Presentations Number systems presentation for a lesson in informatics and ict (Grade 10) on the topic. History of number systems presentation, report Presentation on the topic Babylonian number system

“Because all shades of meaning

smart number conveys ”

Nikolai Gumilyov.

Number systems

Editor of the material, ICT teacher MBOU CO - gymnasium No. 11 in Tula Akimov D.F.

What is a number?

Number is a written symbol representing a number.

Numbering system- a way to connect numbers to represent large numbers.

Consider the numbering systems of some peoples.

Ancient Greek Attic numbering

The numbers 1,2,3,4 were denoted by dashes I, II, III, IIII, and the number 5 was written with the sign G (the ancient inscription of the letter “Pi”, with which the word “pente” begins - five.

The numbers 6,7,8,9 were denoted by ГI, ГII, ГIII, ГIIII, and the number 10 was denoted by ▲ (the initial letter in the word “ten”)

The numbers 100,1000 and 10000 were denoted by H, X, M - the initial letters of the corresponding words.

The numbers 50,500 and 5000 were denoted by combinations of characters 5 and 10, 5 and 100, 5 and 1000, namely

The remaining numbers within the first ten thousand were written as follows:

H H GI = 256; XXI = 2051;

H H H ▲ ▲ ▲ I I = 382; X X H H H= 7800 etc.

Ionian numbering

In the third century BC. Attic numbering was supplanted by the so-called Ionian system. In it, the numbers 1-9 are denoted by the first nine letters of the alphabet:

the numbers 10, 20, 30,…, 90 with the following nine letters:

numbers 100, 200, 300,…, 900 with the last nine letters:

To designate thousands and tens of thousands, they used the same numbers with the addition of a special icon ’ on the side:

’ α=1000 ’ β=2000 etc.

Ionian numbering

To distinguish numbers from letters that make up words, they wrote dashes above the numbers.

Ιη=18; μζ=47; υζ=407; χκα=621; χκ=620, etc.

α=1 β=2 γ=3 δ=4 ε=5 ς =6 ζ=7 η=8 θ=9

Alpha beta Gamma delta epsilon fau zeta eta theta

ι=10 κ=20 λ=30 μ=40 ν=50 ξ=60 ο=70 π=80 Ϥ=90

iota kappa lambda mu nu xi omicron pi kappa

ρ=100 σ=200 τ=300 υ=400 φ=500 χ=600 ψ=700 ω=800 ϡ=900

ro sigma tau upsilon fi chi psi omega sampy

The Jews, Arabs and many other peoples of the Middle East had the same alphabetical numbering in antiquity, and it is not known which people first had it.

Slavic numbering

The southern and eastern Slavs used alphabetical numbering to write numbers. Among the Russian peoples, not all letters played the role of numbers, but only those that are in the Greek alphabet. Above the letter denoting the letter was placed special. icon - " title ”.

In Russia, Slavic numbering survived until the end of the 17th century. Under Peter I, Arabic numbering prevailed (we use it now). Slavic numbering was preserved only in liturgical books. Here are the Slavic numbers:

A

• 2 3 4 5 6 7 8 9 10 20 30 40 50 60 70 80 90 100 200 300 400 500 600 700 800 900

Κ Α =21 ΜΕ=45 ΨΒ=702 СΒ=202

In ancient Babylon, ≈ 40 centuries before our time, local (positional) numbering was created, i.e. such a way of representing numbers, in which the same digit can denote different numbers, depending on the place occupied by this digit. In the Babylonian system, the role that the number 10 plays for us was played by the number 60, so this numbering is called sexagesimal .

Numbers less than 60 were denoted using two signs: for one and for ten.

They had a wedge-shaped appearance, because. The Babylonians wrote on clay tablets with triangular sticks. These signs were repeated the required number of times

Babylonian local numbering

The way to designate numbers greater than 60 is shown in Fig:

5*60+2=302 21*60+35=1295

1*60*60 + 2*60 +5 =3725

Babylonian local numbering

In the absence of an intermediate digit, a sign was used that played the role of zero.

