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Presentation - Number systems

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Topic "Number systems"

Introduction
Modern man constantly encounters numbers and numbers in everyday life - they are with us everywhere. Various number systems are used whenever there is a need for numerical calculations, from pencil-on-paper calculations by elementary school students to calculations performed on supercomputers.

A number system is a certain way of representing numbers and the corresponding rules for operating on them. The purpose of creating a number system is to develop the most convenient way to record quantitative information.
History of number systems
Number systems
Positional
Non-positional

Ancient number systems:
Unit system Ancient Greek numbering Slavic numbering Roman numbering

Positional and non-positional number systems
Non-positional systems Positional systems
The position of the digit in the notation of the number does not determine the value it represents. The value denoted by a digit in a number notation depends on its position. The base is the number of digits used. Position is the location of each digit.

Writing a number in the positional number system
Any integer in the positional system can be written in polynomial form: Xs=An Sn-1 + An-1 Sn-2 + An-2 Sn-3 +...+ A2 S1 + A1 S0 where S - the base of the number system, A – the digits of the number written in this number system, n – the number of digits of the number. So, for example, the number 629310 will be written in polynomial form as follows: 629310 = 6 103 + 2 102 + 9 101 + 3 100

Examples of positional number systems:
Binary number system with base 2, uses two symbols - 0 and 1.
Octal number system with base 8, numbers from 0 to 7 are used.
The base 10 decimal system is the most common number system in the world.
Duodecimal System with base 12. The numbers used are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B.
Hexadecimal Base 16, uses the numbers 0 to 9 and the Latin letters A to F to represent the numbers 10 to 15.
The sexagesimal system, with base 60, is used in the measurement of angles and, in particular, longitude and latitude.

History of the binary number system
The binary number system was invented by mathematicians and philosophers even before the advent of computers (XVII - XIX centuries). The promoter of the binary system was the famous G.V. Leibniz. He noted the particular simplicity of the algorithms for arithmetic operations in binary arithmetic in comparison with other systems and gave it a certain philosophical meaning. In 1936 - 1938, American engineer and mathematician Claude Shannon found remarkable applications of the binary system in the design of electronic circuits.

Binary number system
Binary number system (binary number system, binary) is a positional number system with base 2. The inconvenience of this number system is the need to convert source data from the decimal system to binary when entering them into the machine and reverse conversion from binary to decimal when outputting the calculation results. The main advantage of the binary system is the simplicity of the addition, subtraction, multiplication and division algorithms.

Addition, subtraction, multiplication and division in the binary number system
Addition Subtraction Multiplication Division
0 + 0 = 0; 0 + 1 = 1; 1 + 0 = 1; 1 + 1 = 10. 0 - 0 = 0; 1 - 0 = 1; 1 - 1 = 0; 10 - 1 = 1. 0 1 = 0; 1 1 = 1. 0 / 1 = 0; 1 / 1 = 1.

Binary coding in a computer
At the end of the twentieth century, the century of computerization, humanity uses the binary system every day, since all information processed by modern computers is stored in them in binary form. In modern computers we can enter text information, numerical values, as well as graphic and audio information. The amount of information stored in a computer is measured by its “length” (or “volume”), which is expressed in bits (from English binary digit).

Converting numbers from one number system to another
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Conclusion
The highest achievement of ancient arithmetic is the discovery of the positional principle of representing numbers. We need to recognize the importance of not only the most common system that we use every day. But each one separately. After all, different areas use different number systems, with their own characteristics and characteristics.

Decimal Binary Octal Hexadecimal
1 001 1 1
2 010 2 2
3 011 3 3
4 100 4 4
5 101 5 5
6 110 6 6
7 111 7 7
8 1000 10 8
9 1001 11 9
10 1010 12 A
11 1011 13 B
12 1100 14 C
13 1101 15 D
14 1110 16 E
15 1111 17 F
16 10000 20 10

Converting binary number to decimal
To convert a binary number to a decimal one, it is necessary to write it in the form of a polynomial, consisting of the products of the digits of the number and the corresponding power of the number 2, and calculate it according to the rules of decimal arithmetic: X10 = An 2n-1 + An-1 2n-2 + An-2 ·2n-3 +…+A2·21 + A1·20
Translation of numbers

