Binary number system in computer presentation. Lesson presentation: Number systems. Converting numbers from decimal to binary. Converting fractional decimal numbers to binary

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

Requirements for knowledge and skills (Slide 3)

Students should know:

decimal and binary number systems;

expanded form of writing a number;

rules for converting from binary to decimal and vice versa;

rules for addition and multiplication of binary numbers.

Students should be able to:

convert binary numbers to decimal system;

convert decimal numbers to the binary system;

add and multiply binary numbers.

Software and didactic equipment: Sem., § 16, p. 96; demo "Binary number system"; projector.(Slide 4)

During the classes

Organizing time

Setting lesson goals

What numbers does the computer work with? Why?

How to operate with them?

Work on the topic of the lesson

(With the help of the "Binary number system" demo, show the expanded form of a number, the conversion 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 computer memory. This idea belongs to John von Neumann(Slide 5) , who formulated in 1946 the principles of the structure and operation of computers. But, contrary to the common misconception, the binary number system was invented not by design engineers of electronic computers, but by mathematicians and philosophers, long before the advent of computers, back in the 17th-19th centuries. The great German scientist Leibniz(Slide 6) considered: “Calculation using twos<...> is basic for science and gives rise to new discoveries ... When numbers are reduced to the simplest principles, which are 0 and 1, a wonderful order appears everywhere. " Later, the binary system was forgotten, and only in 1936-1938 American engineer and mathematician Claude Shannon(Slide 7) found remarkable applications of the binary system in the design of electronic circuits.

What is a number system? These are the rules for writing numbers and related ways of performing calculations.

The number system we are all used to is 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 digits determines the base of the number system. If the number of digits is ten, then the base is ten. In the binary system, there are only two digits: 0 and 1. The base is two. The question arises whether it is possible to represent any quantity with just two digits. It turns out you can!

Expanded form of writing a number (Slide 9)

Let's recall the principle of writing numbers in decimal notation. The meaning of a digit in a number recording depends not only on the digit itself, but also on the location of this digit in the number (they say: from the position of the digit). For example, in the number 555, the first digit on the right denotes: three units, the next three tens, the next three hundred. This fact can be expressed as the sum of the bit terms:

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

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

Converting binary numbers to decimal

And here is an example of a multi-digit binary number: 1110112 ... The two at the bottom right indicates the base of the number system. This is necessary in order not to confuse a binary number with a decimal number. After all, there is a decimal number 111011! The weight of each next digit in a binary number doubles as you move from right to left. The expanded form of writing this binary number looks like this:

111011 2 \u003d 1 x 25 + 1 x 24 + 1 x 23 + 0x 22 + 1 x 21 + 1 x 2 ° \u003d 6710 .

In this way, we converted the binary number to the decimal system.

Let's convert a few more binary numbers to the decimal system(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: a rapid increase in the number of digits with an increase in the value of the number.

Exercise 1. (Slide 11) Write the beginning of a natural series of numbers in decimal (A10 ) and binary (A2 ) number systems.

Task 2. Convert the following binary numbers to decimal.

101 ; 11101 ; 101010 ; 100011 ; 10110111011 .

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

Converting decimal numbers to binary (Slide 12)

How to convert a binary number to an equal decimal number, you should be clear from the examples discussed above. And how to carry out the reverse translation: from the decimal system to the binary? To do this, you need to be able to decompose the decimal number into terms that are powers of two. For instance:

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, which we will now get to know.

Let the number 234 be converted to the binary system. Let's divide 234 sequentially by 2 and remember the remainders, not forgetting about the zero ones:

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

Writing out all the remainders, starting with the last one, we get the 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 .

Binary arithmetic (Slide 14)

Binary arithmetic rules are much simpler than decimal arithmetic rules. Here are all the possible options for adding and multiplying single-digit 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

The binary number system attracted the inventors of the computer with its simplicity and consistency with the bit structure of computer memory. It is much easier to implement technically than the decimal system.

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

+ 1011011101

111010110

10010110011

Now take a close look at the following example of multiplying multi-digit binary numbers:

x 1101101

101

1101101

1101101

1000100001

Task 4. (Slide 16) Perform the addition in binary notation.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.

