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-

• Decimal to Binary Conversion
• Decimal to Octal Conversion
• Decimal to Hexadecimal Conversion

 

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

 

Also Read- http://www.exam-labs.com

 

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)₂ = (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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