Complex Numbers: How do I know which plane to use? I'm new to complex numbers and I want to know precisely when I need to use which plane for graphing and a general idea of what the plot would look like.



$$z=a+bi$$ 

When it's in the form above, I know to interpret that as the point $(a,b)$. I would plot this on the complex plane using the Imaginary and Real axes as a point.



$$z=re^{i\theta}$$

When it's in this form, I assume that I'd have to also use the complex plane. I know that the radius is $r$ and that you go counter-clockwise around the origin according to the value of $\theta$. However, I'm not sure how to actually draw it. Would it be a ray or a line? And for either one, does it include the origin?
Point $z=re^{i\theta}$ is given in polar coordinates. It means that you have a radius $r$ and an angle $\theta$. You can always go back to cartesian coordinates using the formula:

$$re^{i\theta}=r\cdot \cos(\theta)+r\cdot i\sin(\theta)$$

There's a nice representations of both coordinate systems at Wikimedia:
$re^{i\theta}=r\cos\theta+ir\sin\theta$

Hence 
$$a=r\cos\theta$$
$$b=r\sin\theta$$
and you can plot it in exactly the same way as the $a+bi$ example, by first calculating the values $a$ and $b$ using the above formulae.