Lesson presentation: Number systems. Converting numbers from decimal to binary number system
, Competition "Presentation for the lesson"
Class: 9
Presentation for the lesson
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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.
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. B. Pascal Learning without reflection is useless, but reflection without learning is also dangerous. Confucius It is better to understand a little than to misunderstand. L. France Everything we know is limited, what we don’t know is infinite. Laplace It is better to know too much than to know nothing. Seneca
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.
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
Lady Programmer Augusta Ada Lovelace
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
Wilhelm Gottfried Leibniz (1646-1716)
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
1 slide
2 slide
* Binary coding in a computer All information that a computer processes must be represented in binary code using two digits: 0 and 1. These two characters are usually called binary digits or bits. Using two numbers 0 and 1 you can encode any message. This was the reason that two important processes must be organized in a computer: encoding and decoding. Coding is the transformation of input information into a form that can be perceived by a computer, i.e. binary code. Decoding is the process of converting data from binary code into a form that can be understood by humans. *
3 slide
* Binary number system The binary number system is a positional number system with base 2. The numbers 0 and 1 are used. The binary system is used in digital devices because it is the simplest and satisfies the requirements: The fewer values there are in the system, the easier it is to manufacture individual elements. The fewer states an element has, the higher the noise immunity and the faster it can operate. Easy to create addition and multiplication tables - basic operations with numbers *
4 slide
* Correspondence between decimal and binary number systems The number of digits used is called the base of the number system. When working simultaneously with several number systems, to distinguish them, the base of the system is usually indicated as a subscript, which is written in the decimal system: 12310 is the number 123 in the decimal system; 11110112 is the same number, but in binary. The binary number 1111011 can be written as: 11110112 = 1*26 + 1*25 + 1*24 + 1*23 + 0*22 + 1*21 + 1*20. p=10 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 p=2 0 1 10 11 100 101 110 111 1000 1001 1010 1011 1100 1101 1110 1111 10000 *
5 slide
* Converting numbers from one number system to another Transfer from a decimal number system to a number system with base p is carried out by sequentially dividing the decimal number and its decimal quotients by p, and then writing out the last quotient and remainders in reverse order. Let's convert the decimal number 2010 to the binary number system (the base of the number system is p=2). As a result, we got 2010 = 101002. *
6 slide
* Converting numbers from one number system to another Transfer from a binary number system to a base 10 number system is carried out by sequentially multiplying the elements of a binary number by 10 to the power of the place of this element, taking into account that the numbering of places goes to the right and begins with the number “0”. Let's convert the binary number 100102 to decimal number systems. As a result, we got 100102 = 1810. 100102=1*24+ 0*23 +0*22+1*21+ 0*20 =16+2=1810 *
Number systems. Translation of numbers from decimal to binary number system.
The presentation was created for 8th grade 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 revision in high school.
Tell me and I'll forget show me and I will remember let me try
and I will learn.
Chinese wisdom
Theory
- Everything is a number... Decimal number system Binary number system Reading numbers
- Everything is a number... Definition of the concept “Number system” Decimal number system Binary number system Reading numbers
- Everything is a number...
- Definition of the concept “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 a number...
- People prefer the decimal number system probably because they have been counting on their fingers since ancient times, and people have 10 fingers and toes.
- The decimal number system came to us from India.
- To communicate with a computer, they use, in addition to decimal, binary, octal and hexadecimal number systems.
- Of all the 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 signs and the corresponding rules for performing actions on numbers.
- All number systems are divided into two large groups
positional
the value that a digit represents in a number depends on the position of the digit in that 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 the entry 36 as the number “thirty-six,” the entry 101 as the number “one hundred and one,” etc.
- But in other number systems, for example, in the binary number system that interests us, we must say this: entry 101 2 – the number “one – zero-one” in the binary number system.
Number conversion method from decimal to binary
Training tasks
- 31, 68, 147
- Convert from decimal to octal system:
- 5, 24, 99
Homework
- Convert from decimal to binary system:
- Convert from decimal to octal system - fill out 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,
There are two in the house, four entrances.
I'm used to eating by the hour -
In the morning at eight, in the afternoon at sixteen.
Will definitely eat it for breakfast
Thirty-two armfuls of hay,
After a morning walk -
Sixty-four rolls.
We bring him for lunch
One hundred twenty-eight cucumbers.
Can eat tomatoes
Two hundred fifty and six
Eat five hundred and twelve pancakes,
That's if you don't try.
And knead it with kefir -
One thousand twenty-four.
