Courses
Courses for Kids
Free study material
Offline Centres
More
Store Icon
Store

Binary Number System

ffImage
Last updated date: 17th Apr 2024
Total views: 365.4k
Views today: 9.65k
hightlight icon
highlight icon
highlight icon
share icon
copy icon

An Introduction to Binary Number System

To define a number in a binary system, the Binary Number System is used. The binary system is the representation of numbers in terms of 0's and 1's. The Binary Number System is generally used in computer languages like Java, C++. This is due to the fact that the computer only understands the binary language that is 0 or 1. All inputs given to a computer are decoded into a series of 0's or 1's. 


Number System is generally used for number representation in computer architecture. The Number System is classified into four types, they are

  • Binary Number System (base 2)

  • Octal Number System (base 8) 

  • Decimal Number System (base 10)

  • Hexadecimal Number System (base 16). 

Both in Mathematics and Digital Electronics, a Binary Number System is a way of presenting numerals that have to base equals to 2 and the same is a combination of 0s and 1s. Here, we have provided a Binary Number System example. Take a look!

110100


History of Binary Number System

Thomas Harriot, Gottfried Leibniz, and Juan Caramuel y Lobkowitz studied the binary system during the 16 and 17th centuries. This is termed the modern binary system. Nevertheless, other representation methods of binary numbers were found in an earlier time in various nations like India, China, Egypt, etc.


Thomas Harriot, Gottfried Leibniz and Juan Caramuel y Lobkowitz studied the binary system during the 16 and 17th centuries. This is termed as the modern binary system. Nevertheless, other representation methods of binary numbers were found in an earlier time in various nations like India, China, Egypt, etc.


In the following table, you will get to see the values of decimal to binary numbers from 1 to 30.


Decimal to Binary Numbers From 1 to 30

Number

Respective Binary Number

Number

Respective Binary Number

Number

Respective Binary Number

1

1

11

1011

21

10101

2

10

12

1100

22

10110

3

11

13

1101

23

10111

4

100

14

1110

24

11000

5

101

15

1111

25

11001

6

110

16

10000

26

11010

7

111

17

10001

27

11011

8

1000

18

10010

28

11100

9

1001

19

10011

29

11101

10

1010

20

10100

30

11110


Arithmetic Calculations in Binary Number System

Almost all kinds of fundamental arithmetical operations like addition, subtraction, multiplication and division are possible on binary digits. Let us study them individually.


Sum binary numbers are the simplest operation that uses a form of carrying.

0 + 0 = 0

0 + 1 = 1

1 + 0 = 1

1 + 1 = 0, here carry 1 as 1 + 1 is 2 which is 0 + (1 x 21)

Addition of two digits (here “1”) gives zero, and the carryover needs to be added with the second number. This is exactly the same as we perform in the decimal system while adding two single-digit numbers. For example:

5 + 5 = 0 and carry over 1

Here, you can check an example showing the addition of two binary expressions.

Add 10101 and 11011


Binary Number System Addition



+

1

1

1

1


1

1

0

1

1

0

0

1

1

1

1

1

0

0

0

0


Similar to addition, subtraction also follows the same procedure:

0 – 0 = 0

0 – 1 = 1, borrow 1

1 – 0 = 1

1 – 1 = 0

When you subtract 1 from 0, it gives out 1, and the same has to be reduced from the next number. This is called borrowing. 


Here, you can check an example showing subtraction of two binary expressions.

Subtract 1010110 – 101010


Binary Number System Subtraction


1


1




1

-

0

1

1

0

0

1

1

0

1

1

0

0


1

0

1

1

0

0


The process to multiply two binary numbers is the same as it is done for decimal numbers. As binary numbers are a combination of two digits only, there will be only two outcomes. By going through the example below, you will get to comprehend it better.

