Quick questions on Complex numbers - TCE Mathematics Specialised (Tasmania)

In Cartesian (rectangular) form we write $z = x + iy$, where $x = \operatorname{Re}(z)$ is the real part and $y = \operatorname{Im}(z)$ is the imaginary part. Addition and subtraction are done component by component:

Use Cartesian form for addition and subtraction, and for stating exact real and imaginary parts. Use polar or exponential form for multiplication, division, powers and roots. The exponential form $z = re^{i\theta}$ is the same information as polar form, and it makes the multiplication and division rules obvious because exponents add and subtract: $r_1 e^{i\theta_1}\cdot r_2 e^{i\theta_2} = r_1 r_2 e^{i(\theta_1 + \theta_2)}$. Switching at the right moment is what separates fast, clean solutions from long algebraic struggles, so practise converting both ways until it is automatic.