Explain why the expression (c+di) ^2 is always a complex number for nonzero, real values of c and d

(c + di) ^2 = c^2 + 2cdi + (di) ^2

= c^2 + 2cdi - d^2

= c^2 - d^2 + 2cdi

The real part is c^2 - d^2. The imaginary part is cdi.

Since c and d are nonzero real numbers, the product cd is also nonzero and real. Even if c and d are equal, and the real part c^2 - d^2 becomes zero, the imaginary part, cdi will always have a nonzero coefficient (cd). Therefore, the imaginary part is always there.