Presentation on the topic binary number system. Presentation - binary number system. Converting decimal numbers to binary

Lesson plan for a computer science lesson in 9th grade on the topic “ Binary number system" (Slide 1)

Target: 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 adding and multiplying binary numbers.

Students should be able to:

convert binary numbers to decimal system;

convert decimal numbers to binary system;

add and multiply binary numbers.

Software and didactic equipment: Sem., § 16, p. 96; demonstration “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 them?

Work on the topic of the lesson

(Using the “Binary Number System” demonstration, show the expanded form of a number, conversion from the binary number system to the decimal number system and vice versa, the arithmetic of binary numbers.)

The binary number system is the main representation systeminformationin the computer's memory. This idea belongs to John von Neumann(Slide 5) , who formulated in 1946 the principles of the design and operation of computers. But, contrary to popular belief, the binary number system was not invented by electronic design engineers. computers, and by mathematicians and philosophers, long before the advent of computers, back in the 17th-19th centuries. The great German scientist Leibniz(Slide 6) believed: “Calculation using twos<...>is fundamental for science and gives rise to new discoveries... When numbers are reduced to the simplest principles, such as 0 and 1, a wonderful order appears everywhere.” Later, the binary system was forgotten, and only in 1936-1938 was the 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 the associated ways of performing calculations.

The number system to which we are all accustomed 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 of the number system is ten. In the binary system, there are only two digits: 0 and 1. The base is equal to two. The question arises whether it is possible to represent any value with just two digits. It turns out that it is possible!

Expanded form of writing a number (Slide 9)

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

555 10 = 5 x 102 + 5 x 101 + 5 x 10° = 500 + 50 + 5.

Thus, as you 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 system

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 so as not to confuse a binary number with a decimal one. After all, there is a decimal number 111011! The weight of each subsequent digit in a binary number increases by 2 times when moving from right to left. The expanded form of writing this binary number looks like this:

111011 2 = 1 x 25 + 1 x 24 + 1 x 23 + 0x 22 + 1 x 21 + 1 x 2° = 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 number! And this is characteristic of the binary system: a rapid increase in the number of digits as the value of the number increases.

Exercise 1. (Slide 11) Write the beginning of the 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.

Translation decimal numbers to binary system (Slide 12)

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

15 10 = 8 + 4 + 2 + 1 = 1 x 2 3 + 1 x 2 2 + 1 x 2 1 + 1 x 2° = 1111 2 . It's complicated. There is another way, which we will now get acquainted with.

Suppose we need to convert the number 234 to the binary number system. We will divide 234 sequentially by 2 and remember the remainders, not forgetting about the zeros:

234 = 2 x 117 + 0 14 = 2 x 7 + 0

Having written out all the remainders, starting from the last, 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 Number Arithmetic (Slide 14)

The rules of binary arithmetic are much simpler than the rules decimal arithmetic. That's all possible options addition and multiplication of 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

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

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

+ 1011011101

111010110

10010110011

Now look carefully at the following example of multiplying multi-digit binary numbers:

X 1101101

101

1101101

1101101

1000100001

Task 4. (Slide 16) Perform addition in 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)

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

Binary numbers are numbers in the binary number system. They are written using two numbers: 0 and 1.

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

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

Homework (Slide 18)

Binary numbers givenX and Y . CalculateX + YAndX- Y , IfX= 1000111, Y = 11010.

Binary numbers givenXAndU. CalculateX + Y - 1001101 ifX = 1010100, Y = 110101.

Perform multiplication: 100110 x 11001.

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

Slide 2

Quotes

All our dignity lies in thought... Let us learn to think well.

B. Pascal Learning without reflection is useless, but reflection without learning is also dangerous.

A number system is a set of techniques and rules for designating numbers.

Number systems A positional number system is a number system in which the same digit receives different quantitative values ​​depending on the place or position it occupies in the record of a given number.

Let's consider decimal numbers. Can we assume that they are the same, since they involve the same numbers - 3 and 4? Don't you agree? Explain why? The positional number system includes the decimal number system and the binary number system. - Positional - Non-positional 43 and 34

Slide 4

A number system is called non-positional if in it the quantitative values ​​of the symbols used to write numbers do not depend on their position (place, position) in the number code. For example, in the Roman numeral system, the entry IX represents the number 9, and the entry XI represents the number 11. The decimal number 28 is represented as follows: XXVIII = 10+10+5+1+1+1 The decimal number 99 is represented as follows: XCIX = -10 +100-1+10

Slide 5

The importance of the binary number system for encoding information

Computers use a binary system because it has a number of advantages over other systems: its implementation requires technical elements with two possible states (there is current, no current; on, off, etc.; one of the states is assigned 1, another - 0), and not ten, as in the decimal system;

presentation of information through only two states is reliable and noise-resistant;

performing arithmetic operations is simplified; the ability to use the apparatus of Boolean algebra to perform logical transformations of information.

