Why can complex numbers be written in exponential form? $z=r(\cos \theta+i\sin \theta)$ is $z=re^{i\theta}$.     Why can complex numbers be written in exponential form? $z=r(\cos \theta+i\sin \theta)$ is $z=re^{i\theta}$.

I have studied that the exponential form of a complex number $z=r(\cos \theta+i\sin \theta)$ is $z=re^{i\theta}$.
Can someone explain why? 

There are several reasons. Even without going into the technical details of why it's correct, here is a small list of reasons for why might be a good idea:


*

*It's easy to use that form to read off the length and angle of your complex number

*It's easy to recognize the form and see that it is indeed meant to convey length and angle as opposed to, for instance, the width and height we see in the $a+bi$ form.

*The rules for complex multiplication means that hijacking the exponential notation and (mis)use the intuition you have from real exponentiation gives the correct results

