What is a set of image rules called? A number system is a set of numbers and rules for designating numbers
A number is a quantitative characteristic of something. At first, numbers were indicated by dashes. But this is inconvenient: try to write two hundred and fifty-five lines accurately on unlined paper. That's it! Fortunately, India came up with a decimal number system that allows you to write any natural number using just ten digits!
Some signs and symbols to indicate something 0 1 2 3 4 5 6 7 8 9 - + × ∙ * : / ∕ ÷ = ≈ ≠ 🙂 🙁 ☀️ 🌥️ 🌧️ 🍎 🍒 🍓 Some mathematical symbols 0 1 2 3 4 5 6 7 8 9 - + × ∙ * : / ∕ ÷ = ≈ ≠ Arabic numerals (10 in total) to represent numbers 0 1 2 3 4 5 6 7 8 9What does a number consist of?
Single-digit numbers consist of only one digit 0 1 2 3 4 5 6 7 8 9 Double-digit numbers consist of only two digits 10 11 12 13 14 15 16 … 97 98 99 Three-digit numbers consist of only three digits 100 101 102 103 104 105 106 … 997 998 999 Four-digit numbers consist of only four digits 1000 1001 1002 1003 1004 1005 1006 … 9997 9998 9999 …To write the number 255 (two hundred fifty-five) you need only two digits: “2” and “5”. The number "5" is used twice. The first right digit in the number indicates the number of units (five lines), the second - the number of tens (five times ten lines), the third - the number of hundreds (two times one hundred lines), the fourth - the number of thousands, etc.
255 (Two hundred fifty five)
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Numbers are made up of more than just digits. Also, for example, the minus or comma symbols are used to separate the fractional part.
Reading and pronouncing whole numbers and decimals
Two hundred fifty-five point one| 2 | 5 | 5 | , | 0 | 1 | |||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| … | Billions | Hundreds of millions | Tens of millions | Millions | Hundreds of thousands | Tens of thousands | Thousands | Hundreds | Dozens | Units | Tenths | Hundredths | thousandths | Ten thousandths | hundred thousandths | Millions | … |
After twenty, numbers have a compound name.
| 2 | 5 | 6 | ( | Two hundred | fifty | six | ) | |
| 2 | 0 | 0 | ( | Two hundred | ) | |||
| 5 | 0 | ( | Fifty | ) | ||||
| 6 | ( | Six | ) |
| 1 | one | 11 | eleven | 10 | ten | 100 | one hundred |
| 2 | two | 12 | twelve | 20 | twenty | 200 | two hundred |
| 3 | three | 13 | thirteen | 30 | thirty | 300 | three hundred |
| 4 | four | 14 | fourteen | 40 | fourty | 400 | four hundred |
| 5 | five | 15 | fifteen | 50 | fifty | 500 | five hundred |
| 6 | six | 16 | sixteen | 60 | sixty | 600 | six hundred |
| 7 | seven | 17 | seventeen | 70 | seventy | 700 | seven hundred |
| 8 | eight | 18 | eighteen | 80 | eighty | 800 | eight hundred |
| 9 | nine | 19 | nineteen | 90 | ninety | 900 | nine hundred |
The number is pronounced in three digits with the corresponding class. Very large numbers can be voiced.
256 (Two hundred fifty-six) 256,000 (Two hundred fifty-six thousand) 256 256 (Two hundred fifty six thousand two hundred fifty six) 2 256 256 (Two million two hundred fifty six thousand two hundred fifty six)
Pronounced in decimals
- number to decimal point
- the word “whole” or “whole” (meaning “whole unit”),
- number after the decimal point,
- digit of the rightmost digit (meaning “part of one”).
In infinite periodic decimal fractions it is pronounced
- number to decimal point
- the word “whole” or “whole”,
- number after the decimal point before the period,
- digit of the rightmost digit before the period,
- the word "and"
- period number,
- the word "in period"
Classical writing of numbers in Roman numerals
=Before Arabic numerals, Roman numerals were used. In order not to lose count when writing lines, first every fifth, and then every tenth line was highlighted. Over time, the entry “| | | | V | | | | X | | | | V | | | | X | | | | V |» decreased to "XXVI".
| I | V | X | L | C | D | M |
| 1 | 5 | 10 | 50 | 100 | 500 | 1000 |
Roman numerals that have a higher value are numbered to the left of those with a lower value. Their values add up (VI = 5 + 1 = 6). The numbers “V”, “L”, “D” are not repeated.
