Multiplying Complex Numbers

Understanding complex numbers: A complex number can be expressed in the form z = a + bi, where a and b are real numbers and i is the imaginary unit with the property that i^2 = -1.
Complex conjugates: The conjugate of a complex number z = a + bi is denoted as z* = a - bi.
Multiplying complex numbers: To multiply complex numbers, take the product of the magnitudes and add the angles (when they are written in polar form).
Distribution law: To multiply complex numbers like (a + bi) and (c + di), follow the distribution law: ac + adi + bci - bd. The last term is real because i^2 = -1.
Real part and imaginary part: The real part of a product of two complex numbers is ac - bd, and the imaginary part is ad + bc.
Application of FOIL method: Multiply complex numbers using the FOIL (First, Outside, Inside, Last) method, clearly separating the real and imaginary components of the result.
Interaction with real numbers: When multiplying a complex number by a real number, distribute over both real and imaginary parts: a (b + ci) = ab + aci.

Complex numbers in the Argand diagram: When multiplying complex numbers, the angles (Arg(z)) add up, while the magnitudes (

) multiply.

Polar form multiplication: Multiplication is significantly easier if the complex numbers are expressed in polar form: r(cos Θ + i sin Θ). The magnitudes are multiplied and the angles are added.
Power of complex numbers: To raise a complex number to a power, raise the magnitude to the power and multiply the angle by the power.