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Many definitions for complex number say

  1. $Re(z) = \frac{1}{2}(z + \bar{z})$
  2. $Im(z) = \frac{1}{2i}(z - \bar{z})$
  3. $|z| = \sqrt{z\cdot\bar{z}}$

I do understand 1. as I can visualize it (the addition will eliminate the value of the y-axis which is just the real part of $z$), but why do the other two apply? Do you know any proof of them?

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1  
$2.$ is not correct, the formula is $Im(z) = \frac 1{2i}(z-\bar z)$... –  abiessu Apr 21 at 19:27
2  
You should clarify what you mean with $|z| = \sqrt{z\cdot\bar{z}}$. I personally translated it as another way of writing $z\overline z=|z|^2$. To prove it just write $z=a+ib$ and compute each side. –  Git Gud Apr 21 at 19:29
    
@abiessu you were right, thank you, I fixed it –  muffel Apr 21 at 19:32

1 Answer 1

up vote 3 down vote accepted

$$z=z_R+iz_I,\quad \bar{z}=z_R-iz_I$$

from which follows

$$z+\bar{z}=2z_R\quad\textrm{and}\quad z-\bar{z}=2iz_I$$

Furthermore you have

$$z\cdot\bar{z}=(z_R+iz_I)(z_R-iz_I)=z_R^2+z_I^2=|z|^2$$

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