Knowledge control
1. Convert from decimal number system to binary : 6 3 , 256, 457, 845
2.Align :
1.Basis 2.Foundation 3.Alphabet
A. set of symbols B. digit weight C. alphabet size
3. Comic task:
P flew somehow to an earthly girl, a written beauty, a suitor from the planet
Onezero ; let's ask her to marry her and brag about what he earns
$1,100,000 per month and his apartments have a total area
10100 sq. m., and he has 10 cars alone.
However, our girl was smart and took into account, that it's all in the binary system.
How long do we think it will be?
Peer review
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 students' attention that
1. if the number we are converting from decimal to binary is 2n - 1, then the answer will be n-units, for example,
31=32-1 =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 = 11111 2
2. if the number we are converting from decimal to binary is 2n, then the answer will be 1 and n zeros, for example,
512=2 9, i.e. Without performing any calculations, when converting the number 512 from decimal to binary SS, we can immediately write down the answer: 512 10 = 1000000000 2
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Slide captions:
Binary number system
Let's repeat the topic "Number systems"
Basic concepts of number systems A number system is a way of writing numbers and associated ways of performing calculations. A number is a certain quantity. A digit is the symbols involved in writing a number. Alphabet is a set of different digits used to write a number.
Unit (“stick”) number system (Paleolithic period, 10-11 thousand years BC) Before a person learned to count or came up with words to denote numbers, he undoubtedly had a visual, intuitive idea of \u200b\u200bnumber. or Designation:
3 4 5 - units - tens - hundreds 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 the decimal number system. Ancient Egyptian number system (c.2850 BC)
2nd digit 1st digit = 60 +20+2 = 82 Babylonian sexagesimal number system (2 thousand years BC) The first number system known to us, based on the positional principle. - units - tens - 60; 60 2 ; 60 3; ... ; 60 n Designation:
X X X I I = 3 2 D X L I I = 542 1000 500 100 50 10 5 1 M D C L X V I Roman numeral system (500 BC) The numbers used in the Roman system are: The value of a digit does not depend on its position in the number. If the smaller number is to the left of the larger one, then it is subtracted, if to the right, it is added. For example, IX = 9 and XI =11. What numbers are written in Roman numerals? The magnitude of a number is defined as the sum or difference of the digits in the number.
– base (p) A set of all digits for writing a number – alphabet Number of digits for writing a number Positional systems can have different alphabet (2,3,4 digits). Positional Number Systems Each positional number system has a specific alphabet and base.
Base Name Alphabet p = 2 Binary 0 1 p = 3 Ternary 0 1 2 p = 8 Octal 0 1 2 3 4 5 6 7 p = 16 Hexadecimal 0 1 2 3 4 5 6 7 8 9 A B C D E F Number system alphabets To write numbers in A positional system with base p needs to have an alphabet of p digits. When p > 10, Latin letters are added to ten Arabic numerals. The position of a digit in a number is called its digit.
Representation of information in a computer Each such “cell” stores only one of two values: zero or one. Each "cell" of computer memory is called a bit. The digits 0 and 1 stored in the cells of a computer are called bit values. 0 1 and It is convenient to imagine machine memory in the form of a sheet of paper in a square.
5555=5000+500+50+5=5*1000+5*100+5*10+5*1=5*10 3 +5*10 2 +5*10 1 +5*10 0 456327=4*100000 +5*10000+6*1000+3*100+2*10+7*1=4*10 5 +5*10 4 +6*10 3 +3*10 2 +2*10 1 +7*10 0 Let's consider the decimal number system. Expanded form of writing a number
The position of a digit in a number is called its digit. A q = 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 base of the system notation (number of digits used) A q - number in the number system with base q a - digits of a multi-digit number A q n (m) - number of integer (fractional) digits of the number A q Expanded form of writing a number
1101 2 =1*2 3 +1*2 2 +0*2 1 +1*2 0 =1*8+1*4+0*2+1*1=13 11100011 2 =? Consider the binary number system Converting binary numbers to decimal
Divide the whole decimal number by 2. Write down the remainder. If the resulting quotient is not less than 2, then continue division. The binary code of a decimal number is obtained by sequentially recording the last quotient and all remainders, starting from the last. Converting integer decimal numbers to binary
Convert decimal numbers to binary 154 10 = 658 10 = 10005 10 = Task
Binary number arithmetic 0+0= 0+1= 1+0= 1+1= 0*0= 0*1= 1*0= 1*1= 0 10 0 0 0 1 1 1
§16 Page 100 task 4, 5 and 6 Homework
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