Two's-complement limits For an 8-bit two's-complement binary number, what is the numerical range represented (inclusive)?

Introduction / Context:
Understanding the representable range of two's-complement integers is fundamental for overflow checks, embedded systems programming, and digital arithmetic design.

Given Data / Assumptions:

• Word size n = 8 bits.
• Two's-complement representation with one sign bit.

Concept / Approach:
The two's-complement range is: minimum = −2^(n−1), maximum = +2^(n−1) − 1. This asymmetry arises because there is only one representation for zero, leaving an extra negative value.

Step-by-Step Solution:
Compute min: −2^(8−1) = −2^7 = −128.Compute max: +2^(8−1) − 1 = +2^7 − 1 = +127.Therefore, range is −128 to +127 (base 10).

Verification / Alternative check:
List extreme bit patterns: 10000000₂ = −128; 01111111₂ = +127, confirming the range.

Why Other Options Are Wrong:
Options showing +128 as representable are incorrect in two's complement; +128 requires 9 bits.

Common Pitfalls:
Confusing sign-magnitude with two's complement or assuming symmetry about zero.

Final Answer:
–12810 to +12710