Describe the closure of the set with formulas

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Homework Help Overview

The discussion revolves around the closure of the set defined by the argument of complex numbers, specifically the condition -π < arg(z) < π, where z ≠ 0. Participants explore the implications of this condition in relation to the closure of the set and the nature of the argument function.

Discussion Character

  • Exploratory, Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants attempt to describe the closure of the set using various formulations and question whether the argument can equal specific values like π. There is discussion about the implications of the argument being an open set and how that relates to closure.

Discussion Status

There is an ongoing exploration of the definitions and properties of the argument function and its implications for the closure of the set. Some participants are clarifying their understanding of the argument and its relationship to the complex plane, while others are questioning the assumptions made about the values that arg(z) can take.

Contextual Notes

Participants note that the closure is not fully defined in their coursework, leading to confusion about the classification of the set as open or closed. The discussion includes references to the boundaries and inner regions of the set, as well as the need for visual aids to better understand the concepts being discussed.

alexcc11
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Homework Statement


-∏<arg(z)<∏ (z≠0)

Homework Equations


arg(z) is the angle from y=0

The Attempt at a Solution



Arg(z) spans the entire graph since -pi to pi is the full 360 degrees so I put:
-∏<arg(z)<∏ -->

0<arg(z)<2∏+k∏, (k ε Z) -->

arg(z) \subset R -->

arg(z) = R: all real numbers

but I don't know if that describes the closure since last time when describing the closure it would be Cl(E)={zεC :-∏<arg(z)<∏}

Would what I did be showing the closure with formulas? Also it's an open set, so I thought it would be an interior as oppose to a closure.
 
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alexcc11 said:

Homework Statement


-∏<arg(z)<∏ (z≠0)

Homework Equations


arg(z) is the angle from y=0

The Attempt at a Solution



Arg(z) spans the entire graph since -pi to pi is the full 360 degrees so I put:
-∏<arg(z)<∏ -->

0<arg(z)<2∏+k∏, (k ε Z) -->

arg(z) \subset R -->

arg(z) = R: all real numbers

but I don't know if that describes the closure since last time when describing the closure it would be Cl(E)={zεC :-∏<arg(z)<∏}

Would what I did be showing the closure with formulas? Also it's an open set, so I thought it would be an interior as oppose to a closure.

arg(z) isn't all real numbers. It can't be equal to π. Which number in (-π,π) is equivalent to π as an argument? Try and describe your set as a subset of the complex numbers.
 
180 is equivalent to pi. Wouldn't it only not be able to be: z≠0+2k*pi? Why can't it equal pi?
 
alexcc11 said:
180 is equivalent to pi. Wouldn't it only not be able to be: z≠0+2k*pi? Why can't it equal pi?

Because there is no number in (-π,π) that differs from π by a multiple of 2π. Notice the 'less than' signs, not 'less than or equal to'.
 
Why would it need to differ from pi by a multiple of 2pi? 2π+π=3π or π and 2π-π=π as well. They are the same thing
 
alexcc11 said:
Why would it need to differ from pi by a multiple of 2pi? 2π+π=3π or π and 2π-π=π as well. They are the same thing

You aren't getting me. -pi and pi aren't in the interval -pi<arg(z)<pi. What point in that interval could be equal to pi if you add 2*pi*k?
 
I see what you're saying. It would be all real numbers excluding any number differing from pi by 2pi*k right?
 
alexcc11 said:
I see what you're saying. It would be all real numbers excluding any number differing from pi by 2pi*k right?

Yes! So what portion of the complex plane is excluded? Can you describe it in words that don't involve 'arg'?
 
Cl(E)={zεC : z=C[STRIKE]ε[/STRIKE]∏+2k∏, k ε Z}

The closure is equivalent to the entire plane excluding z=pi+2k*pi where k is in integer.
 
  • #10
alexcc11 said:
Cl(E)={zεC : z=C[STRIKE]ε[/STRIKE]∏+2k∏, k ε Z}

The closure is equivalent to the entire plane excluding z=pi+2k*pi where k is in integer.

No, there's a big difference between z=pi and arg(z)=pi. Do you know what arg is? And we haven't even gotten to the closure Cl(E) yet, I'm still trying to get you describe the set E.
 
  • #11
arg(z) is the angle from the real axis to z. So arg(z) inplace of z in the closure? I just started this class this week, so sorry about that.
 
  • #12
alexcc11 said:
arg(z) is the angle from the real axis to z. So arg(z) inplace of z in the closure? I just started this class this week, so sorry about that.

That's ok, take it one step at a time. What complex numbers have an angle from the real axis of pi?
 
  • #13
We haven't really covered what arg(z) is equivalent too. I know it's the angle, but we never went over how to find the z value within arg(z). I looked it up and it says it's equivalent to atan (imag(z)/real(z)), but I've never heard of atan, so I'm lost there too. Can't you specify the closure with arg(z) instead of specifically using z?
 
