Numbers in the binary system. History of the binary number system

A number system is a way of displaying numbers on paper. They are used in calculations on equipment and digital apparatus. The binary number system is now one of the most popular tools in computing devices. Consider the features of working with this number system.

The history of the emergence of the binary number system

Scientists ancient world proposed to make calculations using only 2 digits, and suggested that this method of calculation is the future. This is due to the simplicity of this calculation method: only 2 positions (0 and 1), 2 positions, for example, there is a signal or there is no signal. The German mathematician Leibniz believed that mathematical operations performed on 2 digits carry a certain order.

Until the 40s of the 20th century, the theory of the binary system did not develop until the American scientist Claude Shannon suggested using it in the operation of electronic circuits. It turned out that their use in a PC is much more preferable, because it is not easy for a person to memorize a cumbersome accumulation of zeros and ones. And in a computer, it is enough to create a device that has logical 0 and 1, that is, it has no more than 2 logical states. It can be a magnetized or demagnetized core, a closed or open transformer, etc. There are only 2 positions, not 10, as it could be when using the decimal system in computer calculations.

Characteristics of the binary number system

The features of the binary number system include:

• Using just a couple of digits (0 and 1). The base of such a system is 2.
• Algebraic operations carried out with numbers of two digits are not very difficult.
• The storage and conversion of signals by video equipment and recording devices is carried out in a code consisting of 0 and 1.
• Digital communication channels exchange data using their representation in the form of 0 and 1.

Counting in binary system

And then for each digit in order there is an increase in the digit:

100 is four.

110 - six.

After 7, the digits are written as 4 digits:

1000 is eight.

1001 - nine.

1010 - ten.

1011 - eleven.

1100 - twelve.

1101 - thirteen.

1110 - fourteen.

Converting numbers from binary to decimal

Representing decimal numbers in binary makes them quite cumbersome. Let's consider how the reverse process occurs: the translation of a number consisting of 0 and 1 into a form convenient for us. For example, you need to convert the binary code 10101110 to decimal.

It can be broken down into powers, as is done in the decimal system. So, the number 1587 can be displayed as:

1000 + 500 + 80 + 7.

Or in another way:

1*10 3 + 5*10 2 + 8*10 1 + 7*10 0 .

In the previous entry, the degrees corresponding to the category of each digit minus 1 are summed up. The number 10 is taken as the base of the degree, because this is a decimal number system. This method can be applied to a number represented in binary. Only the number 2 should be taken as the basis of the degree. It turns out:

10101110 = 1*2 7 + 0*2 6 + 1*2 5 + 0*2 4 + 1*2 3 + 1*2 2 + 1*2 1 + 0*2 0 = 128 + 0 + 32 + 0 + 8 + 4 + 2 + 0 = 174.

The powers of two are chosen according to the following principle: you need to count the digit of the digit and subtract 1 from this value. It should be remembered that the discharge increases from right to left. So, the very first unit has the eighth digit, then it must be multiplied by 2 7, etc.

So the binary form of 10101110 is 174 in decimal. The correct entry looks like this:

10101110 2 = 174 10 .

There is a need for the reverse process: to convert the decimal notation into a sequence of 0 and 1. This is done by dividing by 2 and forming a binary number from the remainder. For example, the number 69.

Let's look at the rest. We get a number in binary form, starting from the last line: 1000101 (these numbers are located in the "Remainder" column, when viewed from the bottom up). You need to check the result:

1000101 = 1*2 6 + 0*2 5 + 0*2 4 + 0*2 3 + 1*2 2 + 0*2 1 + 1*2 0 = 64 + 4 +1 = 69.

Mathematical operations with binary numbers

Addition.

This is the basic arithmetic operation in computers. The basic principles of adding binary numbers are based on the rules:

Thus, adding 1101 2 and 110 2 in a column, we get 10011 2 or 19 10 .

Subtraction.

This operation is identical to addition, if you imagine that one of the binary numbers is negative. In this case, you need to take into account the modules of the added numbers.

Rules used in subtraction:

0 - 1 = 1 (borrow from the high order).

For example, if we subtract the number 101 2 from 1110 2 , we get 1001 2 or 9 10 .

Multiplication.

