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

[Rubik]

Homework Statement

Find all complex solutions to $\bar{z}$ = z

Homework Equations

z = x + iy and $\bar{z}$ = x - iy

The Attempt at a Solution

What does it mean by find all complex solutions?

$\bar{z}$ = z
0 = x + iy - x + iy
0 = 2iy

 

[aanandpatel]

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.

 

[SammyS]

Homework Statement

Find all complex solutions to $\bar{z}$ = z

Homework Equations

z = x + iy and $\bar{z}$ = x - iy

The Attempt at a Solution

What does it mean by find all complex solutions?

$\bar{z}$ = 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?​

 

[aanandpatel]

2y = 0 and x=0

 

[HallsofIvy]

How do you arrive at "x= 0" from an equation that does not have an "x" in it??

 

[NewtonianAlch]

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.

 

[SammyS]

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 ?

 

[aanandpatel]

Yes when I say x=0 it means that the 'real part' of the solution is 0

 

[HallsofIvy]

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 $\overline{z}$?

 

[HallsofIvy]

Good point. I really hadn't thought of that! Okay, how about a simple sequence of real numbers?