14.5 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:=r0eθ0W:=r_{0}e^{\theta_{0}} and Z:=r1eθ1Z:=r_{1}e^{\theta_{1}} then we can perform the operation using the laws of indices

WZ\displaystyle WZ =(r0eθ0)(r1eθ1)\displaystyle=\left(r_{0}e^{\theta_{0}}\right)\left(r_{1}e^{\theta_{1}}\right) (14.53)
=r0r1eθ0+θ1\displaystyle=r_{0}\cdot r_{1}\cdot e^{\theta_{0}+\theta_{1}} (14.54)

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=eπ2i=e^{\frac{\pi}{2}}, then we are effectively just rotating that complex number by π2 rad\frac{\pi}{2}\text{ rad} (aka 9090^{\circ}).