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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; record 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.

The emergence of numbers It is difficult to say when, and most importantly, how a person learned to count (just as it is impossible to find out for certain when, and most importantly, how language arose). It is only known that all ancient civilizations already had their own counting systems, which means that the history of numbers and the number system originated in pre-civilization times. The history of numbers and number systems began with the separation of the concepts of “one”, “two”, “many”. People, having learned to distinguish one object from all the others, pronounced: “one,” and if there were more objects, “many.” However, already in the oldest known civilizations, more detailed number systems were developed. Over time, the development of civilized settlements “forced” people to engage in writing and mathematics, since more and more information appeared in life and it needed to be mastered more effectively, rather than counting to two. Special signs were invented to write numbers. They acted as numbers and were easy to read, but writing them down took a lot of time.

Babylonian number system The Babylonian (Mesopotamian) number system is sexagesimal. There are still 60 minutes in an hour and 60 seconds in a minute. Therefore, the year is divided by the number of months, which is a multiple of 60, and the day is divided by the same number of hours. Originally it was a sundial, that is, each of them was 1/12 of a day of light. Much later, the duration of the hour began to be determined not by the sun and 12 night hours were added. Babylonian numerals were composite and were written as numbers in a decimal non-positional number system. The Mayans used a similar principle in their base-20 positional number system. To understand the notation of a number, “spaces” are needed between the Babylonian numerals.

Ancient Egyptian Number System The Ancient Egyptian number system, which arose in the second half of the third millennium BC, used special numerals to represent the numbers 1, 10, 102, 103, 104, 105, 106, 107. The numbers in the Egyptian number system were written as combinations of these numbers, in which each of them was repeated no more than nine times. The ancient Egyptian number system was 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. The ancient Egyptians wrote the number 345 as follows: , where - units, - tens, - hundreds

Roman numeral system The Roman numeral system is a non-positional number system in which letters of the Latin alphabet are used to write numbers. To write large numbers, you must first write the number of thousands, then hundreds, then tens, and finally ones. If a larger number is in front of a smaller one, then they are added (the principle of addition), but if a smaller number is in front of a larger one, then the smaller one is subtracted (the principle of subtraction). For example, VI = 5 + 1 = 6 IV = 5 - 1 = 4 XIX = 10 + 10 – 1 = 19 XXI = 10 + 10 + 1 = 21 Currently, the Roman number system is used to designate: centuries (XV century, etc. .d.), AD e. (MCMLXXVII, etc.) and months when indicating dates (for example, 1. V. 1975) ordinal derivatives of large orders: yIV, yV, etc. valency of chemical elements

Cyrillic (Slavic) number system - a separate letter corresponded to each number (from 1 to 9), each ten (from 10 to 90) and each hundred (from 100 to 900). So that the reader understands that there are numbers in front of him, a special sign is used - title. It was depicted as a wavy line and placed above the letter. It was called “az under titlo” and meant one. Cyrillic number system Not all letters of the alphabet were used as numbers. For example, “B” and “F” were not used as numbers, because they were not in the ancient Greek alphabet, which was the basis of the digital system. Until the 17th century, this form of recording numbers was official in the territory of modern Russia, Belarus, Ukraine, Bulgaria, Hungary, Serbia and Croatia. Orthodox church books still use this numbering.

Arabic number system The Arabic number system consists of ten symbols: 0 1 2 3 4 5 6 7 8 9, with which any number is written in the decimal number system. Arabic numerals originated in India in the 10th-13th centuries. were brought to Europe by the Arabs (hence the name). "Arabic" numerals are a glazier's invention - Geometrics. He believed that the nine numbers needed to be given a shape that would correspond to their meaning and proposed figures with the appropriate number of angles for this. If you make certain movements of these figures, then together they will form an Arabic expression: My goal is calculation (Arabic). Europeans borrowed these symbols and the method of using them in the Middle Ages from Muslim mathematicians (the level of mathematics in Arab countries at that time was higher than that of Europeans ), hence the name Arabic numerals. In fact, the Arabs adopted them from the Indians. The Arabic number system is positional - the weight of each digit is determined by its position in the number.

Number systems A number system is a recording of numbers using some alphabet, the symbols of which are called numbers (a method of encoding numerical information). Number systems are divided into: positional non-positional. Positional number systems include binary, decimal, octal, hexadecimal. Here, any number is written as a sequence of digits from the corresponding alphabet, and the meaning of each digit depends on the place (position) it occupies in this sequence. For example, in the entry 555, made in the decimal number system, one digit 5 ​​is used, but depending on the place it occupies, it has a different quantitative value - 5 units, 5 tens or 5 hundreds. Non-positional number systems are those systems in which the value of a digit does not depend on its position in the number (Roman numeral system).

