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Non-positional number systems A non-positional number system is a number system in which the position of a digit in a number entry does not depend on the value that it denotes. The system may impose certain restrictions on the order of the digits (ascending or descending order). An example of a non-positional number system is the Roman system, which uses Latin letters as digits. Presentation made by: Nikita Astashov and Danila Darakhovich

In ancient Babylon, whose culture, including mathematics, was quite high, there was a very complex Sexagesimal system. Historians disagree on exactly how such a system arose. One of the hypotheses, otherwise not particularly reliable, is that there was a mixture of two tribes, one of which used the hexadecimal system, and the other - the decimal system. The sexagesimal system arose as a compromise between these two systems. The Babylonian sexagesimal number system, based on the positional principle, used two symbols, two types of wedges, which are the "numbers" in this number system

The non-positional number system that was used in ancient Egypt until the beginning of the 10th century AD. In this system, hieroglyphic symbols were digits; they denoted the numbers 1, 10, 100, etc. up to a million. Egyptian number system

The unary (single, different) number system is a non-positional number system with a single digit denoting 1. The only “number” is “1”, a dash (|), a pebble, a knuckle, a score, a knot, a notch, etc. In this system, the number written in units. For example, 3 in this system would be written as |||. Apparently, this is chronologically the first number system of each nation that has mastered the account. Unary number system

Roman numerals are numbers used by the ancient Romans in their non-positional number system. Natural numbers are written by repeating these digits. At the same time, if a large number is in front of a smaller one, then they are added (the principle of addition), if the smaller one is in front of a larger one, then the smaller one is subtracted from the larger one (the principle of subtraction). The last rule applies only to avoid the fourfold repetition of the same figure. Roman numerals appeared 500 BC from the Etruscans, who could borrow some of the numbers from the proto-Celts

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Non-positional number systems Completed by: Loginov Vladislav

Non-positional number systems A non-positional number system is a number system in which the position of a digit in a number entry does not depend on the value that it denotes. The system may impose certain restrictions on the order of the digits (ascending or descending order).

Roman numeral system The Roman numeral system is a non-positional number system in which the letters of the Latin alphabet are used to write numbers: 1 - I, 5 - V, 10 - X, 50 - L, 100 - C, 500 - D and 1000 - M.

Greek Numeral System The Greek numeral system, also known as Ionic or Modern Greek, is a non-positional numeral system. Alphabetical notation of numbers, in which the letters of the classical Greek alphabet are used as symbols for counting, as well as some letters of the pre-classical era, such as ϛ (stigma), ϟ (koppa) and ϡ (sampi).

Mayan numerals Mayan numerals are a number notation based on the vigesimal positional number system used by the Mayan civilization in pre-Columbian Mesoamerica.

Babylonian numerals Babylonian numerals are the numerals used by the Babylonians in their sexagesimal number system. Babylonian numbers were written in cuneiform - on clay tablets, while the clay was still soft, signs were squeezed out with a wooden stick for writing or a pointed reed.

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The work was done by Tatyana Mikhaleva, a student of class 10 A Non-positional number systems

A non-positional number system is a number system in which the position of a digit in a number entry does not depend on the value that it denotes. The system may impose certain restrictions on the order of the digits (ascending or descending order).

Single (unary) system In ancient times, when people began to count, there was a need to record numbers. The number of objects, for example, bags, was depicted by drawing dashes or serifs on some solid surface: stone, clay, wood (it was still very far before the invention of paper). Each bag in such a record corresponded to one dash. Archaeologists have found such "records" during excavations of cultural layers dating back to the Paleolithic period (10-11 thousand years BC). The essence of the system. Scientists called this method of writing numbers the unit (stick) number system. In it, only one type of sign was used to write numbers - a stick. Each number in such a number system was designated using a string made up of sticks, the number of which was equal to the designated number.

