Presentation on the topic "Number systems" in computer science in powerpoint format. The voluminous presentation for schoolchildren contains 41 slides, which discuss issues such as what positional and non-positional number systems are, an algorithm for converting numbers from one number system to another, and the representation of numbers in a computer. Author of the presentation: Ivanova Galina Anatolyevna.
Fragments from the presentation
Number systems
Notation– a set of rules for naming and representing numbers using a set of symbols called numbers.
Positional
The quantitative value of each digit of a number depends on the place (position or digit) in which this or that digit is written. 0.7 7 70
Non-positional
The quantitative value of a digit of a number does not depend on in what place (position or digit) this or that digit is written. XIX
Positional number systems
- The first positional number system was invented in Ancient Babylon, and the Babylonian numbering was sexagesimal, i.e. it used sixty digits!
- In the 19th century, the duodecimal number system became quite widespread.
- Currently, the most common number systems are decimal, binary, octal and hexadecimal.
Radix
- The number of different symbols used to represent a number in positional number systems is called the base of the number system.
- The positions of the digits are called digits.
- The base of the number system shows how many times the quantitative value of a digit changes when it is moved to an adjacent position
- Any natural number at least 2 can be taken as the base of the system.
Computers use the binary system because
- To implement it, technical devices with two stable states are needed,
- presentation of information using only two states is reliable and noise-resistant,
- it is possible to use the apparatus of Boolean algebra to perform logical transformations,
- binary arithmetic is much simpler than decimal arithmetic
The binary system, convenient for a computer, is inconvenient for a person due to its bulkiness and unusual notation. In order to understand the computer word, octal and hexadecimal number systems have been developed. Numbers in these systems require 3/4 times fewer digits than in the binary system.
Converting whole numbers from decimal number system
Translation algorithm:
- Consistently divide with the remainder the given number and the resulting integer quotients based on the new number system until the quotient becomes equal to zero.
- Express the resulting remainders in numbers from the alphabet of the new number system
- Write down the number in the new number system from the resulting remainders, starting with the last one.
Converting a correct decimal fraction from the decimal number system
Translation algorithm:
- Consistently multiply the decimal fraction and the resulting fractional parts of the products by the base of the new number system until the fractional part becomes zero or the required translation accuracy is achieved.
- The resulting whole parts of the works are expressed in numbers from the alphabet of the new number system.
- Write the fractional part of the number in the new number system starting from the integer part of the first product.
- Converting real numbers from the decimal number system
- When translating mixed fractions, the whole and fractional parts are translated separately according to their own rules, the translation results are separated by a comma.
Arithmetic operations in positional number systems
- The rules for performing basic arithmetic operations in any positional number system are subject to the same laws as in the decimal system.
- When adding, the digits are summed by digits, and if a digit overflow occurs, then they are transferred to the most significant digit. A digit overflow occurs when the value of the number in it becomes equal to or greater than the base of the number system.
- When subtracting a larger digit from a smaller digit, a unit is taken up in the most significant digit, which, when moving to the lowest digit, will be equal to the base of the number system
- If a digit overflow occurs when multiplying single-digit numbers, then a number that is a multiple of the base of the number system is transferred to the most significant digit. When multiplying multi-digit numbers in various positional systems, the column multiplication algorithm is used, but the results of multiplication and addition are written taking into account the base of the number system.
- Division in any positional system is carried out according to the same rules as division by angle in the decimal system, that is, it comes down to the operations of multiplication and subtraction.
Representing numbers in a computer
- Numbers in a computer can be stored in fixed point format (integers) and in floating point format (real numbers).
- Unsigned integers take up one or two bytes in memory.
- Signed integers occupy one, two, or four bytes in computer memory, with the leftmost (most significant) bit containing information about the sign of the number
- Three forms of recording (coding) of signed integers are used: direct code, reverse code and complementary code.
- Real numbers are stored and processed in a computer in floating point format. This format is based on scientific notation, in which any number can be represented.
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Slide 2
Nowadays, modern man constantly comes across numbers, figures... they are with us everywhere. And 2 thousand years ago, what did people know about numbers? And 5 thousand years ago? The question is not simple, but very interesting. Historians have proven that even 5 thousand years ago people could write down numbers and could perform arithmetic operations on them. But they wrote numbers according to different principles than we do now. The appearance of fractional numbers was associated with the need to make measurements. But since the unit of measurement did not always fit an integer number of times in the measured value, a practical need arose to introduce “smaller” numbers than natural ones. When presenting the material, by number we will understand its value, and not its symbolic notation. Today, humanity mainly uses the decimal number system to record numbers.
Slide 3
The position of the sign in the image of the number does not depend on the value it represents. The value denoted by a digit in a number notation depends on its position.
Slide 4
In non-positional number systems, the position of the digit in the notation of the number does not depend on the value it represents. An example is the Roman system. In the Roman system, Latin letters are used as numbers: I V X L C M D 1 5 10 50 100 500 1000 The number 32 in the Roman number system has the form: XXXII = (X+X+X)+(I+I)= 30+2 The number 444, which has in decimal notation there are 3 identical digits, in the Roman number system it will be written as: CDXLIV=(D-C)+(L-X)+(V-I)= 400+40+4. The number 1974 in the Roman numeral system looks like MCMLXXIV= M+(M-C)+L+(X+X)+(V-I)=1000+900+50+20+4.