Exceptions: since the 19th century, the combinations “IV”, “IX”, “XL”, “XC”, “CD”, “CM”. To avoid repetition of one digit four times (incorrect: “IIII”), in them the digit with a larger value is placed to the right of the digit with a smaller value and the smaller one is subtracted from the larger value (IV = 5 - 1 = 4).
| I | one | X | ten | C | one hundred | M | one thousand |
| II | two | XX | twenty | CC | two hundred | MM | two thousand |
| III | three | XXX | thirty | CCC | three hundred | MMM | three thousand |
| IV | four | XL | fourty | CD | four hundred | ||
| V | five | L | fifty | D | five hundred | ||
| VI | six | LX | sixty | DC | six hundred | ||
| VII | seven | LXX | seventy | DCC | seven hundred | ||
| VIII | eight | LXXX | eighty | DCCC | eight hundred | ||
| IX | nine | XC | ninety | C.M. | nine hundred |
| CC | L | VI | ( | Two hundred | fifty | six | ) | |
| CC | ( | Two hundred | ) | |||||
| L | ( | Fifty | ) | |||||
| VI | ( | Six | ) |
What are numbers (school curriculum)
Natural numbers are positive integer numbers that arose when counting objects 1 2 3 … 98 99 100 … Prime numbers are natural numbers that are divisible without a remainder by only two natural numbers: 1 and itself (one is not a prime number) 2 (2/2 = 1 2/1 = 2) 3 5 … 83 89 97 … Composite numbers are natural numbers that are divisible without remainder into three or more natural numbers (one is not a composite number) 4 (4/4 = 1 4/2 = 2 4/1 = 4) 6 8 … 98 99 100 … Round numbers are natural numbers that end in 0 10 20 30 … 100 … Integers are natural numbers, zero and numbers opposite to natural numbers (negative) … -100 -99 -98 … -2 -1 0 1 2 … 98 99 100 … Even numbers are integers that are divisible by the number 2 without a remainder … -100 -98 -96 … -4 -2 0 2 4 … 96 98 100 … Odd numbers are integers , which are not divisible by the number 2 without a remainder ... -99 -97 -95 ... -3 -1 1 3 ... 95 97 99 ... Real numbers are rational and irrational numbers ... -100.5 ... -5,(6) ... - 3 ... -2, where the numerator m is an integer, and the denominator n is a natural number ... -100.5 ... -5,(6) ... -3 ... -2 or ±m/n, where n ≠ 0 ... -| 201 |
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Doctor of Philology Natalia Chernikova
The concept of number originated in ancient times, when man learned to count objects: two trees, seven bulls, five fish. At first they counted on their fingers. In colloquial speech, we still sometimes hear: “Give me five!”, that is, give me your hand. And before they said: “Give me a hand!” Pastern- this is a hand, and there are five fingers on the hand. Once upon a time, the word five had a specific meaning - five fingers of the metacarpus, that is, the hand.
Later, instead of fingers, they began to use notches on sticks for counting. And when writing arose, letters began to be used to represent numbers. For example, among the Slavs the letter A meant the number “one” (B had no numerical value), B - two, G - three, D - four, E - five.
Gradually, people began to be aware of numbers, regardless of objects and persons who could be counted: simply the number “two” or the number “seven”. In this regard, the Slavs had the word number. In the meaning of “count, magnitude, quantity” it began to be used in Russian from the 11th century. Our ancestors used the word number and to indicate the date, year. Since the 13th century, it also began to mean tribute, tax.
In the old days, in book Russian, along with the word number circulated noun number, as well as adjective clean. In the 16th century the verb appeared count- "count".
In the second half of the 15th century, special signs denoting numbers became widespread in European countries: 1, 2, 3, 4, 5, 6, 7, 8, 9, 0. They were invented by the Indians, and they came to Europe thanks to the Arabs, which is why got the name Arabic numerals.
In our country, Arabic numerals appeared in the Peter the Great era. At the same time, the word entered the Russian language number. Arabic in origin, it also came to us from European languages. The Arabs have the original meaning of the word number- this is zero, empty space. It is in this meaning that the noun number entered into many European languages, including Russian. From the middle of the 18th century the word number acquired a new meaning - the sign of a number.