Ancient Egyptian decimal non-positional system The ancient Egyptian decimal non-positional system arose in the second half of the third millennium BC. The paper was replaced by a clay tablet, and that is why the numbers have such a mark. The Egyptians came up with their own number system, in which to designate the key numbers 1, 10, 100, etc. used special icons - hieroglyphs. All other numbers were compiled from these key numbers using the addition operation. For example, to depict 3252, three lotus flowers (three thousand), two folded palm leaves (two hundreds), five arcs (five tens) and two poles (two units) were drawn. The value of the number did not depend on the order in which the signs that made it up were located: they could be written from top to bottom, from right to left, or interspersed. In the ancient Egyptian number system, special signs (numbers) were used to denote the numbers 1, 10, 102, 103, 104, 105, 106, 107. Numbers in the Egyptian number system were written as combinations of these "numbers", in which each "number" was repeated no more than nine times. Both stick and ancient Egyptian numeral 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.

Roman system An example of a non-positional system that has survived to this day is the number system, which was used more than two and a half thousand years ago in Ancient Rome. The Roman system familiar to us is fundamentally not much different from the Egyptian one. But it is more common these days: in books, in films. Roman numerals have been used for a very long time. Even 200 years ago, in business papers, numbers should have been indicated by Roman numerals (it was believed that ordinary Arabic numerals were easy to fake). The Roman numeral system is used today mainly for naming significant dates, volumes, sections and chapters in books. It uses the capital Latin letters I, V, X, L, C, D and M (respectively), which are the "digits" of this number system, to denote the numbers 1, 5, 10, 50, 100, 500 and 1000. The basis of the Roman numeral system was the signs I (one finger) for the number 1, V (open palm) for the number 5, X (two folded palms) for 10, and the first letters of the corresponding Latin words began to be used to denote the numbers 100, 500 and 1000 (Centum - one hundred, Demimille - half a thousand, Mille - a thousand). To write down a number, the Romans decomposed it into the sum of thousands, half a thousand, hundreds, half hundred, tens, heels, units. To write intermediate numbers, the Romans used not only addition, but also subtraction. In this case, the following rule was applied: each smaller sign placed to the right of the larger one is added to its value, and each smaller sign placed to the left of the larger one is subtracted from it.

Alphabetical system Better non-positional number systems were alphabetic systems. Such number systems included Slavic, Ionian (Greek), Phoenician and others. In them, numbers from 1 to 9, integer numbers of tens (from 10 to 90) and integer numbers of hundreds (from 100 to 900) were denoted by letters of the alphabet. The alphabetic system was also adopted in ancient Russia. This way of writing numbers, as in the alphabetical system, can be considered as the beginnings of a positional system, since it used the same symbols to designate units of different digits, to which only special characters were added to determine the value of the digit. Alphabetical number systems were not very suitable for operating with large numbers. In the course of the development of human society, these systems gave way to positional systems. Among the Slavic peoples, the numerical values ​​of the letters were established in the order of the Slavic alphabet, which used first the Glagolitic alphabet and then the Cyrillic alphabet. Numbers from 1 to 10 were written as follows: a special sign was placed above the letters denoting numbers - a title. This was done in order to distinguish numbers from ordinary words: Interestingly, numbers from 11 (one - by ten) to 19 (nine -I by ten) were written in the same way as they said, that is, the “number” of the units was put before the “number » tens. If the number did not contain tens, then the “number” of tens was not written.

Ancient Egyptian system The ancient Egyptians came up with their own number system, in which to designate the key numbers 1, 10, 100, etc. used special icons - hieroglyphs. All other numbers were compiled from these key numbers using the addition operation.

Roman system The Roman numeral system was based on the signs I (one finger) for the number 1, V (open palm) for the number 5, X (two folded palms) for 10, and for the numbers C-100, D-500 and M- 1000 began to use the first letters of the corresponding Latin words.

