Show that $|z| = 1$ if and only if $\bar{z} = \frac{1}{z}$. Maybe a very stupid question but I am stuck. Show that $|z| = 1$ if and only if $\bar{z} = \frac{1}{z}$.
Is it enough to simply multiply, i.e. $z\bar{z} = \frac{1\times z}{z} = 1$? Showhow I feel this is not correct. I know that if $z = \pm 1$ or $z \pm i$ then $|z| = 1$. Am I supposed to draw the circle $|z| = 1$? But what does $\frac{1}{z}$ represent?
If someone could give me a hint. 

The non-trivial direction: If $|z|=1$ you have $$z=|z|e^{i\arg z}=e^{i\arg z}$$
$$\bar{z}=e^{-i\arg z}$$
$$\frac{1}{z}=\frac{1}{e^{i\arg z}}=e^{-i\arg z}$$