Lesson summary (Slide 17)

A number system is a set of rules for writing numbers and related ways of performing calculations. The base of the number system is equal to the number of digits used in it.

Binary numbers are numbers in the binary number system. They use two digits: 0 and 1.

The expanded form of writing a binary number is its representation as the sum of powers of two multiplied by 0 or 1.

The use of binary numbers in a computer is related to the bit structure of computer memory and to the simplicity of binary arithmetic.

Homework (Slide 18)

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

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

Perform multiplication: 100110 x 11001.

Answers: 1.1100001 and 101101; 2.111100; 3.110110110.

, Competition "Presentation for the lesson"

Class: 9

Lesson presentation

Back forward

Attention! The slide preview is used for informational purposes only and may not represent all the presentation options. If you are interested in this work, please download the full version.

Goal: to form the concept of "binary number system" and the basics of arithmetic calculations in the binary system.

Requirements for knowledge and skills

Students should know:

• decimal and binary number systems;
• expanded form of writing a number;
• rules for converting from binary to decimal and vice versa;
• rules for addition and multiplication of binary numbers.

Students should be able to:

• convert binary numbers to decimal system;
• convert decimal numbers to the binary system;
• add and multiply binary numbers.

Software and didactic support: presentation "Binary number system"; textbook Semakin I.G. Informatics and information and communication technologies. Basic course: Textbook for grade 9; projector.

DURING THE CLASSES

1. Organizational moment

2. Setting lesson goals

- What numbers does the computer work with? Why?
- How to operate with them?

3. Course of the lesson

(The lesson is accompanied by the presentation "Binary number system")

The binary number system is the main system for representing information in computer memory. This idea belongs to John von Neumann, who formulated in 1946 the principles of the structure and operation of computers.
Number systems
What is a number system? These are the rules for writing numbers and related ways of performing calculations.
The number system we are all used to is called decimal. This name is explained by the fact that only 10 digits are used in it: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. The number of digits determines the base of the number system. In the binary system, there are only two digits: 0 and 1. The base is two.
Let's recall the principle of writing numbers in decimal notation. The meaning of a digit in a number recording depends not only on the digit itself, but also on its location in the number (on the position of the digit). For example, in 473, the first digit on the right denotes units, the next one denotes tens, and the next one denotes hundreds. This fact can be expressed as the sum of the bit terms:

473 10 = 4 * 100 + 7 * 10 + 3 * 1 = 4 * 10 2 + 7 * 10 1 + 3 * 10 0 .

In the same way, you can write a number in the binary number system:

101 2 = 1 * 2 2 + 0 * 2 1 + 1*2 0 .

This is called the expanded form of the number.

Exercise 1.

Write down the expanded form of writing numbers:

5 789 = 5 * 10 3 + 7 * 10 2 + 8 * 10 1 + 9 * 10 0
51,89 = 5 * 10 1 + 1 * 10 0 + 8 * 10 –1 + 9 * 10 –2
32 478 = 3 * 10 4 + 2 * 10 3 + 4 * 10 2 + 7 * 10 1 + 8 * 10 0
26,378 = 2 * 10 1 + 6 * 10 0 + 3 * 10 –1 + 7 * 10 –2 + 8 * 10 –3

Number translation

One of the ways to convert numbers from decimal to binary is to divide by a column by the base of the system, i.e. by 2. Division is performed until the remainder is 1. The answer in the binary number system is written by the remainders of the division from the end.
So 1910 \u003d 100112.

Conversion from binary number system to binary is performed using expanded notation of the number.

101 2 = 1 * 2 2 + 0 * 2 1 + 1 * 2 0 = 4 + 0 + 1 = 5 10 .

Task 2.

Translate the numbers:

37 10 = 100101 2
11101 2 = 29 10

Binary arithmetic

Binary arithmetic rules are much simpler than decimal arithmetic rules. Here are all the possible options for adding and multiplying single-digit binary numbers:

The binary system attracted the inventors of the computer with its simplicity and consistency with the bit structure of computer memory. It is much easier to implement technically than the decimal system.

Here is an example of adding two multi-digit binary numbers in a column:

Task 3.

Perform the addition in binary notation:

101101 2 + 11111 2 ; 10111 2 + 101110 2 (answer:1001100 2 ; 1000101 2).

