Presentation on the topic "Binary Number System". Binary number system. Basics of binary arithmetic Download presentation on the topic Binary number system

To enjoy previewing presentations, create yourself an account (account) Google and log in to it: https://accounts.google.com

Signatures for slides:

Binary number system

We repeat the topic "Number Systems"

The main concepts of the number systems of the number is a method of recording numbers and associated methods for performing calculations. Number is some number of digit - these are symbols involved in the number of alphabet numbers - a set of different numbers used to record the number

A single ("stick") system of number (Paleolithic period, 10-11 thousand years BC) Before a person learned to consider or invented words to designate numbers, he undoubtedly owned a visual, intuitive idea of \u200b\u200bthe number. or designation:

3 4 5 - units - dozens - hundreds of designation: the hieroglyphic inscriptions of the ancient Egyptians were carefully carved on stone monuments. From these inscriptions, we know that the ancient Egyptians used only a decimal number system. Ancient Egyptian number system (OK.2850 BC)

2nd category 1st category \u003d 60 + 20 + 2 \u003d 82 Babylonian sixteent number system (2 thousand years BC) First known to us, based on the positional principle. - units - tens - 60; 60 2; 60 3; ...; 60 N Designation:

X x x i i \u003d 3 2 d x L i i \u003d 542 1000 500 100 50 10 5 1 m d c l x v i rima number system (500 years BC) as numbers in the Roman system: the value of the number does not depend on its position among its position. If a smaller figure stands to the left of greater, it will be deducted if the right is added. For example, IX \u003d 9, and xi \u003d 11. What numbers are recorded by Roman numbers? The value of the number is defined as the amount or difference of numbers among the number.

- Base (P) Set of all numbers for recording numbers - Alphabet The number of digits for recording numbers Positional systems may have a different alphabet (2,3,4 characters). Positional viewing systems Each positional numbering system has a certain alphabet and base.

Base title alphabet P \u003d 2 binary 0 1 p \u003d 3 Tropic 0 1 2 P \u003d 8 octal 0 1 2 3 4 5 6 7 p \u003d 16 hexadecimal 0 1 2 3 4 5 6 7 8 9 ABCDEF Alphabets of number systems for recording numbers in The position system with the base p must have an alphabet from p numbers. With p\u003e 10 to ten Arabic figures add Latin letters. The position of the numbers is among the category.

The presentation of information in the computer in each such "cell" is stored only one of two values: zero or unit. Each "Cell" of the computer's memory is called a bit. The numbers 0 and 1 stored in the "cells" of the computer are called bits. 0 1 and machine memory is conveniently present in the form of a sheet into a cell.

5555 \u003d 5000 + 500 + 50 + 5 \u003d 5 * 1000 + 5 * 100 + 5 * 10 + 5 * 1 \u003d 5 * 10 3 + 5 * 10 2 + 5 * 10 1 + 5 * 10 0 456327 \u003d 4 * 100000 + 5 * 10000 + 6 * 1000 + 3 * 100 + 2 * 10 + 7 * 1 \u003d 4 * 10 5 + 5 * 10 4 + 6 * 10 3 + 3 * 10 2 + 2 * 10 1 + 7 * 10 0 Consider a decimal number system disclosed number recording form

The position of the numbers is among the category. A q \u003d a n-1 q n-1 + ... + a 1 q 1 + a 0 Q 0 + a -1 Q -1 + ... + a -m q -m, where q is the basis of the system Number (number of used numbers) a q - number in the number system with the base Qa - numbers of a multi-digit number a Qn (M) - the amount of integers (fractional) discharges of the number a q Deployed form recording form

1101 2 \u003d 1 * 2 3 + 1 * 2 2 + 0 * 2 1 + 1 * 2 0 \u003d 1 * 8 + 1 * 4 + 0 * 2 + 1 * 1 \u003d 13 11100011 2 \u003d? Consider a binary number system for a binary number to decimal

