard surface: stone, clay, or wood (the invention of paper was still a long way off). In this system one line corresponded to each sack. Archaeologists have found such “records” from excavations of cultural layers from the Paleolithic period (10,000-11,000 BC).э.).

Scientists have called this way of writing numbers the unit (stick) number system. It used only one kind of sign for writing numbers. the stick. Each number in this notation was represented by a line of sticks equal in number to the number to be represented.

The inconveniences of this system of recording numbers and the limitations of its use are obvious: the larger the number to be written, the longer the row of sticks, when writing a large number it is easy to make a mistake by putting too many sticks or, conversely, by not writing enough sticks.

We can assume that in order to make counting easier, people began to group objects by 3, 5, or 10. And when writing, they began to use signs corresponding to a group of several objects. This way, more convenient systems of number writing evolved.

Echoes of the unit notation system are still to be found today. For example, children without realizing it themselves show their age on their fingers, and counting sticks are used to teach numeracy to 1st grade pupils.

Ancient Egyptian decimal nonpositional notation (Presentation 1, slide 9)

The ancient Egyptian nonpositional decimal system emerged in the second half of the third millennium BC.э. Paper was replaced by a clay board, and that is why the numbers have this type of writing.

This **numeral** system used special signs (digits) to represent the numbers 1, 10, 100, 1000, etc.д. and were written in special characters.

Egyptian numeration was recorded as a combination of these “digits” in which each “digit” was repeated up to nine times.

Why? (Answer: because ten identical numbers in a row can be replaced by one number, but one digit higher)

All other numbers were made up of these key numbers by adding them together. They began by writing higher numbers and then lower numbers.

Let’s look at the following example. What number is written? (Answer: 345).

The Roman number system (Presentation 1, slide 10)

I think you are very familiar with this number system. Where did you encounter it?? (Answer: chapters in books, hours).

The Roman system we know is not much different from the Egyptian. But it’s more common nowadays: in books, in movies

Let us remember the symbols used for Roman numerals.

It uses capital Latin signs I (one finger) for 1, V (open palm) for 5, X (two folded hands) for 10, and for numbers 50, 100, 500 and 1000 capital Latin letters of corresponding Latin words are used (Centrum, a hundred, Demimille, half a thousand, Mille. thousand) L, C, D, and M (respectively), which are the “digits” of that number system. A number in the Roman numeral system is indicated by a set of consecutive “digits”.

Note that while in ancient Egypt numbers were written using only addition, the ancient Romans used not only addition but subtraction as well. The following rule was applied: the value of each less significant digit, placed to the left of the greater, is subtracted from the value of the greater. (Presentation 1, Slide 11)

Now let’s look at how numbers are written in the Roman **numeral** system. There are certain rules. (Presentation 1, slide 12)

Rules for making up numbers in the Roman **numeral** system:

1) the sum of the values of several consecutive identical “digits” (let’s call them a group of the first kind);

2) the differences in the values of two “digits” if the larger digit has the smaller digit on its left. The value of the smaller digit is deducted from the value of the larger digit. Together they form a group of the second kind.

3) The sum of values of groups and “digits” not included in groups of the first or second kind.

Number 32 in the Roman number system is XXXII = (X X X) (I I) = 30 2 (two groups of the first kind). (Presentation 1, slide 13)

Let us have a look at an example. Let’s convert the number 1974 to the Roman **numeral** system. (Presentation 1, slide 14)

The Roman number 1974 has the form MSMLXXXIV = M (M. C) L (X X) (V. I) = 1000 900 50 20 4 (along with groups of both types, some “digits” are involved in forming the number.).

We will now pause for a short physical activity break (Presentation 1, slide 15):

So, guys, we have talked about nonpositional number systems for so long. Now, let’s give a definition of the positional numbering system (Answer: a numbering system in which the quantitative value of a digit depends on the place (position) it occupies in a number).

The numbering systems based on the positional principle emerged independently of one another in ancient Mesopotamia (Babylon), the Maya, and finally, in India. The prerequisites for the positional number system were the shortcomings of nonpositional. These included the following: in recording numbers a large number of digits are involved, it is inconvenient to perform arithmetic operations, it is impossible to represent negative and fractional numbers. (Presentation 1, slide 16)

Now we will watch a short video about the decimal number system.