Multiply 10111 by 1101


Binary Number System Multiplication


X

1

1

0

1

1

0

1

0



0

0

0

0

1

1

0

1




0

0

0

0


1

1

0

1


1

0

0

0

0

0

1

0


The binary division is again the same way as it is done for decimal numbers. Check the example below:


Binary Number System Division

10

)1

0

1

0(

101


1

0






0

1

0





1

0






0



Solved Binary Number System Problems

Problem 1. Convert the Following Binary Number to a Decimal Number.

(a) 10112

(b) 1012

(c) 101012

Solutions:

(a) 

\[2^{4}\]

\[2^{3}\]

\[2^{2}\]

\[2^{1}\]

\[2^{0}\]


1

0

1

1


\[1011_{2} = 1 \times 2^{3} + 0 \times 2^{2} + 1 \times 2^{1} + 1 \times 2^{0}\]

= 8 + 0 + 2 + 1

\[ = 11_{10}\]

(b) 

\[2^{4}\]

\[2^{3}\]

\[2^{2}\]

\[2^{1}\]

\[2^{0}\]



1

0

1


\[101_{2} = 1 \times 2^{2} + 0 \times 2^{1} + 1 \times 2^{0}\]

= 4 + 0 + 1

\[5_{10}\]

(c) 

\[2^{4}\]

\[2^{3}\]

\[2^{2}\]

\[2^{1}\]

\[2^{0}\]

1

0

1

0

1


\[10101_{2} = 1 \times 2^{4} + 0 \times 2^{3} + 1 \times 2^{2} + 0 \times 2^{1} + 1 \times 2^{0}\]

= 16 + 0 + 4 + 0 + 1

\[  = 21_{10} \]


Vedantu provides the above discussion about the Binary Number System of computers and its arithmetical calculations. If you are looking for study materials for other Number Systems, please download the app immediately.


Binary Number System

A number that can be expressed in the binary system or with the base 2 numeral system is called a binary number. It only has two numerical values such as 1 (one) and 0 (zero). 


The binary system is an internal application used by almost every latest computer and computer-based device due to the direct implementation of electronic circuits that use logic gates. Every digit is denoted as a bit. 


Binary Number Table

Number

Binary Number

Number

Binary Number

Number

Binary Number

1

1

11

1011

21

10101

2

10

12

1100

22

10110

3

11

13

1101

23

10111

4

100

14

1110

24

11000

5

101

15

1111

25

11001

6

110

16

10000

26

11010

7

111

17

10001

27

11011

8

1000

18

10010

28

11100

9

1001

19

10011

29

11101

10

1010

20

10100

30

11110

 

Binary Arithmetic Operations

It is the same as the arithmetic operations in numerals. We can perform addition, subtraction, multiplication, and division operations on Binary numbers in a similar way. 

 

Adding two binary numbers will give a binary number itself. It is the simplest method compared to other arithmetic operations. The addition of two single-digit binary numbers is as follows.


Binary Addition

Binary Numbers

Addition

0

0

0

0

1

1

1

0

1

1

1

0; Carry →1


For Example: Add  \[1101_{2}\], and  \[1001_{2}\].

Solution: 1101 + 1001 = 10110

 

Subtracting two binary numbers will give a binary number itself. It is one of the straightforward methods. The subtraction of two single-digit binary numbers is as follows.


Binary Subtraction

Binary Numbers

Subtraction

0

0

0

0

1

1; Borrow 1

1

0

1

1

1

0


For Example: Subtract  \[1101_{2}\], and  \[1010_{2}\].

Solution:  \[1101_{2} - 1010_{2}\]=  \[0011_{2}\]


Binary Multiplication

The multiplication process is exactly similar to binary numbers and numerals. 

For example Multiply 1101₂ and 1010₂.

Solution: 1101 x 1010 = 10000010

 

Binary Division

The binary division is the same as the decimal number division method. 

