# Complex Solutions

## 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

## The Attempt at a Solution

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
Staff Emeritus
Homework Helper
Gold Member

## 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?​

2y = 0 and x=0

HallsofIvy
Homework Helper
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.

SammyS
Staff Emeritus
Homework Helper
Gold Member
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

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 $\overline{z}$?
Good point. I really hadn't thought of that! Okay, how about a simple sequence of real numbers?