What number systems the video was about? (Answer: decimal, octal).

After watching the video, you should become familiar with the concept of SS base. The basis of the positional CC is the number of digits or other characters that are used to store numbers. (Presentation 1, slide 17)

The main advantages of any positional MSS are simplicity of arithmetic operations and a limited number of characters required to record any numbers.

Let’s record data on some number systems in the table.

There are positional systems with different bases. Let’s take a look at the binary SS. Binary number **system**. a number system based on the positional principle of recording numbers, with a base of 2. This SS uses only two characters. the digits 0 and 1. This number system is used in computers. (Presentation 1, Slide 19)

Recall how information is encoded in a computer? (Answer: with binary coding, t.е. Any information is represented as a sequence of 0 and 1).

Testing. Review of what you’ve learned. (10 min.)

*Introductory instruction. Clarification of the task in order to prevent possible mistakes.*

So now we will consolidate a little bit of the material we covered in class. You will now take a test on the **computer**. There are eight questions in total. I’m sure that each of you will be able to do the following. (Presentation 1, slide 20)

This activity will be timed. You have 8 minutes in total. There are three choices for each question. You will need to left click on the correct answer.

What is the name of the sign system in which numbers are written according to certain rules using signs of some alphabet??

What groups are divided into number systems??

What are the symbols used to write numbers and which make up a particular alphabet??

What is the peculiarity of non-positional number systems??

What is the basis of the positional notation system?

Which numeral system is used to record numbers: I. V. X. L. C. D. M ?

What number system is used in the computer?

What is the base of the number system for the following number 428?

*Review the safety rules. Frontal questioning.*

At home, learn the notation. Fill in the table about number systems. In the table, some cells are missing and some are full. You must fill in the entire table based on the cells you fill in. Let’s look now at the line where base 10 is written. What is a system with base 10?? (Answer: decimal). What digits are used to write numbers in decimal notation? (Answer: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9). Use an analogy to fill in the rest of the table. Next task. Write the expanded form of the number A10=3642. Also review §2.3 pp. 21-23 in the textbook. (Presentation 1, slide 21)

*Instructions for homework.*

*Summary of the lesson. Marking with reasoning.*

So today in the lesson we learned about the concepts of numbers, numbers and number systems and we found out which number systems are called positional and which are called non-positional. You’ve learned to write numbers in different number systems, find the base of a number system, give examples of numbers in different number systems, write numbers in expanded form.

Now let’s do this activity. on your desks you can see three different colored squares: green, yellow and red. Each of these colors means the following: green. you liked the lesson very much, everything was clear; yellow. the lesson was good, but you did not understand all the concepts, there were difficulties; red. you did not like the lesson, it was uninteresting and boring. Pin the square of the color on your monitor screen to give **your** grade for the lesson.

So, for the lesson I have given marks After the bell, give me **your** journals. This concludes our lesson. Goodbye!

## Numbering Systems for Your Computer

There are positive and non-positive numeric systems.

В *non-positional* Number systems the weight of a digit determining the value of a number is independent of its position in the number entry. For example in the Roman number system in the number XXX (thirty ) the weight of digit X in any position equals ten.

В *positional* **numeral** systems, the weight of each digit varies depending on its position in the sequence of digits representing the number. For example, in the number 757.7 the first seven places represent seven hundredths, the second seven places seven one, and the third seven places seven tenths of one. The number 757.7 is essentially an abbreviated notation of the expression:

700 50 7 0,7 = 710 2 510 1 710 0 710.1 = 757,7.

Every positional notation is characterized by its base.

The basis of a positional number system is determined by the number of digits used to record numbers in that system. |

The decimal system uses ten different digits. However,

countless positional systems are possible: binary, ternary, quadruple, etc.д. Numerical notation with base. *q*means an abbreviated notation of an expression in general form:

where *ai*. the digits of a number in a number system; *n* и *m*. the number of integer and fractional digits, respectively.