Slide 5
He was an Italian mathematician. Thanks to his book Liber Abaci, Europe learned the Indo-Arabic number system, which later replaced the Roman numerals.
Slide 6
A positional number system is called traditional if its basis is formed by the terms of a geometric progression, and the values of the digits are non-negative integers. The basis is a sequence of numbers, each of which specifies the weight of the corresponding digit. The denominator P of a geometric progression, the terms of which form the basis of the traditional number system, is called the base of this number system. Traditional number systems with base P are otherwise called P-ary.
Slide 7
A number system or numbering is a way of writing numbers. The symbols with which numbers are written are called digits, and their combination is called the alphabet of the number system. The number of digits that make up an alphabet is called its dimension. A number system is called positional if the quantitative equivalent of a digit depends on its position in the notation of the number. In the decimal system we are familiar with, the value of a number is formed as follows: the value of the digits is multiplied by the “weight” of the corresponding digits and all the resulting values are added up. For example, 5047=5*1000+0*100+4*10+7*1. This method of forming the value of a number is called additive-multiplicative.
Slide 8
Where A is the number itself, q is the base of the number system, a is the digits of the given number system, n is the number of digits of the integer part of the number, m is the number of digits of the fractional part of the number. Example: units tens hundreds thousands
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315 24 75 72 3 8 32 7 8 4 315 16 9 16 155 144 11 (B) 16 3 16 1 15 2 2 2 14 1 7 6 1 3 2 1 1 Binary Octal Hexadecimal 39 1
Slide 12
3750 5000 0000 0 1 x 2 0 1875 7500 1 0 x 2 x 2 x 2 0 1875 0000 x 16 3 0 1875 0000 1 x 8 x 8 4 5000 Binary Octal Hexadecimal
Slide 13
1 1 0 1 1 1 0 1 1 0 1 1 0 0 1 0 0 0 + 1 0 0 1 0 0 0 1 0 1 1 0 1 1 1 0 1 1 _ 1 1 0 0 1 1 0 0 0 1 * 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 + 1 1 0 1 0 1 0 0 1
Slide 14
She was 1100 years old. She went to class 101. She carried 100 books in her briefcase. This is all true, not nonsense. When there are ten feet of dust. She walked along the road, A puppy with only one tail, but a hundred-legged one, always ran after her, She caught every sound with Her ten ears, And 10 tanned hands held the briefcase and the leash. And 10 dark blue eyes looked around the world as usual. But everything will become completely ordinary, When you understand our story. ANSWER
Slide 15
She was 12 years old. She went to 5th grade. She carried 4 books in her briefcase. This is all true, not nonsense. When there are ten feet of dust. She walked along the road, A puppy with one tail, but a hundred-legged one, always ran after her, She caught every sound with Her ten ears, And 2 tanned hands held the briefcase and the leash. And 2 dark blue eyes looked around the world as usual. But everything will become completely ordinary, When you understand our story.
Slide 16
OBJECTIVES: To familiarize students with one of the sections of the school computer science course, the history of development and classification of various number systems, with the algorithm for converting from the decimal number system to others (binary, octal, hexadecimal). Information products used: Microsoft Power Point - for creating and demonstrating presentations; Microsoft Word - for typing; Paint - for creating graphic objects; Adobe Photoshop - for editing graphic objects; System requirements: The presentation can be performed on a computer of any class that contains Win98/ME/2000/XP Microsoft Power Point program of any version. There are NO special restrictions. Project content: Main topics: History of the number system Non-positional number systems Positional number systems Binary arithmetic Algorithm for converting numbers from one number system to another
Slide 17
LITERATURE: Computer science and information technology. Textbook for 10-11 grades. N.D. Ugrinovich - Moscow - publishing house "BINOM. Knowledge Laboratory", 2005. Number systems and computer arithmetic. Tutorial. E. V Andreeva. Moscow - publishing house "BINOM. Knowledge Laboratory", 2004. Computer science. Structured summary of a basic computer science course. I.G. Semakin. Moscow - publishing house "BINOM. Knowledge Laboratory", 2001. Problem book - workshop. I.G. Semakin. Moscow - publishing house "BINOM. Knowledge Laboratory", 2001. Mathematical foundations of computer science. Elective course: Textbook. E. V Andreeva. Moscow - publishing house "BINOM. Knowledge Laboratory", 2005.
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A little history The account appeared when a person needed to inform his relatives about the number of objects he discovered, animals killed and defeated enemies. In different places, different ways of transmitting numerical information were invented: from notches according to the number of objects to ingenious signs - numbers.
Slide no. 3
“number” of ancient people Initially, the concept of an abstract number was absent; the number was “tied” to those specific objects that were being counted. The abstract concept of a natural number appeared along with the development of writing.
Slide no. 4
Number systems A number system is a set of rules for designating and naming numbers. Number systems are divided into positional and non-positional. The signs used to write numbers are called digits.