On paper, multiplication is a collection of addition operations. For example, you need to multiply 10 10 by 40 10 .

Let's transform them into a set of 0 and 1:

10 10 =00001010 2

40 10 = 00101000 2

Both numbers in binary form have several zeros on the left and right, which do not play a role in the multiplication operation. Significant parts are 101 in 10 and 101 in 40, located between zeros. They need to be multiplied, and zeros are simply added to the final result:

We multiply the left and right units of the second factor by the first factor, then sum the intermediate result obtained. We add the zeros and rewrite them into the final result of the multiplication, which in binary form looks like this: 000000110010000 2 (bottom line from left to right).

Checking, we get:

1 * 2 8 + 1 * 2 7 + 1 * 2 4 = 256 + 128 + 16 = 400.

Division.

Consider the simplest example of division without a remainder. Divide 1410 by 210. In binary it looks like this:

14 10 = 1110 2 .

We divide 1110 2 by 10 2 in a column:

1110 |10

We get the number 111 2 , which is equal to 7 in decimal notation. When checking by multiplication, we prove the accuracy of the result:

We look at the bottom line from left to right, the result of multiplication is 1110 2 . The answer is correct.

Lesson Plan

Here you will learn:

♦ how it works with numbers;
♦ what is a spreadsheet;
♦ how computational problems are solved;
♦ using spreadsheets;
♦ how to use spreadsheets for information modeling.

Binary number system

The main topics of the paragraph:

♦ decimal and binary number systems;
♦ extended form of the number;
♦ conversion of binary numbers to decimal system;
♦ conversion of decimal numbers to the binary system;
♦ binary arithmetic.

In this chapter, we will focus on the organization of calculations on computer. Computing is about storing and processing numbers.

The computer works with numbers in the binary system.

This idea belongs to John von Neumann, who formulated the principles of the design and operation of computers in 1946. Let's find out what a number system is.

Decimal and Binary Number Systems

The number system or in the abbreviated form SS is a system of notation of numbers that has a certain set of digits.

You learned about the history of various number systems when you studied chapter 7 of the textbook. And today we will turn our attention to such number systems as binary and decimal SS.

As you already know from the material studied earlier, one of the most commonly used number systems is the decimal SS. And this system is called so because the basis of this word formation is the number 10. That is why the number system is called decimal.

You already know that ten digits such as 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 are used in this system. But the number ten has an exceptional role, since there are ten fingers on our hands . That is, ten digits are the base of this number system.

But in the binary number system, only two digits are involved, such as 0 and 1, and the base of this system is the number 2.

Now let's try to figure out how to represent some value with just two numbers.

Expanded form of writing a number

Let's turn to our memory and remember what principle of writing numbers exists in decimal SS. That is, it will no longer be a secret for you that in such an SS the entry of a number depends on the location of the number, that is, on its position.

So, for example, the figure that is on the far right tells us about the number of units of this number, following this figure, as a rule, indicates the number of twos, etc.

If you and I, for example, take a number such as 333, we will see that the rightmost digit means three ones, then three tens, and after it - three hundreds.

Now let's represent it in the form of such an equation:

Here we see equality, in which the expression located on the right side of the equal sign is provided as an expanded form of notation of this multi-digit number.

Consider another example of a multi-digit decimal number, which is also presented in expanded form:

Converting binary numbers to decimal

Now let's take a meaningful binary number as an example:

In this significant number, we see a deuce on the right side below, which indicates to us the base of the number system. That is, we understand that we have a binary number in front of us and we can no longer confuse it with a decimal number.

And the value of each next digit in the binary number increases by 2 with each step from right to left. Now let's see how the expanded form of this binary number will look like:

In this example, we see how you can translate a converted binary number into a decimal system.

Now let's give some more examples of converting binary numbers to decimal number system:

This example shows us that a two-digit decimal number, in this case, corresponds to a six-digit binary. The binary system is characterized by such an increase in the number of digits with an increase in the value of the number.