Divide the whole decimal number at 2. Record balance. If the received private is at least 2, then continue division. The binary code of the decimal number is obtained with a consistent recording of the last private and all residues starting from the latter. Translation of whole decimal numbers in binary system

Translate decimal numbers in binary 154 10 \u003d 658 10 \u003d 10005 10 \u003d task

Arithmetic binary numbers 0 + 0 \u003d 0 + 1 \u003d 1 + 0 \u003d 1 + 1 \u003d 0 * 0 \u003d 0 * 1 \u003d 1 * 0 \u003d 1 * 1 \u003d 0 10 0 0 0 1 1 1

§16 p. 100 Task 4, 5 and 6 homework

On the topic: Methodical development, presentations and abstracts

Number system. Basic concepts. Binary number system

The multimedia presentation contains the basic concepts on the topic of the "System Challenge". The binary number system is presented in the presentation according to the following scheme: base, assembly and algorithmic numbers, ...

Number System Number System This set of receptions and rules to designate and naming numbers. The positional number is called because the same figure receives various quantitative values \u200b\u200bdepending on the place, or the position it takes in the number of numbers. For example, in the recording of the number 555, the number 5, standing in the first place on the right, denotes 5 units, on the second 5 dozen, on the third 5 hundred.

Positional Surface Systems The base of the positioning system is the number of different characters or characters used for the image of the numbers in this system. For the base of the system, you can take any natural number two, three, four, etc. Consequently, countless positions are possible: binary, ternary, quiet, etc.

Positional Number Systems Example: Binary System Release System Number, 1 2 \u003d 1 * 2 3 + 0 * 2 2 + 1 * 2 1 + 0 * 2 0 + 1 * 2 -1 octal discharge number system Number2 7 6, 5 2 \u003d 2 * 8 2 + 7 * 8 1 + 6 * 8 0 + 5 * * 8 -2

Positional viewing systems The binary system, convenient for computers, is inconvenient for a person because of its bulkness and unusual recording. In this regard, the octal and hexadecimal systems have been developed. The numbers in these systems are read almost as easily as decimal, they require three (octal) and four (hexadecimal) times less discharges, respectively than in the binary system (after all, the number 8 and 16 - respectively, the third and fourth degrees of the number 2) . -Full (numbers are used 0, 1); -Bellic (numbers are used 0, 1, ..., 7); -The property (for the first integers from zero to nine, numbers 0, 1, ..., 9 are used, and for the following numbers from ten to fifteen - as numbers, symbols A, B, C, D, E, F) are used as numbers.

Writing numbers in 10-y2-y8-y16-ya2-ya2-ya8-y16-i2-ya2-ya8-y16-Я2-ya8-y16-i A B C D E F

How information is in a computer, or digital data to understand how the most diverse information is presented in the computer, "disconnect" inside the machine memory. It is convenient to present in the form of a sheet into a cage. In each such "cell" is stored only one of two values: zero or unit. Two digits are convenient for electronic data storage, since they require only two states of the electronic circuit "on" (this corresponds to the number 1) and "off" (this corresponds to the figure 0). Each "Cell" of the computer's memory is called a bit. The numbers 0 and 1 stored in the "cells" of the computer's memory are called bits.

Using the bits sequence, you can submit the most different information. Such a presentation of information is called binary or digital encoding. The advantage of digital data is that they are relatively easy to copy and change. They can be stored and transmitted using the same methods, regardless of the type of data. Methods of digital coding of texts, sounds (voices, music), images (photos, illustrations) and image sequences (cinema and video), as well as three-dimensional objects were invented in the 80s of the last century.

Binary coding of numeric information is known many ways to write numbers. We use a decimal positioning system. It is called decimal because in this number system, ten units of one discharge are one unit of the next older discharge. The number 10 is called the base of the decimal number system. To record numbers in a decimal system, ten digits are used: 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9.

Binary coding of numerical information Consider two numeric rows: 1, 10, 100, 1000, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, both of these rows begin with a unit. Each next number of the first row is obtained by multiplying the previous number by 10. Each next number of the second row is obtained by multiplying the previous number by 2.

