1. ## complex 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.

2. Originally Posted by saha.subham
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.

Start by saying that z1=a+ib and z2=c+id.

If z=x+iy then:
The real part of z is:
Re(z)=x

and the imaginary part is:
Im(z)=y

3. $\displaystyle z_1 = x_1 + iy_1, z_2 = x_2 + iy_2$

\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 $\displaystyle z_1z_2$?

4. Originally Posted by Prove It
$\displaystyle z_1 = x_1 + iy_1, z_2 = x_2 + iy_2$

\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 $\displaystyle z_1z_2$?
ok i have edited the question after consulting with my friend, it seems i have to remove the last part from ur answer, can u please guide me?

5. You can write any complex number as $\displaystyle Z = X + iY$, where $\displaystyle X$ is the real part of $\displaystyle Z$ and $\displaystyle Y$ is the imaginary part of $\displaystyle Z$.

You have $\displaystyle z_1z_2 = x_1x_2 - y_1y_2 + i(x_1y_2 + x_2y_1)$. If you're going to rewrite this as $\displaystyle Z= X + iY$, what is $\displaystyle X$ (the real part of $\displaystyle z_1z_2$)?

6. Originally Posted by Prove It
You can write any complex number as $\displaystyle Z = X + iY$, where $\displaystyle X$ is the real part of $\displaystyle Z$ and $\displaystyle Y$ is the imaginary part of $\displaystyle Z$.

You have $\displaystyle z_1z_2 = x_1x_2 - y_1y_2 + i(x_1y_2 + x_2y_1)$. If you're going to rewrite this as $\displaystyle Z= X + iY$, what is $\displaystyle X$ (the real part of $\displaystyle z_1z_2$)?
How thus is in the form
re(z1) * re(z2) - im(z1) * im(z2) ???

7. Because $\displaystyle x_1$ is the real part of $\displaystyle z_1$, $\displaystyle x_2$ is the real part of $\displaystyle z_2$, $\displaystyle y_1$ is the imaginary part of $\displaystyle z_1$ and $\displaystyle y_2$ is the imaginary part of $\displaystyle z_2$.

8. Originally Posted by Prove It
Because $\displaystyle x_1$ is the real part of $\displaystyle z_1$, $\displaystyle x_2$ is the real part of $\displaystyle z_2$, $\displaystyle y_1$ is the imaginary part of $\displaystyle z_1$ and $\displaystyle y_2$ is the imaginary part of $\displaystyle z_2$.
then it will be,
x1x2 - y1y2,
but how can i cancel out i(x1y2+ x2y1)??

9. You DON'T. You are asked merely to state the REAL part of $\displaystyle z_1z_2$, in other words, everything that's not attached to an $\displaystyle i$.