Slide no. 5
Positional number systems The most advanced are positional number systems, i.e. systems for writing numbers in which the contribution of each digit to the value of the number depends on its position (position) in the sequence of digits representing the number. For example, our familiar decimal system is positional. In the number 34, the number 3 indicates the number of tens, and the number 4 indicates the number of ones. The number of digits used is called the base of the positional number system. Advantages of positional number systems Ease of performing arithmetic operations. A limited number of characters (digits) for writing any numbers. .
Slide no. 6
Non-positional number systems Unit system The number of objects, for example sheep, was depicted by drawing lines or notches on any hard surface: stone, clay, wood. Scientists called this method of writing numbers the unit (“stick”) number system. In it, only one type of sign was used to record numbers - “stick”. Each number in such a number system was designated using a line made up of sticks, the number of which was equal to the designated number. I I I I I I I I I I I I I I I I I I I I I I I I The inconveniences of such a system for writing numbers and the limitations of its application are obvious: the larger the number you need to write, the longer the string of sticks. And when writing down a large number, it’s easy to make a mistake by adding an extra number of sticks or, conversely, not writing them down.
Slide no. 7
The Roman system The Roman system is familiar to us from first grade. It uses capital Latin letters I, V, X, L, C, D and M to denote the numbers 1, 5, 10, 50, 100, 500 and 1000, respectively, which are the digits of this number system. A number in the Roman numeral system is designated by a set of consecutive digits. The value of a number is equal to: the sum of the values of several identical digits in a row (let’s call them the group of the first type); the difference between the values of two digits if the smaller digit is to the left of the larger digit. In this case, the value of the smaller digit is subtracted from the value of the larger digit (let's call them a group of the second type) Example 1. The number 32 in the Roman number system has the form XXXII=(X+X+X)+(I+I)=30+2 (two groups of the first type). Example 2. The number 444, which has 3 identical digits in its decimal notation, will be written in the Roman number system as CDXLIV=(D-C)+(L-X)+(V-I)=400+40+4 (three groups of the second type).
Slide no. 8
Ancient Egyptian decimal system The ancient Egyptian numeral system, which arose in the second half of the third millennium BC, used special numerals to represent the numbers 1, 10, 100, 1000, etc. Numbers in the Egyptian numeral system were written as combinations of these digits, in which each of them was repeated no more than nine times. Example. The ancient Egyptians wrote down the number 345 as follows: Both the stick and ancient Egyptian number systems were based on the simple principle of addition, according to which the value of a number is equal to the sum of the values of the digits involved in its recording. Scientists classify the ancient Egyptian number system as non-positional decimal.
Slide no. 9
The ancient Egyptians used tens hundreds of thousands tens of thousands hundreds of thousands millions
Slide no. 10
Babylonian sexagesimal system Numbers in the Babylonian number system were composed of two types of signs: a straight wedge served to designate units; a lying wedge - to designate tens. To determine the value of a number, it was necessary to divide the image of the number into digits from right to left. A new discharge began with the appearance of a straight wedge after a recumbent one, if we consider the number from right to left. For example: The number 32 was written like this:
Slide no. 13
Slavic number system This number system is alphabetic i.e. Letters of the alphabet are used instead of numbers. This number system was used by our ancestors and was quite complex, because uses 27 letters as numbers.
Slide no. 14
Mathematicians argue with historians Considering that in the Slavic number system large numbers had the following names: darkness 10,000 crows 10^ 48 legion 100,000 deck 10^50 leodr 1,000,000 let’s solve the problem of the number of Batu’s troops during the campaign against Rus'. According to the chronicles, the Mongols were in “darkness.” That is, 10,000 10,000 = 100,000,000 people. In fact, Batu had 11 temnik military leaders subordinate to him, each of whom had “darkness” of soldiers subordinate to him, a total of 11 10 000 = 110 000, a total of 110 thousand people. Therefore, there was no trace of the 100,000,000 people that historians talk about!
Slide no. 15
Disadvantages of non-positional number systems 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. Until the end of the Middle Ages, there was no universal system for recording numbers. Only with the development of mathematics, physics, technology, trade and economics did the need for a single universal number system arise.
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Presentation - Number systems
Text of this presentation
Topic "Number systems"
Introduction
Modern man constantly encounters numbers and figures in everyday life - they are with us everywhere. Various number systems are used whenever there is a need for numerical calculations, from pencil-on-paper calculations by elementary school students to calculations performed on supercomputers.
A number system is a certain way of representing numbers and the corresponding rules for operating on them. The purpose of creating a number system is to develop the most convenient way to record quantitative information.
History of number systems
Number systems
Positional
Non-positional
Ancient number systems:
Unit system Ancient Greek numbering Slavic numbering Roman numbering
Positional and non-positional number systems
Non-positional systems Positional systems
The position of the digit in the notation of the number does not determine the value it represents. The value denoted by a digit in a number notation depends on its position. The base is the number of digits used. Position is the location of each digit.
Writing a number in the positional number system
Any integer in the positional system can be written in polynomial form: Xs=An Sn-1 + An-1 Sn-2 + An-2 Sn-3 +...+ A2 S1 + A1 S0 where S - the base of the number system, A – the digits of the number written in this number system, n – the number of digits of the number. So, for example, the number 629310 will be written in polynomial form as follows: 629310 = 6 103 + 2 102 + 9 101 + 3 100
Examples of positional number systems:
Binary number system with base 2, uses two symbols - 0 and 1.
