Complex Numbers: How do I know which plane to use?

Question

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?

Answer 1

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:

Answer 2

Note that the two planes represent the same thing, with points drawn in different methods. This is expected since $re^{i \theta}$ and $a+bi$ both just represent complex numbers.

A point on the plane can be identified by its real and imaginary coordinates - this is the case of $a + bi$ since we identify it as the point $(a, b)$.

A point on the plane can also be identified by its distance $r$ from the origin and the angle $\theta$ it makes with the positive $x$-axis (can you see why this uniquely determines every point on the plane except for $(0,0)$?). This is the case for $r e^{i \theta}$.