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
In computers they use 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, the other - 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
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
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* 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. *
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* 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 *
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* 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 *
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* 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. *
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* 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 translate 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 A number system is a set of techniques and rules for designating and naming numbers. The number system is called a positional one because the same number receives different quantitative values depending on the place or position it occupies in the number record. For example, in the number 555, the number 5 in the first place on the right means 5 units, the second 5 tens, and the third 5 hundreds.
Positional Number Systems The base of a positional number system is the number of different signs or symbols used to represent numbers in a given system. Any natural number two, three, four, etc. can be taken as the base of the system. Consequently, an infinite number of positional systems are possible: binary, ternary, quaternary, etc.
Positional number systems Example: Binary number system Places Number, 1 2 = =1*2 3 +0*2 2 +1*2 1 +0*2 0 +1*2 -1 Octal number system Places Number2 7 6, 5 2 =2*8 2 +7*8 1 +6*8 0 +5* *8 -2
Positional number systems The binary system, convenient for computers, is inconvenient for humans due to its bulkiness and unusual notation. In this regard, octal and hexadecimal systems were developed. Numbers in these systems are almost as easy to read as decimal ones; they require, respectively, three (octal) and four (hexadecimal) times fewer digits than in the binary system (after all, the numbers 8 and 16 are, respectively, the third and fourth powers of the number 2) . – binary (digits 0, 1 are used); –octal (digits 0, 1,..., 7 are used); – hexadecimal (for the first integers from zero to nine, the digits 0, 1,..., 9 are used, and for the next numbers from ten to fifteen, the symbols A, B, C, D, E, F are used as digits).
Writing numbers in number systems 10-ya2-ya8-ya16-ya10-ya2-ya8-ya16-ya A B C D E F
How information is presented in a computer, or digital data In order to understand how a wide variety of information is presented in a computer, let’s “look” inside computer memory. It is convenient to imagine it in the form of a checkered sheet. Each such “cell” stores only one of two values: zero or one. Two digits are useful for electronic data storage because they require only two states of the electronic circuit, “on” (this corresponds to the digit 1) and “off” (this corresponds to the digit 0). Each "cell" of computer memory is called a bit. The numbers 0 and 1 stored in computer memory cells are called bit values.
A sequence of bits can represent a variety of information. This representation of information is called binary or digital coding. The advantage of digital data is that it is relatively easy to copy and change. They can be stored and transmitted using the same methods, regardless of the data type. Methods for digitally encoding texts, sounds (voices, music), images (photos, illustrations) and sequences of images (cinema and video), as well as three-dimensional objects, were invented in the 1980s.
Binary coding of numerical information There are many known ways to write numbers. We use the decimal positional number system. It is called decimal because in this number system, ten units of one digit constitute one unit of the next highest digit. The number 10 is called the base decimal number system. Ten digits are used to write numbers in the decimal system: 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9.
Binary coding of numerical information Consider two number series: 1, 10, 100, 1000, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024. Both of these series begin with one. Each next number in the first row is obtained by multiplying the previous number by 10. Each subsequent number in the second row is obtained by multiplying the previous number by 2.
Binary encoding of numerical information Any integer can be represented as a sum of digit terms of units, tens, hundreds, thousands, and so on, written in the first row. Moreover, each member of this series can either not be included in the sum or be included in it from 1 to 9 times. Example: 1409 = The numbers 1, 4, 0, 9, by which the terms of the first row are multiplied, form the original number.
Converting integer decimal numbers into binary code Let's try to represent the number 1409 as the sum of the terms of the second row. This method of obtaining the binary code of a decimal number is based on writing the remainders from dividing the original number and the resulting quotients by 2, continued until the next quotient is equal to 0. Example:
Converting integer decimal numbers into binary code The first cell of the top line contains the original number, and each next cell contains the result of integer division of the previous number by 2. The cells of the bottom line contain the remainders from dividing the numbers in the top line by 2. The last cell of the bottom line remains empty . The binary code of the original decimal number is obtained by sequentially recording all remainders, starting from the last: =
Converting whole decimal numbers into binary code The first 20 terms of the natural series in the binary number system are written as follows: 1, 10, 11, 100, 101, 110, 111, 1000, 1001, 1010, 1011, 1100, 1101, 1110, 1111, 10000 , 10001, 10010, 10011,
Using the Calculator 2. Make sure the Calculator is set to work in the decimal number system. Using the keyboard or mouse, enter a random two-digit number in the input field. Activate the Bin switch and watch the changes in the input window. Return to the decimal number system. Clear the input field. 3. Repeat step 2 several times for other decimal numbers. 4. Set up the Calculator to work in the binary number system. Pay attention to which Calculator buttons and number keys on your keyboard are available to you. Alternately enter the binary codes of the 5th, 10th and 15th terms of the natural series and use the Dec switch to convert them to the decimal number system.
