Complex Plane ( $\arg(z)$) Sketch the following regions of the complex plane. For each, say whether it is open, closed, or neither, and whether it is connected. No proofs necessary.
$$\left\{z \in \mathbb{C}\mid -\dfrac{\pi}{2} < \arg(z) < \dfrac{\pi}{4}\right\}$$
So, for this problem I am not sure what $\arg(z)$ means because our teacher never went over it. So I have no idea how to attack this problem. Could someone help explain to me what $\arg(z)$ means and how I would go about drawing this in the complex plane please?
Thank You
 A: $\arg(z)$ is the angle between the line joining your point $z=a+ib$ and the origin, in the complex plane. It becomes visually striking when you are introduced to the polar form of complex numbers, but from the situation you describe, you have just started working on complex numbers.
So, you're looking at the region of all the points in the complex plane that are form an angle between $-\frac{\pi}{2}$ and $\frac{\pi}{4}$ with the origin. Basically, you're going counter-clockwise from the negative imaginary axis (the vertical axis), to the line making a 45° ($\frac{\pi}{4}$) angle with respect to the origin.
This region is (simply) connected, as you can partition it in as many subsets as you want, and it is open, as it extends to infinity (i.e. it is not completely bounded).
A: It's complicated to explain without drawing; the best, it seems to me, is to draw a picture and then invite people to invent their own examples.
There are many notions required in the exercise but they are all very intuitive and easy to understand intuitively with the support of drawing.

*

*An argument of $Z'$ is $\color{red}\alpha$ expressed in radians; An argument of $Z''$ is $\color{green}{-\beta}$ expressed in radians;

*$S:=\left\{z \in \mathbb{C}\mid -\dfrac{\pi}{2} < \arg(z) < \dfrac{\pi}{4}\right\}$ is all in one piece: it is said to be "connected" in mathematics.

*Since the inequalities are strict and we cannot speak of the argument of $0$, for each point $Z$ of $S$, we have a disk with center $Z$, which is completely contained in $S$: we says $S$ is "open" in mathematics.

These notions (connected, open, closed, ...) are basic notions of the part of mathematics called "general topology" and the complex plane is the ideal place to become familiar with.