A set of numbers in Russian was called digit(in the old spelling tsyfir). Children learning to count said: learning numbers, I'm writing numbers. (Remember the teacher by last name Tsyfirkin from Denis Ivanovich Fonvizin’s comedy “The Minor,” who taught the careless Mitrofanushka digits, that is, arithmetic.) Under Peter I, Russia opened digital schools- primary state general educational institutions for boys. In addition to other disciplines, children were taught digital science- arithmetic, mathematics.
So the words number And number differ in meaning and origin. Number- a unit of counting that expresses the quantity ( one house, two houses, three houses etc.). Number- a sign (symbol) indicating the value of a number. To record numbers, we use Arabic numerals - 1, 2, 3... 9, 0, and in some cases Roman numerals - I, II, III, IV, V, etc.
These days words number And number are also used in other meanings. For example, when we ask “What date is today?”, we mean the day of the month. Combinations " including», « from the number someone", " among someone" denote a composition, a collection of people or objects. And if we prove something with numbers in hands, then we must use numerical indicators. In a word number also called a sum of money ( income figure, fee figure).
In colloquial speech the words number And number often replace each other. For example, we call a number not only a quantity, but also a sign that expresses it. Numerically very large quantities are spoken of astronomical numbers or astronomical figures.
Word quantity appeared in Russian in the 11th century. It came from the Old Church Slavonic language and was formed from the word colic- "How many". Noun quantity used to refer to everything that can be counted and measured. These can be people or objects ( number of guests, number of books), as well as the amount of substance that we do not count, but measure ( amount of water, amount of sand).
People learned to count a long time ago, back in the Stone Age. At first, people simply distinguished whether there was one object in front of them or more. After some time, a word appeared that denoted two objects. And some tribes of Polynesia and Australia, until very recently, had only two numerals: “one, two.” And all other numbers were named as a combination of these two numerals. For example, the number four: two, two,” three: one, two,” six: two, two, two.” And of course, as people learned to count, they developed a need to write down these numbers. Findings by archaeologists at the sites of primitive people prove that initially the number of objects was displayed by an equal number of some symbols: lines, notches, dots. This system of writing numbers is called UNIT (UNARY) because. Any number in it is formed by repeating the same sign, symbolizing one.
Fingers are the first computing device because the number of objects or years can be shown on the fingers. Thus, echoes of the unit number system are still found today. For example, to find out what course a military school cadet is studying in, you need to count the number of stripes sewn on his sleeve. Children also use this system, showing their age on their fingers. The unit system is not the most convenient way to write numbers. Recording large quantities in this way is tedious, and the records themselves are very long. Over time, other, more economical number systems emerged.
Around the third millennium BC, one of the oldest numberings that came down to us in ancient papyri and drawings appeared in Egypt - EGYPTIAN. To record numbers, the Egyptians used special symbols - HIEROGLYPHS. Hieroglyphs were used both for writing and to designate key symbols. At first, the icons had a complex appearance, and over time they became simpler..
All other numbers were made up by adding certain hieroglyphs, and the total number was determined by the sum of the values of all the icons. The Egyptians practiced adding numbers to each other, that is, ADDITION (by adding a second term to the existing hieroglyph number of the hieroglyph). Moreover, the size of the number did not depend on the order in which its constituent signs were located on papyrus, that is, the NON-POSITIONAL NUMERAL SYSTEM. (As they wrote and read, in a row). The signs could be written: Top to Bottom, Right to Left or Shuffled. If the number decreased, then when quickly counting, the corresponding sign was crossed out or erased. For example, X L D M stands for: Two thousand, Two hundred, five tens and three units.
