Proving Equivalence Relation: z~w $\in$ $\mathbb{C}$

[latentcorpse]

I'm having a bit of difficulty in showing the following is an equivalence relation:

z~w iff \frac{z}{w} \in \mathbb{R} \forall z,w \in \mathbb{C}*=\mathbb{C} \backslash \{0\]

clearly z~z is ok as 1 is real

then i considered \frac{z}{y} \in \mathbb{R} and tried to show y divided by z is real. i decided to multiply through by \frac{\bar{y}}{\bar{y}} to get \frac{z \bar{y}}{|y|^2} \in \mathbb{R}
then clearly z \bar{y} \in \mathbb{R}
and dividing by |z|^2 we get \frac{\bar{y}}{\bar{z}} \in \mathbb{R}
but now what?

 

[xepma]

1) If a number w is real, what does that tell you about the inverse w^{-1}? (remember: w * w^{-1} = 1)

2) For the final step, use the fact that you can write a fraction as: \frac{a}{b} = \frac{a}{c}\frac{c}{b}

 

[HallsofIvy]

(z/w)(w/z)= 1. Therefore, w/z is the multiplicative inverse of z/w. But z/w is real so its multiplicative inverse is ...

 

[latentcorpse]

erm.. just w/z?

 

[HallsofIvy]

Yes, but what I was looking for is "since z/w is real, its multiplicative inverse, w/z, is real". That was what xepema meant.