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Table of contents Numeral systems of anatomical origin Pentary number system Decimal number system Indian place numbering Duodemal number system Dudecimal number system Hexadecimal number system Alphabetic number systems Roman number system Slavic number system “Machine” number systems
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History of the emergence and development of number systems Five-fold number system According to the testimony of the famous African explorer Stanley, a number of African tribes had a five-fold number system. For a long time they used the five-digit number system in China. The connection between this number system and the structure of the human hand is obvious.
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Number systems of anatomical origin Decimal number system The language of numbers, like any other, has its own alphabet. In the language of numbers that we usually use, the alphabet is ten digits - from 0 to 9. This is the decimal number system. The reason why the decimal number system became generally accepted is not at all mathematical. Ten fingers are a counting apparatus that man has used since prehistoric times. The ancient image of decimal digits is not accidental: each digit represents a number by the number of angles in it. For example, 0 - no corners, 1 - one corner, 2 - two corners, etc. The writing of decimal numbers has undergone significant changes. The form we use was established in the 16th century. Historically, the decimal number system emerged and developed in India. The Europeans borrowed the Indian number theme from the Arabs, calling it Arabic, a historically incorrect name that continues to this day. The emergence and development of the decimal number system was one of the most important achievements of human thought (along with the advent of writing). However, people did not always use the decimal number system. In different historical periods, many peoples used other number systems.
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Indian Place Numbering Various numbering systems existed in different regions of India. One of them has spread throughout 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 (Devangari 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 out of use, and the “devangari” numbering turned into a decimal place system. How and when this transition took place is still unknown. History of the emergence and development of number systems
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By the middle of the 8th century. The positional numbering system is widely used in India. Around this time, it penetrates into other countries (Indochina, China, Tibet, the territory of our Central Asian republics, Iran, etc.). A manual compiled at the beginning of the 9th century played a decisive role in the spread of Indian numbering in Arab countries. Muhammad from Khorezm (now Khorezm region of Uzbekistan). It was translated into Latin in Western Europe in the 12th century. In the 13th century Indian numbering takes precedence in Italy. In other countries of Western Europe it was established in the 16th century. The Europeans, who borrowed Indian numbering from the Arabs, called it Arabic. This historical misnomer continues to this day. History of the emergence and development of number systems
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Number systems of anatomical origin The duodecimal number system The duodecimal number system was quite widespread. The origin is also connected with counting on fingers. The thumb and phalanges of the other four fingers were counted: there are 12 in total (see figure). Elements of the duodecimal number system were preserved in England in the system of measures (1 foot = 12 inches) and in the monetary system (1 shilling = 12 pence). Often in everyday life we come across the duodecimal number system; tea and table sets for 12 people, a set of handkerchiefs - 12 pieces.
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History of the emergence and development of number systems The 20-number system The Aztecs and Mayans, the peoples who inhabited vast areas of the American continent for many centuries and created the highest culture there, including mathematics, adopted the 20-number system. Also, the 20-digit number system was adopted by the Celts, who inhabited Western Europe starting from the 2nd millennium BC. The basis for counting in this number system was the fingers and toes. Some traces of the Celtic base-20 number system survive in the French monetary system: the basic unit of currency, the franc, is divided by 20 (1 franc = 20 sous).
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History of the emergence and development of number systems Sexagesimal number system Of particular interest is the so-called “Babylonian” or sexagesimal number system, a very complex system that existed in Ancient Babylon. Historians have differing opinions about exactly how this number system came into being. There are two hypotheses. The first is based on the fact that there was a merger of two tribes, one of which used the six-digit system, the other - the decimal system. The sexagesimal number system in this case could have arisen as a result of a kind of political compromise. The essence of the second hypothesis is that the ancient Babylonians considered the length of the year to be 360 days, which is naturally associated with the number 60. Echoes of the use of this number system have survived to this day. For example: 1 hour = 60 minutes, 1° = 60‘. In general, the sexagesimal number system is cumbersome.
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History of the emergence and development of number systems Roman number system This number system appeared in Ancient Rome. The recording of numbers in the Roman numeral system is shown in the figure. 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. Examples of writing numbers: XXVIII -28, MCMXXXV – 1935. The difficulty of performing arithmetic operations with these numbers is illustrated. For this reason, the Roman numeral system is currently used where it is convenient in literature (chapter numbering), in documents (passport series, securities, etc.), for decorative purposes - on a watch dial and in a number of other cases. Try to count! Is it easy to get the result of arithmetic operations in the Roman number system?