Now let's see what the beginning will look like natural series numbers in decimal (A10) and binary (A2) SS:

Converting Decimal Numbers to Binary

Having considered the examples above, I hope you now understand how the binary number is converted to an equal decimal number. Well, now let's try to make a reverse translation. Let's see what we need to do for this. For such a translation, we need to try to decompose the decimal number into terms, which are powers of two. Here's an example:

As you can see, this is not so easy to do. Let's try to consider another, simpler method of converting from decimal SS to binary. This method consists in the fact that a known decimal number, as a rule, is divisible by two, and its resulting remainder will act as the least significant digit of the desired number. This, the newly obtained number, we again divide by two and get the next digit of the desired number. We will continue such a division process until the quotient becomes less than the base of the binary system, that is, less than two. This is the resulting quotient and will be the highest digit of the number we were looking for.

Let's now look at the methods for writing division by the number two. For example, let's take the number 37 and try to translate it into a binary system.

In these examples, we see that a5, a4, a3, a2, a1, a0 are the designation of digits in the notation of a binary number, which are carried out in order from left to right. As a result, we will get:

Binary arithmetic

Based on the rules in arithmetic, it is easy to see that in the binary system, they are much simpler than in decimal.

Now let's remember the options for adding and multiplying single-digit binary numbers.

Due to this simplicity, which is easily consistent with the bit structure of computer memory, the binary number system has attracted the attention of the creators of the computer.

Pay attention to how the example of adding two multi-valued binary numbers using a column is performed:

And here is an example of multiplying multi-valued binary numbers in a column:

You have noticed how easy and simple it is to follow these examples.

Briefly about the main

Number system - certain rules for writing numbers and methods for performing calculations associated with these rules.

The base of the number system is equal to the number of digits used in it.

Binary numbers are numbers in the binary number system. They use two digits: 0 and 1.

An expanded form of writing a binary number is its representation as the sum of powers of two multiplied by 0 or 1.

The use of binary numbers in a computer is due to the bit structure of computer memory and the simplicity of binary arithmetic.

Advantages of the binary number system

And now let's look at the advantages of the binary system of calculus:

Firstly, the advantage of the binary number system is that with its help it is quite easy to carry out the processes of storing, transmitting and processing information on a computer.
Secondly, for its implementation, not ten elements are enough, but only two;
Thirdly, displaying information using only two states is more reliable and more resistant to various interferences;
Fourth, there is the possibility of using the algebra of logic to implement logical transformations;
Fifth, binary arithmetic is still simpler than decimal arithmetic, and therefore is more convenient.

Disadvantages of the binary number system

The binary number system is less convenient, as a person is used to using the decimal system more, which is much shorter. But, in the binary system, large numbers have a rather large number of digits, which is its significant drawback.

Why is the binary number system so common?

The binary number system is popular because it is a computer language, where each digit must be represented in some way on a physical medium.

After all, it is easier to have two states in the manufacture of a physical element than to invent a device in which ten different states. Agree that it would be much more difficult.

In fact, this is one of the main reasons for the popularity of the binary number system.

The history of the emergence of the binary number system

The history of the creation of the binary number system in arithmetic is quite bright and swift. The famous German scientist and mathematician G. W. Leibniz is considered the founder of this system. He published an article in which he described the rules by which it was possible to perform all kinds of arithmetic operations on binary numbers.

Unfortunately, until the beginning of the twentieth century, the binary number system was hardly noticeable in applied mathematics. And after simple counting mechanical devices began to appear, scientists began to pay more attention to the binary number system and began to actively study it, since it was convenient and indispensable for computing devices. It is the minimum system with which you can fully implement the principle of positionality in the digital form of writing numbers.

Questions and tasks

1. What are the advantages and disadvantages of the binary number system compared to the decimal.
2. Which binary numbers correspond to the following decimal numbers:
128; 256; 512; 1024?
3. What are the following binary numbers in decimal:
1000001; 10000001; 100000001; 1000000001?
4. Convert the following binary numbers to decimal:
101; 11101; 101010; 100011; 10110111011.
5. Convert the following decimal numbers to binary:
2; 7; 17; 68; 315; 765; 2047.
6. Perform binary addition:
11 + 1; 111 + 1; 1111 + 1; 11111 + 1.
7. Perform binary multiplication:
111 10; 111 11; 1101 101; 1101 1000.

