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.
Best Answer
Write $z = x + i y $. Suppose $|z| = \sqrt{x^2+y^2} =1$. Then,
$$ \bar{z} = x - iy = \frac{ x^2 + y^2 }{x + iy} = \frac{1}{x+iy} = \frac{1}{z}$$
Now, suppose $ x - iy = \frac{1}{x + iy } $. Then,
$$ x^2 + y^2 = 1 \implies |z|=1$$