For example, the entry meant 2*60*60 + 0*60 +3 = 7203

The 60-decimal notation of integers did not become widespread outside the Assyro-Babylonian kingdom, but 60-decimal fractions penetrated far beyond: to the countries of the Middle East, Central Asia, to the North. Africa and Western Europe. Traces of 60-decimal fractions are still preserved in the division of the angular and arc degrees by 60 minutes. and minutes to 60 seconds.

Roman numerals

The ancient Romans used numbering, which is preserved to this day under the name of "Roman numbering". We use it to designate anniversaries, name congresses, number chapters in books, and so on.

In its later form, Roman numerals look like this:

I=1 V=5 X=10 L=50 C=100 D=500 M=1000

There is no reliable information about the origin of Roman numerals. The number V could serve as an image of a hand, and the number X could be made up of two fives.

In Roman numeration, traces of the fivefold system clearly affect. In the language of the Romans (Latin), there are no traces of the 5-ary system. This means that these figures were borrowed by the Romans from another people (probably from the Etruscans).

Roman numerals

All whole numbers (up to 5000) are written by repeating the above digits. At the same time, if a large number is in front of a smaller one, then they are added, if the smaller one is in front of a larger one (in this case it cannot be repeated), then the smaller one is subtracted from the larger one. For example:

VI=6, i.e. 5+1 IV=4, i.e. 5-1

XL=40 i.e. 50-10 LX=60, i.e. 50+10

The same number is placed no more than 3 times in a row.

LXX=70;LXXX=80;number 90 is written XC (not LXXXX).

Examples: XXVIII=28; XXXIX=39; CCCXCVII=397;

MDCCCXVIII=1818.

Performing multi-digit arithmetic in this system is very difficult. However, Roman numeration prevailed in Italy until the 13th century, and in other countries of Western Europe until the 16th century.

Indian local numbering

There were different systems in different parts of India. One of them has spread all over the world and is now generally accepted. In it, the numbers looked like the initial letters of the corresponding numerals in the ancient Indian language - Sanskrit ("Devanagari" alphabet).

Initially, these signs represented the numbers 1,2,3,…9,10,20,30,…90,100,1000; with their help other numbers were written down.

Subsequently, a special sign (bold dot, circle) was introduced to indicate an empty digit; signs for numbers greater than 9 fell into disuse, and the Devanagari numbering turned into a 10-ary local system.

How and when this transition took place is still unknown. In the middle of the 8th century, the positional numbering system was widely used in India.

Indian local numbering

Around this time, it penetrates into other countries (Indochina, China, Tibet, Iran, the territory of the Central Asian republics). A decisive role in the spread of the Indian system was played by the manual compiled at the beginning of the 9th century by the Uzbek scholar Al-Khwarizmi (Kitab al-jabr v’alnukabala). This guide is in Zap. Europe was translated into lat. language in the 12th century. In the 13th century, Indian numbering takes over in Italy. In other countries, Zap. Europe, it is approved in the 16th century.

Europeans who borrowed Ind. numbering from the Arabs, called it "Arab". This historically incorrect name is retained to this day.

Indian local numbering

The word digit (in Arabic “syfr”) was also borrowed from the Arabic language, meaning literally “empty space”.

This word was originally used to name the sign of an empty discharge and retained this meaning as early as the 18th century, although the Latin term “zero” (nullum - nothing) appeared already in the 15th century.

The form of Indian numerals has undergone many changes. The form in which we write them now was established in the 16th century.

A number system is a way of writing numbers using numbers and symbols.

C.C. divided into positional and non-positional

In positional S.S. the weight of a digit depends on its location, “position” in the number (Babylonian 60, our 10)

The basis (basis) of S.S. is the number of digits and symbols used in it. Foundation S.S. shows how many times the numerical value of the unit of the given digit is greater than the numerical value of the unit of the previous digit.

