Equation of a Line: Solving Complex & Real Parts for x, y

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In summary, the conversation discusses a complex equation and its solutions, which can result in either an empty set, a point, or a line. The example given of z+z*=0 is clarified to show that the solution is not just a point but the entire imaginary axis. The conversation also mentions a possible error in the book and asks for clarification on that.
  • #1
loonychune
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Just want a check of this please:

We have a complex equation of the form az+bz*+c=0
where a, b and z are complex #s, c is real...
If you take the real and imaginary parts of such an equation you obtain two linear equations in x and y, whose solutions of each gives rise to a line (L_1 and L_2 respectively)...
The set then, of solutions, is L_1 unison L_2

Now, the set of solutions of the complex equation is either empty, a point, or a line......the book gives these 3 examples as each case:
z + z* = i

z+2z* = 0

z+z* = 0


I don't understand how z+z*=0 is a line... for we in fact have
RE(Z+Z*)=2x=0
IM(Z+Z*)= 0y = 0
which then gives rise to a point solution does it not??

if it was z+z* = c say, then i could see that having a line of solutions but as it is, i reckon the book has made an error... is this the case?

(perhaps i have confused the issue and if that is the case maybe then you coudl point out how my thinking is wrong)

THANKYOU :)
 
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  • #2
x=0 and y=0 refer not to points but to the y- and x- axis respectively, which you can see are lines, it's like saying for every value of y, x=0 or any constant so it's not a point but infact a line. I hope this explanation was coherent.
 
  • #3
loonychune said:
Just want a check of this please:
I don't understand how z+z*=0 is a line... for we in fact have
RE(Z+Z*)=2x=0
IM(Z+Z*)= 0y = 0
which then gives rise to a point solution does it not??
:)

It does not.

Yes 2x = 0 only has the solution that x=0. However 0y = 0 does not just have the solution that y=0, rather it has the solution y=anything. This makes the solutuion the entire imaginary axis (x=0), does that make sense.
 
  • #4
Yeah it makes sense, thanks a lot the both of you...
Physicsforums again proves a real gem..
 

1. What is the equation of a line?

The equation of a line is a mathematical representation of a straight line on a graph. It is usually written in the form y = mx + b, where m is the slope of the line and b is the y-intercept (the point where the line crosses the y-axis).

2. How do you solve for the complex parts of an equation of a line?

To solve for the complex parts of an equation of a line, you can use the quadratic formula or complete the square. If the equation is in standard form (Ax + By = C), you can use the formula x = (-B ± √(B^2 - 4AC)) / 2A to find the x-intercepts of the line. The complex parts will be the imaginary solutions of this equation.

3. What do the x and y values represent in an equation of a line?

The x and y values in an equation of a line represent the coordinates of any point on the line. The x-value tells you the horizontal position of the point, and the y-value tells you the vertical position.

4. How do you solve for the real parts of an equation of a line?

To solve for the real parts of an equation of a line, you can use algebraic methods such as substitution or elimination. These methods involve manipulating the equation to isolate the variable and solve for its value. Sometimes, graphing the line can also help visualize the real parts of the equation.

5. Can an equation of a line have both real and complex parts?

Yes, an equation of a line can have both real and complex parts. This usually happens when the line intersects with the imaginary axis on a graph. In this case, the equation will have both real and imaginary solutions for the x-intercepts, and the line will have a complex slope.

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