Converting an octal number to decimal
To convert an octal number to a decimal one, it is necessary to write it in the form of a polynomial, consisting of the products of the digits of the number and the corresponding power of the number 8, and calculate it according to the rules of decimal arithmetic: X10 = An 8n-1 + An-1 8n-2 + An-2 8n-3 +…+A2 81 + A1 80
Translation of numbers

Convert hexadecimal number to decimal
To convert a hexadecimal number to a decimal one, it is necessary to write it in the form of a polynomial, consisting of the products of the digits of the number and the corresponding power of the number 16, and calculate it according to the rules of decimal arithmetic: X10 = An 16n-1 + An-1 16n-2 + An-2 ·16n-3 +…+A2·161 + A1·160
Translation of numbers

Converting a decimal number to binary
To convert a decimal number to the binary system, it must be sequentially divided by 2 until a remainder less than or equal to 1 remains. A number in the binary system is written as a sequence of the last division result and the remainders from the division in reverse order. Example: Convert the number 2210 to the binary number system: 2210=101102
Translation of numbers

Converting a decimal number to octal
To convert a decimal number to the octal system, it must be successively divided by 8 until a remainder less than or equal to 7 remains. A number in the octal system is written as a sequence of digits of the last division result and the remainders of the division in reverse order. Example: Convert the number 57110 to the octal number system: 57110=10738
Translation of numbers

Converting a decimal number to hexadecimal
To convert a decimal number to the hexadecimal system, it must be successively divided by 16 until a remainder less than or equal to 15 remains. A number in the hexadecimal system is written as a sequence of digits of the last division result and the remainders of the division in reverse order. Example: Convert the number 746710 to hexadecimal number system: 746710=1D2B16
Translation of numbers

Converting numbers from binary to octal
To convert a number from binary to octal, it must be divided into triads (triples of digits), starting with the least significant digit, adding zeros to the leading triad if necessary, and replacing each triad with the corresponding octal digit. When translating, you must use the binary-octal table: Example: Convert the number 10010112 to the octal number system: 001 001 0112 = 1138
8th 0 1 2 3 4 5 6 7
Translation of numbers

Converting from binary to hexadecimal
To convert a number from binary to hexadecimal, it must be divided into tetrads (four digits). Binary hexadecimal table: Example: Convert the number 10111000112 to hexadecimal number system: 0010 1110 00112=2E316
16th 0 1 2 3 4 5 6 7
16th 8 9 A B C D E F
Translation of numbers

Converting an octal number to binary
To convert an octal number to binary, you need to replace each digit with its equivalent binary triad. Example: Convert the number 5318 to the binary number system: 5318=101 011 0012
2nd 000 001 010 011 100 101 110 111
8th 0 1 2 3 4 5 6 7
Translation of numbers

Convert hexadecimal number to binary
To convert a hexadecimal number to binary, you need to replace each digit with its equivalent binary tetrad. Example: Convert the number EE816 to the binary number system: EE816=1110111010002
2nd 0000 0001 0010 0011 0100 0101 0110 0111
16th 0 1 2 3 4 5 6 7
2nd 1000 1001 1010 1011 1100 1101 1110 1111
16th 8 9 A B C D E F
Translation of numbers

Converting from octal to hexadecimal and vice versa
When moving from the octal number system to the hexadecimal number system and vice versa, an intermediate conversion of numbers to the binary system is necessary. Example 1: Convert the number FEA16 to the octal number system: FEA16=1111111010102=111 111 101 0102=77528 Example 2: Convert the number 66358 to the hexadecimal number system: 66358=1101100111012=1101 1001=D 9D16
Translation of numbers

Unit system
In ancient times, when there was a need to record numbers, the number of objects was depicted by drawing dashes or serifs on some hard surface. Archaeologists have found such “records” during excavations of cultural layers dating back to the Paleolithic period (10–11 thousand years BC). In such a system, only one type of sign was used - a stick. Each number was designated using a line made up of sticks, the number of which was equal to the designated number.
Ancient number systems