So familiar to us 10 S.S. turned out to be inconvenient for a computer (it is difficult to implement an element with 10 states, and easy with two). Therefore, in the computer memory, information is represented in binary S.S.

Binary number system

IN 2 s.s. only two digits are used: 0 and 1. Base 2 s.s. written as 10. For example, the representation of the number 8 in 2 s.s. looks like this: 1000 2 =8 10

1*2 3 +0*2 2 +0*2 1 +0*2 0 =8

Arithmetic operations in 2 s.s. performed according to the same rules as in 10 s.s. , only in 2 s.s. the transfer of units to the highest digit occurs more often than in 10 s.s.

Addition table Subtraction table Multiplication table

0+0=0 0-0=0 0*0=0

0+1=1 1-0=1 0*1=0

1+0=1 1-1=0 1*0=0

1+1=10 10-1=1 1*1=1

Decimal Binary

Decimal Binary

Binary Number System Examples

1. Because the base 2 s.s. small, to write even not very large numbers, you have to use a lot of characters. For example, the number 1000 is written in 2 s.s. with ten digits:

1000 10 = 1111101000 2 = 2 9 + 2 8 + 2 7 + 2 6 + 2 5 +2 3

However, this disadvantage is compensated by the advantages associated with hardware implementation (all semiconductor elements work according to the “Yes-No” principle).

2. The natural possibilities of human thinking do not allow to quickly and accurately estimate the value of a number represented, for example, by a combination of 16 zeros and ones.

Disadvantage of the binary number system

To facilitate the perception of a binary number by a person, it was decided to break it into groups of digits, for example, 3 or 4 digits each. This idea turned out to be successful, because. a 3-bit sequence has 8 combinations, and a 4-bit sequence has 16 combinations. The numbers 8 and 16 are powers of two, so it will be easy to match with binary numbers.

Having developed this idea, we came to the conclusion that groups of digits can be encoded, while reducing the length of the character sequence. To encode three bits (triads), 8 digits are required, and therefore the numbers from 0 to 7 decimal ss were taken. To encode four bits (tetrads), 16 characters are needed; for this, 10 digits of the decimal ss were taken. and 6 letters of lat. alphabets A, B, C, D, E, F. The resulting systems were called 8-ary and 16-ary.

Decimal

8-digit number

number

Sequence of triads

hexadecimal number

Sequence from tetrads

Method of triads and tetrads

To convert dv. numbers into an octal number, it is necessary to divide the binary sequence into triads from right to left and replace each triad with the corresponding 8-digit digit. Similarly, when converting to a hexadecimal code, only the binary sequence is divided into tetrads, and for replacement we use hexadecimal characters.

For example:

you need to translate 1101011101 from dv. to 8-ary s.s.

• We break it into triads from right to left.

2. We replace each triad with the corresponding 8-digit number 1 5 3 5. This will be the answer.

001 101 011 101 2 =1535 8

Method of triads and tetrads

The reverse conversion is just as easy - for this, each digit of an 8 or hexadecimal number is replaced by a group of 3 or 4 bits. For example:

AB51 16 =1010 1011 0101 0001 2

177204 8 = 1 111 111 010 000 100 2

Performing arithmetic operations

When working in 8- and hexadecimal s.s. it must be remembered that if there is a transfer, then it is not 10 that is transferred, but 8 or 16. Examples:

27,2643 8 _ 115,3564 8

46,1154 8 55,7674 8

75,4017 8 37,3670 8

287,AB _ EC2A,82

2ED,0D 16 2EAD,E8

Converting numbers from one number system to another

So, we have mastered 4 number systems”

"machine" - binary;

“human” - decimal

and two intermediate - 8 and 16-ary.

Each of them is used in various processes associated with a computer:

2 s.s. - to organize machine operations for information conversion;

8 and 16 s.s. - to represent machine codes in a form convenient for the work of professional users (programmers and apparatchiks);

10 s.s. – to present the results of the computer activity displayed on the input/output devices.

