1. ## complex numbers

when does this relation holds?
[z1]-[z2]=[z1+z2] ; [z1] & [z2] represents the modulus of two complex numbers.
to me only when z2=0?
is there any other possibility?

2. Originally Posted by u2_wa
when does this relation holds?
[z1]-[z2]=[z1+z2]
to me only when z2=0?
is there any other possibility?
$\displaystyle [z1]-[z2]=[z1+z2]$

let's say

$\displaystyle z1=any integer$(e.g let say 2)

$\displaystyle z2=0$

$\displaystyle 2-0=0+2$

$\displaystyle 2=2$

make sense.. i don't think there's any more possibility for other values other than zero

3. There are other possibilities

$\displaystyle z_1 = \alpha \: e^{i\theta}$ and $\displaystyle z_2 = \beta \: e^{i(\theta+\pi)} = - \beta \: e^{i\theta}$ with $\displaystyle \alpha \geq \beta \geq 0$

Then
$\displaystyle |z_1| = \alpha$ and $\displaystyle |z_2| = \beta$

$\displaystyle z_1 + z_2 = (\alpha - \beta) \: e^{i\theta}$

$\displaystyle |z_1 + z_2| = \alpha - \beta = |z_1| - |z_2|$

For instance
$\displaystyle |2i + (-i)| = |i| = 1 = |2i| - |-i|$

4. Originally Posted by running-gag
There are other possibilities

$\displaystyle z_1 = \alpha \: e^{i\theta}$ and $\displaystyle z_2 = \beta \: e^{i(\theta+\pi)} = - \beta \: e^{i\theta}$ with $\displaystyle \alpha \geq \beta \geq 0$

Then
$\displaystyle |z_1| = \alpha$ and $\displaystyle |z_2| = \beta$

$\displaystyle z_1 + z_2 = (\alpha - \beta) \: e^{i\theta}$

$\displaystyle |z_1 + z_2| = \alpha - \beta = |z_1| - |z_2|$

For instance
$\displaystyle |2i + (-i)| = |i| = 1 = |2i| - |-i|$
Thanks for it. Can you please give me graphical proof for it!

5. Here it is