Decimal fraction to binary conversion Convert the decimal number 151.75 (base 10) to binary (base 2).

Difficulty: Medium

Correct Answer: 10010111.11

Explanation:


Introduction / Context:
Converting mixed decimal numbers (integer plus fraction) to binary involves two coordinated processes: repeated division by 2 for the integer part and repeated multiplication by 2 for the fractional part. Mastery of both is essential for representing fixed-point values in binary.


Given Data / Assumptions:

  • Decimal input: 151.75.
  • We want an exact finite binary since 0.75 is a dyadic fraction.
  • No rounding is required for this case.


Concept / Approach:
Split the number into 151 (integer) and 0.75 (fraction). Convert 151 using division-by-2; convert 0.75 by multiplying by 2 repeatedly and recording integer bits produced at each step.


Step-by-Step Solution (Integer):

1) 151 decimal → binary: 151 = 128 + 16 + 4 + 2 + 1.2) Place bits: 2^7=128 → 1; 2^6=64 → 0; 2^5=32 → 0; 2^4=16 → 1; 2^3=8 → 0; 2^2=4 → 1; 2^1=2 → 1; 2^0=1 → 1.3) Integer part = 10010111.


Step-by-Step Solution (Fraction):

1) 0.75 * 2 = 1.5 → record 1, keep 0.5.2) 0.5 * 2 = 1.0 → record 1, fraction ends.3) Fractional part = .11.


Verification / Alternative check:
Reconstruct: 10010111.11₂ = 151 + 0.75 = 151.75, exact.


Why Other Options Are Wrong:

  • 10000111.11: equals 135.75, not 151.75.
  • 11010011.01: equals 211.25 in decimal.
  • 00111100.00: equals 60 exactly.


Common Pitfalls:
Placing 1s for the wrong powers of 2, or mishandling the fractional multiply-by-2 steps.


Final Answer:
10010111.11

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