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

  • Thread starter Rubik
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In summary: If you had a sequence of real numbers like this:1, 3, 5, 7, 9, 11, 13, 15, etc...would the sequence satisfy the equation?Yes, it would.
  • #1
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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  • #2
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.
 
  • #3
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?​
 
  • #4
2y = 0 and x=0
 
  • #5
How do you arrive at "x= 0" from an equation that does not have an "x" in it??
 
  • #6
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.
 
  • #7
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 ?
 
  • #8
Yes when I say x=0 it means that the 'real part' of the solution is 0
 
  • #9
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?
 

What are complex solutions?

Complex solutions refer to solutions of a mathematical equation that involve the use of imaginary numbers (numbers containing the imaginary unit, denoted by "i"). These solutions are often used in fields such as engineering, physics, and mathematics.

How are complex solutions different from real solutions?

Complex solutions differ from real solutions in that they involve the use of imaginary numbers. Real solutions, on the other hand, only involve the use of real numbers (numbers that can be plotted on a number line).

Why are complex solutions important?

Complex solutions are important because they allow for the representation of certain mathematical phenomena that cannot be represented using only real numbers. They are also essential in solving equations and problems in various fields, such as electrical engineering and quantum mechanics.

What are some applications of complex solutions?

Complex solutions have numerous applications, such as in AC circuit analysis, signal processing, and control systems. They are also used in solving differential equations, which are commonly used in physics and engineering problems.

How can one solve for complex solutions?

To solve for complex solutions, one can use tools such as the quadratic formula or the cubic formula. These formulas involve the use of imaginary numbers and can be used to find both real and complex solutions to an equation. Additionally, some mathematical software programs also have capabilities to solve for complex solutions.

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