Binary fraction to decimal — Convert the binary number 1001.0010 (base 2) into its decimal (base-10) equivalent.

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:

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