Question

find the distance between -1 - 2i and 4 + 4i \sqrt{\boxed{?}}

Explanation:

Step1: Identify real and imaginary parts

For complex number \( z_1 = -1 - 2i \), real part \( x_1=-1 \), imaginary part \( y_1 = -2 \).
For complex number \( z_2 = 4 + 4i \), real part \( x_2 = 4 \), imaginary part \( y_2 = 4 \).

Step2: Apply distance formula for complex numbers

The distance between two complex numbers \( z_1 = x_1 + y_1i \) and \( z_2 = x_2 + y_2i \) is \( \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \).
Substitute values:
\( x_2 - x_1 = 4 - (-1) = 5 \)
\( y_2 - y_1 = 4 - (-2) = 6 \)

Step3: Calculate the expression inside the square root

Compute \( (x_2 - x_1)^2 + (y_2 - y_1)^2 \):
\( 5^2 + 6^2 = 25 + 36 = 61 \)

Answer:

\( 61 \)