for any two complex no. z1 and z2 prove that re(z1 z2) = re(z1) * re(z2) - im(z1) * im(z2)

my incomplete soln-

don't have any idea about it.

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- May 29th 2011, 04:52 AMsaha.subhamcomplex number
for any two complex no. z1 and z2 prove that re(z1 z2) = re(z1) * re(z2) - im(z1) * im(z2)

my incomplete soln-

don't have any idea about it. - May 29th 2011, 05:02 AMAlso sprach Zarathustra
- May 29th 2011, 05:13 AMProve It
$\displaystyle \displaystyle z_1 = x_1 + iy_1, z_2 = x_2 + iy_2$

$\displaystyle \displaystyle \begin{align*}z_1z_2 &= (x_1 + iy_1)(x_2 + iy_2) \\ &= x_1x_2 + ix_1y_2 + ix_2y_1 + i^2y_1y_2 \\ &= x_1x_2 - y_1y_2 + i(x_1y_2 + x_2y_1)\end{align*}$

So what is the real part of $\displaystyle \displaystyle z_1z_2$? - May 29th 2011, 07:04 AMsaha.subham
- May 29th 2011, 07:11 AMProve It
You can write any complex number as $\displaystyle \displaystyle Z = X + iY$, where $\displaystyle \displaystyle X$ is the real part of $\displaystyle \displaystyle Z$ and $\displaystyle \displaystyle Y$ is the imaginary part of $\displaystyle \displaystyle Z$.

You have $\displaystyle \displaystyle z_1z_2 = x_1x_2 - y_1y_2 + i(x_1y_2 + x_2y_1)$. If you're going to rewrite this as $\displaystyle \displaystyle Z= X + iY$, what is $\displaystyle \displaystyle X$ (the real part of $\displaystyle \displaystyle z_1z_2$)? - May 29th 2011, 09:51 PMsaha.subham
- May 29th 2011, 11:00 PMProve It
Because $\displaystyle \displaystyle x_1$ is the real part of $\displaystyle \displaystyle z_1$, $\displaystyle \displaystyle x_2$ is the real part of $\displaystyle \displaystyle z_2$, $\displaystyle \displaystyle y_1$ is the imaginary part of $\displaystyle \displaystyle z_1$ and $\displaystyle \displaystyle y_2$ is the imaginary part of $\displaystyle \displaystyle z_2$.

- May 30th 2011, 03:54 AMsaha.subham
- May 30th 2011, 04:13 AMProve It
You DON'T. You are asked merely to state the REAL part of $\displaystyle \displaystyle z_1z_2$, in other words, everything that's not attached to an $\displaystyle \displaystyle i$.