Alphabetical systems These number systems included Greek, Slavic, Phoenician and others. In them, numbers from 1 to 9, integer numbers of tens (from 10 to 90) and integer numbers of hundreds (from 100 to 900) were denoted by letters of the alphabet. Among the Slavic peoples, the numerical values ​​of the letters were established in the order of the Slavic alphabet, which used first the Glagolitic alphabet and then the Cyrillic alphabet.

Maya numerals Recording numbers based on the vigesimal positional number system used by the Mayan civilization in pre-Columbian Mesoamerica.

Babylonian Numerals Numerals used by the Babylonians in their sexagesimal number system. Babylonian numbers were written in cuneiform - on clay tablets, while the clay was still soft, signs were squeezed out with a wooden stick for writing or a pointed reed.

Thanks for watching 

The Roman system of writing numbers has come down to us

Used for over 2500 years.

It uses Latin letters as numbers:

For example:

CXXVIII = 100 +10 +10 +5 +1 +1 +1=128

A positional number system is one 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 made up of signs of two types:

Units - straight wedge

Tens - lying wedge

Positional number systems

The most common at present are

Decimal - Binary

octal

-hexadecimal positional systems

reckoning.

Decimal system

reckoning

We can write any number with 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 on our hands not ten fingers, but eight, then humanity would use the octal number system.

Decimal number system

Although the decimal number system is usually called Arabic, but it originated in India, in the 5th century.

In Europe, this system was learned in the 12th century from Arabic scientific treatises, which were translated into Latin.

This explains the name "Arabic numerals".

However, the decimal number system was widely used in science and in everyday life only in the 16th century. This system allows you to easily perform any arithmetic calculations, write 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 didnumerals used by arabsuntil they took their 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 considered the rules for performing arithmetic operations on binary

numbers. Its disadvantage is the “long” notation of numbers.

At the moment - the most common number system in computer science, computer technology and related industries. Uses two numbers:

0 and 1

Collapsed number notation: 101 2

Extended form: 101 =1*22 +0*21 +1*20

All numbers in a computer are represented

using zeros and ones, that is, in the binary number system.

Positional number system

Any natural number greater than one can be taken as the basis 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 : Base decimal =10

1 of 31

Presentation - Number systems

The text of this presentation

Topic "Number systems"

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

The 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 value that it denotes does not depend on the position of the digit in the notation of the number. The value denoted by a digit in a number entry depends on its position. The base is the number of digits used. Position - the location of each digit.

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

Examples of positional number systems:
Binary number system with base 2, two symbols are used - 0 and 1.
Octal number system with base 8, digits 0 to 7 are used.
Decimal System with base 10, 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 letters A to F to represent the numbers 10 to 15.
Hexadecimal, base 60, is used to measure 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 propagandist of the binary system was the famous G.V. Leibniz. He noted the particular simplicity of the algorithms of arithmetic operations in binary arithmetic in comparison with other systems and gave it a certain philosophical meaning. In 1936-1938, the 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 the source data from decimal to binary when entering them into the machine and reverse translation from binary to decimal when outputting calculation results. The main advantage of the binary system is the simplicity of addition, subtraction, multiplication and division algorithms.

Addition, subtraction, multiplication and division in the binary 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 20th century, the century of computerization, humanity uses the binary system every day, since all the information processed by modern computers is stored in them in binary form. In modern computers, we can enter textual information, numerical values, as well as graphic and sound information. The amount of information stored in a computer is measured by its "length" (or "volume"), which is expressed in bits (from the English binary digit).

Converting numbers from one number system to another
8
16

Conclusion
The highest achievement of ancient arithmetic is the discovery of the positional principle of representing numbers. It is necessary to recognize the importance of not only the most common system that we use on a daily basis. But also each separately. Indeed, in different areas different number systems are used, with their own characteristics and characteristic properties.

