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. 

$z = a + bi$
for $\frac{1}{z} = \bar{z}$
$\frac{1} {a + bi} = a - bi$
$\frac{1} {(a + bi)} \frac{a - bi}{a - bi} = a - bi$
$\frac{a - bi}{a^2 + b^2} = a - bi$
cancel and rearrange
$(a^2 + b^2) = 1$
recalling the formula for |z|
$|z| = \sqrt{a^2 + b^2} = \sqrt 1 =  1$
the other way
if |z| = 1 then $a^2 + b^2 = 1$
$z = a + bi$
again 
$\frac{1}{z} = \frac{a - bi} {a^2 + b^2} = \frac{a - bi}{|z|^2}$
$= \frac{a - bi}{1^2} = a - bi = \bar z$