Positional number systems In positional number systems, the value denoted by a digit in the notation of a number depends on its position. The number of digits used is called the base of the number system. The place of each digit in a number is called position. Binary, decimal, octal and hexadecimal systems with bases two, ten, eight and sixteen are positional number systems. Promotion of a digit refers to its replacement with the next highest one. To advance the number 1 means to replace it with 2, to advance the number 2 means to replace it with 3. To advance the highest digit in the decimal system (this is the number 9) means to replace it with 0. Examples of the first ten digits in different number systems: Binary: 0, 1, 10, 11, 100, 101, 110, 111, 1000, 1001. Decimal: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Octal: 0, 1, 2, 3, 4, 5, 6, 7, 10, 11. Hexadecimal: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 (numbers from 10 to 15 in hexadecimal are represented by the letters A, B, C, D, E, F). Binary, octal and hexadecimal number systems belong to the class of machine number systems.

History of numbers and number systems Number systems A number system is a way of writing numbers using special characters - numbers. Numbers: 123, 45678, 1010011, CXL Digits: 0, 1, 2, ... I, V, X, L, ... The alphabet is a set of numbers. (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) Types of number systems: – non-positional – the value of a digit does not depend on its place (position) in the number record; – positional – the meaning of a digit depends on its place (position) in the number notation; Non-positional number systems Unary number system Unary - one digit denotes one (1 day, 1 stone, 1 ram, ...) At excavations of sites of ancient people, archaeologists find images in the form of serifs, dashes on hard surfaces: stone, clay, wood - that’s what they thought our ancestors some objects, bags, livestock. Ancient Egyptian decimal non-positional system Try to recognize and read this number? 2521 Roman numeral system I – 1 (finger), V – 5 (open palm, 5 fingers), X – 10 (two palms), L – 50, C – 100 (Centum), D – 500 (Demimille), M – 1000 (Mille) Rules: – (usually) do not put more than three identical digits in a row – if the lowest digit (only one!) is to the left of the highest one, it is subtracted from the sum (partially non-positional!) Example: 2381 = M M C C C L X X X I Alphabetic number systems Slavic system numbering Positional number systems Duodecimal system In Rus', counting was done by dozens, remember what a DOZEN is equal to? 12 Where else do we find the duodecimal number system? A year is 12 months, half a day is 12 hours, sets and cutlery are designed for 12 people. Babylonian sexagesimal system Numbers in this number system were composed of two types of signs: a straight wedge served to designate units, and a recumbent wedge - to designate tens. The number 32, for example, was written like this: The signs and served as numbers in this system. The number 60 was again denoted by the same sign as 1, and the numbers 3600, 216000 and all other powers of 60 were denoted by the same sign. Therefore, the Babylonian number system was called sexagesimal. 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. The decimal system appeared in India in \/ century AD. and it arose after the appearance of the number 0, which was invented by Greek astronomers to indicate the missing quantity. Subsequently, the Arabs became acquainted with this number system. They appreciated it, began to use it, and brought it to Europe in the 12th century. And since that time, humanity has been using this number system. Decimal 0,1,2,3,4,5,6,7,8,9 Binary system With the advent of information science and computer technology, the 2nd number system, whose roots go back to ancient China, found its application. What is the base of this number system? What numbers are used in the recording? 2, numbers – 0 and 1. Why is it used in computer science? Associated with information encoding: recording on disk, transmission of electrical signals. Binary 2 0.1 Clock in the binary number system “BREAKING” your head Read the poem by A.N. Starikov: She was 1100 years old, She went to the 101st grade, She carried 100 books in her briefcase All this is true, not nonsense. When, dusting with a dozen legs, She walked along the road, A puppy with one tail, but 100 legs, always ran behind her. She caught every sound with Her 10 ears, And 10 tanned hands held the briefcase and leash. And 10 dark blue eyes looked at the world as usual... But everything will become completely ordinary, When you understand our story. Did you understand the poet's story? 11002 =1210; 1012 = 510 1002 = 410 102 = 210 Entertaining problem The monkey hangs on its tail and chews bananas. There are 101 bananas in each hand, and 1 more banana in each leg than in the hand. How many bananas does a monkey have? Thank you for your attention

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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 indicated 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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Babylonian number system