Binary coding of numerical information Any integer can be represented as the sum of the discharged components of units, tens, hundreds, thousands and so on, recorded in the first row. At the same time, each member of this series can either not be included in the amount, or to choose it from 1 to 9 times. Example: 1409 \u003d Numbers 1, 4, 0, 9, for which members of the first row are multiplied, are the initial number.

Transfer of whole decimal numbers to binary code Let's try to present the number 1409 as the sum of the members of the second row. This method of obtaining a binary code of the decimal number is based on the records of residuals from dividing the initial number and obtained by private on 2, which is continued until the next private will be equal to 0. Example:

The translation of the whole decimal numbers into the binary code in the first cell of the top line is recorded in the original number, and in each the following result of the integer division of the previous number by 2. In the cells of the bottom line, the remains of dividing in the top line of numbers are recorded on 2. The last cell of the bottom line remains empty . The binary code of the original decimal number is obtained with a consistent recording of all residues, starting from the last: \u003d

Translation of whole decimal numbers in binary code The first 20 members of the natural row in the binary number system are written as follows: 1, 10, 11, 100, 101, 110, 111, 1011, 1100, 1010, 1011, 1100, 1010, 1110, 1100, 1010, 1110, 1100, 1010, 1011 10001, 10010, 10011,

Using Calculator 2. Make sure the calculator is configured to work in a decimal number system. Using the keyboard or mouse, enter an arbitrary two-digit number in the input field. Activate the BIN switch and follow the changes in the input window. Return to a decimal number system. Clean the input field. 3. Repeat item 2 several times for other decimal numbers. 4. Set the calculator to work in a binary number system. Pay attention to what the Calculator buttons and the Digital Keyboard Keys are available to you. Alternately, enter the binary codes of the 5th 10th and 15th members of the natural row and with the help of the DEC switch to transfer them to a decimal number system.

Purpose: form the concept of "binary number system"and the foundations of arithmetic calculations in the binary system. (Slide 2)

Requirements for knowledge and skills (Slide 3)

Students should know:

decimal and binary number system;

the detailed form of the number of the number;

translation rules from binary numbering system in decimal and vice versa;

rules of addition and multiplication of binary numbers.

Students should be able to:

translate binary numbers into a decimal system;

translate decimal numbers into a binary system;

fold and multiply binary numbers.

Software-Didactic equipment: Sea., § 16, p. 96; Demonstration "Binary Number System"; projector.(Slide 4)

During the classes

Organizing time

Setting the goals of the lesson

What numbers does a computer work with? Why?

How to operate?

Work on the lesson

(With the help of the "Binary Number System" demonstration, show the detailed form of the number, the translation from the binary number system to decimal and vice versa, the arithmetic of binary numbers.)

Binary number system is the main representation systeminformation In the memory of the computer. This idea belongs to John von Neumanan(Slide 5) Formulated in 1946 the principles of the device and work of the computer. But, contrary to common misconception, the binary number system was invented by non-engineers-designers of electronic computing machines, but mathematicians and philosophers, long before the appearance of computers, in the XVII-XIX centuries. Great German Scientist Leibniz(Slide 6) considered: "Calculation with bobs<...> It is the main one for science and gives rise to new discoveries ... when the numbers are checked to the simplest beginning, what are 0 and 1, a wonderful order appears everywhere. " Later, the binary system was forgotten, and only in 1936-1938. American Engineer and Mathematics Claude Shannon(Slide 7) found the wonderful applications of the binary system when designing electronic circuits.

What is the number system? These are the rules for recording numbers and associated methods for performing calculations.

The number system to which we are all accustomed, called decimal. This name is explained by the fact that it uses ten digits: 0,1,2, 3,4, 5, 6, 7, 8,9. (Slide 8) The number of numbers determines the base of the number system. If the number of numbers is ten, then the base of the number system is ten. In the binary system there are only two digits: 0 and 1. The base is two. The question arises, is it possible to present any value using only two digits. It turns out, you can!