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History of the emergence and development of number systems Slavic number systems Alphabetic number systems represent a special group. They used the alphabetic alphabet to write numbers. An example of an alphabetic number system is Slavic. Among some Slavic peoples, the numerical values of letters were established in the order of the letters of the Slavic alphabet, while among others, in particular among the Russians, not all letters played the role of numbers, but only those that are in the Greek alphabet. A special sign was placed above the letter denoting the number - “titlo”. The Slavic number system was preserved in liturgical books. The alphabetic number system was common among the ancient Armenians, Georgians, Greeks (Ionic number system), Arabs, Jews and other peoples of the Middle East.
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History of the emergence and development of number systems “Machine” number systems Before mathematicians and designers in the 50s. The problem arose of finding such number systems that would meet the requirements of both computer developers and software creators. It turned out that arithmetic calculation, which humanity has used since ancient times, can be improved, sometimes quite unexpectedly and surprisingly effectively. Experts have developed the so-called “machine” group of number systems and developed methods for converting numbers from this group. The “machine” group of number systems includes: binary; octal; hexadecimal. 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. History of the emergence and development of number systems Table of contents Numeral systems of anatomical origin Pentary number system Pentary number system Decimal number system Decimal number system Indian place numbering Indian place numbering Duodenum number system Duodenum number system Duodenum number system Codec number system Hexadecimal number system Hexadecimal number system Alphabetic number systems Roman system numerals Roman numeral system Slavic number system Slavic number system "Machine" number systems "Machine" number systems Exit
History of the emergence and development of number systems Five-fold number system According to the testimony of the famous African explorer Stanley, a number of African tribes had a five-fold number system. For a long time they used the five-digit number system in China. The connection between this number system and the structure of the human hand is obvious. Exit
Number systems of anatomical origin Decimal number system The language of numbers, like any other, has its own alphabet. In the language of numbers that we usually use, the alphabet is ten digits from 0 to 9. This is the decimal number system. The reason why the decimal number system became generally accepted is not at all mathematical. Ten fingers are the counting apparatus that man has used since prehistoric times. The ancient image of decimal digits is not accidental: each digit represents a number by the number of angles in it. For example, 0 there are no corners, 1 one corner, 2 two corners, etc. The writing of decimal numbers has undergone significant changes. The form we use was established in the 16th century. Historically, the decimal number system emerged and developed in India. The Europeans borrowed the Indian number theme from the Arabs, calling it Arabic, a historically incorrect name that continues to this day. The emergence and development of the decimal number system was one of the most important achievements of human thought (along with the advent of writing). However, people did not always use the decimal number system. In different historical periods, many peoples used other number systems. Exit
Indian Place Numbering Various numbering systems existed in different regions of India. One of them has spread throughout 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 (Devangari alphabet). Initially, these signs represented the numbers 1, 2, 10, 20, 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 out of use, and the “devangari” numbering turned into a decimal place system. How and when this transition took place is still unknown. History of the emergence and development of number systems Exit
By the middle of the 8th century. The positional numbering system is widely used in India. Around this time, it penetrates into other countries (Indochina, China, Tibet, the territory of our Central Asian republics, Iran, etc.). A manual compiled at the beginning of the 9th century played a decisive role in the spread of Indian numbering in Arab countries. Muhammad from Khorezm (now Khorezm region of Uzbekistan). It was translated into Latin in Western Europe in the 12th century. In the 13th century Indian numbering takes precedence in Italy. In other countries of Western Europe it was established in the 16th century. The Europeans, who borrowed Indian numbering from the Arabs, called it Arabic. This historical misnomer continues to this day. History of the emergence and development of number systems Exit
The duodecimal number system The duodecimal number system was quite widespread. The origin is also connected with counting on fingers. The thumb and phalanges of the other four fingers were counted: there are 12 in total (see figure). Elements of the duodecimal number system were preserved in England in the system of measures (1 foot = 12 inches) and in the monetary system (1 shilling = 12 pence). Often in everyday life we come across the duodecimal number system; tea and table sets for 12 persons, a set of handkerchiefs 12 pieces. Number systems of anatomical origin Output
History of the emergence and development of number systems The base-20 number system The Aztec and Mayan peoples, who inhabited vast areas of the American continent for many centuries and created the highest culture there, including mathematics, adopted the base-20 number system. Also, the 20-digit number system was adopted by the Celts, who inhabited Western Europe starting from the 2nd millennium BC. The basis for counting in this number system was the fingers and toes. Some traces of the Celtic base-20 number system survive in the French monetary system: the basic unit of currency, the franc, is divided by 20 (1 franc = 20 sous). Exit