The number 2 and its powers played a special role among the Egyptians. They carried out multiplication and division by sequentially doubling and adding numbers. Such calculations looked rather cumbersome. For example, to multiply 15 by 24, the following table was compiled: Here the results of doubling one are written in the left column, and the number 24 is written in the right column. The entries did not end until it was possible to create a multiplier (1 * 2) from the numbers in the left column 48 4(2*2) 96 8(4*2) (8*2) =15. After this, the numbers from the right column were added up =360
When dividing, the Egyptians repeatedly doubled the divisor in the right column and, accordingly, 1 in the left column, until the numbers in the right column remained no more than the dividend. Next, they tried to create a dividend from the numbers in the right column, and if this was successful, then the sum of the corresponding numbers in the left column gave the desired quotient. If the dividend was not evenly divisible by the divisor, then the quotient and remainder were obtained. For example, to divide 541 by 12, you had to create a table:
The idea of assigning different values to numbers depending on what position they occupy in the number record first appeared IN ANCIENT BABYLON around the third millennium BC. Many clay tablets of ANCIENT BABYLON have survived to this day, on which complex problems were solved, such as calculating roots, finding the volume of a pyramid, etc. To record numbers, the Babylonians used only two signs: a vertical wedge (units) and a horizontal wedge (tens). All numbers from 1 to 59 were written using these signs, as in the usual hieroglyphic system. Example:
Alphabetical numbering was also used by the southern and eastern Slavic peoples. Among some Slavic peoples, the numerical values of letters were established in the order of the Slavic alphabet, while for others (including Russians), the role of numbers was played not by all the letters of the Slavic alphabet, but only by those that were present in the Greek alphabet. A special “TITLO” icon was placed above the letter indicating the number. At the same time, the numerical values of the letters increased in the same order as the letters in the Greek alphabet. (The order of the letters of the Slavic alphabet was slightly different) The southern and eastern Slavic peoples also used alphabetical numbering. Among some Slavic peoples, the numerical values of letters were established in the order of the Slavic alphabet, while for others (including Russians), the role of numbers was played not by all the letters of the Slavic alphabet, but only by those that were present in the Greek alphabet. A special “TITLO” icon was placed above the letter indicating the number. At the same time, the numerical values of the letters increased in the same order as the letters in the Greek alphabet. (The order of the letters of the Slavic alphabet was slightly different) In Russia, Slavic numbering was preserved until the end of the Seventeenth century. Under Peter the Great, the so-called ARAB NUMERATION prevailed and was preserved only in liturgical books. In Russia, Slavic numeration was preserved until the end of the Seventeenth century. Under Peter the Great, the so-called ARAB NUMERATION prevailed and was preserved only in liturgical books.
Some letters are used as numbers. I(1), V(5), X(10), L(50), C(100), D(500), M(1000). The meaning of a digit does not depend on its position in the number. for example, in the number XXX, the digit X occurs three times, and in each case it denotes the same value 10, and in the sum XXX is 30. The value of a number in the Roman numeral system is defined as the sum or difference of numbers. If the smaller number is to the left of the larger one, then it is subtracted; if it is to the right, it is added. For example: 1998=MCMXCVIII=1000+()+()
..
Hieroglyphic and alphabetic number systems have one significant drawback - it was very difficult to perform arithmetic operations in them. In a positional number system, the quantitative value of a digit depends on its position in the number. The position of the digit is called the digit. The digit of the number increases from right to left. The most common currently are decimal, binary, octal and hexadecimal positional number systems. In a positional number system, the base of the system is equal to the number of digits used by it and determines how many times the values of the digits of adjacent digits of numbers differ. The main advantages of any positional number system are the ease of performing arithmetic operations and the limited number of symbols required to write any numbers.
The French mathematician Pierre Simon Laplace (). With these words he assessed the “OPENING” of the positional number system: “The idea of expressing all numbers with a few signs, giving them meaning in form, and also meaning in place, is so simple that it is precisely because of this simplicity that it is difficult appreciate how amazing she is..."
Its widespread use in the past is clearly indicated by the names of numerals in many languages, as well as the methods of counting time, money, and the relationship between certain units of measurement that have been preserved in a number of countries. A year consists of 12 months, and half a day consists of 12 hours. In Russian, counting is often done in dozens, slightly less often in grosses (144 = 12 2 each), but in the old days the word for 1728 = 12 3 was also used. In the English language there are special (and not formed according to the general rule) words eleven (11) and twelve (12). The English pound is divided into 12 shillings.
In 595 (already AD), the decimal number system familiar to all of us today first appeared in India. (Thanks to the Indians, what would we do today without it?) The famous Persian mathematician Al-Khwarizmi published a textbook in which he outlined the basics of the Hindu decimal system. After its translation into Latin and the publication of the book by Leonardo Pisano (Fibonacci), this system became available to Europeans.