Therefore, the processes of converting numbers from one s.s. are constantly taking place in the machine. to another.

Converting numbers to 10 s.s. is performed by the summation method, taking into account the weight of the digits

1101,011 2 =1*2 3 +1*2 2 +1*2 0 +1*2 -2 +1*2 -3 = =8+4+1+0,25+0,125= 13,375

142,4 8 =1*8 2 +4*8 1 +2*8 0 +4*8 -1 = =64+32+2+0,5= 98,5

12E.6 16 =1*16 2 +2*16 1 +14*16 0 +6*16 -1 = =256+32+14+0.375= 302.375

Translation of numbers from 10 s.s. to another system

Usually performed by the method of successive division of the original number by the base of the s.s. The resulting remainder after the first division is the least significant digit of the new number. The resulting quotient is again divided by this base. From the remainder we get the next digit of the new number, and so on.

Example: _212 2 212 10 =11010100 2

Let's translate the decimal number 31318 into 8 s.s.

Example2: _31318 8 31318 10 =75126 8

Let's translate the decimal number 286 into 16 s.s.

Example 3: _286 16 286 10 = 11E 16

List of used literature

• S.I. Fomin. Popular lectures in mathematics. Issue 40. Number systems. Moscow: Nauka, 1980.
• M.Ya. Vygodsky. Handbook of mathematics.

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-civilizational 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, said: “one”, and if there were more objects - “many”. However, already in the most ancient known civilizations, more detailed number systems were developed. Over time, the development of civilized settlements “forced” people to engage in writing and mathematics, as more and more information appeared in life and it needed to be mastered more efficiently, and not counted to two. Special signs were invented to write numbers. They served as numbers and were easy to read, but it took a lot of time to write them down.

Babylonian number system The Babylonian (Mesopotamian) number system is sexagesimal. Until now, there are 60 minutes in an hour and 60 seconds in a minute. Therefore, the year is divisible by the number of months, a multiple of 60, and the day is divisible by the same number of hours. Initially, it was a sundial, that is, each of them was 1/12 of a daylight hours. 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. A similar principle was used by the Maya Indians in their vigesimal positional number system. To understand the writing of the number between the Babylonian numerals, "gaps" are needed.

Ancient Egyptian number system In the ancient Egyptian number system, which arose in the second half of the third millennium BC, special 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 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 attribute the ancient Egyptian number system to decimal non-positional. The ancient Egyptians wrote the number 345 like this: where - units, - tens, - hundreds

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. To write large numbers, you must first write down the number of thousands, then hundreds, then tens, and finally units. If the larger number is in front of the smaller one, then they are added (the principle of addition), if the smaller one is in front of the 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 .d.), years b.c. 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 digit (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, they used a special sign - a title. It was depicted as a wavy line and placed above the letter. It was called "az under the title" and meant a unit. Cyrillic number system Not all letters of the alphabet were used as numbers. For example, "B" and "F" did not turn into 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 writing numbers was official on the territory of modern Russia, Belarus, Ukraine, Bulgaria, Hungary, Serbia and Croatia. Until now, Orthodox church books use this numbering.

Arabic numeral system The Arabic numeral system consists of ten characters: 0 1 2 3 4 5 6 7 8 9, with the help of which any number is written in the decimal number system. Arabic numerals originated in India and in the 10th-13th centuries. were brought to Europe by the Arabs (hence the name). "Arabic" numerals are an invention of the glazier - Geometrics. He believed that nine figures should be given a form that would correspond to their meaning and proposed figures for this with the appropriate number of angles. If you make certain movements of these figures, then together they will form an Arabic expression: My goal is calculation (Arab.) Europeans borrowed these symbols and the way they were used 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 the position in the number.

