Consider the unsigned 8-bit fixed point binary number representation below, b7…
2018
Consider the unsigned 8-bit fixed point binary number representation below,
b7 b6 b5 b4 b3 . b2 b1 b0
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
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Correct answer: C
Key facts: The binary point is between b3 and b2, so there are 5 integer bits (b7..b3) and 3 fractional bits (b2..b0).
Resolution and representable fractional parts:
Smallest fractional step = 2⁻³ = 0.125.
Representable fractional parts are multiples of 0.125: 0, 0.125, 0.25, 0.375, 0.5, 0.625, 0.75, 0.875.
Evaluate each number:
31.500: Integer part 31 is the maximum representable with 5 bits, fractional 0.5 = 4 × 0.125, so 31.500 is exactly representable (binary example: 11111.100).
0.875: This equals 7/8 = 0.875 = 7 × 0.125, so it is exactly representable (fractional bits 111).
12.100: Fractional part 0.100 (decimal) is not a multiple of 0.125 (0.100/0.125 = 0.8), so it cannot be represented exactly.
3.001: Fractional part 0.001 is not a multiple of 0.125, so it cannot be represented exactly.
Conclusion: Only 12.100 and 3.001 cannot be exactly represented in this fixed-point format.
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