Introduction / Context:
Converting from decimal (base-10) to binary (base-2) is foundational in computer architecture and number systems. The standard approach repeatedly divides the decimal number by 2, tracking remainders, or uses powers of 2 decomposition.
Given Data / Assumptions:
- Decimal input: 71₁₀.
- We want an unsigned binary representation.
- Binary weights: 1, 2, 4, 8, 16, 32, 64, ...
Concept / Approach:
- Power-of-two decomposition: represent 71 as a sum of distinct powers of two.
- Alternatively, use successive division by 2 and read remainders in reverse.
Step-by-Step Solution:
Find largest power of 2 ≤ 71 → 64 = 2^6.Compute remainder: 71 - 64 = 7.Express 7 as 4 + 2 + 1 → 2^2 + 2^1 + 2^0.Set bits at positions 6, 2, 1, 0 → 1 000 111.Final binary: 1000111₂.
Verification / Alternative check:
Convert back: 164 + 032 + 016 + 08 + 14 + 12 + 1*1 = 64 + 0 + 0 + 0 + 4 + 2 + 1 = 71₁₀.
Why Other Options Are Wrong:
- 110011₂: Equals 51₁₀, not 71.
- 1110011₂: Equals 115₁₀, not 71.
- 0110011₂: Equals 51₁₀ with a leading zero.
- None of the above: Incorrect because 1000111₂ is correct.
Common Pitfalls:
- Reversing the order of remainders when using the division method.
- Dropping significant leading 1 in the highest place (2^6 here).
Final Answer:
1000111₂
Discussion & Comments