Converting to base 10 | Number System Conversions

Conversion to base 10-

 

In number system, it is very important to have a good knowledge of how to convert numbers from one base to another base.

In this article, we will learn how to convert any given number in any base to a number in base 10.

 

(Given Number)any base   (?)10

 

Expansion Method-

 

We can easily convert any given number in base x to a number in base 10 using expansion method.

If abc.de is any given number in base x, then using expansion method, its value in base 10 will be given by-

 

(abc.de)x = (ax2 + bx + c + dx-1 + ex-2)10

 

Explanation-

 

  • Assign position number to each digit of the number given in base x as shown.
  • Digits to the left of decimal are numbered starting from 0.
  • Digits to the right of decimal are numbered starting from -1.
  • Write a term for each digit as digit x (base)position number of digit
  • Then, perform the addition of those terms as shown to obtain the number in base 10.

 

 

We can expand if the number of digits are more in the given number.

 

Special Case-

 

If abc is any given number in base x, then using expansion method, its value in base 10 will be given by-

 

(abc)x = (ax2 + bx + c)10

 

PRACTICE PROBLEMS BASED ON CONVERSION TO BASE 10-

 

Problem-

 

Convert the following numbers to base 10-

  1. (10010)2
  2. (254)8
  3. (AC)16
  4. (10010.101)2
  5. (254.7014)8
  6. (AC.FBA5)16
  7. (0.1402)8
  8. (0.ABDF)16

 

Solution-

 

1. (10010)2

 

(10010)2 → ( ? )10

 

Using expansion method, we have-

(10010)2

= ( 1 x 24 + 0 x 2+ 0 x 22 + 1 x 21 + 0 x 20 )10

= ( 16 + 0 + 0 + 2 + 0 )10

= ( 18 )10

 

2. (254)8

 

(254)8 → ( ? )10

 

Using expansion method, we have-

(254)8

= ( 2 x 82 + 5 x 8+ 4 x 80 )10

= ( 128 + 40 + 4 )10

= ( 172 )10

 

3. (AC)16

 

(AC)16 → ( ? )10

 

Using expansion method, we have-

(AC)16

= ( A x 161 + C x 160 )10

= ( 10 x 16 + 12 x 1 )10

= ( 160 + 12 )10

= ( 172 )10

 

4. (10010.101)2

 

(10010.101)2 → ( ? )10

 

Using expansion method, we have-

(10010.101)2

= ( 1 x 24 + 0 x 2+ 0 x 22 + 1 x 21 + 0 x 20 + 1 x 2-1 + 0 x 2-2 + 1 x 2-3 )10

= ( 16 + 0 + 0 + 2 + 0 + 0.5 + 0.125 )10

= ( 18.625 )10

 

5. (254.7014)8

 

(254.7014)8 → ( ? )10

 

Using expansion method, we have-

(254.7014)8

= ( 2 x 82 + 5 x 8+ 4 x 80 + 7 x 8-1 + 0 x 8-2 + 1 x 8-3 + 4 x 8-4 )10

= ( 128 + 40 + 4 + 0.875 + 0.0019 + 0.0009 )10

= ( 172.8778 )10

 

6. (AC.FBA5)16

 

(AC.FBA5)16 → ( ? )10

 

Using expansion method, we have-

(AC.FBA5)16

= ( A x 161 + C x 160 + F x 16-1 + B x 16-2 + A x 16-3 + 5 x 16-4 )10

= ( 10 x 16 + 12 x 1 + 15 x 16-1 + 11 x 16-2 + 10 x 16-3 + 5 x 16-4 )10

= ( 160 + 12 + 0.9375 + 0.0429 + 0.0024 + 0.0001 )10

= ( 172.9829 )10

 

7. (0.1402)8

 

(0.1402)8 → ( ? )10

 

Using expansion method, we have-

(0.1402)8

= ( 0 x 80 + 1 x 8-1 + 4 x 8-2 + 0 x 8-3 + 2 x 8-4 )10

= ( 0 + 0.125 + 0.0625 + 0 + 0.0005 )10

= ( 0.188 )10

 

8. (0.ABDF)16

 

(0.ABDF)16 → ( ? )10

 

Using expansion method, we have-

(0.ABDF)16

= ( 0 x 160 + A x 16-1 + B x 16-2 + D x 16-3 + F x 16-4 )10

= ( 0 x 1 + 10 x 16-1 + 11 x 16-2 + 13 x 16-3 + 15 x 16-4 )10

= ( 0 + 0.625 + 0.0429 + 0.0032 + 0.0002 )10

= ( 0.6713 )10

 

Get more notes and other study material of Number System.

Watch video lectures by visiting our YouTube channel LearnVidFun.

Summary
Converting to base 10 | Number System Conversions
Article Name
Converting to base 10 | Number System Conversions
Description
In this article, we will learn how to convert any given number in any base to a number in base 10 using expansion method and then we will solve problems based on number conversions to base 10.
Author
Publisher Name
Gate Vidyalay
Publisher Logo
Liked this article? Share it with your friends and classmates now-