Complex numbers - describe geometrically

In summary, the conversation discusses a complex numbers homework problem involving an equation in complex form. The question asks to describe geometrically the relationship between a line and the points (2,0) and (-2,0). The solution involves finding the equation of the line and understanding the concept of distance between points. The solution should be presented in words, describing the line as the perpendicular bisector of the line segment between the given points.
  • #1
4
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Hi!

I was just wondering if anyone would be able to help me with a question I received recently as complex numbers homework and didn't quite understand.

There was an equation given in complex form ie. something along the lines of |z-2|=|zconjugate=2| (I cannot remember this exactly now, which I know doesn't help, but nor do I know how to convert the y=-x equation back into the given form!)

The question then stated: "describe geometrically the relation between the line and the points (2,0) and (-2,0)" There was also a dotted line drawn through the points.

Homework Statement



"Describe geometrically the realtionship between the line and the points (2,0) and (0,-2)"


Homework Equations





The Attempt at a Solution



Well, I didn't actually understand what 'desribe geometrically' meant, but it was obvious that the lines intersected at the point (1,-1) and that they were perpendicular to each other. The equation of the line passing through the two points is y=x-2, and I also found that if the line y=-x was reflected about the line x=1 the other graph was obtained.

Aside from this, though, I don't actually understand what the question is asking or how I should answer it. Any help given would be greatly appreciated.

:):):)
 
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  • #2
jason1989 said:
Hi!

I was just wondering if anyone would be able to help me with a question I received recently as complex numbers homework and didn't quite understand.

There was an equation given in complex form ie. something along the lines of |z-2|=|zconjugate=2| (I cannot remember this exactly now, which I know doesn't help, but nor do I know how to convert the y=-x equation back into the given form!)
Since you don't give the precise equation, I can't say exactly what the line should be. |z- 2| can be interpreted geometrically as the distance from z= x+ iy interpreted as the point (x,y) to (2,0). |zconjugate- 2| is the distance from (x,-y) to (2,0). Unfortunately, It is true that the distance from any point (x,y) to (2, 0) is the same as the distance from (x,-y) to (2, 0). The equation you write doesn't say anything- it is true for all points.


The question then stated: "describe geometrically the relation between the line and the points (2,0) and (-2,0)" There was also a dotted line drawn through the points.

Homework Statement



"Describe geometrically the realtionship between the line and the points (2,0) and (0,-2)"


Homework Equations



If the equation were, for example, |z-2|= |z+ 2|, that is the set of points whose distance from (2,0) is the same as the distance from (-2, 0). That, geometrically, is the perpendicular bisector of the line segment from (2, 0) to (-2, 0).

The Attempt at a Solution



Well, I didn't actually understand what 'desribe geometrically' meant, but it was obvious that the lines intersected at the point (1,-1) and that they were perpendicular to each other. The equation of the line passing through the two points is y=x-2, and I also found that if the line y=-x was reflected about the line x=1 the other graph was obtained.

Aside from this, though, I don't actually understand what the question is asking or how I should answer it. Any help given would be greatly appreciated.

:):):)
 
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  • #3
ahhhh okay; thanks!

but I'm still not sure what I should present as a solution. Does 'describe geometrically' mean in terms of equations or words?

:D
 
  • #4
Words. In the example I gave, the description would be "the perpendicular bisector of the line segment between (-2, 0) and (2, 0)."
 

What are complex numbers?

Complex numbers are numbers that contain both a real and imaginary component. They are represented in the form a + bi, where a is the real part and bi is the imaginary part, with i being the imaginary unit (√-1).

How are complex numbers represented geometrically?

Complex numbers can be represented on a two-dimensional plane known as the complex plane. The real component of the complex number is plotted on the x-axis, while the imaginary component is plotted on the y-axis. The resulting point is known as the complex number's coordinates.

What is the geometric interpretation of the real and imaginary parts of a complex number?

The real part of a complex number represents the horizontal distance from the origin on the complex plane, while the imaginary part represents the vertical distance. Together, they form the coordinates of the complex number.

How do operations on complex numbers affect their geometric representation?

Operations such as addition, subtraction, multiplication, and division on complex numbers result in corresponding geometric transformations on the complex plane. Addition and subtraction correspond to translations, multiplication corresponds to scaling and rotation, and division corresponds to a combination of scaling and rotation.

What are some real-life applications of complex numbers and their geometric representation?

Complex numbers and their geometric representation are used in fields such as engineering, physics, and economics to model and solve problems involving alternating current, oscillations, and electromagnetic waves. They are also used in computer graphics and signal processing.

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