Question

A decimal number has 64 digits. The number of bits needed for its equivalent binary representation is
1. 200
2. 213
3. 246
4. 277

Answer 1

"Correct Answer - Option 2 : 213

The correct answer is “option 2”.

CONCEPT :

The highest decimal number in n digit number is = Maximum binary number in x bit

CALCULATION:
The highest decimal number in n digit number is (10ⁿ - 1). 

Example: 

For  n = 3, highest decimal number is (10³ - 1) = 999.

The maximum binary number in x bit is (2^x -1). 

Example :  For  x = 3, maximum binary number in 3 bit is (2³ - 1) = 7.

The Highest decimal number in n digit number is = Maximum binary number in x bit

(10n - 1) = (2x -1)

10n = 2x
For n =  64 :

10 = 2^x            

log₂(10⁶⁴ )  =  log₂(2^x)        { taking log both sides }

64 * log₂(10) =  x * log₂(2) 

64 * 3.219 = x * 1            { log10 = 3.219 , log2 = 1 }

x = 212.6 

x = 213

Hence, the number of bits needed for its equivalent binary representation is 213 bits. "