How Do You Solve Complex Number Equations in Quadratics and Exponentials?

[jisbon]

You don't have to assume $b = 0$. You should be able to show that $b = 0$ and $a \ge 0$.

Showing b=0 is done on the very first part where there is no imaginary number on the right is it?

 

[PeroK]

Showing b=0 is done on the very first part where there is no imaginary number on the right is it?

Yes.

 

[jisbon]

Yes.

So since I managed to show, I can just sub in:
a+bi+7(a−bi)=∣(a+4)+bi∣
where b =0,
4/7 + 7(4/7) = (4/7) +4
which is true.
Hence z = 4/7 + 0i?

 

[PeroK]

So since I managed to show, I can just sub in:
a+bi+7(a−bi)=∣(a+4)+bi∣
where b =0,
4/7 + 7(4/7) = (4/7) +4
which is true.
Hence z = 4/7 + 0i?

What I would have done is noted that $|z + 4| = c$, where $c \ge 0$.

That shows that $b = 0$ and $a \ge 0$ (do the algebra). You don't have to guess or assume that $b = 0$.

Also, a little trick, as $a \ge 0$, we have $|z + 4| = |a + 4| = a + 4$

And that avoids having to square the equation.

 

[jisbon]

What I would have done is noted that $|z + 4| = c$, where $c \ge 0$.

That shows that $b = 0$ and $a \ge 0$. You don't have to guess or assume that $b = 0$.

Also, a little trick, as $a \ge 0$, we have $|z + 4| = |a + 4| = a + 4$

And that avoids having to square the equation.

Oh okay. So by proving b=0, we can now assume z = 4/7 +0i right?
Also for the next question, I've edited to include my initial answers. Do you think they are correct? Thanks.

 

[PeroK]

Oh okay. So by proving b=0, we can now assume z = 4/7 +0i right?
Also for the next question, I've edited to include my initial answers. Do you think they are correct? Thanks.

I think I'd post the remaining questions in a new thread.

 

[jisbon]

I think I'd post the remaining questions in a new thread.

I could post them. Oh and for the previous question, it will be z=4/7+0i right? Thanks

 

[PeroK]

I could post them. Oh and for the previous question, it will be z=4/7+0i right? Thanks

You should be able to check that for yourself. Once you know that $z$ is real, it's a fairly simple question.

 

[jisbon]

You should be able to check that for yourself. Once you know that $z$ is real, it's a fairly simple question.

Yep checked it's correct.
Here's the new link: https://www.physicsforums.com/threads/help-with-complex-numbers-2.976744/
Attempted 2 of them, I really do not know how to proceed with the last question unfortunately

 

[scottdave]

Homework Statement: NIL
Homework Equations: NIL

Hello all!
Thanks for helping me out so far :) Really appreciate it.
I don't seem to understand some of the questions presented to me, so if anyone has an idea on how to start the questions, please do render your assistance :)
7)
Let
$\sum_{k=0}^9 x^k = 0$
Find smallest positive argument. Same thing as previous question, but I guess I can expand to
z+z2+z3+...+z9=0z+z2+z3+...+z9=0
$z=re^iθ$
$rei^θ+re^2iθ+re^3iθ+...$
What do I do to proceed on?
Cheers

This would be easier if you ask only 1 (or 2 max, if they are pretty simple) questions at a time.
On your summation, you start with x^k then switch to x^k when you expand it out.
Also in the sum, it's k = 0 to 9, so what happened to the zero power? $z^0 = 1$
If $z=re^{iθ}$ then shouldn't $z^2 = (re^{iθ})^2 = r^2e^{i2θ}$

Latex tip: use curly braces { } to put multiple characters in the exponent.