Whole numbers in any number system are generated using the general Counting Rule:

To form an integer following any given integer, the rightmost digit of the number must be incremented by one; if this operation results in any digit being zero, then the digit to its left must be incremented by one. |

Applying this rule, we can write the first ten integers

In the binary system: 0, 1, 10, 11, 100, 101, 110, 111, 1000, 1001;

in the ternary system: 0, 1, 2, 10, 11, 12, 20, 21, 22, 100;

in the pentatonic system: 0, 1, 2, 3, 4, 10, 11, 12, 13, 14;

octal system: 0, 1, 2, 3, 4, 5, 6, 7, 10, 11.

Besides decimal numeration systems are widely used with a base that is an *integer degree of number 2*, *namely*:

octal (digits 0, 1. 7);

hexadecimal (for the first ten digits from zero to nine the digits 0, 1 are used). 9, and for the next digits. from ten to fifteen. the symbols A, B, C, D, E, F are used as digits).

For technical implementation in computers binary notation is used, because it is much easier to implement than decimal notation:

a) it requires technical devices with only two stable states (current. no current, magnetized. not magnetized, etc.).п.);

b) it is possible to use the apparatus of Boolean algebra to perform logical transformations of information.

A disadvantage of the binary system is the rapid increase in the number of digits needed to write the numbers.

Integer numbers in positional notation systems.

Arithmetic basics of computers

A number system is a way of writing numbers using a given set of digits. |

There are positive and non-positional notation systems.

В *nonpositional* Numbering systems. number systems, where the weight of a digit determining the value of the number is independent of its position in the number entry. So, in the Roman notation system in number XXX (thirty ) the weight of digit X in any position equals ten.

В *positional* Numeration systems the weight of each digit changes depending on its position in the sequence of digits representing the number. For example, in the number 757.7 the first seven means 7 hundreds, the second seven means 7 units, and the third seven means 7 tenths of a unit. The number 757.7 is essentially a shortened notation of an expression:

700 50 7 0,7 = 710 2 510 1 710 0 710.1 = 757,7.

Any positional number system is characterized by its base.

The basis of a positional number system is determined by the number of digits used to write numbers in that system. |

The decimal system uses ten different digits. However,

an infinite number of positional systems are possible: binary, ternary, quadruple, etc.д. Writing numbers in a number system with a base. *q*means an abbreviated notation of the expression in general form:

where *ai*. digits of a number in a number system; *n* и *m*. is the number of integer digits and fractional digits, respectively.

Whole numbers in any number system are generated using the general Counting Rule:

To form an integer following any given integer, the rightmost digit of the number must be increased by one; if as a result of this operation any digit becomes zero, then the digit to its left must be increased by one. |

Applying this rule, we can write the first ten integers

in binary system: 0, 1, 10, 11, 100, 101, 110, 111, 1000, 1001;

In the ternary system: 0, 1, 2, 10, 11, 12, 20, 21, 22, 100;

in hex system: 0, 1, 2, 3, 4, 10, 11, 12, 13, 14;

octal system: 0, 1, 2, 3, 4, 5, 6, 7, 10, 11.

In addition to the decimal, systems with a basis which is *the integer degree of the number 2*, *namely*:

octal (digits 0, 1 are used). 7);

hexadecimal (for the first ten digits from zero to nine the digits 0, 1. 9, and for the following digits. from ten to fifteen. the symbols A, B, C, D, E, F are used as digits).

For technical implementation in computers, the binary number system is used because it is much easier to implement than the decimal system:

(a) it requires technical devices with only two steady states (there is current. no current, magnetized. not magnetized, etc.п.);

b) it is possible to use the apparatus of Boolean algebra to perform logical transformations of information.

A disadvantage of the binary system is the rapid increase in the number of digits needed to write the numbers.