The detailed form of the number of numbers (Slide 9)

Recall the principle of recording numbers in a decimal number system. The number of numbers in the number of numbers depends not only on the number itself, but also from the location of this figure in number (they say: from the number of numbers). For example, among the 555 first right of the figure indicates: three units, the following - three dozen, the next one is three hundred. This fact can be expressed as the sum of the discharge terms:

555 10 \u003d 5 x 102 + 5 x 101 + 5 x 10 ° \u003d 500 + 50 + 5.

Thus, with the promotion from the figure to the digit to the right to the left "weight" of each figure increases 10 times. This is due to the fact that the base of the number system is ten.

Translation of binary numbers in a decimal system

But an example of a multi-valued binary number: 1110112 . Two right to the right indicates the base of the number system. It is necessary in order not to confuse a binary number with decimal. After all, there is a decimal number 111011! The weight of each number of the next digit in the binary number in the promotion of the right to the right increases by 2 times. The detailed form of the record of this binary number looks like this:

111011 2 \u003d 1 x 25 + 1 x 24 + 1 x 23 + 0x 2.2 + 1 x 21 + 1 x 2 ° \u003d 6710 .

In this way, we transferred a binary number into a decimal system.

We translate into the decimal system a few more binary numbers(Slide 10).

10 2 = 2 1 =2; 100 2 = 2 2 = 4; 1000 2 = 2 3 = 8;

10000 2 = 2 4 = 16; 100000 2 = 2 5 = 32 etc.

Thus, it turned out that a two-digit decimal number corresponds to a six-digit binary! And this is typical for the binary system: the rapid increase in the number of numbers with increasing the value of the number.

Exercise 1. (Slide 11) Write down the beginning of the natural range of numbers in decimal10 ) and binary (and2 ) Number systems.

Task 2. Translate the following binary numbers into the decimal system.

101 ; 11101 ; 101010 ; 100011 ; 10110111011 .

Answer: 5; 29; 42; 35; 1467.

Translation of decimal numbers in binary system (Slide 12)

How to translate a binary number to an decimal equal to him, you should be clear from the examples discussed above. And how to make a reverse translation: from the decimal system to binary? To do this, you need to be able to decompose the decimal number on the components, which are deductible degrees. For example:

15 10 \u003d 8 + 4 + 2 + 1 \u003d 1 x 2 3 + 1 x 2 2 + 1 x 2 1 + 1 x 2 ° \u003d 1111 2 . It's complicated. There is another way to whom we will meet now.

Let us be translated into a binary number 234 number 234. We will divide 234 sequentially 2 and memorize the remains, not forgetting about zero:

234 \u003d 2 x 117 + 0 14 \u003d 2 x 7 + 0

After writing all the remnants, starting with the last, we get a binary decomposition of the number: 23410 = 11101010 2 .

Task 3. (Slide 13) What binary numbers correspond to the following decimal numbers?

2; 7; 17; 68; 315; 765; 2047.

Answer: 10 2 ; 111 2 ; 10001 2 ; 1000100 2 ; 100111011 2 ; 1011111101 2 ; 11111111111 2 .

Arithmetic binary numbers (Slide 14)

The rules of binary arithmetic is much easier than the rules of decimal arithmetic. Here are all possible options for addition and multiplication of unambiguous binary numbers:

0+0=0

0+1=1

1+0=1

1+1=10

0*0=0

0*1=0

1*0=0

1*1=1

With its simplicity and consistency with the bit structure of computer memory, a binary number system and attracted computer inventors. It is much easier to implement technical means than the decimal system.

Here is an example of addition of two multi-valued binary numbers(Slide 15) :

+ 1011011101

111010110

10010110011

And now look carefully on the following example of multiplying multivalued binary numbers:

h. 1101101

101

1101101

1101101

1000100001

Task 4. (Slide 16) Perform addition in the binary number system.11 + 1; 111 + 1; 1111 + 1; 11111 + 1.

