The Roman system of recording numbers has reached 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 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!

The numbers were made up of two types of signs:

Units – straight wedge

Tens - lying wedge

Positional number systems

The most common currently are

Decimal - binary

Octal

-hexadecimal positioning systems

Reckoning.

Decimal system

dead reckoning

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 they changednumbers used by Arabs, 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

Collapsed form of writing the number: 101 2

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

All numbers in the computer are represented

using zeros and ones, i.e. in the binary number system.

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: base 10 = 10

| Computer Science and Information and Communication Technologies | Lesson planning and lesson materials | 6th grade | Material for the curious | Babylonian number system

Material
for the curious

Babylonian number system

The idea of ​​assigning different values ​​to numbers depending on their position in the number record first appeared in Ancient Babylon around the 3rd millennium BC.

Many clay tablets of Ancient Babylon have survived to this day, on which complex problems were 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 entire number as a whole was written in the positional number system with base 60. Let us explain this with examples.

Record denoted 6 60 + 3 = 363, just as our notation 63 denotes 6 10 + 3.

Record designated 32 60 + 52 = = 1972; recording meant 1 60 60 + 2 60 + + 4 = 3724.

The Babylonians also had a sign that played the role of a zero. They denoted the absence of intermediate categories. But the absence of junior ranks was not indicated in any way. So, the number could mean 3, and 180 = 3 60 and 10 800 = 3 60 60 and so on. Such numbers could be distinguished only by meaning.

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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 the notation of a number does not depend on the value it denotes. The system may impose certain restrictions on the order of numbers (arrangement in ascending or descending order). An example of a non-positional number system is the Roman system, which uses Latin letters as numbers. Presentation by: Nikita Astashov and Danila Darakhovich

In ancient Babylon, whose culture, including mathematics, was quite high, there was a very complex sexagesimal system. Historians have differing opinions about exactly how such a system arose. One of the hypotheses, which is not particularly reliable, is that there was a mixture of two tribes, one of which used the sixfold system, and the other the decimal one. The sexagesimal system arose as a compromise between these two systems. In the Babylonian sexagesimal number system, based on the positional principle, two symbols were used, two types of wedges, which are the “digits” in this number system Babylonian number system

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

Unary (single, different) number system is a non-positional number system with a single digit denoting 1. The only “digit” is “1”, a dash (|), a pebble, a knuckle, a knot, a notch, etc. In this system, the number written using units. For example, 3 in this system would be written as |||. Apparently, this is chronologically the first number system of every people who mastered counting. 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 numbers. Moreover, if a larger number is in front of a smaller one, then they are added (the principle of addition), but if a 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 repeating the same number four times. Roman numerals appeared 500 BC among the Etruscans, who could have borrowed some of the numerals from the proto-Celts. Roman number system

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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 the notation of a number does not depend on the value it denotes. The system may impose certain restrictions on the order of numbers (arrangement in 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 number system The Greek number system, also known as Ionian or Modern Greek, is a non-positional number system. An alphabetical notation of numbers in which the letters of the classical Greek alphabet are used as counting symbols, as well as some letters of the pre-classical era, such as ϛ (stigma), ϟ (coppa) and ϡ (sampi).

Mayan numerals Mayan numerals are a notation of numbers based on the base-20 positional number system used by the Mayan civilization in pre-Columbian Mesoamerica.

Babylonian numerals Babylonian numerals are the numbers 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 writing stick or a pointed reed.

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

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

Unit (unary) system In ancient times, when people began to count, there was a need to write down numbers. The number of objects, for example, bags, was depicted by drawing dashes or serifs on any hard surface: stone, clay, wood (the invention of paper was still very far away). Each bag in such a record corresponded to one line. 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 record numbers - a 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.

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 an outline. The Egyptians came up with their own number system, in which the key numbers were 1, 10, 100, etc. special icons were used - hieroglyphs. All other numbers were composed from these key numbers using the operation of addition. For example, to depict 3252, three lotus flowers (three thousand), two rolled palm leaves (two hundreds), five arcs (five tens) and two poles (two units) were drawn. The size of the number did not depend on the order in which its constituent signs were located: they could be written from top to bottom, from right to left, or interspersed. In the ancient Egyptian number system, special signs (digits) were used to designate the numbers 1, 10, 102, 103, 104, 105, 106, 107. Numbers in the Egyptian number system were written as combinations of these “digits”, in which each “digit” was repeated no more than nine times. Both the rod 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.

