Complex Numbers Proof

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TheWrathOfMath

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Z1, Z2, Z3, Z4, Z5 are complex numbers such that |Z1|=|Z2|=|Z3|=|Z4|=|Z5|=1.
Prove that |Z1+Z2+Z3+Z4+Z5|=|1/Z1+1/Z2+1/Z3+1/Z4+1/Z5|.
 
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TheWrathOfMath said:
Z1, Z2, Z3, Z4, Z5 are complex numbers such that |Z1|=|Z2|=|z3|=|z4|=|z5|=1.
Prove that |Z1+Z2+z3+z4+z5|=|1/Z1+1/Z2+1/z3+1/z4+1/z5|.
Click to expand...
You have (small) 'z' and (Capital) 'Z' - in the expression above. What is/are the difference/s between those two ('z' & 'Z') ?

Please show us what you have tried and exactly where you are stuck.

Please follow the rules of posting in this forum, as enunciated at:


Please share your work/thoughts about this problem.
 
Subhotosh Khan said:
You have (small) 'z' and (Capital) 'Z' - in the expression above. What is/are the difference/s between those two ('z' & 'Z') ?

Please show us what you have tried and exactly where you are stuck.

Please follow the rules of posting in this forum, as enunciated at:


Please share your work/thoughts about this problem.
Click to expand...
It is a mistake. They should all be either upper or lower case.
 
TheWrathOfMath said:
z1, z2, z3, z4, z5 are complex numbers such that |z1|=|z2|=|z3|=|z4|=|z5|=1.
Prove that |z1+z2+z3+z4+z5|=|1/z1+1/z2+1/z3+1/z4+1/z5|.
Click to expand...
Can you express 1/z differently? What is it equal to if |z| = 1?

Please show some effort, so we can see where you need help. You're been told that at least twice already.
 
Subhotosh Khan said:
You have (small) 'z' and (Capital) 'Z' - in the expression above. What is/are the difference/s between those two ('z' & 'Z') ?

Please show us what you have tried and exactly where you are stuck.

Please follow the rules of posting in this forum, as enunciated at:


Please share your work/thoughts about this problem.
Click to expand...
I do not know where to start.
This feels like an extremely easy problem, though.
Shame on me.
 
TheWrathOfMath said:
1/z=1 as well, I believe.
Click to expand...
Really? So 1/i = 1 ??

I'm not just asking about |1/z|, if that's what you were thinking.

And don't just answer the specific question; apply it to your problem. It's true that |1/z| = 1, but that wouldn't be useful, would it?
 
TheWrathOfMath said:
Never mind. I see that you are not interested in helping me solve this question.
Thank you anyways.
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I don't see how you can say that @Dr.Peterson isn't interested in HELPING you solve the problem. He asked you guided questions and provided constructive criticism of your thinking. IMO, you're not interested in doing the problem.
 
BigBeachBanana said:
I don't see how you can say that @Dr.Peterson isn't interested in HELPING you solve the problem. He asked you guided questions and provided constructive criticism of your thinking. IMO, you're not interested in doing the problem.
Click to expand...
I do not understand this topic at all.
My teacher finished the topic of complex numbers after only three hours of teaching it, with a break in-between.
Therefore, I do not even know where to start.
@Dr.Peterson, I apologize if my above comment may have come off as impertinent . I am simply extremely frustrated and disheartened at the moment, though this is not an excuse.
 
TheWrathOfMath said:
I do not understand this topic at all.
My teacher finished the topic of complex numbers after only three hours of teaching it, with a break in-between.
Therefore, I do not even know where to start.
@Dr.Peterson, I apologize if my above comment may have come off as impertinent . I am simply extremely frustrated and disheartened at the moment, though this is not an excuse.
Click to expand...
So please cooperate! Help us help you, by making an effort, both to use our hints, and to let us know what sort of help will work.

Do you know how to divide by a complex number? If no, then look it up; it is obviously necessary for this problem! If yes, think about how the answer might be relevant; and at the least, TELL US what you know, so we can help you see how it is relevant, or suggest an alternative that is.

More generally, you might list the facts you were taught in those three hours. Tell us the context -- is this part of a course you're taking? What course? What else do you know?
 
Dr.Peterson said:
So please cooperate! Help us help you, by making an effort, both to use our hints, and to let us know what sort of help will work.

Do you know how to divide by a complex number? If no, then look it up; it is obviously necessary for this problem! If yes, think about how the answer might be relevant; and at the least, TELL US what you know, so we can help you see how it is relevant, or suggest an alternative that is.

More generally, you might list the facts you were taught in those three hours. Tell us the context -- is this part of a course you're taking? What course? What else do you know?
Click to expand...
I believe that in order to divide by a complex number, both numerator and denominator should be multiplied by the conjugate of the denominator.

The facts that I were taught include algebraic and polar form of complex numbers and identities, such as:
|z1|*|z2|=|z1*z2|, z*z(conjg)=|z|^2, z is a purely real number if z=z{conjg), z is a purely imaginary number if z=-z(conjg), and so on.
 
TheWrathOfMath said:
I believe that in order to divide by a complex number, both numerator and denominator should be multiplied by the conjugate of the denominator.

The facts that I were taught include algebraic and polar form of complex numbers and identities, such as:
|z1|*|z2|=|z1*z2|, z*z(conjg)=|z|^2, z is a purely real number if z=z{conjg), z is a purely imaginary number if z=-z(conjg), and so on.
Click to expand...
Okay. Keep thinking: When |z| = 1, what does 1/z equal? So what is another way to write 1/z1+1/z2+1/z3+1/z4+1/z5?

I don't need to drag you all the way to the answer; I want to give you a little nudge and see you roll as far as you can.

I'd also like answers to my other questions. Is this part of a college course, or a random video you looked at because you were bored? Context can make a big difference in our knowledge of what we can expect of you.
 
Hint: Complex numbers can be written as [imath]z_1 = r_1 e^{i \theta _1}[/imath]. It's much easier to attack this problem with polar form instead of using z = x + iy.

As I said elsewhere, try this with just two terms. The overall proof is almost identical to this.

-Dan
 
Dr.Peterson said:
Okay. Keep thinking: When |z| = 1, what does 1/z equal? So what is another way to write 1/z1+1/z2+1/z3+1/z4+1/z5?

I don't need to drag you all the way to the answer; I want to give you a little nudge and see you roll as far as you can.

I'd also like answers to my other questions. Is this part of a college course, or a random video you looked at because you were bored? Context can make a big difference in our knowledge of what we can expect of you.
Click to expand...
|z|=1, which means that every z we have is located on the unit circle.
If z=r*cis(thea), then 1/z=1/r*cis(360-thea).

Since r=1, 1/z will have the same radius (also on the unit circle), the same cos(theta) value, but the opposite sin(theta) value.

In other words, if |z|=1, then 1/z=z(conjg).
 
TheWrathOfMath said:
In other words, if |z|=1, then 1/z=z(conjg).
Click to expand...
Correct, though this really isn't much more than what you already said about division:

[math]\frac{1}{z}=\frac{1\cdot \bar{z}}{z\cdot \bar{z}}=\frac{\bar{z}}{|z|^2} = \bar{z}[/math]
Please don't stop at every step and wait for approval! Keep thinking.

How does z1+z2+z3+z4+z5 relate to 1/z1+1/z2+1/z3+1/z4+1/z5?

Then, how does |z1+z2+z3+z4+z5| relate to|1/z1+1/z2+1/z3+1/z4+1/z5|?
 
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