Slide 6

Charles Babbage (1791-1871), English mathematician and engineer who developed the principles on which all modern computers are designed. Analytical Engine

Slide 7

From his student years until the end of his life, the great European, German scientist Wilhelm Gottfried Leibniz studied the properties of the binary number system, which later became the main one in the creation of computers. Image of the medal by W. Leibniz

Slide 9

10  2 2  10 19 2 9 18 1 2 4 8 1 2 2 4 0 2 1 2 0 2 0 0 1 19 = 100112 number system 100112 4 3 2 1 0 digits = 1·24 +0·23+0· 22+1·21+1·20 = 16 + 2 + 1 = 19 Number conversion 1 1 0 0 1 Number systems

Slide 1

Binary number system
GBOU secondary school No. 1167

Slide 2

Quotes
All our dignity lies in thought... Let us learn to think well.

Slide 3

A number system is a set of techniques and rules for designating numbers.
Number systems A positional number system is a number system in which the same digit receives different quantitative values ​​depending on the place or position it occupies in the record of a given number.
Let's consider decimal numbers. Can we assume that they are the same, since they involve the same numbers - 3 and 4? Don't you agree? Explain why? The positional number system includes the decimal number system and the binary number system.

- Positional - Non-positional

43 and 34
Slide 4

A number system is called non-positional if in it the quantitative values ​​of the symbols used to write numbers do not depend on their position (place, position) in the number code.

A number system is called non-positional if in it the quantitative values ​​of the symbols used to write numbers do not depend on their position (place, position) in the number code. For example, in the Roman numeral system, the entry IX represents the number 9, and the entry XI represents the number 11. The decimal number 28 is represented as follows: XXVIII = 10+10+5+1+1+1 The decimal number 99 is represented as follows: XCIX = -10 +100-1+10
For example, in the Roman numeral system, the entry IX represents the number 9, and the entry XI represents the number 11. The decimal number 28 is represented as follows: XXVIII = 10+10+5+1+1+1 The decimal number 99 is represented as follows: XCIX = -10 +100-1+10

Slide 5

Computers use a binary system because it has a number of advantages over other systems: its implementation requires technical elements with two possible states (there is current, no current; on, off, etc.; one of the states is assigned 1, another - 0), and not ten, as in the decimal system;
presentation of information through only two states is reliable and noise-resistant;

performing arithmetic operations is simplified; the ability to use the apparatus of Boolean algebra to perform logical transformations of information.

Slide 6
The essence and purpose of the machine will change depending on what information we put into it. The machine will be able to write music, draw pictures and show science ways that we have never seen anywhere. Ada Lovelace
Ada Lovelace suggested that Charles Babbage use the binary number system. She wrote several programs for the Analytical Engine and developed programming theory.

Slide 8

Slide 7
From his student years until the end of his life, the great European, German scientist Wilhelm Gottfried Leibniz studied the properties of the binary number system, which later became the main one in the creation of computers. Image of the medal by W. Leibniz

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Target: form the concepts 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 adding and multiplying binary numbers.

Students should be able to:

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

Software and didactic support: presentation “Binary number system”; textbook Semakin I.G. Computer science and information and communication technologies. Basic course: Textbook for 9th grade; projector.

DURING THE CLASSES

1. Organizational moment

2. Setting lesson goals

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

3. Lesson progress

(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 the principles of the design and operation of computers in 1946.
Number systems
What is a number system? These are the rules for writing numbers and the associated ways of performing calculations.
The number system to which we are all accustomed is called decimal. This name is explained by the fact that it uses only 10 digits: 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 equal to two.
Let us recall the principle of writing numbers in the decimal number system. The meaning of a digit in a number depends not only on the digit itself, but also on its location in the number (on the position of the digit). For example, in the number 473, the first digit on the right stands for units, the next for tens, and the next for hundreds. This fact can be expressed as a sum of 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 notation is called the expanded form of writing a 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

Translation of numbers

One of the ways to convert numbers from the decimal number system to the binary system is to divide by a column into the bases of the system, i.e. by 2. Division is carried out until the remainder is 1. The answer in the binary number system is written using the remainders of the division from the end.
Thus, 1910 = 100112.

Conversion from the binary number system to the binary number system 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.

Convert the numbers:

37 10 = 100101 2
11101 2 = 29 10

Binary Number Arithmetic

The rules of binary arithmetic are much simpler than the rules of decimal arithmetic. 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 2

0 x 0 = 0 0 x 1 = 0 1 x 0 = 0 1 x 1 = 1

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

Here is an example of column addition of two multi-digit binary numbers:

Task 3.

Perform addition in binary number system:

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

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

Task 4.

Perform multiplication in binary number system:

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 associated methods of performing calculations)
– What digits are used to write binary numbers? ( 0 and 1)

5. Homework

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