Doubt about condition solutions of complex line equation

In summary, the conversation discusses doubts about Theorem 3.3.1 and the conditions for satisfying equations involving complex numbers. The first attachment mentions 8 cases for the condition |a| = |b|, but the second attachment shows that this is not always the case. The conversation also addresses the meaning of "in the direction of b" and requests examples of complex numbers b.
  • #1
Dear All,
Please help me clear some doubts about Theorem 3.3.1 in the 1st attachment.
The condition ## |a| = |b| ## has only 8 cases right? ## { x+iy. x - iy, -x + iy, -x - iy, y + ix, y - ix, -y + ix, -y - ix } ##

so for the condition ## |a| = |b| ## and ## b \bar c = \bar a c ## in (2) and (3) in the attachement, what must ##b## and ##\bar a## be for them to to satisfy this equation given that C is a complex number of the form ## Cx + iCy ##.

why is there a line of solutions in (3)? usually you only get one imaginary and one real value for z right??

In the 2nd attachement, I have tried to do question 3. what does in the direction of b mean? does it pass through b also?? c =0 right? can you please give examples of the complex number b?

Danke...
 

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  • #2
PcumP_Ravenclaw said:
Dear All,
Please help me clear some doubts about Theorem 3.3.1 in the 1st attachment.
The condition ## |a| = |b| ## has only 8 cases right? ## { x+iy. x - iy, -x + iy, -x - iy, y + ix, y - ix, -y + ix, -y - ix } ##
No, and how are x + iy, x - iy, etc. conditions? These are complex numbers. I don't see that they are conditions in any way. For example, consider a = 0 + 1i and b = -1/2 + (√3/2)i. These two complex numbers have the same magnitude, but I don't see how they are included in your eight cases, whatever it is you mean by them.

PcumP_Ravenclaw said:
so for the condition ## |a| = |b| ## and ## b \bar c = \bar a c ## in (2) and (3) in the attachement, what must ##b## and ##\bar a## be for them to to satisfy this equation given that C is a complex number of the form ## Cx + iCy ##.

why is there a line of solutions in (3)? usually you only get one imaginary and one real value for z right??

In the 2nd attachement, I have tried to do question 3. what does in the direction of b mean?
b is a complex number, so you can think of it as being a vector, with its tail at the origin and its head at the point in the complex point (b1, b2). What direction does it point?
PcumP_Ravenclaw said:
does it pass through b also?? c =0 right? can you please give examples of the complex number b?

Danke...
 
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Likes PcumP_Ravenclaw

1. What is a complex line equation?

A complex line equation is an equation that describes a line in a two-dimensional complex plane. It is represented in the form of z = a + bi, where z is a complex number, a and b are real numbers, and i is the imaginary unit (equal to the square root of -1).

2. How do you solve a complex line equation?

To solve a complex line equation, you need to find the values of a and b that satisfy the equation. This can be done through algebraic manipulation and using the properties of complex numbers, such as conjugation and multiplication by the imaginary unit. Alternatively, you can also plot the equation on a complex plane and find the points that lie on the line.

3. Can a complex line equation have more than one solution?

Yes, a complex line equation can have infinitely many solutions. This is because any point on the line can be represented as a complex number z = a + bi, where a and b satisfy the equation. Additionally, a complex line equation can also have no solutions if the line does not intersect with the complex plane.

4. What is the difference between a real line equation and a complex line equation?

A real line equation describes a line in a two-dimensional real plane, while a complex line equation describes a line in a two-dimensional complex plane. Real numbers are only composed of a single dimension, while complex numbers have both a real and an imaginary component.

5. What is the importance of solving complex line equations in scientific research?

Complex line equations are used in many scientific fields, including physics, engineering, and mathematics. They are particularly useful in analyzing systems with complex variables, such as electrical circuits, quantum mechanics, and fluid dynamics. Solving complex line equations allows for a better understanding and prediction of these systems, leading to advancements in technology and scientific knowledge.

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