GATE | GATE CS 2018 | Question 53

Consider the unsigned 8-bit fixed point binary number representation, below,

b₇b₆b₅b₄b₃ ⋅ b₂b₁b₀

where the position of the binary point is between b₃ and b₂ . Assume b₇ is the most significant bit. Some of the decimal numbers listed below cannot be represented exactly in the above representation:

(i) 31.500 (ii) 0.875 (iii) 12.100 (iv) 3.001

Which one of the following statements is true?

(A) None of (i), (ii), (iii), (iv) can be exactly represented
(B) Only (ii) cannot be exactly represented
(C) Only (iii) and (iv) cannot be exactly represented
(D) Only (i) and (ii) cannot be exactly represented


Answer: (C)

Explanation:
(i) 31.500 can be represented as :
(31.5)₁₀ = (11111.100)₂

24 + 23 22 + 21 + 20 + 2-1
  = 16 + 8 + 4 + 2 + 1 + 0.5 = (31.05)10

(ii) 0.875 can be represented as :
(0.875)₁₀ = (00000.111)₂

 2-1 + 2-2 2-3
   = 0.50 + 0.25 + 0.125
   = (0.875)10

(iii) and (iv) can not be be represented.

Therefore, option (C) is Correct.

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