Teacher of Informatics
MKOU "Kaltutska ZOSh"
Pershikh Evgeniya Ivanivna
dodavannya
saving
processor
vector
broadcast
History of the development of number systems. Non-positional and positional number systems.
Rakhunok vinik then, if people needed to inform their relatives about the number of objects revealed to them.
On the back of the head, people simply divided one object in front of them. If there is more than one subject, then they said “bagato”.
The simplest tool for the rahunka is the fingers on the hands of a person
One of these systems rahunku with a year and became zagalnozhivanoy - tens.
For a long time people went barefoot. So the stench could koristuvatisya for the rahunka with fingers like hands, so і nіg. In such a rank, the stench could, it would have been used, rahuvati up to twenty.
But for the help of a “barefoot car”, people could reach significantly larger numbers.
1 person - tse 20,
2 individuals - tse dvіchі 20 skinny.
It was important to memorize the great numbers, the hands did not give mechanical attachments to the “lichilnoї machine”.
Ways of rahunka were found chimalo: In different places, different ways of transmitting numerical information were seen:
For example, the Peruvians victorious for memorizing the numbers of different colors of cords tying them into knots.
To memorize the numbers, stones, grains, and turtles were victorious.
Archaeologists have found such "records" during the excavation of cultural spheres that lie until the Paleolithic period (10 - 11 thousand years BC)
Whose way of writing numbers is called
solitary
("stick", "unary")
number system
Be it a number, it will be settled
repetitions of one sign - one.
For cadet training courses
5 course 4 course 3 course 2 course 1 course
In the course of a single system, the numbers are increasing and today. So, in order to find out, on which course a cadet of a military school is studying, it is necessary to frighten, a little smuzhok is sewn on his sleeve. Without noticing it themselves, children are shriveled with a single system of numbers, showing their age on their fingers, and the sticks are zastosovuyutsya for teaching the 1st class rahunka.
Number system- this is a sign system, which has adopted the same rules for writing numbers. Signs, for the help of which they write down numbers, are called figures, And їhnya sukupnіst - number system alphabet.
number systems
Positional
Non-positional
Non-positional number systems: Non-positional with. – the same number system, for any value of the digit, to lie in the її position of the number entry. Egyptian numbering10000 100000 1000000 10000000
Vinykla 5000 years ago
Non-positional number systems: Old Greek numbering Roman number system Before us, the Roman system of numbers has reached us. Її mi, as before, victorious for the recognition of divisions, century:- VI = 6, tobto. 5+1,
- LX = 60, tobto. 50 + 10,
- IV = 4, then. 5 - 1,
- XL = 40, then 50 - 10.
The numbers are written in ascending order to the right. Їх value add up. Like a lefthander, there is a smaller number, and on the right - a larger one, its value are visible
Task 1. Convert numbers from the Roman number system to the tenth number system:
LXXVI=50+10+10+5+1=76
XLIX = (50-10) + (10-1) = 49
Task 2. Write down the tens of numbers in the Roman number system:
463=500-100+50+10+5-2=CDLXIIV
Non-positional systems of numbers may have a low number of shortcomings:- There is a constant demand for new signs to record great numbers.
- It is impossible to represent fractional and negative numbers.
- It is easy to win arithmetic operations, there are no shards of algorithms to beat them.
Position with. – the same number system, for any value of the digit, to lie in the її position of the number entry.
For example, Changing the position of the number 2 in the tenth system of numbers, you can write down the difference for the value of the tenth number: 2; 20; 200; 2000 etc.
Setting up the number system– the number (p) of different symbols, which are chosen for the image of the number in the positional number system. The basis of the system is the old number of digits in the її alphabet.
The main advantages of any positional number system:- the number of symbols for writing numbers has been delimited;
- simplicity of arithmetic operations. For example: in the Arabic decimal number system, digits are used to write numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 . There are 10 such numbers, then 10 is the basis of the Arabic number system. That її і is called the tenth system of numbers.
