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. 

Take $z$ in the unit circle. $\bar{z}$ is the reflection of $z$ with respect to the real axis. Therefore, $\bar{z}$ has modulus $1$ and argument the negative of the argument of $z$. Since we multiply complex numbers by multiplying their modulus and adding their arguments, we have the $z\bar{z}$ has modulus $1$ and argument $0$ and so $z\bar{z}=1$.
