Binary to decimal conversion Convert the binary number 11001001₂ to its decimal (base 10) value.
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A201
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B2001
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C20
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D210
Answer
Correct Answer: 201
Explanation
Introduction / Context:Converting a binary number to decimal reinforces the idea that each bit position represents a power of 2. Understanding positional weights lets you evaluate any base-2 number quickly and accurately.
Given Data / Assumptions:
- Binary number: 11001001₂.
- Bit weights from left (MSB) to right (LSB) for 8 bits: 128, 64, 32, 16, 8, 4, 2, 1.
- Only positions with a 1 contribute to the sum.
Concept / Approach:Write the binary digits aligned with their powers of two and add the weights where the bit is 1. This is the standard positional notation method for base conversion.
Step-by-Step Solution:
List bits: 1 1 0 0 1 0 0 1.Associate weights: 128, 64, 32, 16, 8, 4, 2, 1.Select weights with bit = 1: 128 + 64 + 8 + 1.Compute sum: 128 + 64 = 192; 192 + 8 = 200; 200 + 1 = 201.Verification / Alternative check:Group into hexadecimal nibbles: 1100 1001₂ = C9₁₆. Convert hex to decimal: C(12)*16 + 9 = 192 + 9 = 201, confirming the result.
Why Other Options Are Wrong:
- 2001: Misplaces the base-10 positional idea; binary 11001001₂ cannot reach thousands.
- 20: Under-counts by ignoring higher-order bits.
- 210: Off by 9 due to arithmetic or bit-weight mistake.
Common Pitfalls:Reading from the wrong end (mixing up MSB and LSB), forgetting a weight, or confusing binary with decimal digit concatenation.
Final Answer:201