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Suppose you have a complex number $z$. How can you draw $\frac{1}{z}$ in the complex plane without calculations? I know $z\cdot \operatorname{conj}(z) = |z|^2$ so $\frac{1}{z} = \frac{\operatorname{conj}(z)}{|z|^2}$

Regards, Kevin

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    http://planetmath.org/encyclopedia/CompassAndStraightedgeConstructionOfInversePoint.html2012-01-14

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You just answered your own question. Since $\frac{1}{z} = \overline{z}/|z|^2$, $\frac{1}{z}$ is the conjugate of $z$, reflected across the unit circle. Alternately, as percusse suggested, if $z=re^{i\theta}$, then $\frac{1}{z} = \frac{1}{r}e^{-i\theta}$.

This should work for sketching or giving you a sense of where $\frac{1}{z}$ lies given any (nonzero) $z$ unless you're looking for something more precise, like a compass-and-straightedge construction. In that case, please make it clear in the question.