At the moment, it is the number system most commonly used in computer science, computer technology and related industries. Uses two digits – 0 and 1, as well as the symbols “+” and “–” to indicate the sign of a number and a comma (period) to separate the integer and fractional parts.
Number systems:
- positional.
- non-positional.
Non-positional number systems are systems in which the symbols used to represent a number do not change their meaning with changes in location. For example, Roman: I, V, X, C (rule: if the number on the left is less than the number on the right, then the left is subtracted from the right. If the number on the right is less than or equal to the number on the left, then these numbers are added).
A positional number system is an ordered set of symbols defined by the alphabet. The number of characters or numbers of the alphabet is called the base of the system.
The equivalent of a 16-digit digit is four-digit 2-digit tetrad number.
| q | ||||||||||||||||
| A | B | C | D | E | F |
Translation of integers.
From 10th to qth. There are 3 translation methods:
1. division by the base of the new s.s. (q)-the original number X and the subsequent quotients obtained are divided by q to obtain. quotient, less than q; received remnants of phenomena digits of the number in the qth s.s.; last quotient senior category new numbers, the last remainder is second, first. rest-last:
2.method of sub-digit “weighing”;
Weighted coding method.
Converting fractional numbers.
From 10th to qth.
When converting fractional numbers, they talk about transferring with a given accuracy and use the method of sequential multiplication by the base of the new s.s.
Ref. the number X (fractional, decimal) and the resulting fractions are successively multiplied by q until the result is obtained. fractional part equal to 0 (with exact translation) or until received. required quantity digits in the qth notation of a number (when translated with a given accuracy). Number X in q s.s. image. as a sequence of entire parts of works.
X 10 =0.875; q=2.
-the fractional part without 1 is equal to 0.
When converting fractional numbers, containing. the denominator is a multiple of a power of two, the numerator is translated according to the rule for integers, and then the point is moved n places to the left (n-power of two, which is a multiple of the denominator):
Translation of mixed numbers.
When translating mixed numbers, its whole. and fractional parts are translated separately according to the rules above; then connected through a dot.
X 10 =15.875; q=2;
[X 10 ]=15= =1111 2
0,875 10 = 2 X 2 =1111.111 2
Transfer from qth to 10th s.s. completed according to the polynomial formula 
.
Translation of numbers from one s.s. in other s.s. with arbitrary grounds ex. through decimal s.s.
Information and data.
Data is a specific implementation of information. They can be presented in numerical, graphical or symbolic form. Data becomes information only when solving a specific problem, that is, during its consumption.
Information is only that data that eliminates uncertainty in resolving an issue and allows you to make an appropriate decision.
The transformation of data into information is carried out by the consumer based on his own information model. An information model of an object is a set of characteristics of an object along with a numerical or other value.
The form of data presentation determines the time and effort that the user needs to expend to obtain information, which affects consumer activity and the cost of information.
Data operations:
Data collection– accumulation of information in order to ensure sufficient completeness for decision-making.
Formalization– reduction of data to one form.
Sorting– ordering of data according to a given characteristic.
Archiving- ordering data according to a given characteristic for the purpose of convenience.
Conversion– transfer of data from one form to another.
Data protection– a set of measures aimed at preventing the loss, reproduction and modification of data.
Transportation-process of information transfer. from the place of its generation to the place of use and storage.
General data transfer scheme:
Processes associated with operations on data are called information processes, and the symbols that implement them are called information systems.
An information system is an organizationally ordered set of documents and information technologies that implement issues.
There are information systems:
Information and reference systems.
Information retrieval systems.
Data processing and transmission systems.
Communication systems.
Control systems.
Quantifying information.
Such an assessment of information is necessary to compare arrays of stored or transmitted information with each other, as well as to estimate the size of the media.
Lecture 1. Number systems
Notation- a set of techniques and rules for naming and designation
a set of certain symbols (letters or numbers), with the help of which, as a result of some operations, any number of them can be represented.
The image of any number of characters is called a number, and the characters of the alphabet are called letters and numbers and. The characters of the alphabet must be different and the meaning of each
Calculus is the development of the most convenient way to write numbers, in particular for the simple and quick solution of logical problems. For “convenience” of use, the number system must have the following properties:
- simplicity of the method of recording on physical media;
- ease of performing arithmetic operations;
- visualization of teaching the basics of working with numbers.