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### Number Systems Introduction. Decimal, Binary, Octal & Hexadecimal

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## Numbering systems used in computers

The computer uses the binary number system to represent information because it has several advantages over other number systems:

To implement it you need technical devices with two steady states (not ten, like in decimal notation). For example: electromagnetic relay (closed/open), which was widely used in the designs of the first computers; surface area of a magnetic data carrier (magnetized/ demagnetized); surface area of a laser disk (reflect/not reflect); a trigger, which can stably be in one of two states;

is widely used in the main memory of a computer;

Representation of information by means of only two states is reliable and noise-resistant;

it is possible to use the apparatus of Boolean algebra to perform logical transformations of information (cf. dl. 3);

Binary arithmetic is much easier than decimal arithmetic. A disadvantage of binary notation is the rapid growth of

It is very difficult for a human to comprehend multibit numbers, t.е. But for computer it doesn’t matter how big the number is, because modern computers can process over 64 binary digits per clock cycle.

Conversion of numbers from decimal to binary and vice versa is done by the machine, but programmers often use octal and hexadecimal number systems during the stages of debugging programs and viewing the contents of files in machine code mode.

Numbers in these number systems are read almost as easily as decimal numbers, requiring three (octal) and four (hexadecimal) times as few digits respectively than in binary (numbers 8 and 16 are respectively the third and fourth powers of 2).

The conversion of octal and hexadecimal numbers into binary is very simple; just replace each digit with its equivalent binary triad (three digits) for the octal number system or tetrad (four digits) for the hexadecimal number system.

To convert a number from binary into octal or hexadecimal, divide it to the left and right of the decimal point into triads or tetrads and replace each such group with the corresponding octal or hexadecimal digit.

10101001, 101112 = 10 101 001, 101 1112 = 251,568; 2 5 1 5 6

10101001, 101112 = Y10 1001, 1011, 10002 = L9, L816. *А* 9 *В* 8

Arithmetic operations in positional notation systems

Consider the basic arithmetic operations: addition, subtraction, multiplication, and division. The rules of these operations in the decimal system are well known: addition, subtraction, column multiplication, and angle division. These rules are also applicable to all other positional notations. Only use the specific addition and multiplication tables for each system.

Addition.During addition the digits are summed up one by one and if there is surplus it is transferred to the left to the higher digit.

Binary addition

Decimal addition: 1510 610.

Binary addition: 11112 110:

Consider some more examples of addition in binary notation:

Subtraction.Always use the subtraction operation to subtract a smaller number from a larger absolute number and assign the appropriate sign. In the subtraction table a point represents a loan in the senior digit which goes into the junior digit as *д* units.

Subtraction in decimal notation: 201.2510.59,7510.

Subtraction in binary: 11001001,01.- 111011,112.

1 100 100 1,0 1 __00111011,11__ 1000 110 1,10

Consider a few more examples of subtraction in the decimal number **system**:

Multiplication.When multiplying multidigit numbers in different positional notation systems you may use the usual columnar multiplication algorithm, but in doing so the results of multiplication and addition of single digit numbers must be taken from the corresponding multiplication and addition tables of the system in question.

*Multiplication in Binary Numerals*

Let’s look at some examples of multiplication in binary notation:

100 1, 1 | 1 1 0 0,0 1 | 1 0 0 0 0 0, 1 |

10,1 | х 10,0 1 | |

100 11 | 110001 | |

1 0 0 11 | 11000 1 | 1 0 0 0 0 0 1 |

101 1 1, 1 1 | 1 0 0 1 0 0, 1 1 | 100 1001,00 1 |

Division.Division in any positional notation **system** follows the same rules as division by an angle in the decimal notation system. Division in binary is especially easy: after all the next digit of the quotient can only be zero or one.

Examples.Divide 5865 by 115.

Division in decimal notation: 586510 : 115ш.

Division (5865 : 115) 10 in binary notation: 10I01I010012:11100112.

## Number Systems

A **computer** is a device that performs a well-defined sequence of operations that is specified by a program. About what role the number systems play, we will describe a little later. A computer combines the concepts of hardware on the one hand and software on the other.

All PC devices are connected to the motherboard and ensure their interaction with the CPU, RAM, BIOS, and each other. All PC devices are divided into internal and external.

Internal devices are installed in the system unit along with the motherboard and external devices are connected to slots on the back of the PC case. Some of the devices may be both external and internal, such as a modem or a magneto-optic disk drive.

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