Ancient Greek numbering

Attic numbering
Ionian system
In the third century BC. Attic numbering was supplanted by the Ionian system.
In ancient times, Attic numbering was common in Greece.
Ancient number systems

Slavic numbering
In Russia, Slavic numbering was preserved until the end of the 17th century. The southern and eastern Slavic peoples used alphabetical numbering to record numbers. Slavic numbering was preserved only in liturgical books. A special icon was placed above the letter indicating the number: (“title”). To indicate thousands, a special sign was placed in front of the number (bottom left).
Z
Ancient number systems

Roman numbering
The ancient Romans used numbering, which remains to this day under the name “Roman numbering.” We use it to designate centuries, anniversaries, names of congresses and conferences, to number chapters of a book or stanzas of a poem.
I - 1 V - 5 X - 10 L - 50 C - 100 D - 500 M - 1000
Writing numbers in Roman numeration:
Ancient number systems

Ionian system
Notation of numbers in the Ionian numbering system

Designation of numbers in the ancient Slavic numbering system
Slavic numbering

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A little history The account appeared when a person needed to inform his relatives about the number of objects he discovered, animals killed and defeated enemies. In different places, different ways of transmitting numerical information were invented: from notches according to the number of objects to ingenious signs - numbers.

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“number” of ancient people Initially, the concept of an abstract number was absent; the number was “tied” to those specific objects that were being counted. The abstract concept of a natural number appeared along with the development of writing.

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Number systems A number system is a set of rules for designating and naming numbers. Number systems are divided into positional and non-positional. The signs used to write numbers are called digits.

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Positional number systems The most advanced are positional number systems, i.e. systems for writing numbers in which the contribution of each digit to the value of the number depends on its position (position) in the sequence of digits representing the number. For example, our familiar decimal system is positional. In the number 34, the number 3 indicates the number of tens, and the number 4 indicates the number of ones. The number of digits used is called the base of the positional number system. Advantages of positional number systems Ease of performing arithmetic operations. A limited number of characters (digits) for writing any numbers. .

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Non-positional number systems Unit system The number of objects, for example sheep, was depicted by drawing lines or notches on any hard surface: stone, clay, wood. Scientists called this method of writing numbers the unit (“stick”) number system. In it, only one type of sign was used to record numbers - “stick”. Each number in such a number system was designated using a line made up of sticks, the number of which was equal to the designated number. I I I I I I I I I I I I I I I I I I I I I I I I The inconveniences of such a system for writing numbers and the limitations of its application are obvious: the larger the number you need to write, the longer the string of sticks. And when writing down a large number, it’s easy to make a mistake by adding an extra number of sticks or, conversely, not writing them down.

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The Roman system The Roman system is familiar to us from first grade. It uses capital Latin letters I, V, X, L, C, D and M to denote the numbers 1, 5, 10, 50, 100, 500 and 1000, respectively, which are the digits of this number system. A number in the Roman numeral system is designated by a set of consecutive digits. The value of a number is equal to: the sum of the values of several identical digits in a row (let’s call them the group of the first type); the difference between the values of two digits if the smaller digit is to the left of the larger digit. In this case, the value of the smaller digit is subtracted from the value of the larger digit (let's call them a group of the second type) Example 1. The number 32 in the Roman number system has the form XXXII=(X+X+X)+(I+I)=30+2 (two groups of the first type). Example 2. The number 444, which has 3 identical digits in its decimal notation, will be written in the Roman number system as CDXLIV=(D-C)+(L-X)+(V-I)=400+40+4 (three groups of the second type).

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Ancient Egyptian decimal system The ancient Egyptian numeral system, which arose in the second half of the third millennium BC, used special numerals to represent the numbers 1, 10, 100, 1000, etc. Numbers in the Egyptian numeral system were written as combinations of these digits, in which each of them was repeated no more than nine times. Example. The ancient Egyptians wrote down the number 345 as follows: Both the stick and ancient Egyptian number systems were based on the simple principle of addition, according to which the value of a number is equal to the sum of the values of the digits involved in its recording. Scientists classify the ancient Egyptian number system as non-positional decimal.

