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Nahhh no such thing as number theory needed.

\[i+(-i)=0 \\ i(-i)=-i^2=-(-1)=1\]

so are we looking for the pair i,-i?

maybe I should also consider
\[(x_1+y_1i)+(x_2+y_2i)=0 \\ (x_1+y_1i)(x_2+y_2i)=1\]

look at a caley table...

Maybe I am confused by the question...

how can a pair of numbers have an inverse?

I am still confused. The quaternians are a group. Ill stop cluttering up the question...

that would be an extension. The quaernains are a 8 element group. By definition.

as another example^^^