- high;
- sixteenth Name number systems indicate the number of digits that are scored when writing a number in a given number system, so that substantiation of the number system (p)
Name the basis of the skin number system
Number system alphabet- the number of symbols used to distinguish numbers in this number system Number system alphabet- the number of characters that are chosen for the identification of numbers in this number system Vyhodyachi z ogo memorably table0,1,2,3,4.5,6,7,8,9
0,1,2,3,4.5,6,7,8,9,10(A),11(B),12(C),13(D),14(E),15(F)
Name the alphabet of the skin number system
Whether or not a number can be written in whether or not a positional number system looks like a sum of positive and negative
powers of the number p (the basis of the number system)
Burnt number form
76510=700+60+5=7*100+6*10+5*1=7*102 +6*101 +5*100
76,5410=7*10+6*1+5*0,1+4*0,01=7*101+6*100+5*10-1+4*10-2
Pervinne comprehension that zakrіplennya vyvchennogo
1. What is the number system?
2. Non-positional number systems - ce ...
3. Positional number systems - ts…
4. What is the basis of the number system?
5. What does the flared form of a number mean?
Write in the fluted form of the number
- 485,2310 =
- 123,4510 = 3. 11011,1012 = 4. 111011,112 =
1 *102+2*101+3*100+4*10-1+5*10-2
5 4 3 2 1 0 -1 -2
1 *25+1*24+1*23+0*22+1*21+1*20+1*2-1+1*2-2
3 *83+4*82+5*81+6*80+6*8-1
3 *162+10*161+15*160+1*16-1+5*16-2
4 *102+8*101+5*100+2*10-1+3*10-2
4 3 2 1 0 -1 -2 -3
1 *24+1*23+0*22+1*21+1*20 +1*2-1+0*2-2+1*2-3
Home tasks:
- Zoshita's notes.
- Card from tasks.
, Competition "Presentation before the lesson"
Class: 6
Presentation before the lesson
Back forward
Respect! The forward review of the slides is scored solely for the purpose of learning and may not give notice about all the possibilities of the presentation. Like you got hooked by this robot, be kind, zavantazhte povnu version.
Objectives of the lesson: Motivation of knowledge activity, which gives the students the opportunity to learn and systematize the lesson of knowledge on the mastery of other systems of numbers, the number ten.
The achievement of the goal was achieved through manager lesson:
- lighting:
- learn about the number systems in different countries and eras;
- obtaining the maximum number of studies, as well as a dialogue from the study of the material covered, and the work of analyzing the introduced new material and reinforcing it;
- consolidation of the theoretical material with different technological skills - a robot from a card and a robot in a graphic editor over the accomplishments of one and the other tasks - "National Day is not in the 10th SS".
- analyze them by making a list about their classification (not positional and positional);
Methodi ta priyomi
- Organizations of educational and educational activity: selection of information and health protection technologies; staging problematic meals, scheduling poshukovyh tasks.
- Self-supporting educational activities of academics: practical work design with elements of programmed learning;
- Control and self control: self-assessment of one's activities by the students
Initially-methodical equipment for the lesson:
- Material and technical base: computer class, multimedia projector, tables for backing up and cards (distribution material), graphic editor Paint.
- Didactic security: author's presentation "History of vinification of numbers and systems of numbers", assistant.
Technical support: Windows or Linux; graphic editor Paint abo...; Power point presentation.
Tsey lesson of calls:
- aktivizuvat pіznavalnu diyalnіst uchnіv;
- develop in them a newcomer to the development of information technologies from an independent development of popular science material;
- expand your speech and bring your point to light;
- to develop in the meantime that victorious naboutih novichok s writing tests in practice.
The maximum number of students is transferred, as a contribution to the new material, and to the robot from the consolidation of yoga.
Epigraph:"The idea of hanging all the numbers with ten signs, giving them a crimson meaning behind the form, more meaning behind the plate, the flooring is simple, which itself through simplicity is important to understand, the scales are wondrous. It was not easy to come to this method, Archimedes and Apolloniya, for whom my thought was lost." P. Laplace
Hid lesson
I. Organizational moment(1 hv)
II. theoretical part. Revision of that robot from the presentation of the lesson: "History of Vinicnennya of Numbers and Number Systems". (20 khvilin.) ( Presentation)
Entry(reader-slide 1 and 2 cm. ( Presentation)): A modern person in everyday life constantly sticks with numbers and numbers - stink with us. And two thousand years that the person knew about numbers? And five thousand years to that? Vcheni assert that even then people could write down numbers and work arithmetic over them, but they tried to stink following other principles, lower work. At once we know about the systems of numbers, which have gone back in time, they have arisen, and yet laid the foundations of the modern systems of numbers.
