The smallest integer that can be represented by an 8-bit number in 2's…

Concept

In an n-bit two's complement representation, the most significant bit is the sign bit, and the representable integers form the range −2n−1 to 2n−1 − 1. The range is asymmetric: there is exactly one extra negative value because the single zero pattern frees one code on the positive side.

Application

1. Here n = 8, so the lower bound is −28−1 = −27.

2. Evaluate the power: 27 = 128, so the smallest (most negative) value is −128.

3. The upper bound is 27 − 1 = 127, giving the full range −128 to 127.

Cross-check

Count the patterns: 8 bits give 28 = 256 codes. These cover −128, …, −1 (128 negatives), 0, and 1, …, 127 (127 positives): 128 + 1 + 127 = 256. The count matches, confirming the smallest value is −128.