Graph of a Complex Number (basic concept)

[DEMJ]

Homework Statement

Sketch $$0 \le arg z \le \frac{\pi}{4}$$ $$(z \not= 0)$$

The Attempt at a Solution

I know from my book that his is a punctured disk aka deleted neighborhood only because it says so and because it is in the form of $$0 < \mid z - z_0 \mid < \epsilon$$. I honestly have no clue how to graph this or even visual how it is a disk with a hole in it. Anyone care to mention or explain anything that will help me understand this concept and eventually be able to graph it. Thank you.

 

[Dick]

It's not a punctured disk. If you can explain to me what you think arg(z) is, I think I can explain what the graph looks like. If you can explain that clearly, you probably won't need me to explain what the graph looks like.

 

[DEMJ]

honestly I always think of it as the angle in radians you get when you graph a complex number z = x + iy but this may be wrong =[

 

[Dick]

honestly I always think of it as the angle in radians you get when you graph a complex number z = x + iy but this may be wrong =[

No, that's exactly right. So the region you are graphing should look more like a wedge, shouldn't it?

 

[DEMJ]

I do not even know how to begin graphing this so I honestly do not see how it looks like a wedge...sigh.

All I can visualize is a vector z = x + iy with theta = pi/4 but this is can't be right and 0 must be involved somehow

 

[Dick]

Sigh. The set of points where arg(z)=0 is the positive x-axis, right? The set of points where arg(z)=pi/4 is a ray at 45 degrees to the x-axis, right? What you want are the points whose angles are in between. I would call that a "wedge". What would you call it?

 

[DEMJ]

Wow I feel embarrassed how straight forward it is yet I could not grasp it by myself LOL, and I would call it Sergeant Dick Amazing (lol just kidding) and thank you for clearing this up for me kind sir.

 

[Dick]

You'll do better next time, right? Not all 'complex' problems are hard.