Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I know that if complex numbers $z$ and $z'$ correspond to opposite points on the Riemann sphere, then it must be the case that $z\bar{z}'=-1$.

Is the converse true, that $z\bar{z}'=-1$ implies that the corresponding points on the Riemann sphere are opposite points?

I associate $(x_1,x_2,x_3)$ with $z$ and $(x'_1,x'_2,x'_3)$ with $z'$. The usual correspondence gives $$ z=\frac{x_1+ix_2}{1-x_3},\;\;\;\;\; z'=\frac{x'_1+ix'_2}{1-x'_3}. $$ Then I reach an equation $$ z\bar{z}'=\frac{x_1+ix_2}{1-x_3}\cdot\frac{x'_1-ix'_2}{1-x'_3}=-1 $$ which implies $$ x_1x'_1+x_2x'_2+x_3x'_3+(x'_1x_2-x_1x'_2)i=-1+x_3+x'_3. $$ Can one conclude that $(x_1,x_2,x_3)=-(x'_1,x'_2,x'_3)$ from this relation? Thank you.

share|improve this question

1 Answer 1

up vote 3 down vote accepted

There's only one possible complex $w$ for which $zw$ can equal $-1$, given by $w = -{1 \over z}$. So since $z\bar{z'} = -1$ for $z$ and $z'$ corresponding to antipodal points on the Riemann sphere, if $z\bar{w} = -1$ as well you'd have to have $\bar{w} = \bar{z'}$ or $w = z'$.

share|improve this answer
    
Oops, that's simple. Thanks. –  Kaila Jan 27 '12 at 3:27

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.