# Complex number equation

• March 12th 2011, 12:13 AM
Glitch
Complex number equation
The question:
Show that $|z_1 + z_2|^2 + |z_1 - z_2|^2 = 2|z_1|^2 + 2|z_2|^2$, for all complex numbers $z_1$ and $z_2$.

I tried getting the LHS to match the RHS, but I've done something wrong. My attempt involved factorising using the difference of two squares, but it doesn't seem to help. Any assistance would be great.
• March 12th 2011, 12:18 AM
alexmahone
Quote:

Originally Posted by Glitch
The question:
Show that $|z_1 + z_2|^2 + |z_1 - z_2|^2 = 2|z_1|^2 + 2|z_2|^2$, for all complex numbers $z_1$ and $z_2$.

I tried getting the LHS to match the RHS, but I've done something wrong. My attempt involved factorising using the difference of two squares, but it doesn't seem to help. Any assistance would be great.

The first page of http://www.math.ca/crux/v31/n8/public_page502-503.pdf contains the expansion of $|z_1+z_2|^2$. Similarly expand $|z_1-z_2|^2$ and add them.
• March 12th 2011, 12:30 AM
Prove It
Quote:

Originally Posted by Glitch
The question:
Show that $|z_1 + z_2|^2 + |z_1 - z_2|^2 = 2|z_1|^2 + 2|z_2|^2$, for all complex numbers $z_1$ and $z_2$.

I tried getting the LHS to match the RHS, but I've done something wrong. My attempt involved factorising using the difference of two squares, but it doesn't seem to help. Any assistance would be great.

Write $\displaystyle z_1$ and $\displaystyle z_2$ in terms of their real and imaginary parts and simplify...
• March 12th 2011, 12:32 AM
Glitch
Ahh, I forgot about that relation. Thank you.
• March 12th 2011, 04:30 AM
Plato
Quote:

Originally Posted by Glitch
The question:
Show that $|z_1 + z_2|^2 + |z_1 - z_2|^2 = 2|z_1|^2 + 2|z_2|^2$, for all complex numbers $z_1$ and $z_2$.

Here is a different approach.
Recall $|z|^2=z\cdot\overline{z}$ and $\overline{z\pm w}=\overline{z}\pm\overline{w}$.
So $|z+w|^2=(z+w)\cdot\overline{(z+w)}=(z+w)\cdot(\ove rline{z}+\overline{w})$.
$=z\cdot\overline{z}+z\cdot\overline{w}+w\cdot\over line{z}+w\cdot\overline{w}$
$=|z|^2+z\cdot\overline{w}+w\cdot\overline{z}+|w|^2$

Do that for $|z-w|^2$ and add.
• March 12th 2011, 04:58 AM
Glitch
Quote:

Originally Posted by Plato
Here is a different approach.

That's what alexmahone suggested. I ended up solving it that way.