Complex Matrices and Unit Circles

[tatianaiistb]

Homework Statement

What can you say about
a. the sum of a complex number and its conjugate?
b. the conjugate of anumber on the unit circle?
c. the product of two numbers on the unit circle?
d. the sum of two numbers on the unit circle?

Homework Equations

The Attempt at a Solution

Here's what I'm thinking:
a. The sum of a complex number and its conjugate is real: (a+bi)+(a-bi)=2a
b. The conjugate of a number on the unit circle LIES ON the unit circle.
c. The product of two numbers on the unit circle also LIES ON the unit circle.
d. The sum of two numbers on the unit circle LIES INSIDE OR OUTSIDE the unit circle.

Am I thinking correctly or am I missing something? I'm unsure on this one. Thanks!

 

[Mark44]

Homework Statement

What can you say about
a. the sum of a complex number and its conjugate?
b. the conjugate of anumber on the unit circle?
c. the product of two numbers on the unit circle?
d. the sum of two numbers on the unit circle?

Homework Equations

The Attempt at a Solution

Here's what I'm thinking:
a. The sum of a complex number and its conjugate is real: (a+bi)+(a-bi)=2a
b. The conjugate of a number on the unit circle LIES ON the unit circle.
c. The product of two numbers on the unit circle also LIES ON the unit circle.
d. The sum of two numbers on the unit circle LIES INSIDE OR OUTSIDE the unit circle.

Am I thinking correctly or am I missing something? I'm unsure on this one. Thanks!

I don't see anything wrong with what you've said, but they might be looking for more than you've said.

For a) Yes, the sum is real, but notice what you have in your formula.
For b) is it possible to say where on the unit circle the conjugate would be? Let z = √2/2 + i√2/2, which is on the unit circle. Where is [itex]\overline{z}[/itex]?
For c), if z₁ and z₂ are on the unit circle, what can you say about z₁z₂?
For d), same question, but about z₁ + z₂.

The key is to think geometrically - draw some pictures.

 

[Deveno]

what about 1 and -1/2 + i√3/2?

 

[tatianaiistb]

Hmmm... I drew some pictures as suggested... I'm still unsure, but here's another attempt:

a) I forgot to mention that the sum is real and is twice the real part
b) 90 degree reflection about the origin (for it's imaginary part)?
For parts c and, after I drew the pictures, I still cannot see it. Are there any particular numbers I should be looking at that are easier to see? Thanks!

 

[Deveno]

Hmmm... I drew some pictures as suggested... I'm still unsure, but here's another attempt:

a) I forgot to mention that the sum is real and is twice the real part
b) 90 degree reflection about the origin (for it's imaginary part)?
For parts c and, after I drew the pictures, I still cannot see it. Are there any particular numbers I should be looking at that are easier to see? Thanks!

a) yes
b) what does "a reflection about the origin" even mean? reflections usually involve a line, or a plane, or some higher-dimensional-thingy
c) my advice: google "de moivre's theorem"
d) do you know how to draw a vector sum?

 

[tatianaiistb]

for part b, is it sufficient to say 90 deg reflection about the Imaginary axis?

I'll follow your advice and google "de moivre's theorem"

For part d, I haven't done that in a long time, so I'll also have to look it up

thanks

 

[Deveno]

degrees normally have to do with rotation (they "twist" or "turn"). think "mirror-like" when thinking about reflections.

to draw a vector sum, make a parallelogram (2 sides of this will be your 2 vectors starting at the origin, the other two sides will be the same two vectors drawn "head to tail"), and draw the diagonal, which represents the vector sum.

 

[tatianaiistb]

Another try:

For part b, it'd be a reflection about the real axis?

For part c, the product of two numbers on the unit circle will still have length 1 and will be located at the sum of their angles? e.g., (0+i)(1+0i) = i, where 0+i is located 90deg and 1+0i at 0 deg, so the product will be at 90 deg.

For part d, their sum will still have length 1?

 

[tatianaiistb]

For part d, I'm thinking it should be still length 1 half way between the two numbers...

 

[Mark44]

Another try:

For part b, it'd be a reflection about the real axis?

Yes, and I think this is what they had in mind.

For part c, the product of two numbers on the unit circle will still have length 1 and will be located at the sum of their angles? e.g., (0+i)(1+0i) = i, where 0+i is located 90deg and 1+0i at 0 deg, so the product will be at 90 deg.

Yes. This is more like what they're looking for, IMO.

For part d, their sum will still have length 1?

For part d, I'm thinking it should be still length 1 half way between the two numbers...

Yes.

 

[tatianaiistb]

Thank you for your help!