Question

complete each of the following statements given ( z = 1 - 2i ) and ( w = -3 + 5i ).( z + w = -2 + square i )( z - w = square - 7i )( w - z = -4 + square i )

Explanation:

Step1: Calculate \( z + w \)

To add complex numbers \( z = 1 - 2i \) and \( w = -3 + 5i \), we add the real parts and the imaginary parts separately.
The real part: \( 1 + (-3) = -2 \)
The imaginary part: \( -2i + 5i = 3i \)
So \( z + w = -2 + 3i \)

Step2: Calculate \( z - w \)

To subtract complex numbers \( z = 1 - 2i \) and \( w = -3 + 5i \), we subtract the real parts and the imaginary parts separately.
The real part: \( 1 - (-3) = 1 + 3 = 4 \)
The imaginary part: \( -2i - 5i = -7i \)
So \( z - w = 4 - 7i \)

Step3: Calculate \( w - z \)

To subtract complex numbers \( w = -3 + 5i \) and \( z = 1 - 2i \), we subtract the real parts and the imaginary parts separately.
The real part: \( -3 - 1 = -4 \)
The imaginary part: \( 5i - (-2i) = 5i + 2i = 7i \)
So \( w - z = -4 + 7i \)

Answer:

For \( z + w = -2 + \square i \), the answer is \( 3 \).
For \( z - w = \square - 7i \), the answer is \( 4 \).
For \( w - z = -4 + \square i \), the answer is \( 7 \).