Binary fraction to decimal — Convert the binary number 1001.0010 (base 2) into its decimal (base-10) equivalent.
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A125
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B12.5
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C90.125
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D9.125
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E9.5
Answer
Correct Answer: 9.125
Explanation
Introduction:Binary numbers with fractional parts are common in fixed-point representations, digital filters, and timing calculations. Converting them into decimal clarifies magnitudes and assists in sanity checks when designing or debugging digital systems.
Given Data / Assumptions:
- Binary number: 1001.0010 (base 2).
- Digits to the left of the point carry weights 2^n, n ≥ 0.
- Digits to the right carry fractional weights 2^(−n), n ≥ 1.
Concept / Approach:
Expand the value as a weighted sum of powers of two. The integer portion 1001_2 equals 8 + 1 = 9. The fractional portion .0010_2 has a single 1 in the 2^(−3) place, equal to 1/8 = 0.125. Summing yields the final decimal result.
Step-by-Step Solution:
Integer side: 12^3 + 02^2 + 02^1 + 12^0 = 8 + 0 + 0 + 1 = 9.Fractional side: 02^(−1) + 02^(−2) + 12^(−3) + 02^(−4) = 0 + 0 + 0.125 + 0 = 0.125.Add: 9 + 0.125 = 9.125.Therefore, 1001.0010_2 = 9.125_10.Verification / Alternative check:
Convert to an exact fraction: 9 + 1/8 = 73/8. Dividing 73 by 8 indeed gives 9.125, confirming the calculation without rounding error.
Why Other Options Are Wrong:
- 125 / 12.5 / 90.125 / 9.5: These result from misplacing the binary point or misweighting fractional bits (e.g., treating 2^(−1) as 1/10).
Common Pitfalls:
- Confusing binary fraction weights with decimal fractions; right of the binary point halves with each step.
- Dropping leading or trailing zeros that actually set bit positions and therefore the correct weights.
Final Answer:
9.125