# Thread: Reciprocal of Complex Number?

1. ## Reciprocal of Complex Number?

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

My attempt:

$\displaystyle \begin{array}{rcrcrc} z=\frac{3+4i}{2-3i}\\ \\ zw=1\\ \\ w=\frac{1}{z}\\ \\ \frac{3+4i}{2-3i}*\frac{2+3i}{2+3i}=\frac{-6+17i}{13} \end{array}$

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?

2. Originally Posted by Viral
$\displaystyle z=\frac{3+4i}{2-3i}$. What is the complex number which satisfies the equation $\displaystyle zw=1$.

My attempt:

$\displaystyle \begin{array}{rcrcrc} z=\frac{3+4i}{2-3i}\\ \\ zw=1\\ \\ w=\frac{1}{z}\\ \\ \frac{3+4i}{2-3i}*\frac{2+3i}{2+3i}=\frac{-6+17i}{13} \end{array}$

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 $\displaystyle 1/(a/b) = b/a$ and so $\displaystyle w = 1/z = \frac{2-3i}{3+4i}$.

Or am I missing something subtle?

3. 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.