In the modern world, the most common is the decimal number system, the origin of which is associated with finger counting. It originated in India
and in the 13th century. was brought to Europe by the Arabs. Therefore, the decimal number system began to be called Arabic, and the numbers used to record numbers that we now use - 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 - Arabic.
Since ancient times, various number systems have been used for calculations and calculations. For example, in the Ancient East the duodecimal system was quite widespread. Many items (knives, forks, plates, etc.) are still counted in dozens. The number of months in a year is twelve. This number system was preserved in the English system of measures (for example, 1 foot = 12 inches) and in the monetary system (1 shilling = 12 pence). In Ancient Babylon there was a very complex 60-digit system. It, like the decimal system, has been preserved to some extent to this day (for example, in the time measurement system: 1 hour = 60 minutes, 1 minute = 60 s). The first numbers (signs to indicate numbers) appeared among the Egyptians and Babylonians. A number of peoples (ancient Greeks, Syrians, Phoenicians) used letters of the alphabet as numbers. A similar system until the 16th century. was also used in Russia. During the Middle Ages in Europe
used a system of Roman numerals, which |
used for |
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designations of chapters, parts, sections in |
various |
documents, books, |
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month designations, etc. |
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All number systems can be divided into positional and non-positional. |
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Non-positional number system- a system in which symbols denoting something |
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or another quantity, do not change their |
values in |
depending on |
locations |
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(positions) in the image of the number. |
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A non-positional number system is the simplest system with an o symbol (stick). To depict any number in this system, you need to write down the number of sticks equal to the given number. This system is ineffective because the recording form is very cumbersome.
The non-positional number system also includes Roman numerals, which are often used to number centuries, volumes, etc. Here, Latin letters are used as numbers
IN In general, non-positional number systems are characterized by complex ways of writing numbers and rules for performing arithmetic operations.
IN Currently, all the most common number systems belong to the category of positional ones.
Positional number systems.
A number system in which the value of a digit is determined by its location (position) in the image of the number is called positional.
An ordered set of characters (letters and numbers) (a0, a1, ..., аn), used to represent any numbers in a given positional number system, is called its alphabet, the number of characters (digits) of the alphabet p=n+1 is its base, and the system itself
numbers: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9., and the base p = 10, i.e. in this system, only ten different symbols (digits) are used to write any numbers. These numbers were introduced to designate the first ten consecutive numbers, and all subsequent numbers, starting from 10, etc., are designated without the use of new numbers. Decimal system
Notation is based on the fact that 10 units of each digit are combined into one unit of the adjacent highest digit, so each digit has a weight equal to a power of 10. Therefore, the value of the same digit is determined by its location in the number image, characterized by a power of 10.
For example, in the image of the number 222.22, the number 2 is repeated 5 times, while the first number 2 on the left means the number of hundreds (its weight is 102); the second is the number of tens (its weight is 10), the third is the number of units (its weight is 100), the fourth is the number of tenths of a unit (its weight is 101) and the fifth digit is the number of hundredths of a unit (its weight is 102), t i.e. the number 222.22 can be expanded into powers of the number 10:
Likewise
Thus, any number A can be represented as a polynomial by expanding it into powers of 10:
the sequence of coefficients of which is a decimal notation
numbers A10: A comma separating the integer part of the number from the fractional part, serves to fix specific
the values of each position in this sequence of numbers is the starting point.
Binary, octal and hexadecimal number systems
implementation requires technical devices with only two stable states, for example: the material is magnetized or demagnetized (magnetic tapes, disks), a hole
application of Boolean algebra apparatus to perform logical transformations of information. In addition, arithmetic operations in the binary number system are performed most simply.
The disadvantage of the binary system is the rapid increase in the number of digits required to record large numbers. This drawback is not significant for a computer. If there is a need to encode information “manually,” for example, when compiling a program in machine language, then octal or hexadecimal number systems are used. Numbers in these systems are read almost as easily as decimal ones; they require, respectively, three (octal) and four (hexadecimal) times fewer digits than in the binary system (the numbers 8 and 16 are the 3rd and 4th powers of the number 2, respectively), and converting them to the binary number system and back is much simpler compared to the decimal number system.
