Solving Inequality With Complex Numbers Question

[sarahs52]

"Solving Inequality With Complex Numbers" Question

Homework Statement

What does the inequality pz + conjugate(pz) + c < 0 represent if |p|^2 >c ?

Homework Equations

p is a constant and a member of the set of complex numbers. c is a constant and a member of the set of real numbers.

The Attempt at a Solution

First, for the EQUALITY pz + conjugate(pz) + c = 0, where p = a+ib, z = x+iy :

pz + conjugate(pz) + c = 0 => (a+ib)(x+iy) + conjugate((a+ib)(x+iy)) + c = 0 => ax + iay + ibx - by + ax - iay -ixb -by + c = 0 => y=((a/b)*x)+ c/(2*b).

So, the inequality is y> ((a/b)*x)+ c/(2*b).

MY QUESTION IS: WHAT IS THE IMPORTANCE OF --> |p|^2 >c WHEN SHOWING THE INEQUALITY --> pz + conjugate(pz) + c < 0.

I know that |p| = sqrt((a^2) + (b^2)), so |p|^2 >c implies ((a^2) + (b^2)) > c but I don't understand how this helps to show the the inequality pz + conjugate(pz) + c < 0... or more so, the inequality y > ((a/b)*x)+ c/(2*b).

ALSO, if I were to graph this inequality, would it be a (diagonal) dotted line (I say dotted because points on the line are not a part of the solution) where the graph is shaded above the dotted line?

Any help would be greatly appreciated.
Thank you.

 

[HallsofIvy]

Homework Statement

What does the inequality pz + conjugate(pz) + c < 0 represent if |p|^2 >c ?

It doesn't represent anything because the complex numbers do NOT form an "ordered field". There is no way to define "<" for complex numbers consistent with their arithmetic.

Homework Equations

p is a constant and a member of the set of complex numbers. c is a constant and a member of the set of real numbers.

The Attempt at a Solution

First, for the EQUALITY pz + conjugate(pz) + c = 0, where p = a+ib, z = x+iy :

pz + conjugate(pz) + c = 0 => (a+ib)(x+iy) + conjugate((a+ib)(x+iy)) + c = 0 => ax + iay + ibx - by + ax - iay -ixb -by + c = 0 => y=((a/b)*x)+ c/(2*b).

So, the inequality is y> ((a/b)*x)+ c/(2*b).

MY QUESTION IS: WHAT IS THE IMPORTANCE OF --> |p|^2 >c WHEN SHOWING THE INEQUALITY --> pz + conjugate(pz) + c < 0.

I know that |p| = sqrt((a^2) + (b^2)), so |p|^2 >c implies ((a^2) + (b^2)) > c but I don't understand how this helps to show the the inequality pz + conjugate(pz) + c < 0... or more so, the inequality y > ((a/b)*x)+ c/(2*b).

ALSO, if I were to graph this inequality, would it be a (diagonal) dotted line (I say dotted because points on the line are not a part of the solution) where the graph is shaded above the dotted line?

Any help would be greatly appreciated.
Thank you.

 

[I like Serena]

It doesn't represent anything because the complex numbers do NOT form an "ordered field". There is no way to define "<" for complex numbers consistent with their arithmetic.

That doesn't seem to be quite right, since the resulting expression is a real number.
And for real numbers the "<" inequality is defined.

pz + conjugate(pz) = 2 Re(pz)

 

[I like Serena]

I don't see any particular relevance either to |p|² > c.

So, the inequality is y> ((a/b)*x)+ c/(2*b).

Not quite.
The direction of the inequality depends on the sign of b.
If b is negative, the inequality is inverted.


And I agree that you get a shaded half surface bounded by your dotted line.
You only have to be careful, which half surface you have exactly.

 

[HallsofIvy]

That doesn't seem to be quite right, since the resulting expression is a real number.
And for real numbers the "<" inequality is defined.

pz + conjugate(pz) = 2 Re(pz)

Right, I completely missed that.