  • #14
alexcc11 said:
We haven't really covered what arg(z) is equivalent too. I know it's the angle, but we never went over how to find the z value within arg(z). I looked it up and it says it's equivalent to atan (imag(z)/real(z)), but I've never heard of atan, so I'm lost there too. Can't you specify the closure with arg(z) instead of specifically using z?

You are making this much more complicated than it has to be. Look at http://mathworld.wolfram.com/ArgandDiagram.html Concentrate on the picture. Not the words. What complex numbers have arg(z)=pi?
 
  • #15
From the real axis to pi wouldn't it just be 0+ik to 180+ik where k is any real number?
 
  • #16
alexcc11 said:
From the real axis to pi wouldn't it just be 0+ik to 180+ik where k is any real number?

Look, 0+ik where k is real is the imaginary axis (x=0). 180+ik is line parallel to the imaginary axis 180 units to the right of it (x=180). I don't think that's what you mean. THINK about the what the symbols you are writing mean, ok? Look at the Argand diagram again.
 
  • #17
From the real axis to the imaginary axis it would be from 0 to 0+90i or 0+pi/2 i? and from the imaginary axis back down to the real axis would it be pi or 180 + 0i?
 
  • #18
alexcc11 said:
From the real axis to the imaginary axis it would be from 0 to 0+90i or 0+pi/2 i? and from the imaginary axis back down to the real axis would it be pi or 180 + 0i?

Angles aren't imaginary. One definition of arg(z) is the angle measured counterclockwise from the POSITIVE part of the real axis. As the picture was supposed to show. To go from that to the POSITIVE part of the imaginary axis is an angle of +pi/2, counterclockwise. What happens if you rotate by another +pi/2 in the same direction. Where do you wind up? And what is the arg of those numbers?
 
  • #19
pi/2 brings the angle to the imaginary axis and another pi/2 you're asking? It would bring it back down to the real axis at +pi. I don't understand what you're saying about the arg of those numbers. Isn't it just 0 to pi/2 to get to the imaginary axis and another pi/2 to get back down to the positive part of the real axis?
 
  • #20
alexcc11 said:
pi/2 brings the angle to the imaginary axis and another pi/2 you're asking? It would bring it back down to the real axis at +pi. I don't understand what you're saying about the arg of those numbers. Isn't it just 0 to pi/2 to get to the imaginary axis and another pi/2 to get back down to the positive part of the real axis?

If you rotate in opposite directions, yes. But you aren't getting the arg thing. Look, if I take a point on the positive real axis and rotate by pi, where do I wind up?
 
  • #21
You would end up on the opposite side rotated 180 degrees...
 
  • #22
alexcc11 said:
You would end up on the opposite side rotated 180 degrees...

Do you mean the NEGATIVE part of the real axis? I sure hope so... If so, then which complex numbers have arg(z)=pi?
 
  • #23
Yes. I get that. But I'm lost as to how use that with the problem?
 
  • #24
alexcc11 said:
Yes. I get that. But I'm lost as to how use that with the problem?

You didn't really answer my question, but I'll guess you do. Assuming you know what E is, then I'll ask, now what does closure (Cl(E)) mean?
 
  • #25
The closure of E is the the boundary of the closed set along with the inner regions, I thought. We haven't fully defined a closure in class, which is why I was unsure as to why this is classified as a closed set if it goes on forever everywhere when arg(z) doesn't equal pi+2kpi.
 
  • #26
alexcc11 said:
The closure of E is the the boundary of the closed set along with the inner regions, I thought. We haven't fully defined a closure in class, which is why I was unsure as to why this is classified as a closed set if it goes on forever everywhere when arg(z) doesn't equal pi+2kpi.

Closure of E is the "inner region" along with the "boundary". If (as I'll hope you'll agree) that -pi<arg(z)<pi=E is the whole complex plane EXCEPT for the negative real axis (which has the whole point so far) then it's NOT CLOSED. If you haven't fully defined closure, except in those terms, this may be a little iffy. You'll have to rely on pictures. What's the "inner region" of E and what's the "boundary" of E?
 
  • #27
I get that it's not closed since the set is open, with < as opposed to <=, but I don't understand how a set that is open, can still have a closure. I have a hard time picturing that. I do agree with the rest however.

I suppose the boundary would be the set it's self, -pi<arg(z)<pi or would that be the "inner region"?
 
  • #28
I'm going to head off to sleep now since it's 1am here, but thank you so much for your help. I get back to you tomorrow. Thanks again so much
 
  • #29
alexcc11 said:
I get that it's not closed since the set is open, with < as opposed to <=, but I don't understand how a set that is open, can still have a closure. I have a hard time picturing that. I do agree with the rest however.

I suppose the boundary would be the set it's self, -pi<arg(z)<pi or would that be the "inner region"?

The set itself is the inner region. The boundary points are the points that are on the edge of your set. They are zero distance from your set but not in the inner region. Can you think of any? I wish you had been given more precise definitions of these terms, are you sure you haven't??
 
  • #30
We haven't been given the definition. One of the other problems left in this problem set is to come up with a definition of a closure interms of a line R where R is all real numbers
 

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