Complex number

Set

 definiendum $\mathbb C \equiv \mathbb R^2$

Discussion

We write the complex numbers as $a+ib\equiv\langle a,b\rangle$, where $a,b\in\mathbb R$. The complex numbers are then set up as a field with $i^2=-1$, see arithmetic structure of complex numbers. We identify the real numbers within $\mathbb C$ as the set of elements of the form $\langle a,0\rangle=a+i0=a$.

Reference

Wikipedia: Complex number