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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.

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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 $w = z'$.

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Oops, that's simple. Thanks. – Kaila Jan 27 '12 at 3:27

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