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 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 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
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HISTORY of number systems
Numbers don't rule the world, but they show how the world is run. Johann Goethe
This is what the Pythagoreans said, emphasizing the extremely important role of numbers in practical activity. “Everything is a number” Every day a modern person remembers car and telephone numbers, calculates the cost of purchases in a store, maintains a family budget...
Numbers... they are with us everywhere and always. But in any case, the number was depicted using one or more symbols - numbers. People have always counted and written down numbers, even five thousand years ago. But they wrote them down completely differently, according to different rules.
Numbers are symbols that make up some alphabet. What is a number then? A number is a certain quantity consisting of numbers added according to certain rules. At different stages of human development, among different peoples, these rules were different, and today we call them number systems.
The number system is a signed system in which all numbers are written according to certain rules using symbols of a certain alphabet, called numbers. Non-positional Positional
So, let's look at various non-positional number systems. Non-positional number systems arose earlier than positional ones.
At first, people simply distinguished between ONE object in front of them or not. If there was more than one item, they said “MANY”
The first concepts of mathematics were “less”, “more”, “same”. >
It was enough to place a knife next to each fish for the exchange between the tribes to take place. If one tribe exchanged caught fish for stone knives made by people of another tribe, there was no need to count how many fish and how many knives they brought.
The account appeared when a person needed to inform his fellow tribesmen about the number of objects he found. And, since many peoples in ancient times did not communicate with each other, different peoples developed different number systems and representations of numbers and numbers.
Numerals in many languages indicate that primitive man's counting tools were primarily fingers. Fingers turned out to be an excellent computing machine.
However, peoples are known whose units of counting were not fingers, but their joints. Therefore, they could use their fingers and toes to count. In ancient times, people walked barefoot. There are still tribes in Polynesia that use the 20th number system.
For example, at the world's largest grain exchange in Chicago, offers and requests, as well as prices, are announced by brokers on their fingers without a single word. Finger counting has survived in some places to this day.
There was a need to write down numbers. It was difficult to remember large numbers, so various devices were added to the “counting machine” of the arms and legs. The number of objects was depicted by drawing dashes or serifs on any hard surface: stone, clay...
Single (“stick”) from the Paleolithic period 10-11 thousand years BC. or Archaeologists found such “records” during excavations of cultural layers related to Any number in it is formed by the repetition of one sign - one.
The more grain people collected from their fields, the more numerous their herds became, the larger numbers they needed. Unit notation for such numbers was cumbersome and inconvenient, so people began to look for more compact ways to represent large numbers.
2.5 thousand years BC Ancient Egyptian decimal = 2342
Number Symbol Designation 1 Like most people, the Egyptians used sticks to count a small number of objects. 10 The Egyptians tied cows with such fetters. 100 This is a measuring rope that was used to measure plots of land after the Nile flood. 1,000 Blooming lotus 10,000 "Be careful in large numbers!" - says the raised index finger. 100,000 Common frog tadpole 1,000,000 Number of pharaohs. Seeing such a number, an ordinary person will be very surprised and raise his hands to the sky. 10,000,000 The Egyptians worshiped Amon Ra, the sun god, and that is probably why they depicted their largest number as the rising sun
What ancient Egyptian number is written down? 5 3 8 6 4 2 1
People dealt with the operations of addition and subtraction long before numbers received names. When several groups of root gatherers or fishermen put their catch in one place, they performed the operation When people began to sow grain and saw that the harvest was several times greater than the number of seeds sown, then they became familiar with the operation When the meat of animals was harvested or the nuts were collected divided equally between all “mouths”, the operation was performed. And the subtraction operation? addition multiplication division
The Egyptians performed multiplication and division by successively doubling numbers. How did the Egyptians count?
Example. 19 * 31 31 62 124 248 496 and added the numbers in the marked lines on the right (31 + 62 + 496 = 589). Then they marked with vertical lines the lines of the left column, from which the factor could be added (19 = 1 + 2 + 16) 1 2 4 16 The Egyptians wrote down the corresponding power of two in the left column, and in the right column they wrote down the results of doubling the number 31.
Egyptian fractions always had one in the numerator (the exception was 2/3). Fractions were written as natural numbers, only a dot was placed over them. Exception: there were special signs for 1/2 and 2/3
Roman decimal I, V, X, L, C, D, M A number in the Roman numeral system is denoted by a set of consecutive “digits”. thousand years BC to this day
In the Roman system, the signs used to represent numbers are: I (one finger) for the number 1, V (open palm) for the number 5, X (two folded palms) for 10, and for other numbers the capital Latin letters of the corresponding Latin words are used 50 - L , 100 – С entum, 500 – D emimille, 1000 – M ille, which are “digits”.
444 400 40 4 Example. Write the number 444 in the Roman system. (D–C) (L–X) (V–I) CDXLIV
444 CDXLIV ATTENTION! All digits of a number in the decimal system are the same, but in the Roman system they are different.
1986 Example. Write the number 1986 in the Roman system. 9 00 80 6 10 00 MCMLXX X VI M (M – C) (V + I) (L + X + X + X)
Alphabetic number systems
The Greeks used several ways to write numbers. The Athenians used the first letters of numeral words to denote numbers: Greek (Ionian) For example, I, II, III, IIII - 1, 2, 3, 4 IIII – 10+10+10+4 = 34 G G five ten N one hundred X thousand M ten thousand
The great Greek mathematician Diophantus of Alexandria wrote fractions approximately as is customary now: the numerator is above the denominator, without a line. This was one of the ways to write fractions in Ancient Greece.