Number system - a way to record (image) numbers.
The simplest number system (SS)(reader):
- Kamintsі, brushes ... (slide 3 and 4 see. ( Presentation)).
- Head 1(Slide 4). Show your people's day for the help of fingers. Post nutrition: How to show the river?
Visnovok(learn): the simplest SS does not allow the use of values greater than 100. - Historical review (slides 5 and 6 see ( Presentation)). Rahunok in the Indians, the peoples of ancient Asia Maya.
Additive number system: The introduction of special symbols for the meaning of great numbers - five, ten and so on.
- Numbering of Maya Indians (slide 7, 8, 9 cm.) Presentation))
- Egyptian numbering (slide 9-14 cm.) Presentation))
Visnovok(learn slide 15 cm. ( Presentation)): short - great and zavzhdi zrozumіla record, foldability for an hour of rozrahunka.
Alphabetical-additive number system: For the recognition of numbers, the key alphabet and title are used.
- Old Greek numbering "Ioniyska" (slide 16 and 17 see. ( Presentation))
- Slovyansk English numbering (slide 18 and 19 see ( Presentation))
Zavdannya-2. (7 min.) We store at the table vikoristovuyuchi cards with Runes of words of the yanskian fiestial alphabet of your national day. Rechecking the hour of the meeting. It is necessary to give out the table “My Day of the Nation” (Addendum 1) and cards of the words of the yansky alphabet (Addendum 2). ).
- Slovian Cyrillic numbering (slide 20-22 cm. Presentation)). Porіvnyaєmo with the "Ionіyskaya" SS near Greece (slide 17 cm. ( Presentation))
Visnovok(uchnі): the same letters of the alphabet are taken as a basis.
- Rimska (Latin slide 23 cm. ( Presentation)) SS. Vikoristovuєtsya dosi.
Visnovok (readers): We looked at earlier number systems were not positional
Multiplicative number system:
- Vikoristannya ієroglіfіv did not allow to create a system of rahunka for other principles, which is the reason for the second step to the formation of numbers - positions. (Slide 24 cm. Presentation))
- Chinese numbering (slide 26-27 cm.) Presentation))
- Indian (Arabic) numbering (slide 28-29 cm.) Presentation))
Visnovok(readers slide 30 cm.) Presentation)):
The Chinese and Indian systems of numbers were positional.
Blitz examination of the class for the analysis of the knowledge of the material covered(3 hv).
- What is the number system? ( Way to write (image) numbers).
- How do you see SS You know, briefly describe them? ( Positional and not positional).
- What positions of the SS did we know earlier?
- What are the symbols for writing numbers? ( Arabic numerals, letters of the English alphabet.).
- (For respect): In any system, the number of babies is shown on the fingers, how many fates? Vidpovid: in the palm (finger) non-positional SS - the value of the number - the number of fingers - is easy to guess.
- Alone (palichna) SS. Old Egyptian dozen is not a position of the SS.
Cards are won for the correct scores of students.
III. Practical piece. (20 khvilin.)
Practical work to be done by a graphic editor. Two blanks are given for learning: Egyptian numbering, Chinese numbering (divine presentation).
Robot consists of two tasks:
- The non-positional number system is the Egyptian numbering.
- Positional number system - Chinese numbering
Manager. It is necessary to use the editing tools of the graphic editor (copying and pasting fragments) to select the date of your birth.
Teacher: “We will immediately defeat a practical robot.
Harvest to know
working style → folder “CLASS” → folder “6_a” → number_systems.jpg
Save with your dad's name: SS_date_of_nation.ipg
Manager:
- Instruct your date of birth (in Arabic numerals).
- Vykoristovuyuchi roztashovani symbols, right-handed choose the date of nationalization from the proponing systems of numbers.
- Choose the type of number system (positional or positional).
Visnovok (uchnіv): Chinese numeral system vicarious, ale handy, lower Egyptian SS, out of position
VI. Pіdbitya pіdbagіv. (2 min.) Exhibition of notes
Teacher: Due to the skin participant of our today's lesson. Only a sleepy robot made the qiu hoarsely more expensive in the past. Badges are awarded for the active participation of those who are correct in their work... Badges are awarded for good independent work from filling the tables …
V. Homework. (2 min.)
Home tasks are asked what is tasked for the decisions of the teacher.
