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Question 8 ( Conversion from base 5 system into decimal system)

Question 8: Convert ​\( (413242)_5 \)​ into equivalent decimal system.

a) 13572                  b) 13762                c)13751                    d) 13753

Solution:

 

Here the number of digits in the given term is 6 – so base 5 will have powers of 5,4,3,2,1 and 0.

To convert given term into a decimal system we multiply first digit i-e 4 with highest power of base 5 i-e with ​\( 5^5 \)​ and plus it with second digit i-e 1 multiplied by second highest power 4 of base 5 i-e with ​\( 5^4 \)​ – plus it with third digit i-e 3 multiplied with third highest power of base 5 i-e with ​\( 5^3 \)​ ………… and so on till – plus the last digit i-e 2 multiplied with last and the least power of base 5 i-e with ​\( 5^0 \)​.

 

Mathematically,

\[ (413242)_5=4\times 5^5+ 1\times 5^4+ 3 \times 5^3+ 2\times 5^2+ 4 \times 5^1 + 2\times 5^0 \]

(Note if any term has power 0 then it is equal to 1, so the last term will be 2 x 1)

Simplifying,

\[ (413242)_5= 4\times 3125+ 1\times 625+ 3\times 125+ 2 \times 25+ 4 \times 5+ 2\times 1 \]

This gives,

\[ (413245)5= 12500+625+375+50+20+20+2 = 13572 \]

 

So the given number is equivalent to 13572 in decimal system

 

 

 

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