History of the emergence and development of number systems Sexagesimal number system Of particular interest is the so-called “Babylonian” or sexagesimal number system, a very complex system that existed in Ancient Babylon. Historians have differing opinions about exactly how this number system came into being. There are two hypotheses. The first is based on the fact that there was a merger of two tribes, one of which used the sixfold system, the other the decimal one. The sexagesimal number system in this case could have arisen as a result of a kind of political compromise. The essence of the second hypothesis is that the ancient Babylonians considered the length of the year to be 360 days, which is naturally associated with the number 60. Echoes of the use of this number system have survived to this day. For example: 1 hour = 60 minutes, 1° = 60. In general, the sexagesimal number system is cumbersome. Exit
History of the emergence and development of number systems Roman number system This number system appeared in Ancient Rome. The recording of numbers in the Roman numeral system is shown in the figure. 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. Examples of writing numbers: XXVIII -28, MCMXXXV – The difficulty of performing arithmetic operations with these numbers is illustrated. For this reason, the Roman numeral system is currently used where it is convenient in literature (chapter numbering), in documents (passport series, securities, etc.), for decorative purposes - on a watch dial and in a number of other cases. Try to count! Is it easy to get the result of arithmetic operations in the Roman number system? Exit
History of the emergence and development of number systems Slavic number systems Alphabetic number systems represent a special group. They used the alphabetic alphabet to write numbers. An example of an alphabetic number system is Slavic. Among some Slavic peoples, the numerical values of letters were established in the order of the letters of the Slavic alphabet, while among others, in particular among the Russians, not all letters played the role of numbers, but only those that are in the Greek alphabet. A special “titlo” sign was placed above the letter indicating the number. The Slavic number system was preserved in liturgical books. The alphabetic number system was common among the ancient Armenians, Georgians, Greeks (Ionic number system), Arabs, Jews and other peoples of the Middle East. Exit
History of the emergence and development of number systems “Machine” number systems Before mathematicians and designers in the 50s. The problem arose of finding such number systems that would meet the requirements of both computer developers and software creators. It turned out that arithmetic calculation, which humanity has used since ancient times, can be improved, sometimes quite unexpectedly and surprisingly effectively. Experts have developed the so-called “machine” group of number systems and developed methods for converting numbers from this group. The “machine” group of number systems includes: – binary; –octal; – hexadecimal. 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. Exit
History of the emergence and development of number systems “Machine” number systems A curious case with the octal number system is known from history. In 1717, the Swedish king Charles XII was fond of the octal number system, considered it more convenient than the decimal number system, and intended to introduce it as generally accepted by royal order. Unexpected death prevented the king from carrying out such an unusual intention. Exit
IT-teacher
MKOU "Kaltukskaya Secondary School"
First Evgenia Ivanovna
addition
storage
CPU
vector
broadcast
History of the development of number systems. Non-positional and positional number systems.
The account appeared when a person needed to inform his relatives about the number of objects he discovered.
At first, people simply distinguished one object in front of them or not. If there was more than one item, they said “many.”
The simplest counting instrument was the fingers of man.
One of these counting systems subsequently became commonly used - decimal.
In ancient times, people walked barefoot. Therefore, they could use their fingers and toes to count. Thus, they could seemingly only count to twenty.
But with the help of this "barefoot machine" people could achieve much larger numbers,
1 person is 20,
2 people is two times 20, etc.
It was difficult to remember large numbers, so mechanical devices were added to the “counting machine” of the arms and legs.
Many methods of counting were invented: In different places, different ways of transmitting numerical information were invented:
For example, the Peruvians used multi-colored cords with knots tied on them to remember numbers.
Pebbles, grains, shells, etc. were used to remember numbers.
Archaeologists have found such “records” during excavations of cultural layers dating back to the Paleolithic period (10 - 11 thousand years BC)
This way of writing numbers is called
single
(“stick”, “unary”)
number system
Any number in it is formed
repetition of one sign - one.
According to cadet training courses
5th course 4th course 3rd course 2nd course 1st course
Echoes of the unit number system are still found today. So, to find out what course a military school cadet is studying in, you need to count how many stripes are sewn on his sleeve. Without realizing it, kids use the unit number system, showing their age on their fingers, and counting sticks are used to teach 1st grade students how to count.
Notation is a sign system in which certain rules for recording numbers are adopted. The signs with which numbers are written are called in numbers, and their totality – number system alphabet.
Number systems
Positional
Non-positional
Non-positional number systems: Non-positional s.s. is a number system in which the value of a digit does not depend on its position in the number record. Egyptian numbering10000 100000 1000000 10000000
Originated 5000 years ago
Non-positional number systems: Ancient Greek numbering Roman number system The Roman number system has reached us. We still use it to designate chapters, centuries:- VI = 6, i.e. 5 + 1,
- LX = 60, i.e. 50 + 10,
- IV = 4, i.e. 5 – 1,
- XL = 40, i.e. 50 – 10.
The numbers are written from left to right in descending order. Their meanings fold up. If there is a smaller number on the left and a larger one on the right, then their meanings are are deducted
Task 1. Convert numbers from the Roman number system to the decimal number system:
LXXVI=50+10+10+5+1=76
XLIX=(50-10)+(10-1)=49
Task 2. Write decimal numbers in the Roman numeral system:
463=500-100+50+10+5-2=CDLXIIV
Non-positional number systems have a number of significant disadvantages:- There is a constant need to introduce new symbols for recording large numbers.