An example of a homemaker.
A small reminder
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Indian local numbering In different regions of India, different numbering systems were used. One of them expanded in the whole world and at the given hour є greedily accepted. The numerals of the numerals looked small as the cob letters of the most recent numerals in the ancient Indian language - Sanskrit (the alphabet is "Devangari"). The numbers 1, 2, 3 ... 9, 10, 20, 30 ... 90, 100, 1000 were represented on the back of the hand with signs; Other numbers were recorded with their help. Years ago, a special sign was introduced (bold dot, circle) to indicate an empty number, signs for numbers greater than 9 were taken out of life, and the numbering of "devangari" turned into a ten-point system. How and if you have passed through the transition, do not know.
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The history of the development of the systems of numbers The sixties system of numbers Of particular interest is the so-called "Babylonian", or the sixties system of numbers, even more collapsible system, which was founded in Old Babylon. Thoughts of historians about how, as the system of numbers itself vindicated, diverge. Establish two hypotheses. The first to go out of that which became the anger of two tribes, one of them was scorched with a cog, and else - with a dozen. The sixty-decade system of numbers in this case could be blamed as a result of a kind of political compromise. The essence of another hypothesis is that the ancient Babylonians respected the trivality of fate, that the old 360 decibels, which is naturally associated with the number 60. For example: 1 year = 60 hvilin, 1 ° = 60 '. Zagalom sixty-decades' number system is cumbersome.
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The history of history and the development of number systems The Slovian number system Alphabetical number systems represent a particular group. Stink for writing down the numbers vicoristovavsya literal alphabet. The butt of the alphabetical system of numbers is words'yanska. In some words of the Yang peoples, in the numerical values of the letters, the letters were established in the order of the direct order of the letters of the words of the Yang alphabet, in others, in the Russian alphabet, the role of numbers was played in the total obsyaza of the letters, only t, but є in the Greek alphabet. Above the letter, which means a number, a special sign is placed - "title". The word system of numbers was preserved in liturgical books. The alphabetic system of numbers was expanded among ancient times, Georgians, Greeks (ionic system of numbers), Arabs, Jews and other peoples of the Near Skhod.
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The presentation on the topic "Number systems" can be downloaded free of charge on our website. Subject of the project: Informatics. Barvy slides and illustrations will help you to win over your classmates and audience. For a review, use the player instead, otherwise you want to add additional text - press the appropriate text under the player. Presentation 14 slides.
slide presentations
slide 1
number systems
Vikonala: student of 10-B class Ovchinnikova Anastasia Perevila: Fedorova E.A., teacher of computer science
slide 2
Positional Babylonian sixteenth system Double system Sixteenth system Decimal system
Non-positional Single (unary) system Roman system Ancient Egyptian tens system Alphabetical system
slide 3
Positional number system
The most thorough є positional number systems are systems for recording numbers, in which the contribution of the skin digit to the value of the number lies in the position of the sequence of digits, which represents the number.
Our zvichna tens system is positional.
slide 4
Sixteenth Babylonian system
The sixty-decade Babylonian system is the first number system, based on the positional principle. Numbers in the second number system were formed from the signs of two types: a straight wedge served for the designation of singles, a recumbent wedge - for the designation of dozens.
slide 5
Double system
Double numbering system is used to encode a discrete signal. In this number system, two signs are used to represent the number - 0 and 1.
slide 6
Sixteen system
Sixteenth numbering system is used to encode a discrete signal. In such a form, it is hoped that there will be a file. The signs that are drawn for the representation of the number are tens of digits from 0 to 9 and letters of the Latin alphabet - A, B, C, D, E, F.
Slide 7
Desyatkov system
Desyat's numbering system is used to encode a discrete signal. The signs that are vindicated for the representation of a number are numbers like 0 to 9.
Slide 8
Non-positional systems
Number systems, which have skin digits with a similar value, such as to lie in a її month in the record of the number, are called non-positional.
Positional number systems are the result of a trivial historical act of non-positional number systems.
Slide 9
alone system
Archaeologists have found "records" during excavations of cultural versions that lie until the Paleolithic period (10-11 thousand years BC). Vcheni called this method of writing numbers a single system of numbers.