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The ancient Egyptians used tens hundreds of thousands tens of thousands hundreds of thousands millions

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Babylonian sexagesimal system Numbers in the Babylonian number system were composed of two types of signs: a straight wedge served to designate units; a lying wedge - to designate tens. To determine the value of a number, it was necessary to divide the image of the number into digits from right to left. A new discharge began with the appearance of a straight wedge after a recumbent one, if we consider the number from right to left. For example: The number 32 was written like this:

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Slavic number system This number system is alphabetic i.e. Letters of the alphabet are used instead of numbers. This number system was used by our ancestors and was quite complex, because uses 27 letters as numbers.

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Mathematicians argue with historians Considering that in the Slavic number system large numbers had the following names: darkness 10,000 crows 10^ 48 legion 100,000 deck 10^50 leodr 1,000,000 let’s solve the problem of the number of Batu’s troops during the campaign against Rus'. According to the chronicles, the Mongols were in “darkness.” That is, 10,000 10,000 = 100,000,000 people. In fact, Batu had 11 temnik military leaders subordinate to him, each of whom had “darkness” of soldiers subordinate to him, a total of 11 10 000 = 110 000, a total of 110 thousand people. Therefore, there was no trace of the 100,000,000 people that historians talk about!

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Disadvantages of non-positional number systems There is a constant need to introduce new symbols for recording large numbers. It is impossible to represent fractional and negative numbers. It is difficult to perform arithmetic operations because there are no algorithms for performing them. Until the end of the Middle Ages, there was no universal system for recording numbers. Only with the development of mathematics, physics, technology, trade and economics did the need for a single universal number system arise.

Presentation on the topic: "Number systems"

The concept of number systems

Representation of numbers in positional number systems

Binary number system

Tasks for consolidation

Representation of numbers in the binary number system

Arithmetic operations in the binary number system

Relationship between binary and decimal systems

Converting a number from binary ss to decimal ss

Conversion from decimal ss to binary number system

Integer conversion

Translation of proper fractions

Converting mixed numbers

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Lesson on computer science Number systems

is a way of writing numbers using a given set of special characters (digits). a number system in which the value of each numerical sign (digit) in the recording of a number depends on its position (digit) the value that the digit denotes does not depend on the position in the number Positional Non-positional Number Systems 22 XXII =20 =2 = 1 0 = 10 Concept about number systems

Non-positional number systems In non-positional number systems, the weight of a digit does not depend on the position it occupies in the number. The Roman number system has survived to this day. In the Roman number system, numbers are designated by letters of the Latin alphabet: I -1; V -5; X -10; L -50; C -100; D – 500; M – 1000; ... So, for example, in the Roman number system in the number XXXII (thirty-two), the weight of the digit X in any position is simply ten.

Positional Number Systems In positional number systems, the weight of each digit varies depending on its position in the sequence of digits representing the number. Any positional system is characterized by its base.

The base of the positional ss is the number of different signs or symbols used to represent numbers in a given system. Any natural number can be taken as a base - two, three, four, sixteen, etc. Consequently, an infinite number of positional systems are possible. back

100101 2 - binary number system, alphabet: 0, 1 base - 2 102 3 - ternary number system, alphabet: 0, 1, 2 base - 3 231 4 - ___________________________________________ 12244 5 - ________________________________________ ??? 6 - ___________________________________________ ??? 7 - ___________________________________________ ??? 8 - ___________________________________________ ??? 9 - ___________________________________________ ??? 16 - _____________________, alphabet 0-9, A, B, C, D, E, F 543210 Digit size Base The base of a number system is ________________________ the number of digits in the alphabet

Representation of numbers in positional ss Let a number in decimal ss be given, in which there are N digits. We will denote the i-th digit by a i. Then the number can be written in the following form: A 10 = a n a n-1 .... a 2 a 1 is a collapsed form of writing a number.

