# Is there a proof for existence of complex and hypercomplex numbers [duplicate]

We can define $\mathbb C$ as the set $\mathbb R^2$ with binary operations $+_C$ and $\times_C$, where $(a,b)+_C(a',b')=(a+a',b+b')$ and $(a,b)\times_C(a'b')=(aa-bb', ab'+ba').$ The reals are then isomorphically embedded in $\mathbb C$ as $\mathbb R\times \{0\}.$