Slide 10
Roman number system
The Roman system is fundamentally similar to the Egyptian one. For the new number, for the recognition of the upcoming numbers: 1, 5, 10, 50, 100, 500,1000, the great Latin letters are used: I, V, X, L, C, D, M, є “numbers” of the numerical system.
slide 11
Ancient Egyptian dozen non-positional system
In the ancient Egyptian system of numbers, which vinicla in the other half of the third millennium BC. special signs (numbers) were won for the numbers 1, 10, 102, 103, 104, 105, 106, 107.
At the basis of a single, and ancient Egyptian system, there is a simple principle of folding, behind which the value of the number is the sum of the values of the numbers that this record has.
slide 12
Alphabet systems
The most advanced non-positional number systems were alphabetic systems. Prior to such systems, numbers lay: slov'yanska; ionic (walnut); Finnish and other.
In the alphabetic word system, the number of "numbers" was 27 Cyrillic letters.
slide 13
The appearance of zero
The current tens of Vinyl number systems are approximately the 5th century A.D. in India. The vindication of the system number became possible after the largest input of the digit “0” for the value of the daily value. To denote the zero value of the order, Greek astronomers use the symbol “0” (the first letter of the Greek word Ouden is nothing). Tsey sign, maybe, і buv the prototype of our zero.
Zmіst Number systems of anatomical movement P'yateric number system P'yateric number system Slov'yan's number system Slov'yan's number system "Machine" number system "Machine" number system Vihid
The history of the development of number systems The fivefold number system According to the famous patriarch of Africa Stanley, a number of African tribes have expanded the fivefold number system. For a long time they were koristuvalis by the fivefold system of numbers in China. Obviously, there is a link between the number system and the everyday human hand. Withdrawal
Number systems of anatomical movement Decimal number system of Mova numbers, like and be-yak іnsha, maє svіy alphabet. In those mov numbers, which we call koristuemosya, the alphabet has ten digits from 0 to 9. Tse tens is a numerical system. The reason, through the yak of a dozen, the number system became zagalnopriynyatoy, zovsі not mathematical. Ten fingers of the hands are like an apparatus for a rahunka, like a person basking in prehistoric hours. The ancient image of ten digits is unviable: the skin number indicates the number for the number of kutіv u nіy. For example, 0 kutiv no, 1 one kut, 2 two kuti, etc. The writing of ten digits recognized the essence of the changes. The form, which we admire, was established in the 16th century. Historically, dozens of numbers have evolved and evolved in India. The Europeans posited the Indian theme of numbers in Arabs, calling it Arabic, and the historically incorrect name is the reduction of dosi. The blame for that development of the tenth system of numbers was one of the most important attainable human thoughts (instruction from the appearance of writing). Prote with a tenth system of numbers, people did not fawn. In different historical periods, a lot of people victorious other systems of numbers. Withdrawal
Indian pomisna numbering In different galleys of India, different numbering systems were used. One of them expanded in the whole world and at the given hour є greedily accepted. The numerals have little look of the letters of the first letters of the ancient numerals in old Indian Sanskrit (the alphabet is "Devangari"). The numbers 1, 2, 10, 20, 100, 1000 were represented on the hand with signs; Other numbers were recorded with their help. Years ago, a special sign was introduced (bold dot, circle) to indicate an empty number, signs for numbers greater than 9 were taken out of life, and the numbering of "devangari" turned into a ten-point system. How and if you have passed through the transition, do not know. History of vinification and development of number systems Vihіd
In the middle of the 8th century the positional numbering system is widely used in India. At about the same hour, it penetrates to other lands (Indo-China, China, Tibet, to the territory of our Central Asian republics, to Iran and other). The dominant role of the expanded Indian numeration in the Arab lands was played by ceramics, folded on the cob of the 9th century. Mohammed from Khorezm (ninth Khorezm region of Uzbekistan). It was translated into Western Europe in Latin language in the XII century. At the XIII century. Indian numbering takes precedence in Italy. In other lands of Western Europe, it will be solidified in the XVI century. The Europeans, yak, put the Indian numbering in the Arabs, called it Arabic. Tsya is historically incorrect name utrimuetsya y dosі. History of vinification and development of number systems Vihіd
