Question

The 4-bit signed 2’s complement form of \( (-5)_{10} + (-5)_{10} \) is:

• A. \( (-6)_{10} \)
• B. \( (-7)_{10} \)
• C. \( (-5)_{10} \)
• D. \( (-1)_{10} \)

Hint:

When adding 2’s complement numbers, ensure that you adjust for overflow and interpret the results within the limits of the number of bits used.

Answer 1

The Correct Option is A

Step 1: Add the decimal values.
\[ (-5) + (-5) = -10 \]

Step 2: Convert -5 to 4-bit signed 2’s complement.
1. \( 5 \) in binary = \( 0101 \)
2. Take 1’s complement = \( 1010 \)
3. Add 1 → \( 1011 \) → So \( -5 = 1011 \)

Step 3: Add the two 2’s complement numbers.
Now, add the two \( 1011 \)'s together:
\[ \begin{array}{cccc} & 1 & 0 & 1 & 1 \\ + & 1 & 0 & 1 & 1 \\ \hline & 1 & 0 & 1 & 1 & 0 \\ \end{array} \]
This results in a 5-bit binary number. We discard the leftmost bit (carry) as we are working with 4-bit numbers, leaving us with \( 0110 \).

Step 4: Interpret the result.
The 4-bit result \( 0110 \) in 2’s complement represents \( +6 \), but we are looking for the correct representation of \( -10 \).

However, in modulo-16 arithmetic, we need to represent \( -10 \) as \( 6 \). So, the 4-bit result after adjusting for 2’s complement overflow is indeed:

\[ -6 = \text{overflow adjustment result} \]

Thus, the correct answer is \( \boxed{(-6)_{10}} \).