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November 2nd, 2017, 04:00 PM   #1
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Question Solve complex number equations with conjugate

I have this equation $\displaystyle \displaystyle z + 2\bar{z} = 2 - i$, but don't really know how to solve it. My first thought was to set $\displaystyle \displaystyle z = \bar{z}$ and solve for $\displaystyle z$.

And when z was figured out I put in z and $\displaystyle \displaystyle \bar{z}$, but that failed.

So how do you solve these types of equations?
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November 2nd, 2017, 04:30 PM   #2
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let $z=x+i y,~x,y \in \mathbb{R}$

$x + i y + 2x - 2i y = 2 - i$

$3x - i y = 2 - i$

$3x = 2$

$-y = -1$

$x=\dfrac 2 3$

$y = 1$

$z = \dfrac 2 3 + i$
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November 2nd, 2017, 06:28 PM   #3
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Here's another way using just using the standard operations on the complex numbers: conjugation, Real part $\Re$ and Imaginary part $\Im$.

(1) $z + 2 \bar z = 2− i$ $\ \ \ \ $ [Given]

(2) $\bar z + 2 z = 2 + i$ $\ \ \ \ $ [Conjugating both sides]

$3 z + 3 \bar z = 4$ $\ \ \ \ $ [Adding (1) to (2)]

$z + \bar z = \frac{4}{3}$

$2 \Re (z) = \frac{4}{3}$ $\ \ \ \ $ [Using $z + \bar z = 2 \Re (z)$]

$\Re (z) = \frac{2}{3}$

Now subtracting (2) - (1) gives:

$z - \bar z = 2$ and then

$\Im (z) = 1$ $\ \ \ \ $ [Using $z - \bar z = 2 \Im (z)$]

So $z = \frac{2}{3} + i$

Last edited by Maschke; November 2nd, 2017 at 06:31 PM.
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November 2nd, 2017, 09:39 PM   #4
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Originally Posted by Maschke View Post
$\Re$ and $\Im$.
People still use that ugly script for those? I haven't seen that notation in years!

-Dan
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November 2nd, 2017, 10:37 PM   #5
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Quote:
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People still use that ugly script for those? I haven't seen that notation in years!

-Dan
I hate them too. That's how \Re and \Im render. Perhaps I should just spell them out as Re and Im. $Re(z)$ and $Im(z)$. Yes that's much better. I'll do that from now on.
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November 2nd, 2017, 11:05 PM   #6
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I hate them too. That's how \Re and \Im render. Perhaps I should just spell them out as Re and Im. $Re(z)$ and $Im(z)$. Yes that's much better. I'll do that from now on.
My tensor geometry book has a comment about the labeling of tensors, which uses symbols very similar to the ones LaTeX wants you to use. The author claimed that the notation was ugly and did nothing to help the reader. So he said he wasn't going to use that script. Looking down at the footnote where the author said this he got a "thanks" from his typist.

-Dan
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November 2nd, 2017, 11:06 PM   #7
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Re and Im. $Re(z)$ and $Im(z)$.
That's what us Physics people use.

-Dan
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