A definition of multiplication on the projective $1$-points $(a:b)$ of $P\_K^1$ with $a$ and $b$ elements of a field $K$ ( e.g. the real or rational numbers ) can be given by mimicking the multiplication of complex numbers.


Is it known whether there a way to define an addition between projective $1$-points ? 



Can the projective line be provided with a ring structure?
The topological space $\mathbb{P}^1\_{\mathbb{R}}$ is a circle, so it has an abelian group structure just defined by adding angles but this is not algebraic, not compatible with the multiplication you describe, and it can't be extended to $\mathbb{P}^1\_{\mathbb{C}}$ (which is a sphere and thus has no topological group structure), nor does it restrict to $\mathbb{P}^1\_{\mathbb{Q}}$. **EDIT**: As Noam Elkies points out, the circle group structure can be made algebraic on $\mathbb{P}^1\_{\mathbb{R}}$ by defining it by $\tan(x+y)=\tan(x)+\tan(y)$, instead of by adding angles. This still has all the other defects listed above. 


More generally, it's impossible to do this algebraically: it's an extremely well known fact that a projective curve with an algebraic group structure (as addition would be) must be an elliptic curve. 


