Question

Let the given numbers 11001, 1001, and 111001 correspond to the 2's complement representation. Then with which one of the following decimal numbers, the given numbers match?

• A. -25, -9, and -57, respectively
• B. -7, -7, and -7, respectively
• C. -6, -6, and -6, respectively
• D. 25, 9, and 57, respectively

Hint:

In 2's complement representation, the first bit indicates the sign. Convert the number by first inverting the bits and adding 1 to the result, then multiply by -1 for negative numbers.

Answer 1

The Correct Option is B

Step 1: Convert each binary number to decimal using 2's complement. 
- For \( 11001 \) in 2's complement:
- The first bit is 1, so it is negative.
- To find the decimal, take the 2's complement of \( 11001 \): \[ 11001 \quad {(invert)} \quad 00110 \quad {(add 1)} \quad 00111 = 7. \] Thus, \( 11001 \) corresponds to \( -7 \). - For \( 1001 \) in 2's complement:
- The first bit is 1, so it is negative.
- To find the decimal, take the 2's complement of \( 1001 \): \[ 1001 \quad {(invert)} \quad 0110 \quad {(add 1)} \quad 0111 = 7. \] Thus, \( 1001 \) corresponds to \( -7 \). - For \( 111001 \) in 2's complement:
- The first bit is 1, so it is negative.
- To find the decimal, take the 2's complement of \( 111001 \): \[ 111001 \quad {(invert)} \quad 000110 \quad {(add 1)} \quad 000111 = 7. \] Thus, \( 111001 \) corresponds to \( -7 \). 
Step 2: Conclusion. Thus, the given binary numbers correspond to \( -7, -7, { and } -7 \) in decimal. \[ \boxed{-7, -7, { and } -7}. \]