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 15D
14 1110 16 E
15 1111 17 F
16 10000 20 10

Binary to decimal conversion
To convert a binary number into a decimal one, it is necessary to write it as a polynomial consisting of the products of the digits of the number and the corresponding power of the number 2, and calculate according to the rules of decimal arithmetic: X10 \u003d An 2n-1 + An-1 2n-2 + An-2 2n-3 +…+А2 21 + А1 20
Number translation

Convert octal to decimal
To convert an octal number to a decimal one, it is necessary to write it as a polynomial consisting of the products of the digits of the number and the corresponding power of the number 8, and calculate according to the rules of decimal arithmetic: X10 \u003d An 8n-1 + An-1 8n-2 + An-2 8n-3 +…+А2 81 + А1 80
Number translation

Convert hexadecimal to decimal
To convert a hexadecimal number to a decimal one, it is necessary to write it as a polynomial consisting of the products of the digits of the number and the corresponding power of the number 16, and calculate according to the rules of decimal arithmetic: X10 \u003d An 16n-1 + An-1 16n-2 + An-2 16n-3 +…+А2 161 + А1 160
Number translation

Decimal to binary conversion
To convert a decimal number to the binary system, it must be successively divided by 2 until there is a remainder less than or equal to 1. A number in the binary system is written as a sequence of the last result of division and the remainder of the division in reverse order. Example: Convert the number 2210 to binary: 2210=101102
Number translation

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

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

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, if necessary, supplementing the highest triad with zeros, and replacing each triad with the corresponding octal digit. When translating, you must use a binary-octal table: Example: Convert the number 10010112 to octal: 001 001 0112=1138
8th 0 1 2 3 4 5 6 7
Number translation

Binary to hexadecimal conversion
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: 0010 1110 00112=2E316
16th 0 1 2 3 4 5 6 7
16-way 8 9 A B C D E F
Number translation

Converting an octal number to binary
To convert an octal number to binary, each digit must be replaced by its equivalent binary triad. Example: Convert the number 5318 to binary: 5318=101 011 0012
2nd 000 001 010 011 100 101 110 111
8th 0 1 2 3 4 5 6 7
Number translation

Converting hexadecimal to binary
To convert a hexadecimal number to binary, each digit must be replaced by its equivalent binary tetrad. Example: Convert the number EE816 to the binary 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
16-way 8 9 A B C D E F
Number translation

Converting from octal to hexadecimal and vice versa
When switching from octal to hexadecimal and vice versa, an intermediate conversion of numbers to binary is required. Example 1: Convert FEA16 to octal: FEA16=1111111010102=111 111 101 0102=77528 Example 2: Convert 66358 to hexadecimal: 66358=1101100111012=1101 1001 11012=D9D16
Number translation

Single 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 denoted using a string 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. Above the letter denoting the number, a special icon was placed: (“titlo”). To denote thousands, a special sign was placed in front of the number (lower left).
Z
ancient number systems

Roman numeration
The ancient Romans used numbering, which is preserved to this day under the name "Roman numbering". We use it to designate centuries, anniversaries, to name congresses and conferences, to number the chapters of a book or the 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 Old Slavic numbering system
Slavic numbering

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for the curious

Babylonian number system

The idea of ​​assigning different values ​​to numbers, depending on what position they occupy in the notation of a number, first appeared in ancient Babylon around the 3rd millennium BC.

Many clay tablets of Ancient Babylon have survived to our time, on which the most complex problems have been solved, such as calculating roots, finding the volume of a pyramid, etc. To record numbers, the Babylonians used only two signs: a vertical wedge (units) and a horizontal wedge (tens). All numbers from 1 to 59 were written using these signs, as in the usual hieroglyphic system.

The whole number as a whole was written in a positional number system with base 60. Let's explain this with examples.

Recording stood for 6 60 + 3 = 363, just as our notation 63 stands for 6 10 + 3.

Recording marked 32 60 + 52 = = 1972; record denoted 1 60 60 + 2 60 + + 4 = 3724.

The Babylonians also had a sign that played the role of zero. They denoted the absence of intermediate digits. But the absence of junior digits was not indicated in any way. So, the number could mean both 3 and 180 = 360 and 10800 = 36060 and so on. It was possible to distinguish such numbers only by meaning.

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