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  1. #1
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    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.
    Last edited by saha.subham; May 29th 2011 at 07:00 AM. Reason: wrng question
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    Quote Originally Posted by saha.subham View Post
    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
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  3. #3
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    $\displaystyle \displaystyle z_1 = x_1 + iy_1, z_2 = x_2 + iy_2$

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

    So what is the real part of $\displaystyle \displaystyle z_1z_2$?
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  4. #4
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    Quote Originally Posted by Prove It View Post
    $\displaystyle \displaystyle z_1 = x_1 + iy_1, z_2 = x_2 + iy_2$

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

    So what is the real part of $\displaystyle \displaystyle z_1z_2$?
    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?
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  5. #5
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    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 $\displaystyle \displaystyle z_1z_2$)?
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  6. #6
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    Quote Originally Posted by Prove It View Post
    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 $\displaystyle \displaystyle z_1z_2$)?
    How thus is in the form
    re(z1) * re(z2) - im(z1) * im(z2) ???
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  7. #7
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    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 $\displaystyle \displaystyle z_2$.
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  8. #8
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    Quote Originally Posted by Prove It View Post
    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 $\displaystyle \displaystyle z_2$.
    then it will be,
    x1x2 - y1y2,
    but how can i cancel out i(x1y2+ x2y1)??
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  9. #9
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    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 $\displaystyle \displaystyle i$.
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