A calculus of the absurd

15.5 Properties of the exponential form of a complex number

It is substantially easier to multiply together complex numbers in exponential form (than their Cartesian counterparts). For example, if we want to multiply \(W := r_0 e ^{\theta _0}\) and \(Z := r_1e^{\theta _1}\) then we can perform the operation using the laws of indices

\begin{align} WZ &= \rbrackets {r_0 e ^{\theta _0}}\rbrackets {r_1e^{\theta _1}} \\ &= r_0\cdot r_1\cdot e^{\theta _0 + \theta _1} \end{align}

This means that when we multiply two complex numbers, we obtain a new complex number whose argument is the sum of the arguments of the two (complex) numbers which we multiplied together, and whose modulus is just the product of the two (complex) numbers we multiplied together.

This allows us to perform some transformations. For example, if we multiply a complex number by \(i = e^{\frac {\pi }{2}}\), then we are effectively just rotating that complex number by \(\frac {\pi }{2} \text { rad}\) (aka \(90^{\circ }\)).