2's complement representation: What decimal number is represented by the binary string 11100101 in 8-bit 2's complement form?
Electronics and Communication Engineering
Digital Electronics
Difficulty: Medium
Choose an option
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A+37
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B-31
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C+27
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D-27
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E-59
Answer
Correct Answer: -27
Explanation
Introduction / Context:Two's complement is the standard representation for signed integers in modern digital systems. Interpreting an 8-bit two's complement binary number requires identifying whether it is positive or negative, and then decoding it properly.
Given Data / Assumptions:
- Binary string: 11100101.
- Word size: 8 bits.
- Representation: 2's complement.
Concept / Approach:In 2's complement, the most significant bit (MSB) is the sign bit. If MSB = 0, the number is positive and interpreted directly. If MSB = 1, the number is negative, and its magnitude is obtained by inverting all bits and adding 1.
Step-by-Step Solution:
Binary given: 11100101.MSB = 1 ⇒ number is negative.Invert bits: 00011010.Add 1: 00011011 (decimal 27).Therefore, value = −27.Verification / Alternative check:
Direct conversion: interpret as unsigned = 229. For 8-bit 2's complement, subtract 256: 229 − 256 = −27. Same result.Why Other Options Are Wrong:
+37, +27: positive, contradict MSB = 1.−31: would correspond to binary 11100001.−59: unrelated to this binary string.Common Pitfalls:
Forgetting to apply 2's complement inversion +1.Confusing sign-magnitude with 2's complement.Final Answer:
-27