In ancient times, number systems reminiscent of the system of Ancient Egypt were widely used in Rus'. With their help, tax collectors filled out tax payment receipts (yasak) and made entries in the tax notebook. Star - one thousand rubles Wheel - one hundred rubles Square - ten rubles X - ruble | - a penny. Ancient Rus' 1232 rub. 24 kopecks
In the 9th century, by the monks brothers Cyril and Methodius, this form of recording numbers became widespread due to the fact that it was completely similar to the Greek recording of numbers. A new numbering was created along with the Slavic alphabetic system for the translation of the sacred biblical books.
We see that the entry is no longer than our decimal. This is because alphabetic systems used at least 27 "digits". Example. Let's write the number 444 in the Slavic system.
This form of recording numbers was official in the territory of modern Russia, Belarus, Ukraine, Bulgaria, Hungary, Serbia and Croatia until the reform of Peter I (until the end of the 17th century). But Orthodox church books still use this numbering.
1 2 3 4 5 6 7 8 9 10 - title “Az” “Lead” “Verb” “Good” “Is” “Zelo” “Earth” “Izhe” “Fita” “I”
Number Image Designation 1000 Thousand 10,000 Darkness 100,000 Legion 1,000,000 Leodr 10,000,000 Raven 100,000,000 Deck
True, the Slavs, like the Greeks, knew how to write down numbers greater than 1000. To do this, new designations were added to the alphabetic system. So, for example, the numbers 1000, 2000, 3000 were written in the same “digits” as 1, 2, 3..., only a special sign was placed in front of the “digit” at the bottom left. Alphabetic systems are only useful for writing numbers up to 1000. Are alphabetic systems convenient?
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.
A non-positional number system is a number system in which the quantitative equivalent (“weight”) of a digit does not depend on its location in the number record.
Disadvantages of the non-positional number system 1. There is a constant need to introduce new symbols for recording large numbers. 2. It is impossible to represent fractional and negative numbers. 3. It is difficult to perform arithmetic operations, since there are no algorithms for performing them.
Next, we will consider positional number systems. But we still use elements of the non-positional number system in everyday speech, in particular, we say one hundred, not ten tens, a thousand, a million, a billion, a trillion.
A positional number system is a number system in which the quantitative equivalent (“weight”) of a digit depends on its location in the number record. Consider two numbers 52 and 25. The numbers are the same - 5 and 2, but how do these numbers differ? Position digits in the number.
Any positional number system is characterized by its base. The base of a positional number system is the number of different digits used to represent numbers in a given number system. You can take any natural number as a base - two, three, four, ..., forming a new positional system: binary, ternary, quaternary and...
2 thousand years BC Babylonian sexagesimal - units - tens numbers: and - 60; 60 2 ; 60 3 ; ... ; 60 n 2nd digit 1st digit = 60 + 20 + 2 = 82 = 33
Traces of counting by six tens have survived to this day. A circle is divided by 360 0, that is, 6 * 60 degrees, a degree is divided into 60 minutes, and a minute is divided into 60 seconds. 1 0 360 0 0 Until now we divide an hour into 60 minutes and a minute into 60 seconds.
Arab scientist mathematician (from the city of Khorezm on the Amu Darya River). Muhammad ben Musa al-Khwarizm ≈ 850 AD he wrote a book about the general rules for solving arithmetic problems using equations. It was called "Kitab al-Jabr". This book gave its name to the science of algebra.
Indian scientists made one of the most important discoveries in mathematics - they invented the positional number system, which is now used by the whole world. Three hundred years later (in 1120) this book was translated into Latin, and it became the first textbook of “Indian” arithmetic for all European cities. Al-Khwarizmi described Indian arithmetic in detail in his book.
10 in the usual decimal number system (ten fingers on the hands). Alphabet: 1, 2, 3, 4, 5, 6, 7, 8, 9, 0. 60 was invented in Ancient Babylon: dividing an hour into 60 minutes, minutes into 60 seconds, and an angle into 360 degrees. 12 were spread by the Anglo-Saxons: there are 12 months in a year, two periods of 12 hours in a day, and 12 inches in a foot. 7 is used to count the days of the week Bases used today
1. What is a number system? 2. Give examples of positional and non-positional number systems. 3. A. S. Pushkin was born in the year MDCCXCIX? 4.What is the base of a number system? 5. The number system with what base was the very first? 6. In which country did special symbols for 100,1000,1000000 first begin to be used? 7. List the disadvantages of non-positional number systems. QUESTIONS TO REVIEW:
1. What numbers are written using Roman numerals: MC I X, L X V? 2. Write down the year of your birth: A) in the ancient Egyptian number system; B) in the Roman number system; B) in the ancient Slavic number system. Homework.
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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.
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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.
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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.
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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 Cyrillic letters were used as “numbers”.
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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.
100, 500 and 1000 - they were designated I, V, X, L, C, D and M, respectively. Of course, it is no coincidence that for many peoples the “key” numbers turned out to be 5 and 10: this is explained by the fact that counting was carried out on fingers (one or two hands). This is probably where the designations of the Roman numerals V (one “five”) and X (two “fives”) came from. The signs C, M and D are simply the first letters of the Latin words centum (one hundred), mille (thousand) and demimille (half a thousand), and L = 50 can be remembered as "half of C = 100", although its real origin seems to be other. Roman numerals can be seen on watch dials; they are often used to indicate centuries and number chapters in a book. Occasionally you can see a large Roman numeral on an old house - the year of construction. The numbers 2, 3, 4, 8, 9, 14, 19, 20, 40, 1989 in this Roman notation look like this: II, III, IV, VIII, IX, XIV, XIX, XX, XL, MCMLXXXIX. As you can see, not only the sum is used here, but also the difference of two “key” numbers - for this, the smaller one is placed in front of the larger one; thanks to this, instead of a long additive notation, say LXXXXIIII, the shorter XCIV is obtained. Interestingly, the most ancient texts used long, additive notation - the “rule of difference” appeared later.
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