Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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

share|cite|improve this question
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
up vote 2 down vote accepted

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.