Converting Bases | Conversion of Bases

Number System Conversions-

 

Before you go through this article, make sure that you have gone through the previous article on Basics of Number System.

 

In number system,

  • It is very important to have a good knowledge of how to convert numbers from one base to another base.
  • Here, we will learn how to convert any given number from any base to any other base.

 

 

Conversion of Bases-

 

A given number in base x can be converted to any other base y using the following steps-

 

Step-01:

 

Convert the number from base x to base 10 using expansion method.

 

Read More- Conversion to Base 10

 

Step-02:

 

Convert the number from base 10 to base y using division & multiplication method.

 

 

Read More-

 

PRACTICE PROBLEMS BASED ON CONVERSION OF BASES-

 

Problem-01:

 

Convert (1056)16 to ( ? )8

 

Solution-

 

Step-01: Conversion To Base 10-

 

(1056)16 → ( ? )10

 

Using Expansion method, we have-

(1056)16

= 1 x 163 + 0 x 162 + 5 x 161 + 6 x 160

= 4096 + 0 + 80 + 6

= (4182)10

 

From here, (1056)16 = (4182)10

 

Step-02: Conversion To Base 8-

 

(4182)10 → ( ? )8

 

Using Division method, we have-

 

 

From here, (4182)10 = (10126)8

Thus, (1056)16 = (10126)8

 

Problem-02:

 

Convert (11672)8 to ( ? )16

 

Solution-

 

Step-01: Conversion To Base 10-

 

(11672)8 → ( ? )10

 

Using Expansion method, we have-

(11672)8

= 1 x 84 + 1 x 83 + 6 x 82 + 7 x 81 + 2 x 80

= 4096 + 512 + 384 + 56 + 2

= (5050)10

 

From here, (11672)8 = (5050)10

 

Step-02: Conversion To Base 16-

 

(5050)10 → ( ? )16

 

Using Division method, we have-

 

 

From here, (5050)10 = (13BA)16

Thus, (11672)8 = (13BA)16

 

Problem-03:

 

Convert (2724)8 to ( ? )5

 

Solution-

 

Step-01: Conversion To Base 10-

 

(2724)8 → ( ? )10

 

Using Expansion method, we have-

(2724)8

= 2 x 83 + 7 x 82 + 2 x 81 + 4 x 80

= 1024 + 448 + 16 + 4

= (1492)10

 

From here, (2724)8 = (1492)10

 

Step-02: Conversion To Base 5-

 

(1492)10 → ( ? )5

 

Using Division method, we have-

 

 

From here, (1492)10 = (21432)5

Thus, (2724)8 = (21432)5

 

Problem-04:

 

Convert (3211)4 to ( ? )5

 

Solution-

 

Step-01: Conversion To Base 10-

 

(3211)4 → ( ? )10

 

Using Expansion method, we have-

(3211)4

= 3 x 43 + 2 x 42 + 1 x 41 + 1 x 40

= 192 + 32 + 4 + 1

= (229)10

 

From here, (3211)4 = (229)10

 

Step-02: Conversion To Base 5-

 

(229)10 → ( ? )5

 

Using Division method, we have-

 

 

From here, (229)10 = (1404)5

Thus, (3211)4 = (1404)5

 

Problem-05:

 

Convert (1001001100)2 to ( ? )6

 

Solution-

 

Step-01: Conversion To Base 10-

 

(1001001100)2 → ( ? )10

 

Using Expansion method, we have-

(1001001100)2

= 1 x 29 + 0 x 28 + 0 x 27 + 1 x 26 + 0 x 25 + 0 x 24 + 1 x 23 + 1 x 22 + 0 x 21 + 0 x 20

= 512 + 64 + 8 + 4

= (588)10

 

From here, (1001001100)= (588)10

 

Step-02: Conversion To Base 6-

 

(588)10 → ( ? )6

 

Using Division method, we have-

 

 

From here, (588)10 = (2420)6

Thus, (1001001100)2 = (2420)6

 

To gain better understanding about Conversion of Bases,

Watch this Video Lecture

 

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Summary
Converting Bases | Conversion of Bases
Article Name
Converting Bases | Conversion of Bases
Description
Conversion of Bases- To convert a number from any base to any other base, it is first converted to base 10 using expansion method and then it is converted from base 10 to required base using division & multiplication method.
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Publisher Name
Gate Vidyalay
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