Additive Group of Complex Numbers is Direct Product of Reals with Reals

Theorem
Let $\struct {\C, +}$ be the additive group of complex numbers.

Let $\struct {\R, +}$ be the additive group of real numbers.

Then the direct product $\struct {\R, +} \times \struct {\R, +}$ is isomorphic with $\struct {\C, +}$.

Proof
We need to show that $\phi: \tuple {x, y} \mapsto x + y i$ is a group isomorphism.

$\phi$ is bijective
We show that $\phi^{-1}: z \mapsto \tuple {\map \Re z, \map \Im z}$ is the inverse of $\phi$.

We have:
 * $\map \phi {\map {\phi^{-1} } z} = \map \phi {\map \Re z, \map \Im z} = \map \Re z + i \map \Im z = z$
 * $\map {\phi^{-1} } {\map \phi {a, b} } = \map {\phi^{-1} } {a + b i} = \tuple {a, b}$

hence $\phi^{-1}$ is the inverse of $\phi$.

Hence $\phi$ is a group isomorphism, and thus the direct product $\struct {\R, +} \times \struct {\R, +}$ is isomorphic with $\struct {\C, +}$.