A complex number #z# is always expressible uniquely as #a+ib#, where #a, b in RR#. That is it is expressible as a point #(a, b)# in #RR xx RR#.

Any such point can also be represented using polar coordinates as #(r cos theta, r sin theta)# for some radius #r >= 0# and angle #theta in RR#.

The point #(r cos theta, r sin theta) corresponds to the complex number:

#r cos theta + r i sin theta = r(cos theta + i sin theta)#

Given #z = a+ib#, we can calculate a suitable #r#, #cos theta# and #sin theta# ...

#r = sqrt(a^2 + b^2)#

#cos theta = a / r#

#sin theta = b / r#

One of the nice things about #cos theta + i sin theta# is Euler's formula:

#cos theta + i sin theta = e^(itheta)#

So polar cis form is equivalent to #re^(i theta)#