Number systems The number system is the recording of numbers using a certain alphabet, the symbols of which are called numbers (a way of encoding numerical information). Number systems are divided into: positional non-positional Number systems include binary, decimal, octal, hexadecimal. Here, any number is written as a sequence of digits of the corresponding alphabet, and the value 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 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 a number entry 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 a position. The binary, decimal, octal, and hexadecimal systems with bases two, ten, eight, and sixteen are positional number systems. The promotion of a number is its replacement with the next largest. Promoting a 1 means replacing it with a 2, advancing a 2 means replacing it with a 3. Promoting the highest digit in the decimal system (which is the number 9) means replacing it with a 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 E, F). Binary, octal and hexadecimal number systems belong to the class of machine number systems.

"Translation of number systems" - Translation of integers into 2, 8, 16th number systems. Decimal. Octal. Translation of numbers from the 2nd number system to the 8th. Translation of numbers from the 16th number system to the 10th. You can perform arithmetic operations on numbers in the binary system. Translation of numbers from the 10th number system to the 8th.

"Numbers and number systems" - Translation of numbers (10) ? (q). Binary arithmetic. Positional number systems. Base 10 in the usual decimal number system (ten fingers on the hands). Example. Disadvantage: the rapid increase in the number of digits required to write numbers. Translation of numbers (2) ? (8), (2) ? (16). counting rule. Binary number system.

"History of numbers and number systems" - History of numbers. Non-positional number systems. For example: 0101101000112 = 0101 1010 0011 = 5A316. Positional number systems. Roman numerals appeared around 500 BC with the Etruscans. Addition of numbers of unlimited length. Numerals used by the ancient Romans in their non-positional number system.

"Babylonian Kingdom" - Slaves were sold, exchanged, given, passed on by inheritance. Slavery. The ancient Babylonian state reached its peak in the reign of Hammurabi (1792-50 BC). Hanging gardens before... Even the images on the bricks were dedicated to cats. The population here was mainly engaged in fishing, cattle breeding and agriculture.

"History of number systems" - The number represented a certain pattern in which the number of angles corresponded to the number. Time flies, everything changes. The usual system of writing numbers that we are accustomed to enjoy life. History of the number system. Secondary school with in-depth study of mathematics MOUSOSH school No. 125. Decimal number system.

"Examples of number systems" - Base (number of digits): 8 Alphabet: 0, 1, 2, 3, 4, 5, 6, 7. Step 2. Divide into triads: Table of hexadecimal numbers. Topic 2. Binary number system. Convert to octal and vice versa. Number systems. Convert to binary and vice versa. Loan. Most fractional numbers are stored in memory with an error.

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Babylonian sixagesimal system Two thousand years before our era, in another great civilization - Babylonian - people wrote numbers differently. The numbers in this number system were composed of two types of signs: Straight wedge (served to denote units) Recumbent wedge (to denote tens) The number 60 was denoted by the sign, which is the same 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 the 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 as follows: 2. The number 444 looked like: FOR EXAMPLE: 444 = 7 * 60 + 24. The number consists of two digits

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Additional information was needed to determine the absolute value of the number. Subsequently, the Babylonians introduced a special character to indicate the missing sixdecimal digit, which corresponds in decimal to the appearance of the digit 0 in the notation of the number. The number 3632 was written like this: At the end of the number, this character was usually not put. The Babylonians never memorized the multiplication table, because it was almost impossible to do so. When calculating, they used ready-made multiplication tables.

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The sixagesimal Babylonian system is the first number system known to us based on the positional principle. The Babylonian system played a large role in the development of mathematics and astronomy, traces of which 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 use the capital Latin 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 denoted by a set of consecutive "numbers".

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Roman numerals table Units Tens Hundreds Thousands I 10 XC 1000 M II XX CC 2000 MM 3 III XXX CCC 3000 MMM IV 40 XL 400 CD V 50 L 500 D VI LX 600 DC VII LXX 700 DCC VIII LXXX 800 DCCC 9 IX XC 900CM

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