The same number can be represented in the following form: A 10 = a n a n-1 …. a 2 a 1 = a n * 10 n-1 + a n-1 *10 n-2 +….+a 2 *10 2 +a 1 *10 0 is an expanded form of writing a number where a i is a character from the set “ 0123456789" Base decimal is 10 back

Binary number system Representation of numbers in the binary number system Arithmetic operations in the binary number system Relationship between binary and decimal systems back

Representation of a number in the binary number system If the base of the number system is 2, then the resulting number system is called binary and the number in it is defined as follows: A 2 = a n a n-1 .... a 2 a 1 = a n * 2 n-1 + a n-1 * 2 n-2 +….+a 2 * 2 2 +a 1 * 2 0 where a i is a character from the set "0 1" This system is the most the simplest of all possible ones, since in it any number is formed only from two digits 0 and 1.

Arithmetic operations in binary ss Arithmetic in binary ss is based on the use of the following addition, subtraction and multiplication tables - 0 1 0 0 ī 1 1 1 0 + 0 1 0 0 1 1 1 10 * 0 1 0 0 0 1 0 1

Addition The binary addition table is extremely simple. Since 1+1=10, then 0 remains in this digit, and 1 is transferred to the next digit. Let's look at a few examples: 1001 1101 11111 1010011.111 1 1011 1 11001.110 10011 11000 100000 1101101.101 Task 1

Subtraction When performing a subtraction operation, the smaller number is always subtracted from the larger number in absolute value and the corresponding sign is placed. In the subtraction table, Ī means a loan in the highest digit 10111001.1 110110101 10001101.1 101011111 00101100.0 001010110 Task 2

Multiplication The multiplication operation is performed using the multiplication table according to the usual scheme used in decimal ss. 11001 11001.01 1101 11.01 11001 1100101 11001 1100101 11001 1100101 101000101 1010010.0001 Task 3

Physical education Exercise 1. Take a deep breath, closing your eyes as tightly as possible. Hold your breath for 2-3 seconds and try not to relax. Exhale quickly, opening your eyes wide, and feel free to exhale loudly. Repeat 5 times. Exercise 2. Close your eyes, relax your eyebrows. Slowly feeling the tension of the eye muscles, move the eyeballs to the extreme left position, then slowly, with tension, move the eyes to the right (you should not squint, the tension of the eye muscles should not be excessive). Repeat 10 times.

Relationship between binary and decimal number systems Converting numbers from binary ss to decimal ss Converting from decimal ss to binary number system Converting whole numbers Converting proper fractions Converting mixed numbers back

Converting a number from binary ss to decimal ss The method of such translation is given by our way of writing numbers. Let's take, for example, the following binary number 1011. Let's expand it into powers of two. We get the following: 1011 2 = 1 * 2 3 + 0 * 2 2 + 1 * 2 1 + 1 * 2 0 We perform all the recorded actions and get: 1 * 2 3 + 0 * 2 2 + 1 * 2 1 + 1 * 2 0 = 8 + 0+ 2 + 1 = 1 1 10 . Thus, we get that 1011 (binary) = 11 (decimal). Task 4

Conversion to decimal number system 101001 2 = 101001 2 = 543210 +1·2 3 +1·2 0 +0·2 4 +0·2 2 +0·2 1 =0 1·2 5 = 41 543210 +1·2 3 +1·2 0 +0·2 4 +0·2 2 +0·2 1 =0 1·2 5 = 41

Converting a number from decimal ss to decimal ss A person is accustomed to working in the decimal number system, but the computer is oriented towards the binary system. Therefore, communication between a person and a machine would be impossible without the creation of simple algorithms for converting numbers from one number system to another. Let's consider separately the translation of integers and proper fractions.

Translation of integers There is a simple algorithm for converting numbers from the decimal number system to the binary system: - Divide the number by 2, fix the remainder (0 or 1) and the quotient - If the quotient is not equal to 0, then divide by 2, etc. - If the quotient is 0, then write down all the resulting remainders, starting from the last, from left to right.