I. Semakin, L. Zalogova, S. Rusakov, L. Shestakova, Informatics, Grade 9
Submitted by readers from Internet sites

Binary system

Binary number system is a positional number system with base 2. In this number system, natural numbers are written using only two symbols (which are usually the numbers 0 and 1).

The binary system is used in digital devices because it is the simplest and meets the requirements:

• How less values exists in the system, the easier it is to make individual elements that operate with these values. In particular, two digits of the binary number system can be easily represented by many physical phenomena: there is current - there is no current, the magnetic field induction is greater than the threshold value or not, etc.
• The lower the number of states for an element, the higher the noise immunity and the faster it can work. For example, to encode three states through the value of the magnetic field induction, it will be necessary to enter two threshold values, which will not contribute to the noise immunity and reliability of information storage.
• Binary arithmetic is pretty simple. Simple are the tables of addition and multiplication - the basic operations on numbers.
• It is possible to use the apparatus of the algebra of logic to perform bitwise operations on numbers.

Links

• Online calculator for converting numbers from one number system to another

Wikimedia Foundation. 2010 .

See what "Binary System" is in other dictionaries:

BINARY, in mathematics, a number system that has BASE 2 (the decimal system has base 10). It is most suitable for working with computers, because it is simple and corresponds to two positions (open 0 and closed ... ... Scientific and technical encyclopedic dictionary

binary system- - Telecommunication topics, basic concepts of EN binary system ... Technical Translator's Handbook

binary system- dvejetainė sistema statusas T sritis automatika atitikmenys: engl. binary system vok. Binärsystem, n rus. binary system, f pranc. système binaire, m … Automatikos terminų žodynas

binary system- dvejetainė sistema statusas T sritis fizika atitikmenys: engl. binary system; dyadic system vok. Binärsystem, n; Dualsystem, n rus. binary system, f pranc. système binaire, m … Fizikos terminų žodynas

Jarg. stud. Shuttle. Strong intoxication. PBS, 2002 ... Big dictionary of Russian sayings

A positional number system with base 2, in which the numbers 0 and 1 are used to write numbers. See also: Positional number systems Financial Dictionary Finam ... Financial vocabulary

BINARY NUMBERS, a way of writing numbers that uses two digits 0 and 1. Two units of the 1st digit (i.e., the place occupied in a number) form a unit of the 2nd digit, two units of the 2nd digit form a unit of the 3rd digit and etc… … Modern Encyclopedia

Binary number system- BINARY NUMBER SYSTEM, a way of writing numbers that uses two digits 0 and 1. Two units of the 1st digit (i.e., the place occupied in the number) form a unit of the 2nd digit, two units of the 2nd digit form a unit of the 3rd digit etc.… … Illustrated Encyclopedic Dictionary

Binary number system- a system that uses sets of combinations of numbers 1 and 0 to represent alphanumeric and other characters, the basis of codes used in digital computers ... Publishing Dictionary

BINARY NUMBER SYSTEM- positional number system with base 2, in which there are two digits 0 and 1, and all natural numbers are written in their sequences. Eg. number 2 is written as 10, number 4 \u003d 22 as 100, number 900 as 11 digit number: 11 110 101 000 ... Great Polytechnic Encyclopedia

We come across the binary number system when studying computer disciplines. After all, it is on the basis of this system that the processor and some types of encryption are built. There are special algorithms for writing a decimal number in binary and vice versa. If you know the principle of building a system, it will not be difficult to operate in it.

The principle of building a system of zeros and ones

The binary number system is built using two digits: zero and one. Why exactly these numbers? This is due to the principle of constructing signals that are used in the processor. At the lowest level, the signal takes only two values: "false" and "true". Therefore, it was accepted that the absence of a signal, “false”, is denoted by zero, and its presence, “true”, by one. This combination is easy to implement technically. Numbers in the binary system are formed in the same way as in decimal. When a bit reaches its upper limit, it is reset to zero and a new bit is added. According to this principle, the transition through a dozen in the decimal system is carried out. Thus, numbers are made up of combinations of zeros and ones, and this combination is called "binary number system".

How to write a binary number as a decimal?

