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2's complement representation: What decimal number is represented by the binary string 11100101 in 8-bit 2's complement form?

Difficulty: Medium

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
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