Example Convert the decimal number 11 to the binary number system. 11 2 1 5 2 1 2 2 0 1 2 1 0 Collecting the remainders from division in the direction indicated by the arrow, we get: 11 10 =1011 2. Task 5

Converting proper fractions Example 1 Convert the decimal fraction 0.5625 to binary ss. The calculations are best done according to the following scheme: 0.5625  2 1 1250  2 0 2500  2 0 5000  2 1 0000 Answer: 0.5625 10 =0.1001 2

Example 2 Convert the decimal fraction 0.7 to binary ss. 0, 7  2 1 4  2 0 8  2 1 6  2 1 2 …… Answer: 0.7 10 =0.1011 2 Task 6 This process can continue endlessly, giving more and more new signs . This process is terminated when it is believed that the required accuracy has been obtained. Calculations are best formatted according to the following scheme:

Translation of mixed numbers Translation of mixed numbers containing integer and fractional parts is carried out in two stages. The whole part is translated separately, and the fractional part separately. In the final recording of the resulting number, the integer part is separated from the fractional part.

Example Convert the integer part: 17 2 1 8 2 0 4 2 0 2 2 0 1 2 1 0 Convert the fractional part: 0. 25  2 0 50  2 1 00 Convert the number 17.25 10 to binary ss Answer: 17.25 10 =10001.01 2 Task 7

Task 1 Perform the addition operation on binary numbers: 1) 1011101+11101101 2) 11010011+11011011 3) 110010.11+110110.11 4)11011.11+101111.11 Answers: 1) 101001010 1) 10101110 3) 1101001.10 4) 1101011.10 back

Task 2 Perform a subtraction operation on binary numbers: 1) 11011011-110101110 2) 110000110-10011101 3) 11110011-10010111 4)1100101,101 - 10101,111 Answers: 1)11010011 2) 11101001 3) 1011100 4) 1001111.110 back

Task 3 Perform the multiplication operation on binary numbers: 1) 100001*1111.11 2) 111110*100010 3) 100011*1111.11 4) 111100*100100 Answers: 1) 1000000111.11 2) 10000011110 0 3) 1000010101.11 4) 100001110000 ago

Task 4 Convert integers from binary to decimal: 1) 1000000001 2) 1001011000 3) 1001011010 4) 1111101000 Answers: 1) 513 2) 600 3) 602 4) 1000 back

Task 5 Convert integers from the decimal number system to binary: 1) 2304 2) 5001 3) 7000 4) 8192 Answers: 1) 100100000000 2) 1001110001001 3) 1101101011000 4) 1000000000 0000 back

Task 6 Convert decimal fractions to binary ss (write the answer with six binary digits): 1) 0.7351 2) 0.7982 3) 0.8544 4) 0.9321 Answers: 1) 0.101111 2) 0.110011 3 ) 0.110110 4) 0.111011 back

Task 7 Convert mixed decimal numbers to binary ss: 1) 40.5 2) 31.75 3) 173.25 4) 124.25 Answers: 1) 101000.1 2) 11111.11 3) 10101101.01 4) 1111100 .01 ago