For example Divide 1010₂ by 10₂

Solution: 1010 / 10 = 0


Uses of Binary Number System

Binary numbers are usually used in computer applications. All the coding and languages in computers use binary digits 0 and 1 to write a program or encode any digital data such as C, C++, Java, etc. This is due to the fact that computers only understand only the coded language. Therefore these 2-digit Number Systems are used to represent a set of data or information in the form of bits of information.

FAQs on Binary Number System

1. What is a Binary Number System?

A number that can be expressed in the binary system or with the base 2 numeral system is called a binary number. It only has two numerical values such as 1 (one) and 0 (zero). The Binary Number System is generally used in computer languages like Java, C++. The binary system is an internal application used by almost every latest computer and computer-based device due to the direct implementation of electronic circuits that use logic gates. Every digit is denoted as a bit.

2. What are the uses of the Binary Number System?

The following are some of the uses of the Binary Number systems, Binary numbers are usually used in computer applications. All the coding and languages in computers use binary digits 0 and 1 to write a program or encode any digital data such as C, C++, Java, etc. This is due to the fact that computers only understand only the coded language. Therefore these 2-digit Number Systems are used to represent a set of data or information in the form of bits of information.

3. Explain Binary Addition and Binary Multiplication?

Binary Addition

Adding two binary numbers will give a binary number itself. It is the simplest method compared to other arithmetic operations. The addition of two single-digit binary numbers is as follows.


Binary Numbers

Addition

0

0

0

0

1

1

1

0

1

1

1

0; Carry →1


For Example: Add 1101₂ and 1001₂.


Solution: 1101 + 1001 = 10110


Binary Multiplication

The multiplication process is exactly similar to binary numbers and numerals.

 

For Example: Multiply 1101₂ and 1010₂.


Solution: 1101 x 1010 = 10000010


4. Explain Binary Subtraction and Binary Division?

Binary Subtraction

Subtracting two binary numbers will give a binary number itself. It is one of the straightforward methods. The subtraction of two single-digit binary numbers is as follows.


Binary Numbers

Subtraction

0

0

0

0

1

1; Borrow 1

1

0

1

1

1

0


For Example: Subtract 1101₂, and 1010₂.


Solution: 1101₂– 1010₂ = 0011₂


Binary Division

The binary division is the same as the decimal number division method.


For Example: Divide 1010₂ by 10₂


Solution: 1010 / 10 = 0


5. What is a Number System?

A Number System is nothing but the representation of numbers using digits or other symbols in a regular manner. The value of any digit can be represented based on the digit, its number position, and the base of the Number System.


Number System is generally used for number representation in computer architecture. The Number System is classified into four types, they are

  • Binary Number System (base 2)

  • Octal Number System (base 8) 

  • Decimal Number System (base 10)

  • Hexadecimal Number System (base 16). 

6.  What are the Applications of a Binary Number System?

There is extensive usage of binary numbers in computer technology. All kinds of computer programming languages are entirely dependent on the 2 digits binary system. For example, the graphics you see on a device’s screen are encoded with a binary line corresponding to every pixel. Furthermore, you all may have heard of digital binary clocks. It comprises LEDs which are segregated as per hours, minutes, and seconds. When an LED light gets illuminated, it denotes Binary 1 and 0 and symbolizes the OFF state.

7. What is Meant by Boolean Logic?

Boolean logic is nothing but an algebraic form consisting of three Boolean operators, namely – AND, OR, and NOT. Firstly, AND indicates that all the conditions have to be fulfilled to get a TRUE result. Secondly, OR denotes that only one condition has to be fulfilled to get a TRUE result. Finally, the NOT operator changes all valid values to false and vice versa.

8. What are the Types of Number Systems?

There are four types of numbers systems in Mathematics. First is the Binary Number System having base 2, and the digits used are 1 and 0. Secondly, the octal Number System has a base 8, and the numbers used are 0, 1, 2, 3, 4, 5, 6, and 7. Thirdly, the decimal Number System has base 10 and the numbers used are 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Lastly, the hexadecimal Number System uses digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F with base 16.