Complex Solutions for \bar{z} = z: Find All Possible Solutions

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Homework Help Overview

The discussion revolves around finding all complex solutions to the equation \(\bar{z} = z\), where \(z\) is expressed in terms of its real and imaginary components.

Discussion Character

  • Exploratory, Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants explore the meaning of finding all complex solutions and question how to equate real and imaginary parts. There is discussion on the implications of the equation \(0 = 2iy\) and what it suggests about the values of \(x\) and \(y\).

Discussion Status

Participants are actively questioning assumptions about the relationship between \(x\) and \(y\) in the context of the equation. Some guidance is offered regarding the interpretation of the absence of \(x\) in certain equations, but there is no explicit consensus on the implications of these interpretations.

Contextual Notes

There is a focus on the equality of real and imaginary parts of complex numbers, and some participants express confusion regarding how to derive values for \(x\) when it does not appear in certain forms of the equation.

Rubik
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Homework Statement


Find all complex solutions to [itex]\bar{z}[/itex] = z


Homework Equations



z = x + iy and [itex]\bar{z}[/itex] = x - iy

The Attempt at a Solution


What does it mean by find all complex solutions?

[itex]\bar{z}[/itex] = z
0 = x + iy - x + iy
0 = 2iy
 
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Two complex numbers are only equal if their real parts are equal and their imaginary parts are equal so you may have to equate real and imaginary parts to find the values of x and y.
 
Rubik said:

Homework Statement


Find all complex solutions to [itex]\bar{z}[/itex] = z


Homework Equations



z = x + iy and [itex]\bar{z}[/itex] = x - iy

The Attempt at a Solution


What does it mean by find all complex solutions?

[itex]\bar{z}[/itex] = z
0 = x + iy - x + iy
0 = 2iy
If 0 = 2iy, then ...
1. What must x be for this to be true?

2. What must y be for this to be true?​
 
2y = 0 and x=0
 
How do you arrive at "x= 0" from an equation that does not have an "x" in it??
 
Well the basic form is x + iy, so we know the x part of the complex number must be equal to zero if it's not there.
 
NewtonianAlch said:
Well the basic form is x + iy, so we know the x part of the complex number must be equal to zero if it's not there.
Look at the equation (from post #1):
0 = x + iy - x + iy​
Is there any x that will not satisfy this, if y=0 ? If there is such an x, what is it ?
 
Yes when I say x=0 it means that the 'real part' of the solution is 0
 
Yes, and as you have been told repeatedly, that is wrong. The equation 2iy= 0 does NOT say "x= 0 because x isn't there". The fact that x is not in that equation means that the equation does not tell you anything about x. Suppose z= 4+ 0i. What is [itex]\overline{z}[/itex]?
 
  • #10
Good point. I really hadn't thought of that!:redface: Okay, how about a simple sequence of real numbers?
 

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