Octal number system with base 8, numbers from 0 to 7 are used.
The base 10 decimal system is the most common number system in the world.
Duodecimal System with base 12. The numbers used are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B.
Hexadecimal Base 16, uses the numbers 0 to 9 and the Latin letters A to F to represent the numbers 10 to 15.
The sexagesimal system, with base 60, is used in the measurement of angles and, in particular, longitude and latitude.
History of the binary number system
The binary number system was invented by mathematicians and philosophers even before the advent of computers (XVII - XIX centuries). The promoter of the binary system was the famous G.V. Leibniz. He noted the particular simplicity of the algorithms for arithmetic operations in binary arithmetic in comparison with other systems and gave it a certain philosophical meaning. In 1936 - 1938, American engineer and mathematician Claude Shannon found remarkable applications of the binary system in the design of electronic circuits.
Binary number system
The binary number system (binary number system, binary) is a positional number system with base 2. The inconvenience of this number system is the need to convert the source data from the decimal system to the binary system when entering them into the machine and the reverse conversion from binary to decimal when outputting the calculation results. The main advantage of the binary system is the simplicity of the addition, subtraction, multiplication and division algorithms.
Addition, subtraction, multiplication and division in the binary number system
Addition Subtraction Multiplication Division
0 + 0 = 0; 0 + 1 = 1; 1 + 0 = 1; 1 + 1 = 10. 0 - 0 = 0; 1 - 0 = 1; 1 - 1 = 0; 10 - 1 = 1. 0 1 = 0; 1 1 = 1. 0 / 1 = 0; 1 / 1 = 1.
Binary coding in a computer
At the end of the twentieth century, the century of computerization, humanity uses the binary system every day, since all information processed by modern computers is stored in them in binary form. In modern computers we can enter text information, numerical values, as well as graphic and audio information. The amount of information stored in a computer is measured by its “length” (or “volume”), which is expressed in bits (from English binary digit).
Converting numbers from one number system to another
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Conclusion
The highest achievement of ancient arithmetic is the discovery of the positional principle of representing numbers. We need to recognize the importance of not only the most common system that we use every day. But each one separately. After all, different areas use different number systems, with their own characteristics and characteristics.
Decimal Binary Octal Hexadecimal
1 001 1 1
2 010 2 2
3 011 3 3
4 100 4 4
5 101 5 5
6 110 6 6
7 111 7 7
8 1000 10 8
9 1001 11 9
10 1010 12 A
11 1011 13 B
12 1100 14 C
13 1101 15 D
14 1110 16 E
15 1111 17 F
16 10000 20 10
Converting binary number to decimal
To convert a binary number to a decimal one, it is necessary to write it in the form of a polynomial, consisting of the products of the digits of the number and the corresponding power of the number 2, and calculate it according to the rules of decimal arithmetic: X10 = An 2n-1 + An-1 2n-2 + An-2 ·2n-3 +…+A2·21 + A1·20
Translation of numbers
Converting an octal number to decimal
To convert an octal number to a decimal one, it is necessary to write it in the form of a polynomial, consisting of the products of the digits of the number and the corresponding power of the number 8, and calculate it according to the rules of decimal arithmetic: X10 = An 8n-1 + An-1 8n-2 + An-2 8n-3 +…+A2 81 + A1 80
Translation of numbers
Convert hexadecimal number to decimal
To convert a hexadecimal number to a decimal one, it is necessary to write it in the form of a polynomial, consisting of the products of the digits of the number and the corresponding power of the number 16, and calculate it according to the rules of decimal arithmetic: X10 = An 16n-1 + An-1 16n-2 + An-2 ·16n-3 +…+A2·161 + A1·160
Translation of numbers
Converting a decimal number to binary
To convert a decimal number to the binary system, it must be sequentially divided by 2 until a remainder less than or equal to 1 remains. A number in the binary system is written as a sequence of the last division result and the remainders from the division in reverse order. Example: Convert the number 2210 to the binary number system: 2210=101102
Translation of numbers
Converting a decimal number to octal
To convert a decimal number to the octal system, it must be successively divided by 8 until a remainder less than or equal to 7 remains. A number in the octal system is written as a sequence of digits of the last division result and the remainders of the division in reverse order. Example: Convert the number 57110 to the octal number system: 57110=10738
Translation of numbers
Converting a decimal number to hexadecimal
To convert a decimal number to the hexadecimal system, it must be successively divided by 16 until a remainder less than or equal to 15 remains. A number in the hexadecimal system is written as a sequence of digits of the last division result and the remainders of the division in reverse order. Example: Convert the number 746710 to hexadecimal number system: 746710=1D2B16
Translation of numbers
Converting numbers from binary to octal
To convert a number from binary to octal, it must be broken down into triads (triples of digits), starting with the least significant digit, adding zeros to the leading triad if necessary, and replacing each triad with the corresponding octal digit. When translating, you must use the binary-octal table: Example: Convert the number 10010112 to the octal number system: 001 001 0112 = 1138
8th 0 1 2 3 4 5 6 7
Translation of numbers
Converting from binary to hexadecimal