The Twelve Number System Pokhodzhennya is also pov'yazane with a rachunk on his fingers. They pressed the thumb of the hand and the phalanx reshti chotiriokh fingers: total їх 12 (div. fig.). The elements of the twelve-digit system of numbers were preserved in England in the system of entries (1 foot = 12 inches) and in the penny system (1 shilling = 12 pence). Quite often mystique is in the presence of the twelve system of numbers; tea and table services for 12 persons, a set of bows for 12 pieces. Number systems of anatomical walking
History and development of number systems Twenty-digit number system The Aztecs and Mayan peoples, who inhabited the rich capital of the great regions of the American continent and created there the greatest culture, among them a mathematical one, adopted a two-digit number system. Also, the dvadtsaterichnaya system of numbers was adopted by the Celts, who inhabited Western Europe, starting from the 2nd millennium BC. The basis for the rahunka in this system of numbers was the fingers and the fingers. The deyakі following the dvadtsaterichny system of the number of the Celts were saved by the French penny system: the main penny unit, the franc, is divided into 20 (1 franc = 20 sous). Withdrawal
The history of the development of the systems of numbers The sixties system of numbers Of particular interest is the so-called "Babylonian", or the sixties system of numbers, even more collapsible system, which was founded in Old Babylon. Thoughts of historians about how, as the system of numbers itself vindicated, diverge. Establish two hypotheses. The first to come out of that which became the anger of two tribes, one of them was scorched with a gear, even a dozen. The sixty-decade system of numbers in this case could be blamed as a result of a kind of political compromise. The essence of another hypothesis is that the ancient Babylonians respected the trivality of fate, that the old 360 decibels, which is naturally associated with the number 60. For example: 1 year = 60 hvilin, 1 ° = 60. The number system is cumbersome. Withdrawal
History of Vincent and Development of Number Systems Roman Number System The record of numbers in the Roman system of numbers is shown in a little. The first 12 natural numbers in the Roman number system are written as follows: I, II, III, IV, V, VI, VII, VIII, IX, X, XI, XII. Apply the notation of numbers: XXVIII -28, MCMXXXV - The problem of solving arithmetic processes with these numbers is illustrated. Therefore, the Roman number system is victorious there, de tse handily in the literature (numbering of divisions), in issued documents (passport series, valuable papers and in), in decorative purposes - on the dial of the anniversary and in a row of other types. Try to freak out! How easy is it to take the result of arithmetic diy in the Roman number system? Withdrawal
The history of history and the development of number systems The Slovian number system Alphabetical number systems represent a particular group. Stink for writing down the numbers vicoristovavsya literal alphabet. The butt of the alphabetical system of numbers is words'yanska. In some words of the Yang peoples, in the numerical values of the letters, the letters were established in the order of the direct order of the letters of the words of the Yang alphabet, in others, in the Russian alphabet, the role of numbers was played in the total obsyaza of the letters, only t, but є in the Greek alphabet. Above the letter, which means a number, a special sign "title" is placed. The word system of numbers was preserved in liturgical books. The alphabetic system of numbers was expanded among ancient times, Georgians, Greeks (ionic system of numbers), Arabs, Jews and other peoples of the Near Skhod. Withdrawal
The history of the development of number systems "Machine" number systems Before mathematicians and designers in the 50s. The problem arose of identifying such systems, the number of which would be given to the helpers as distributors of the EOM, and the creators of software security. It appeared that the arithmetical rahunok, like the people of the most recent hours, can get better, at times even unsettled and completely effective. Fahіvtsі created the so-called “machine” group of numbers and developed the ways of converting the numbers of this group. Up to the "machine" group of systems, the following can be seen: - Dviykov; -visimkova; -sixteendatsyatkov. The official people of two arithmetic are explained by the names of G. V. Leibnitz, who, having published a statute in 1703, having looked at the rules of vikonnanny of arithmetic diy over two numbers. Withdrawal
The history of the development of the number systems "Machine" number system In the history of history, there is a curious twist from the octal number system. Swedish King Charles XII in 1717. choking on the octal system of numbers, respecting the її ruchnіshoy, lower than tens, and maw usir by royal order to send її like a zagalnopriynyat. Unstoppable death made the king create such an unimaginable world. Withdrawal