Roman system An example of a non-positional system that has survived to this day is the number system that was used more than two and a half thousand years ago in Ancient Rome. The Roman system we are familiar with is not fundamentally 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 had to be denoted by Roman numerals (it was believed that ordinary Arabic numerals were easy to counterfeit). The Roman numeral system is used today mainly for naming significant dates, volumes, sections and chapters in books. It uses 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 Roman number 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 the first letters of the corresponding Latin words began to be used to designate 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 thousand, hundreds, fifty, tens, heels, units. To record 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.

Alphabetic system More advanced non-positional number systems were alphabetic systems. Such number systems included Slavic, Ionian (Greek), Phoenician and others. In them, numbers from 1 to 9, whole numbers of tens (from 10 to 90) and whole numbers of hundreds (from 100 to 900) were designated by letters of the alphabet. The alphabetic system was also adopted in ancient Rus'. This method of writing numbers, as in the alphabetic system, can be considered as the beginnings of a positional system, since in it the same symbols were used to designate units of different digits, to which only special signs were added to determine the value of the digit. Alphabetic number systems were not very suitable for handling large numbers. During 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 like this: above the letters denoting the numbers, a special sign was placed - a title. This was done in order to distinguish numbers from ordinary words: It is interesting that the numbers from 11 (one - ten) to 19 (nine -I by ten) were written in the same way as they were spoken, that is, the “digit” of the units was placed before the “digit” » dozens. If the number did not contain tens, then the tens digit was not written.

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

Roman system The Roman number 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 to designate the numbers C-100, D-500 and M- 1000 began to use the first letters of the corresponding Latin words.

Alphabetic systems Such number systems included Greek, Slavic, Phoenician and others. In them, numbers from 1 to 9, whole numbers of tens (from 10 to 90) and whole numbers of hundreds (from 100 to 900) were designated 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.

Mayan numerals A notation of numbers based on the base-20 numeral system used by the Maya civilization in pre-Columbian Mesoamerica.

Babylonian numerals 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 writing stick or a pointed reed.

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HISTORY OF NUMERAL SYSTEMS

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Babylonian sexagesimal system

Two thousand years BC, in another great civilization - Babylonian - people wrote down numbers differently.
Numbers in this number system were composed of two types of signs:
Straight wedge (used to indicate units)

Recumbent wedge (to indicate tens)

The number 60 was denoted by the same sign as 1

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To determine the value of a number, it was necessary to divide the image of the number into digits from right to left. The alternation of groups of identical characters (“digits”) corresponded to the alternation of digits:

The value of a number was determined by the values ​​of its constituent “digits,” but taking into account the fact that the “digits” in each subsequent digit meant 60 times more than the same “digits” in the previous digit.

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1. The number 92 = 60 + 32 was written like this:

2. The number 444 looked like:

FOR EXAMPLE:

444 = 7*60 + 24. The number consists of two digits

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To determine the absolute value of a number, additional information was required.
Subsequently, the Babylonians introduced a special symbol to indicate the missing sixdecimal place, which corresponds in the decimal system to the appearance of the number 0 in the number notation.

The number 3632 was written like this:

This symbol was usually not placed at the end of the number.
The Babylonians never memorized the multiplication tables, because... it was almost impossible to do this. When making calculations, they used ready-made multiplication tables.

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The Babylonian sexagesimal system is the first number system known to us based on the positional principle.

The Babylonian system played a major role in the development of mathematics and astronomy, and traces of it have survived to this day. So, we still divide an hour into 60 minutes, and a minute into 60 seconds.
We divide the circle into 360 parts (degrees).

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ROMAN SYSTEM

In the Roman system, the numbers 1, 5, 10, 50, 100, 500 and 1000 are represented by the capital letters I, V, X, L, C, D and M (respectively), which are the “digits” of this number system. A number in the Roman numeral system is designated by a set of consecutive “digits”.

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Calendar on a stone slab (3rd – 4th centuries), found in Rome