To convert a number from binary to hexadecimal, it must be divided into tetrads (four digits). Binary hexadecimal table: Example: Convert the number 10111000112 to hexadecimal number system: 0010 1110 00112=2E316
16th 0 1 2 3 4 5 6 7
16th 8 9 A B C D E F
Translation of numbers
Converting an octal number to binary
To convert an octal number to binary, you need to replace each digit with its equivalent binary triad. Example: Convert the number 5318 to the binary number system: 5318=101 011 0012
2nd 000 001 010 011 100 101 110 111
8th 0 1 2 3 4 5 6 7
Translation of numbers
Convert hexadecimal number to binary
To convert a hexadecimal number to binary, you need to replace each digit with its equivalent binary tetrad. Example: Convert the number EE816 to the binary number system: EE816=1110111010002
2nd 0000 0001 0010 0011 0100 0101 0110 0111
16th 0 1 2 3 4 5 6 7
2nd 1000 1001 1010 1011 1100 1101 1110 1111
16th 8 9 A B C D E F
Translation of numbers
Converting from octal to hexadecimal and vice versa
When moving from the octal number system to the hexadecimal number system and vice versa, an intermediate conversion of numbers to the binary system is necessary. Example 1: Convert the number FEA16 to the octal number system: FEA16=1111111010102=111 111 101 0102=77528 Example 2: Convert the number 66358 to the hexadecimal number system: 66358=1101100111012=1101 1001 11012=D9 D16
Translation of numbers
Unit system
In ancient times, when there was a need to record numbers, the number of objects was depicted by drawing dashes or serifs on some hard surface. Archaeologists have found such “records” during excavations of cultural layers dating back to the Paleolithic period (10–11 thousand years BC). In such a system, only one type of sign was used - a stick. Each number was designated using a line made up of sticks, the number of which was equal to the designated number.
Ancient number systems
Ancient Greek numbering
Attic numbering
Ionian system
In the third century BC. Attic numbering was supplanted by the Ionian system.
In ancient times, Attic numbering was common in Greece.
Ancient number systems
Slavic numbering
In Russia, Slavic numbering was preserved until the end of the 17th century. The southern and eastern Slavic peoples used alphabetical numbering to record numbers. Slavic numbering was preserved only in liturgical books. A special icon was placed above the letter indicating the number: (“title”). To indicate thousands, a special sign was placed in front of the number (bottom left).
Z
Ancient number systems
Roman numbering
The ancient Romans used numbering, which remains to this day under the name “Roman numbering.” We use it to designate centuries, anniversaries, names of congresses and conferences, to number chapters of a book or stanzas of a poem.
I - 1 V - 5 X - 10 L - 50 C - 100 D - 500 M - 1000
Writing numbers in Roman numeration:
Ancient number systems
Ionian system
Notation of numbers in the Ionian numbering system
Designation of numbers in the ancient Slavic numbering system
Slavic numbering
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“Everything is a number,” the sages said, emphasizing the extremely important role of numbers in people’s lives. There are many ways to represent numbers. In any case, a number is represented by a symbol or group of symbols (word) of some alphabet. Such symbols are called numbers. A number system is a set of techniques and rules for designating and naming numbers. People learned to count a long time ago, back in the Stone Age. At first they simply distinguished whether there was one object in front of them or more. After some time, a word appeared to denote two objects... and... at the moment there are more than 30 different number systems. Some of them have already lost their original meaning and possibility of use in the modern world. But, despite this, they form a significant part of the history of the emergence of number systems.
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Number system alphabet. The alphabet forms the basis of the number system. The characters of the alphabet are called numbers. Number systems differ in their alphabet and the rules for forming other numbers from the base digits. Any number system intended for practical use must provide: the ability to represent any number in the range of values under consideration, uniqueness of representation (each combination of symbols must correspond to one and only one value), ease of handling numbers.
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Main types of number systems. Necessary definitions. A positional number system is a number system in which the weight of a digit changes with the position of the digit in the number, but is completely determined by the spelling of the digit and the place it occupies. In particular, this means that the weight of a digit does not depend on the values of the surrounding digits. A non-positional number system is a number system in which the weight of a digit does not depend on its position. A universal number system is a number system that allows you to write any real number (in a finite or infinite sequence of digits). A non-universal number system is a number system that allows you to write only relatively small numbers, sometimes only integers (or vice versa, only smaller units). The basic number system is a positional number system in which the weight of each digit changes the same number of times when it is transferred from any digit to its adjacent one. A minor number system is a positional number system in which the ratio of the weights of adjacent digits can change. A binary number system is a non-primary positional number system in which a number is actually represented in a larger-radix number system, but instead of using a corresponding set of digits, their representation in sets of signs in a lower-base number system is used.
