# Draw $\frac{1}{z}$ in the complex plane

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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Use $z=re^{i\theta}$ – user13838 Jan 14 '12 at 16:53
So, $\frac{1}{z}=\frac{1}{r}e^{-i \theta}$ means that the angle is opposite, and the radius is the inverse multiplicative. How can I draw this radius? – Kevin Jan 14 '12 at 17:00
On a number line, if I give you $n\neq 0$, can you pinpoint $\frac{1}{n}$? There's no difference between that case and this. – Neal Jan 14 '12 at 17:03
No, that's my problem :(.. – Kevin Jan 14 '12 at 17:18

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.