Question

The 8-bit signed 2's complement representation of -26 is

• A. 01100110
• B. 11100101
• C. 11100110
• D. 01100101

Hint:

A quick shortcut to find 2's complement:
Write down the positive binary number: \( 00011010 \).
Scanning from right to left, keep all bits the same up to and including the first '1':
The rightmost part is '10'.
Invert all the bits to the left of this first '1':
\( 000110 \rightarrow 111001 \).
Combine them to get: \( 11100110 \).

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The question requires us to find the 8-bit signed 2's complement representation of the decimal integer \( -26 \).

Step 2: Key Formula or Approach:
The steps to find the 2's complement representation of a negative number \( -X \) are:
1. Find the 8-bit binary representation of the corresponding positive number \( +X \).
2. Perform the 1's complement by inverting all the bits ($0 \to 1$ and $1 \to 0$).
3. Add 1 to the least significant bit (LSB) of the 1's complement result.

Step 3: Detailed Explanation:

• Convert positive 26 to 8-bit binary:
\[ 26 = 16 + 8 + 2 = 2^4 + 2^3 + 2^1 \] Thus, in 8-bit binary:
\[ +26 = 00011010_2 \]
• Find the 1's complement of \( 00011010_2 \):
Invert each bit:
\[ \text{1's complement} = 11100101_2 \]
• Find the 2's complement by adding 1:
\[ \text{2's complement} = 11100101_2 + 1 \] Performing binary addition:
\[ 11100101 + 00000001 = 11100110_2 \]
• Verify the result using weights:
The weights of the bits from MSB to LSB are \( -128, 64, 32, 16, 8, 4, 2, 1 \).
\[ \text{Value} = -128(1) + 64(1) + 32(1) + 16(0) + 8(0) + 4(1) + 2(1) + 1(0) \] \[ \text{Value} = -128 + 64 + 32 + 4 + 2 = -26 \] This confirms that \( 11100110_2 \) is correct.