Answer: 100; 1000; 10000; 100000.

Task 5. Perform multiplication in the binary number system.

111 x 10; 111 x 11; 1101 x 101; 1101 x 1000.

Answer: 1110; 10101; 1000001; 1101000.

Summing up the lesson (Slide 17)

The number system is certain rules for the recording of numbers and these rules associated with the calculation methods. The base of the number system is equal to the number of numbers used in it.

Binary numbers are numbers in a binary number system. Two digits are used in their records: 0 and 1.

The detailed form of a binary number record is its representation in the form of the amount of detects twice multiplied by 0 or 1.

The use of binary numbers in the computer is connected with the bit structure of computer memory and with simplicity of binary arithmetic

Homework (Slide 18)

Binary numbers are specifiedX I. Y. . CalculateX. + Y. andX- Y. , if aX \u003d 1000111, Y. = 11010.

Binary numbers are specifiedX. andW. CalculateX. + Y. - 1001101, ifX \u003d. 1010100, Y. = 110101.

Multiplying: 100110 x 11001.

Answers: 1.1100001 and 101101; 2. 111100; 3. 1110110110.

1 Slide

2 Slide

* Binary encoding in the computer All information that the computer processes must be represented by binary code using two digits: 0 and 1. These two characters are called binary numbers or bits. Using two digits 0 and 1, you can encode any message. This was the reason that two important processes must be organized in the computer: coding and decoding. Coding - converting input information into a form perceived by a computer, i.e. Binary code. Decoding - transformation of data from binary code into a form, understandable person. *

3 Slide

* Binary counting system Binary counting system - POSITIONING SYSTEM SYSTEM SYSTEM 2. Figures 0 and 1. The binary system is used in digital devices, because it is the easiest and satisfying the requirements: the fewer values \u200b\u200bexists in the system, the easier it is to make individual elements. The smaller the number of states at the element, the higher the noise immunity and the faster it can work. Easy to create tables of addition and multiplication - basic actions on numbers *

4 Slide

* Compliance with decimal and binary number systems The number of numbers used is called the base of the number system. With simultaneous operation with several number systems, the base of the system is usually indicated as a lower index, which is recorded in the decimal system: 12310 is the number 123 in a decimal number system; 11110112 - the same number, but in the binary system. Binary number 1111011 can be painted as: 11110112 \u003d 1 * 26 + 1 * 25 + 1 * 24 + 1 * 23 + 0 * 22 + 1 * 21 + 1 * 20. P \u003d 10 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 P \u003d 2 0 1 10 11 100 101 110 111 1000 1001 1010 1011 1100 1101 1110 1111 10000 *

5 Slide

* Translation of numbers from one number system to another Transfer from a decimal number system to the number system with a base P is carried out by the sequential division of the decimal number and its decimal private on P, and then by writing out the last private and residues in the reverse order. We transfer the decimal number 2010 to binary number systems (base of the number system p \u003d 2). As a result, they received 2010 \u003d 101002. *

6 Slide

* Translation of numbers from a single number system to another transition from a binary number system to the counting system with a base 10 is carried out by the consistent multiplication of the binary elements by 10 to the degree of place of this element, when registering that the numbering of seats is on the right and starts with the number "0". We translate binary number 100102 in decimal systems of the number systems. As a result, 100102 \u003d 1810 were obtained. 100102 \u003d 1 * 24 + 0 * 23 + 0 * 22 + 1 * 21 + 0 * 20 \u003d 16 + 2 \u003d 1810 *

Number system. Translation of numbers from decimal to binary number system.

The presentation was created for grade 8 students who only get acquainted with the concepts: a system of number, decimal, binary, positional, non-procurement; And, which, in my opinion, should master the rules for the transfer of numbers from decimal to binary SS and vice versa.

Presentation can be used to repetition in high school.

