- #1
latentcorpse
- 1,444
- 0
I'm having a bit of difficulty in showing the following is an equivalence relation:
[itex]z~w iff \frac{z}{w} \in \mathbb{R} \forall z,w \in \mathbb{C}*=\mathbb{C} \backslash \{0\][/itex]
clearly z~z is ok as 1 is real
then i considered [itex]\frac{z}{y} \in \mathbb{R}[/itex] and tried to show y divided by z is real. i decided to multiply through by [itex]\frac{\bar{y}}{\bar{y}}[/itex] to get [itex]\frac{z \bar{y}}{|y|^2} \in \mathbb{R}[/itex]
then clearly [itex]z \bar{y} \in \mathbb{R}[/itex]
and dividing by [itex]|z|^2[/itex] we get [itex]\frac{\bar{y}}{\bar{z}} \in \mathbb{R}[/itex]
but now what?
[itex]z~w iff \frac{z}{w} \in \mathbb{R} \forall z,w \in \mathbb{C}*=\mathbb{C} \backslash \{0\][/itex]
clearly z~z is ok as 1 is real
then i considered [itex]\frac{z}{y} \in \mathbb{R}[/itex] and tried to show y divided by z is real. i decided to multiply through by [itex]\frac{\bar{y}}{\bar{y}}[/itex] to get [itex]\frac{z \bar{y}}{|y|^2} \in \mathbb{R}[/itex]
then clearly [itex]z \bar{y} \in \mathbb{R}[/itex]
and dividing by [itex]|z|^2[/itex] we get [itex]\frac{\bar{y}}{\bar{z}} \in \mathbb{R}[/itex]
but now what?