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Main types of number systems. Necessary definitions. (continued...) The traditional number system is a number system in which the notation of a number consists of two parts - an integer and a fraction. The number of digits before the comma (dot) separating these parts is not known in advance and can be as large as desired. In fact, writing a number forms two sequences of numbers, running to the left and to the right of the decimal point. Information number system is a number system in which the recording of a number (unlike the traditional one) consists of a single sequence of digits. In this case, each successive digit (bit) specifies the value of the number (its position on the axis). Let the first few digits indicate that the number t we are interested in is contained in some subset U of the number axis, which, in turn, is divided into several disjoint subsets V1, ..., Vk. Then the choice of one of the k possible values of the next digit indicates one of these subsets. Interval number system is an information number system in which all subsets of the number axis, defined by the first few digits of any number, are intervals. A non-interval number system is an information number system in which among the subsets of the number axis, defined by the first few digits of a number, not all are intervals. Iterative number system is an interval number system in which the roots of successive iterations of a certain monotonic function are chosen as partition points (interval boundaries). The tower number system is an iterative number system in which each successive bit in the number record has the meaning of the sign of the logarithm of the absolute value of the mantissa obtained at the previous step.
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Non-positional (non-universal) number systems. A non-positional number system is a number system in which the weight of a digit does not depend on its position. Unit number system; Ancient Egyptian decimal number system; Roman number system; Slavic number systems (glagolic and Cyrillic).
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The simplest, but absolutely inconvenient number system. Based on a single digit - one (stick). Allows you to write only natural numbers. To represent a number in this number system, you need to write down as many sticks as the number itself. It was used by uncivilized tribes, whose counting needs, as a rule, did not go beyond the first ten. Purely formally, the unit number system can be classified as basic (with base 1). But, unlike other basic number systems, it can only be considered positional with a very strong stretch, and it is not universal at all (zero, fractions and negative numbers cannot be represented in it). 1 I 2 II 3 III 4 IIII 5 IIIIII, etc. Unit number system.
Slide no. 8
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Around the third millennium BC, the ancient Egyptians came up with their own numerical system, in which the key numbers were 1, 10, 100, etc. special icons were used - hieroglyphs. All other numbers were composed from these key numbers using the operation of addition. The number system of Ancient Egypt is decimal, but non-positional and additive. 1. Like most people, the Egyptians used sticks to count small numbers of objects. If several sticks need to be depicted, then they were depicted in two rows, and the bottom row should have the same number of sticks as the top, or one more. 10. The Egyptians tied cows with such fetters. If you need to depict several dozen, then the hieroglyph was repeated the required number of times. The same applies to other hieroglyphs. 100. This is a measuring rope that was used to measure plots of land after the Nile flood. 1,000. Have you ever seen a blooming lotus? If not, then you will never understand why the Egyptians assigned such significance to the image of this flower. 10,000. “Be careful in large numbers!” - says the raised index finger. 100,000. This is a tadpole. Common frog tadpole. 1,000,000. Seeing such a number, an ordinary person will be very surprised and raise his hands to the sky. This is what this hieroglyph 10,000,000 represents. The Egyptians worshiped Amon Ra, the sun god, and that is probably why they depicted their largest number as the rising sun. The Ancient Egyptian decimal number system.
Slide no. 9 col-6 col-last"> Slide description:
Roman number system. Using seven digits - I=1, V=5, X=10, L=50, C=100, D=500, M=1000 - you can very successfully and quite expressively represent natural numbers in the range of up to several thousand. Continues to be used to a limited extent to indicate ordinal numbers (hours, centuries, numbers of congresses or conferences, etc.). Numbers in this system, just like ours, were written from left to right, from largest to smallest. For example, XI = 11, XII = 12, XIII = 13, but the next number is already special, since such a number “XIII” is not convenient to write, the Romans came up with abbreviations, they began to write it like this: XIV = 14, i.e. 10+5-1 = 14. That is if a digit with a smaller value was written before a digit with a larger value, then it was subtracted. The number 9 = IX was also written. And besides this, it was impossible to write four identical numbers in a row, for example, “XXXX” = XL (50-10) = 40.
Slide no. 10
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Slavic Glagolitic number system. This system was created to designate numbers in the sacred books of the Western Slavs. It was used infrequently, but for quite a long time. In terms of organization, it exactly repeats the Greek numbering. It was used from the 8th to the 13th centuries. Numbers were written from digits the same way from left to right, from large to small digits. If there were no tens, units, or some other digit, then it was skipped. This notation of a number is additive, that is, it uses only addition: In order not to confuse letters and numbers, titles were used - horizontal lines above the numbers, or dots.