Tell me, and I will forget, show me and I remember give me try,

and I will learn.

Chinese wisdom

Theory

• Everything is the number ... Decimal number system Binary number system Reading numbers
• Everything is the number ... Definition of the concept of "Number System" Decimal number system Binary number system Reading numbers
• Everything is the number ...
• Definition of the concept of "Number System"
• Decimal number system
• Binary number system
• Reading numbers

Training tasks

• Training tasks
• Training tasks

• Practice Knowledge control
• Translation from a decimal SS into binary (theory) Practice Knowledge control
• Translation from a decimal SS into binary (theory) Practice Knowledge control
• Translation from a decimal SS into binary (theory)
• Practice
• Knowledge control

Everything is the number ...

• People prefer a decimal number system probably because from ancient times they thought to fingers, and people had 10 fingers on their hands and legs.
• A decimal number system came to us from India.
• To communicate with computers, except for decimal, binary, octal and hexadecimal number systems are used.
• Of all the number systems are particularly simple and therefore interesting for technical implementation in computer binary number system.

Definition of concept "Notation"

• The number system is a method of recording numbers using a specified set of special characters and the corresponding rules for performing actions over numbers.
• All number systems are divided into two large groups.

positional

the value indicates the number in the recording of the number depends on the position of the number in this number

non-aposition

the value that the number indicates in the number of numbers does not depend on the number of numbers in this number

Decimal notation

Binary notation

Reading numbers

• In the decimal system, you can read the record 36 - as the number of "thirty six", record 101 - as the number "one hundred one", etc.

• But in other surcharge systems, for example, in binary you are interested in, it is necessary to say this: Record 101 2 - The number "one - zero one" in the binary number system.

The method of translation of the number from the decimal system in binary

Training tasks

• 31, 68, 147
• Translate from decimal to an ecrossar system:
• 5, 24, 99

Homework

• Translate from decimal to binary system:

• Translate from decimal to an eight-lit system - Fill out a table.

Remember

2 0

2 1

2 2

2 3

2 4

2 5

2 6

2 7

2 8

2 9

2 10

Elephant lives in our apartment

In the house two, the entrance four.

By the hour I got used to eat -

In the morning at eight, in the afternoon at sixteen.

Eat for breakfast certainly

Thirty-two hay oyans,

After the morning walk -

Sixty four boots.

For lunch he bring

Cucumbers one hundred twenty eight.

Tomatoes can eat

Two hundred fifty and six

Eat pancakes five hundred twelve

This is if not try.

And beware of kefir -

Thousand twenty-four.

Knowledge control

1. Transfer from a decimal number system to binary : 6 3 , 256, 457, 845

2. In compliance :

1.Bases 2.Ontion 3.Alphabet

AM Specifications B. Wes discharge Vortha Alphabet

3.Conal task:

P rivel somehow to the earth girl, beautiful writer, worker from the planet

Onezero. ; let's marry her to call and boast me that he earns

1100000 dollars a month and his apartment has a total area

10100 square meters. m., and one cars have 10 pieces.

However, our maiden was with the mind and took into account what is all in the binary system.

And how much will it be?

Motion

1. 63 10 = 111111 2

256 10 = 100000000 2

457 10 = 111001001 2

845 10 = 1101001101 2

3. 1100000 2 = 96 10

10100 2 = 20 10

10 2 = 2 10

Pay students attention to

1. If the number that we translate from the decimal to the binary number system is 2 n - 1, then the answer will be equal to N-units, for example,

31 \u003d 32-1 \u003d 2 5 -1, i.e. Without performing no computing, when transferring a number 31 of the decimal to binary SS, we can immediately write out the answer: 31 10 \u003d 11111 2

2. If the number that we translate out of the decimal to the binary number system is 2 n, then the answer will be 1 and n zeros, for example,

512 \u003d 2 9, i.e. Without performing no calculations, when transferring the number 512 of the decimal to binary SS, we can immediately write out the answer: 512 10 \u003d 1000000000 2