Thread: Complex number equation

The question:
Show that $\displaystyle |z_1 + z_2|^2 + |z_1 - z_2|^2 = 2|z_1|^2 + 2|z_2|^2$, for all complex numbers $\displaystyle z_1$ and $\displaystyle z_2$.

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.

Originally Posted by Glitch

The question:
Show that $\displaystyle |z_1 + z_2|^2 + |z_1 - z_2|^2 = 2|z_1|^2 + 2|z_2|^2$, for all complex numbers $\displaystyle z_1$ and $\displaystyle z_2$.

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 $\displaystyle |z_1+z_2|^2$. Similarly expand $\displaystyle |z_1-z_2|^2$ and add them.

Originally Posted by Glitch

The question:
Show that $\displaystyle |z_1 + z_2|^2 + |z_1 - z_2|^2 = 2|z_1|^2 + 2|z_2|^2$, for all complex numbers $\displaystyle z_1$ and $\displaystyle z_2$.

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 $\displaystyle \displaystyle z_1$ and $\displaystyle \displaystyle z_2$ in terms of their real and imaginary parts and simplify...

Ahh, I forgot about that relation. Thank you.

Originally Posted by Glitch

The question:
Show that $\displaystyle |z_1 + z_2|^2 + |z_1 - z_2|^2 = 2|z_1|^2 + 2|z_2|^2$, for all complex numbers $\displaystyle z_1$ and $\displaystyle z_2$.

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

Do that for $\displaystyle |z-w|^2$ and add.

Originally Posted by Plato

Here is a different approach.

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