Slide no. 11
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Slavic Cyrillic number system. This numbering was created together with the Slavic alphabetic system to translate the sacred biblical books for the Slavs by the Greek monks brothers Cyril and Methodius in the 9th century. This form of writing numbers became widespread due to the fact that it was completely similar to the Greek notation of numbers. Until the 17th century, this form of recording numbers was official in the territory of modern Russia, Belarus, Ukraine, Bulgaria, Hungary, Serbia and Croatia. Until now, Orthodox church books use this numbering. Numbers were written from digits in the same way from left to right, from large to small. Numbers from 11 to 19 were written in two digits, with the unit coming before the ten: We read literally “fourteen” - “four and ten.” As we hear, we write: not 10+4, but 4+10, - four and ten. Numbers from 21 and above were written in reverse, with the full tens sign written first. The number notation used by the Slavs is additive, that is, it only uses addition. In order not to confuse letters and numbers, titles were used - horizontal lines above the numbers, which we see in the figure. Slavic numbering existed until the end of the 17th century, until the positional decimal number system came to Russia from Europe with the reforms of Peter I. To indicate numbers greater than 900, special icons were used that were added to the letter. This is how the numbers were formed:
Slide no. 12
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Number systems in modern high technologies. Positional number systems. A positional number system is a number system in which the weight of a digit changes with the position of the digit in the number, but is completely determined by the spelling of the digit and the place it occupies. In particular, this means that the weight of a digit does not depend on the values of the surrounding digits. Non-universal number systems (Formats for representing numbers in microcalculators and computers) Recording in fixed point format; Normalized (engineering, scientific) form of writing numbers; Byte number system. "Machine" number systems. Binary number system; Octal number system; Hexadecimal number system.
Slide no. 13
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Record in fixed point format. Recording in fixed point format was used in the first electronic computers (in particular, in the Soviet Ural-1). It allows you to represent numbers whose absolute value does not exceed one, and, moreover, only those of them that have a given fixed number of binary or binary decimal digits.
Slide no. 14
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Normalized (engineering, scientific) form of writing numbers. The normalized (engineering, scientific) form of writing numbers is now used by most microcalculators, computers and other computing devices. The notation of a number consists of two parts - the mantissa and the exponent, each of which has its own sign and a strictly defined number of decimal (binary or other) digits. The range for the mantissa is determined by one of two rules. Most often, it is less than one, but greater than one of the next least significant digit of the corresponding number system (usually decimal or binary). The opposite rule: the mantissa is greater than one, but less than one of the next highest digit (only one of these two rules can be in effect at the same time, but not both at once).
Slide no. 15
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Byte number system. The contents of a file, in a certain sense, do not depend on its type and purpose. In terms of internal structure, a file is a finite sequence of bytes. Each byte is 8 bits, which in the binary number system can be read as an integer from 0 to 255. Each such number (code) can be considered as a digit in the base 256 number system. Since the file is a single sequence of bytes (and in Unlike the traditional recording of a number, it is not divided into integer and fractional parts), then there are two options for reading the file as a number. First, you can treat the file as an integer. Secondly, you can, on the contrary, treat the integer part as zero (as in fixed-point notation). Along with the byte, larger derived units are used to measure the amount of information: 1 KB = 2 bytes = 1024 bytes (kilobytes) 1 MB = 2 bytes = 1024 KB (megabytes) 1 GB = 2 bytes = 1024 MB (gigabytes) 1 TB = 2 byte = 1024 GB (terabyte) 10 20 30 40
Slide no. 16
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"Machine" number systems. Binary number system. At the end of the 20th century, the century of computerization, Humanity uses the binary system every day, since all information processed by modern computers is stored in them in binary form. How is this storage carried out? Each register of a computer's arithmetic device, each memory cell is a physical system consisting of a certain number of homogeneous elements. Each such element is capable of being in several states and serves to represent one of the digits of a number. That is why each cell element is called a digit. The numbering of digits in a cell is usually carried out from right to left, the leftmost digit has a serial number 0. If, when writing numbers in a computer, we want to use the usual decimal number system, then we should get 10 stable states for each digit, as in an abacus using dominoes. Such machines exist. However, the design of the elements of such a machine is extremely complex. The most reliable and cheapest is a device, each digit of which can take two states: magnetized - not magnetized, high voltage - low voltage, etc. In modern electronics, the development of computer hardware is moving precisely in this direction. Consequently, the use of the binary number system as an internal system for presenting information is caused by the design features of the elements of computers. Advantages of the binary number system: Simplicity of the operations performed. The ability to automatically process information, realizing only two states of computer elements. Disadvantage of the binary number system: Rapid growth in the number of digits in the notation representing the binary number
Slide no. 17
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Octal number system. Octal number system. Uses eight digits—0, 1, 2, 3, 4, 5, 6, and 7—as well as the symbols “+” and “–” to indicate the sign of a number and a comma (period) to separate the integer and fractional parts of the number. Widely used in programming in the 1950s-70s. To date, it has been almost completely replaced by the hexadecimal number system, however, the functions of converting a number from the decimal system to octal and vice versa are preserved in microcalculators and many programming languages. Also used to write number codes and machine instructions.
Slide no. 18
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Hexadecimal number system. Hexadecimal number system. Uses sixteen digits - 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 in their usual sense, followed by A=10, B=11, C=12, D=13, E=14. F=15. Also uses the symbols “+” and “–” to denote the sign of a number and a comma (period) to separate the integer and fractional parts of a number. Implemented by the American corporation IBM. Widely used in programming for IBM-compatible computers. On the other hand, some languages have preserved traces of the use of this number system in the past. For example, in Romance languages (Spanish, French, etc.), numerals from 11 to 16 are formed according to one rule, and from 17 to 19 - according to another. And in the Russian language a pud is known, equal to 16 kilograms. The hexadecimal system is also used to record command addresses.