Lesson on the topic: Lesson objectives: To learn the definition of the following concepts: Numeral system, digit, number, base of the number system, place, alphabet, non-positional number system, positional number system, unit (unary) number system. Learn to write: a decimal number in the Roman number system, any number in a positional number system in expanded form Be able to: determine the base of a number system give examples of numbers of different positional number systems explain the difference between a number and a digit positional and non-positional number system - Said the ancient Greek philosophers, students of Pythagoras , emphasizing the important role of numbers in practical activities. - This is a sign system in which numbers are written according to certain rules using symbols of a certain alphabet, called numbers. Number system - This is a set of techniques and rules by which numbers are written and read. Positional non-positional number systems A non-positional number system is a number system in which the quantitative value of a digit does not depend on its position in the number. Examples of non-positional number systems are: unit decimal ancient Egyptian alphabetic number system (Roman) unit number system In ancient times, when people began to count, there was a need to write numbers. Initially, the number of objects was displayed by an equal number of some icons: notches, dashes, dots. + + = Decimal Ancient Egyptian number system (Second half of the third millennium) To designate key numbers, special hieroglyphs were used: Alphabetic system for writing numbers Until the end of the 17th century in Rus', the following Cyrillic letters were used as numbers if a special sign was placed above them - title. For example: Roman number system The Roman number system has reached us. It has been used for more than 2500 years. It uses Latin letters as numbers: I 1 V 5 X 10 L C 50 100 D M 500 1000 For example: CXXVIII = 100 +10 +10 +5 +1 +1 +1=128 Positional is a number system in which the quantitative value of a digit depends on its position in the number. Babylonian number system The first positional number system was invented in ancient Babylon, and the Babylonian numbering was sexagesimal, that is, it used sixty digits! Numbers were composed of two types of signs: Units - straight wedge Tens - recumbent wedge Hundreds 10 + 1 = 11 Positional number systems The most common at present are -decimal -binary -octal -hexadecimal positional number systems. Decimal number system We can write any number using ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 That is why our modern number system is called decimal. The famous Russian mathematician N.N. Luzin put it this way: “The advantages of the decimal number system are not mathematical, but zoological. If we had not ten fingers on our hands, but eight, then humanity would use the octal number system.” Decimal number system Although the decimal number system is usually called Arabic, it originated in India, in the 5th century. In Europe, they learned about this system in the 12th century from Arabic scientific treatises, which were translated into Latin. This explains the name “Arabic numerals”. However, the decimal number system became widespread in science and in everyday life only in the 16th century. This system makes it easy to perform any arithmetic calculations and write down numbers of any size. The spread of the Arabic system gave a powerful impetus to the development of mathematics. Arabic numbering Prevailed under Peter I How the numbers used by the Arabs changed until they took modern forms: It was invented long before the advent of computers. The official birth of binary arithmetic is associated with the name of G. W. Leibniz, who published an article in 1703 in which he examined the rules for performing arithmetic operations on binary numbers. Its disadvantage is the “long” recording of numbers. At the moment, it is the number system most commonly used in computer science, computer technology and related industries. Uses two digits: 0 and 1 Example: Collapsed form of a number: 1012 2 1 0 Expanded form: 101 =1*22 +0*21+1*20 All numbers in a computer are represented using zeros and ones, i.e. in the binary system Reckoning. Positional Number System The number of digits used is called the base of the positional number system. Any natural number greater than one can be taken as the base of a positional system. The base of the system to which a number belongs is indicated by a subscript to that number. 1110010012 356418 43B8D16 Example: decimal base = 10 The position of a digit in a number is called the digit. The number 555 is a collapsed form. 2 1 0 555=5*10+5*10+5*10 - expanded form of the number. Alphabets of several systems Base System Alphabet n=2 Binary 01 n=3 Ternary 012 n=8 Octal 01234567 n=16 hexadecimal 0123456789ABCDEF Independent work 1. Read carefully the algorithm for completing tasks; 2. Complete the task in Card No. 1 in your notebook and hand it in to the teacher for checking. 3. Read carefully everything about the Roman number system in the task in Card No. 2. Complete No. 1 and No. 2 on the same form without fail, and No. 3 (+) if you can. Exchange tasks with forms for mutual checking with your desk neighbor. 3. Read carefully everything about positional number systems in Card No. 3 and complete tasks on the same form: No. 1 - fill out table No. 2 - the first task is mandatory. With a sign (+) - additionally, if you can. Exchange tasks with your desk neighbor for mutual checking. Card No. 1: Write down in a notebook the basic definitions of concepts, given in explicit and implicit form: 1. Number system 2. Digit 3. Number 4. Base of the number system 5. Place 6. Alphabet 7. Non-positional number system 8. Positional number system 9 Unit (unary) number system Card No. 2: Write down the numbers in the Roman number system: 1. 9= 12 = 2778 = 2. Which numbers are written using Roman numerals: LXV= MCMLXXXVI = __________________________+ (optional) Correct the incorrect equations by rearranging from one place to another only one stick: VII –V = XI IX – V = VI Card No. 3: (done on the same form) Task No. 1: Fill out the table: Task No. 2: Write down the numbers in expanded form: 5.1610 = 1001.012 = __________________________+ (optional) Think and try to explain how the positional number system differs from the non-positional number system. Homework: §4.1.1, tasks for independent completion: 4.1, 4.2, 4.3, 4.4, 4.5 Creative task: Compose and format a crossword puzzle on the topic “Number systems” in MS Word

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