There are online services that convert a number to a binary system and vice versa, but it's better to be able to do it yourself. The binary system in translation is denoted by the subscript 2, for example, 101 2 . Each number in any system can be represented as a sum of numbers, for example: 1428 = 1000 + 400 + 20 + 8 - in the decimal system. This is how a number is represented in binary. Take an arbitrary number 101 and consider it. It has 3 digits, so we decompose the number in order in this way: 101 2 \u003d 1 × 2 2 + 0 × 2 1 + 1 × 2 0 \u003d 4 + 1 \u003d 5 10, where the index 10 denotes the decimal system.

How to write a prime number in binary?

It is very easy to convert to binary by dividing a number by two. It is necessary to divide as long as it is possible to complete it completely. For example, let's take the number 871. We start dividing, be sure to write down the remainder:

871:2=435 (remainder 1)

435:2=217 (remainder 1)

217:2=108 (remainder 1)

The answer is written according to the received balances in the direction from the end to the beginning: 871 10 \u003d 101100111 2 . You can check the correctness of the calculations using the reverse translation described earlier.

Why do you need to know the rules of translation?

The binary number system is used in most disciplines related to microprocessor electronics, encoding, transmission and encryption of data, in various areas of programming. Knowledge of the basics of converting from any system to binary will help the programmer develop various microcircuits and control the operation of the processor and other similar systems programmatically. The binary number system is also necessary to implement methods for transmitting data packets over encrypted channels and creating client-server software projects based on them. In a school computer science course, the basics of converting to a binary system and vice versa are the basic material for learning programming in the future and creating simple programs.

Numbers are the second most common after the decimal, which is familiar to everyone, although few people think about it. The reason for this demand is that it is used in We will talk about this later, but for starters, a few words about the number system in general.

This phrase refers to a system of writing or other visual representation of numbers. This is a dry definition. Unfortunately, not everyone understands what lies behind these words. However, everything is quite simple, and the first number system appeared at the same time when a person learned to count. The easiest way to represent numbers is to identify some objects with others, well, at least the fingers on the hands and the number of fruits collected in a certain time. However, there are much fewer fingers on the hands than there can be countable objects. They began to be replaced with sticks or dashes on sand or stone. This was the very first number system, although the concept itself appeared much later. It is called non-positional, because each digit in it has a strictly defined value, regardless of what position it occupies in the record.

But such a recording is extremely inconvenient, and later the idea came to group objects and designate each group with a stone, and not with a stick, or, well, with a drawing of a different shape when recording. This was the first step towards the creation of positional systems, which include the binary number system. However, they finally formed only after the invention of numbers. Due to the fact that it was initially more convenient for people to count on their fingers, which normal person 10, it was the decimal system that became the most common. At the disposal of a person using this system, the numbers are from 0 to 9. Accordingly, when a person reaches 9 when counting, that is, he runs out of numbers, he writes a unit to the next digit, and resets the units to zero. And this is the essence of positional number systems: the value of the digits in a number directly depends on what position it occupies.

The binary number system provides only two digits for calculations, it is easy to guess that these are 0 and 1. Accordingly, new digits appear during writing in this case much more often: the first register transition occurs already at the number 2, it is the binary system that is designated as 10.

Obviously, this system is also not very convenient in writing, why is it so in demand? The thing is that when building computers, the decimal system turned out to be extremely inconvenient and unprofitable, since the production of a device that has ten different states is quite expensive, and they take up a lot of space. So they adopted the binary system invented by the Incas.

Converting to the binary number system is unlikely to cause difficulties for anyone. The easiest and most understandable way to do this is to divide the number by two until the answer is zero. In this case, the residuals are recorded separately from right to left sequentially. Consider an example, take the number 73: 73 \ 2 = 36 and 1 in the remainder, write the units in the extreme right position, write all further residues to the left of this unit. If you did everything correctly, then you should have the following number: 1001001.

How does a computer translate a number into a binary number system, because we enter decimal numbers from the keyboard? Does it also divide by 2? Naturally, no. Each button on the keyboard corresponds to a certain line in the encoding table. We press a button, a program called a driver sends a certain sequence of signals to the processor. That, in turn, sends a request to the table, which character corresponds to this sequence, and displays this character on the screen, or performs an action, if necessary.

Now you know the importance of the binary number system in our lives. After all, a lot of things in our world are now being done with the help of electronic computing systems, which, in turn, would be completely different if this system did not exist.