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Math Help - Reciprocal of Complex Number?

  1. #1
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    Reciprocal of Complex Number?

    z=\frac{3+4i}{2-3i}. What is the complex number which satisfies the equation zw=1.

    My attempt:

    \begin{array}{rcrcrc}<br />
z=\frac{3+4i}{2-3i}\\<br />
\\<br />
zw=1\\<br />
\\<br />
w=\frac{1}{z}\\<br />
\\<br />
\frac{3+4i}{2-3i}*\frac{2+3i}{2+3i}=\frac{-6+17i}{13}<br />
\end{array}

    Is it right up to there? If so, I see I need to get the reciprocal of that (which will be the value of w). How would I do that?
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  2. #2
    Super Member Matt Westwood's Avatar
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    Quote Originally Posted by Viral View Post
    z=\frac{3+4i}{2-3i}. What is the complex number which satisfies the equation zw=1.

    My attempt:

    \begin{array}{rcrcrc}<br />
z=\frac{3+4i}{2-3i}\\<br />
\\<br />
zw=1\\<br />
\\<br />
w=\frac{1}{z}\\<br />
\\<br />
\frac{3+4i}{2-3i}*\frac{2+3i}{2+3i}=\frac{-6+17i}{13}<br />
\end{array}

    Is it right up to there? If so, I see I need to get the reciprocal of that (which will be the value of w). How would I do that?
    My initial utterly naive approach would be to say that 1/(a/b) = b/a and so w = 1/z = \frac{2-3i}{3+4i}.

    Or am I missing something subtle?
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  3. #3
    Member
    Joined
    Sep 2009
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    Ahh, I was trying to do that after working it out >< . I didn't think it would make a difference. Thanks, I'll try that out.
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