- It is impossible to represent fractional and negative numbers.
- It is difficult to perform arithmetic operations because there are no algorithms for performing them.
Positional s.s. is a number system in which the value of a digit depends on its position in the number record.
For example By changing the position of the number 2 in the decimal number system, you can write decimal numbers of different sizes: 2; 20; 200; 2000, etc.
Radix– the number (p) of different symbols used to represent a number in the positional number system. The base of a system is equal to the number of digits in its alphabet.
The main advantages of any positional number system:- limited number of characters for writing numbers;
- ease of performing arithmetic operations. For example: The Arabic decimal system uses digits to write numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 . There are 10 such numbers in total, i.e. 10 is the base of the Arabic number system. That's why it is called the decimal number system.
- octal;
- hexadecimal Name number system corresponds to the number of digits used when writing a number in a given number system, that is number system base (p)
Name the base of each number system
Number system alphabet is a set of symbols used to represent numbers in a given number system Number system alphabet is a set of symbols used to represent numbers in a given number system. The alphabet of number systems consists of numbers from 0 to p-1, where p is the base of the number system. Based on this, fill out the 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 each number system
Any real number can be written in any positional number system as a sum of positive and negative
powers of the number p (radix of the number system)
Expanded form of the number
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
Primary comprehension and consolidation of what has been learned
1. What are number systems?
2. Non-positional number systems are...
3. Positional number systems are...
4. What is the base of a number system?
5. What does the expanded form of a number mean?
Write the numbers in expanded form
- 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
Homework:
- Notebook entries.
- Task card.
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The presentation on the topic “Number systems” can be downloaded absolutely free on our website. Project subject: Computer science. Colorful slides and illustrations will help you engage your classmates or audience. To view the content, use the player, or if you want to download the report, click on the corresponding text under the player. The presentation contains 14 slide(s).
Presentation slides
Slide 1
Number systems
Completed by: 10-B grade student Anastasia Ovchinnikova Checked by: E.A. Fedorova, computer science teacher
Slide 2
Positional Babylonian sexagesimal system Binary system Hexadecimal system Decimal system
Non-positional Unit (unary) system Roman system Ancient Egyptian decimal system Alphabetic systems
Slide 3
Positional number system
The most advanced are positional number systems - systems for writing numbers in which the contribution of each digit to the value of the number depends on its position in the sequence of digits representing the number.
Our familiar decimal system is positional.
Slide 4
Babylonian sexagesimal system
The Babylonian sexagesimal system is the first known number system based on the positional principle. Numbers in this number system were composed of two types of signs: a straight wedge served to designate units, a recumbent wedge - to designate tens.
Slide 5
Binary system
The binary number system is used to encode a discrete signal. In this number system, two signs are used to represent numbers - 0 and 1.
Slide 6
Hexadecimal system
Hexadecimal number system is used to encode a discrete signal. The contents of any file are represented in this form. The characters used to represent the number are decimal digits from 0 to 9 and letters of the Latin alphabet - A, B, C, D, E, F.
Slide 7
Decimal system
The decimal number system is used to encode a discrete signal. The symbols used to represent a number are numbers from 0 to 9.
Slide 8
Non-positional systems
Number systems in which each digit corresponds to a value that does not depend on its place in the number are called non-positional.
Positional number systems are the result of a long historical development of non-positional number systems.
Slide 9
Unit system
Archaeologists have found “records” during excavations of cultural layers dating back to the Paleolithic period (10–11 thousand years BC). Scientists called this method of writing numbers the unit number system.
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Roman number system
The Roman system is fundamentally not much different from the Egyptian one. It uses capital Latin letters to denote the following numbers: 1, 5, 10, 50, 100, 500, 1000: I, V, X, L, C, D, M, which are the “digits” of this number system.
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Ancient Egyptian decimal non-positional system
In the ancient Egyptian number system, which arose in the second half of the third millennium BC. special signs (numbers) were used to indicate the numbers 1, 10, 102, 103, 104, 105, 106, 107.
Both the unit and ancient Egyptian 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.
Slide 12
Alphabetic systems
Alphabetic systems were more advanced non-positional number systems. Such number systems included: Slavic; Ionic (Greek); Phoenician and others.
In the alphabetic Slavic number system, 27 letters of the Cyrillic alphabet were used as “numbers”.
Slide 13
The appearance of zero
The modern decimal number system arose around the 5th century AD. in India. The emergence of this system became possible after the great discovery of the number “0” to indicate a missing quantity. To indicate the zero value of the digit, Greek astronomers began to use the symbol “0” (the first letter of the Greek word Ouden - nothing). This sign, apparently, was the prototype of our zero.
