Binary Number System

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Both in mathematics and digital electronics, a binary number system is a way of presenting numerals which has 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 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.


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.

  • Binary Number System Addition

Sum binary numbers are the simplest operation which 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 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



+

1

1

1

1

1

1

0

1

1

0

0

1

1

1

1

1

0

0

0

0


  • Binary Number System Subtraction

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


1


1

1

-

0

1

1

0

0

1

1

0

1

1

0

0


1

0

1

1

0

0


  • Binary Number System Multiplication

The process to multiply two binary numbers is 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


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


  • Binary Number System Division

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


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 Decimal Number.

(a) 10112

(b) 1012

(c) 101012

Solutions:

(a) 

24

23

22

21

20


1

0

1

1


10112 = 1 x 23 + 0 x 22 + 1 x 21 + 1 x 20

= 8 + 0 + 2 + 1

= 1110


(b) 

24

23

22

21

20



1

0

1


1012 = 1 × 22 + 0 × 21 + 1 × 20

= 4 + 0 + 1

= 510


(c) 

24

23

22

21

20

1

0

1

0

1


101012 = 1 × 24 + 0 × 23 + 1 × 22 + 0 × 21 + 1 × 20

= 16 + 0 + 4 + 0 + 1

= 2110


Vedantu provides the above discussion about the binary number system of computer and its arithmetical calculations. If you are looking for study materials of other number systems, please download the app immediately.

FAQ (Frequently Asked Questions)

1. What are the Applications of Binary Number System?

Ans. There is the extensive usage of binary numbers in computer technology. All kinds of computer programming languages are entirely dependent on the 2 digits binary system. Like 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 symbolises the OFF state.

2. What is Meant by Boolean Logic?

Ans. Boolean logic is nothing but an algebraic form comprising 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.

3. What are the Types of Number Systems?

Ans. There are four types of numbers systems in mathematics. First is the binary number system having base 2, and the used digits are 1 and 0. Secondly, octal number system having base 8, and the used numbers are 0, 1, 2, 3, 4, 5, 6, and 7. Thirdly, decimal number system having base 10 and the numbers used are 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Lastly, the hexadecimal number system using digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F with base 16.