Introduction / Context:
Binary-to-hexadecimal conversion is a core skill in computer organization and digital electronics. It relies on grouping binary digits in sets of four (nibbles) because each nibble maps directly to one hexadecimal digit. This question checks your fluency with base conversions and nibble mapping.
Given Data / Assumptions:
- Binary number: 101000010111₂
- No sign bit is implied; treat as an unsigned binary value.
- Hexadecimal digits map from 0000₂→0 to 1111₂→F.
Concept / Approach:
- Pad the left of the binary string with zeros, if needed, so its length is a multiple of 4.
- Split into 4-bit groups (nibbles) from left to right.
- Convert each nibble to its hexadecimal digit and concatenate the results.
Step-by-Step Solution:
Start with 101000010111₂.Group into nibbles: 1010 0001 0111.Map each nibble: 1010₂ → A; 0001₂ → 1; 0111₂ → 7.Concatenate: A17₁₆.
Verification / Alternative check:
Convert A17₁₆ back to binary: A → 1010, 1 → 0001, 7 → 0111, yielding 1010 0001 0111, which matches the original.
Why Other Options Are Wrong:
- D8F9₁₆ and D9F8₁₆: These would imply binary nibbles beginning with 1101 (D) and include F (1111) which are not present in the original grouping.
- A8B9₁₆: Would require nibbles 1010 1000 1011 1001, which do not match.
- None of the above: Incorrect because A17₁₆ exactly matches.
Common Pitfalls:
- Grouping from the right but forgetting to pad on the left to complete the leftmost nibble.
- Confusing nibble values such as 1010 (A) versus 1001 (9).
- Dropping leading zeros within a nibble (e.g., 0001 must remain 0001).
Final Answer:
A17₁₆
Discussion & Comments