Question

if this is true prove it, if false make a counter
z bar = mod(z) ^ 2 / z, where z can't equal 0

Answer 1

\[\overline{z}=\frac{|z|^2}{z}\]
If \(z=x+iy\), then
\[\overline{z}=x-iy\\
|z|=\sqrt{x^2+y^2}\]
So you have
\[\begin{align*}x-iy&=\frac{x^2+y^2}{x+iy}\\
(x-iy)&=\frac{x^2+y^2}{x+iy}\cdot\frac{x-iy}{x-iy}\\
(x-iy)&=\frac{x^2+y^2}{x^2-i^2y^2}(x-iy)\\
(x-iy)&=\color{red}{\frac{x^2+y^2}{x^2+y^2}}(x-iy)
\end{align*}\]
Clearly, the red part is 1, so the statement is true.

Answer 2

thank you this was so helpful