Slide no. 19
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More complex positional number systems. Date Time; Babylonian cuneiform (decimal/sexagesimal) number system; Mayan base numeral system or long counting; Ancient Chinese decimal number system;
Slide no. 20
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Date Time. The traditional way of representing moments and large periods of time combines the use of several different units of measurement. When moving from millennia to centuries, from them to decades, and then to years, the weight of the digit in the date record changes 10 times. A year consists of 12 months, a month - of 4 weeks, a week - of 7 days. A day is made up of 24 hours, an hour is made up of 60 minutes, and a minute is made up of 60 seconds. Smaller time intervals are most often measured in tenths, hundredths, thousandths, etc. fractions of a second (although it is also known that the sexagesimal division of the second and its subsequent fractions are used). Thus, we are dealing here with a number system that combines six different bases at once: 4, 7, 10, 12, 24 and 60.
Slide no. 21
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Babylonian cuneiform (decimal/sexagesimal) number system. In ancient Babylon around the 2nd millennium BC there was such a number system - numbers less than 60 were indicated using two signs: for one and for ten. They had a wedge-shaped appearance, since the Babylonians wrote on clay tablets with triangular sticks. These signs were repeated the required number of times, for example -3 -20 -32 Numbers greater than 60 were written in digits, with small spaces between them: - this is how the number 302 is written, that is, 5*60+2. - and this is 1*60*60+2*60+5 = 3725. But the representation of some numbers in this system will be the same, for example, the number 302 may also be equal to 5*60*60 + 2 = 18002. Since there is no icon to indicate zero. Only in the 5th century BC was a special sign introduced - an inclined wedge to indicate missing digits, which played the role of a zero. is a recording of the number 7203 (2*60*60+3). However, the absence of a lower rank was not indicated, and therefore the number 180 = 3*60 was written like this, and this entry could mean 3, 180, 10800 (3*60*60), etc. It is believed that the decimal system was the Sumerians, and after they were conquered by the Semites, their system was adapted to the sexagesimal system of the Semites. The sexagesimal notation of integers was not widely used outside the Assyro-Babylonian kingdom, but sexagesimal fractions are still used in measuring time. For example, one minute = 60 seconds, one hour = 60 minutes.
Slide no. 22
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Mayan decimal number system or "long count". This system is very interesting because its development was not influenced by any of the civilizations of Europe and Asia. This system was used for the calendar and astronomical observations. Its characteristic feature was the presence of a zero (an image of a shell). The base of this system was the number 20, although traces of the fivefold system are strongly visible. The first 19 numbers were obtained by combining dots (one) and dashes (five). The number 20 was depicted with two digits, zero and one at the top, and was called uinalu. The numbers were written down in a column, with the smallest digits at the bottom and the largest at the top, resulting in a “bookcase” with shelves. If the number zero appeared without a unit at the top, this meant that there were no units for this digit. But, if at least one unit was in this digit, then the zero sign disappeared, for example, the number 21, this will be. Also in our number system: 10 – with zero, 11 – without it. Here are some example numbers:
Slide no. 23
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Mayan decimal number system or "long count". (continued...) There is an exception in the ancient Mayan 20-count system: if you add only one first-order unit to the number 359, this exception immediately comes into force. Its essence boils down to the following: 360 is an initial number of the third order and its place is no longer on the second, but on the third shelf. But then it turns out that the initial number of the third order is not twenty times greater than the initial number of the second (20x20 = 400, not 360!), but only eighteen! This means that the principle of twenty-fold has been violated! That's right. This is the exception. The fact is that among the Mayan Indians, 20 kin days formed a month or uinal. 18 months-uinals formed a year or tuna (360 days a year) and so on: K"in = 1 day. Vinal = 20 k"in = 20 days. Tun = 18 Vinal = 360 days = about 1 year. K"atun = 20 tun = 7200 days = about 20 years. Bak"tun = 20 k"atun = 144,000 days = about 400 years. Pictun = 20 bak"tun = 2,880,000 days = about 8,000 years. Kalabtun = 20 pictuns = 57,600,000 days = about 160,000 years. K"inchiltun = 20 kalabtun = 1152000000 days = about 3200000 years. Alavtun = 20 k"inchiltun = 23040000000 days = about 64000000 years. This is a rather complex number system, mainly used by priests for astronomical observations; another Mayan system was additive, similar to the Egyptian one, and was used in everyday life.
Slide no. 24
Slide description:
Ancient Chinese decimal number system. This system is one of the oldest and most progressive, since it contains the same principles as the modern “Arab” one that we use. This system arose about 4,000 thousand years ago in China. O 1 2 3 4 5 6 7 8 9 0 Numbers in this system, just like ours, were written from left to right, from large to small. If there were no tens, units, or some other digit, then at first they did not put anything and moved on to the next digit. (During the Ming Dynasty, a sign for an empty digit was introduced - a circle - an analogue of our zero). In order not to confuse the digits, several service hieroglyphs were used, written after the main hieroglyph, and showing what value the hieroglyph-digit takes in a given digit. 10 100 1 000 10 000 - 1*1 000 = 1000; - 5*100 + 4*10 +8 = 548 This is a multiplicative notation because it uses multiplication. It is decimal, it has a zero sign, and besides this it is positional. Those. it almost corresponds to the “Arabic” number system.
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