1. **State the problem:** Perform the subtraction $-35 - 27$ using 8-bit 2's complement representation. 2. **Understand 2's complement:** To represent negative numbers in 8 bits, we use 2's complement. The range is from $-128$ to $127$. 3. **Convert $-35$ to 8-bit 2's complement:** - First, convert $35$ to binary: $35_{10} = 00100011_2$ - Find 1's complement: invert bits $11011100$ - Add 1 to get 2's complement: $$11011100 + 1 = 11011101$$ So, $-35$ in 8-bit 2's complement is $11011101$. 4. **Convert $27$ to 8-bit binary:** $27_{10} = 00011011_2$ 5. **Perform subtraction $-35 - 27$ as addition:** $-35 - 27 = -35 + (-27)$ We need $-27$ in 2's complement: - $27$ in binary: $00011011$ - 1's complement: $11100100$ - Add 1: $$11100100 + 1 = 11100101$$ So, $-27$ is $11100101$. 6. **Add $-35$ and $-27$ in binary:** $$ \begin{array}{c} 11011101 \ + 11100101 \ \hline 110000010 \end{array} $$ Since we are using 8 bits, discard the 9th bit (carry out): Result = $10000010$ 7. **Interpret the result $10000010$:** - The leading bit is 1, so it's negative. - Find its magnitude by taking 2's complement: - 1's complement: $01111101$ - Add 1: $$01111101 + 1 = 01111110$$ - $01111110_2 = 126_{10}$ - So the result is $-126$. 8. **Check for overflow:** - Overflow occurs if adding two numbers with the same sign produces a result with a different sign. - Here, both operands are negative, and the result is negative, so no overflow. **Final answer:** $$-35 - 27 = -126$$