Now take a close look at the following multi-digit binary multiplication example:

Task 4.

Perform the multiplication in binary:

101101 2 x11 2; 10101 2 x11 2 ( answer: 10000111 2 ; 111111 2).

4. Summing up the lesson

- What is a number system? ( these are the rules for writing numbers and related ways of performing calculations)
- What numbers are used in writing binary numbers? ( 0 and 1)

5. Homework

• §16 of the textbook;
• P. 104 questions 2-7 in writing.

Number systems. Number translation from decimal to binary system.

The presentation was created for grade 8 students who are just getting acquainted with the concepts: number system, decimal, binary, positional, non-positional; and, which, in my opinion, should master the rules for converting numbers from decimal to binary SS and vice versa.

The presentation can be used for repetition in high school.

Tell me and I will forget show me and I will remember let me try

and I will learn.

Chinese wisdom

Theory

• Everything is number ... Decimal number system Binary number system Reading numbers
• Everything is number ... Definition of the concept of "Number system" Decimal number system Binary number system Reading numbers
• Everything is 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
• Converting from decimal SS to binary (theory) Practice Knowledge control
• Converting from decimal SS to binary (theory) Practice Knowledge control
• Converting from decimal SS to binary (theory)
• Practice
• Knowledge control

Everything is number ...

• People prefer the decimal number system, probably because since ancient times they counted with their fingers, and people have 10 fingers and toes.
• The decimal number system came to us from India.
• To communicate with a computer, in addition to decimal, binary, octal and hexadecimal number systems are used.
• Of all number systems, the binary number system is especially simple and therefore interesting for technical implementation in a computer.

Definition of the concept "Notation"

• A number system is a way of writing numbers using a given set of special characters and the corresponding rules for performing actions on numbers.
• All number systems are divided into two large groups

positional

the value denoted by a digit in a number recording depends on the position of the digit in this number

non-positional

the value denoted by a digit in a number notation does not depend on the position of the digit in this number

Decimal notation

Binary notation

Reading numbers

• In the decimal system, you can read entry 36 as the number "thirty-six", entry 101 as the number "one hundred and one", and so on.

• But in other number systems, for example, in the binary of interest to us, we must say this: notation 101 2 - the number "one - zero - one" in the binary system.

Number translation method decimal to binary

Training tasks

• 31, 68, 147
• Convert from decimal to octal system:
• 5, 24, 99

Homework

• Convert from decimal to binary:

• Convert from decimal to octal system - fill in the table.

Remember

2 0

2 1

2 2

2 3

2 4

2 5

2 6

2 7

2 8

2 9

2 10

The elephant lives in our apartment,

The house has two, entrance four.

I used to eat by the hour -

At eight in the morning, sixteen in the afternoon.

Eat for breakfast without fail

Thirty-two armfuls of hay

After a morning walk -

Sixty-four rolls.

We bring him for lunch

Ogurtsov one hundred twenty eight.

Tomatoes can eat

Two hundred fifty and six,

Eat pancakes five hundred and twelve

If you don't try.

And knead on kefir -

One thousand twenty four.

Knowledge control

1.Convert from decimal to binary : 6 3 , 256, 457, 845

2.Adjust :

1.Basis 2.Foundation 3.Alphabet

A. set of characters B. digit weight C. alphabet size

3.Comic task:

P flew somehow to an earthly girl, a beauty written, boyfriend from the planet

Onezero ; let's call her married and boast that he earns

$ 1,100,000 per month and his apartments with a total area

10100 sq. m., and he has 10 cars alone.

However, our girl was wise and took into account, that it's all in binary.

And how much will it be in our opinion?

Mutual verification

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

Draw the attention of students that

1.if the number that we convert from decimal to binary is equal to 2 n - 1, then the answer will be n-units, for example,

31 \u003d 32 - 1 \u003d 2 5 - 1, i.e. without performing any calculations, when converting the number 31 from decimal to binary SS, we can immediately write down the answer: 31 10 \u003d 11111 2

2.if the number that we convert from decimal to binary is equal to 2 n, then the answer will be equal to 1 and n zeros, for example,

512 \u003d 2 9, i.e. without performing any calculations, when converting the number 512 from decimal to binary SS, we can immediately write the answer: 512 10 \u003d 1,000,000,000 2