Complex Numbers under Addition form Abelian Group

Theorem

Let $\C$ be the set of complex numbers.

The structure $\struct {\C, +}$ is an infinite abelian group.

Proof

Taking the group axioms in turn:

G0: Closure

$\Box$

G1: Associativity

$\Box$

G2: Identity

From Complex Addition Identity is Zero, we have that the identity element of $\struct {\C, +}$ is the complex number $0 + 0 i$:

$\paren {x + i y} + \paren {0 + 0 i} = \paren {x + 0} + i \paren {y + 0} = x + i y$

and similarly for $\paren {0 + 0 i} + \paren {x + i y}$.

$\Box$

G3: Inverses

From Inverse for Complex Addition, the inverse of $x + i y \in \struct {\C, +}$ is $-x - i y$.

$\Box$

C: Commutativity

$\Box$

Infinite

$\blacksquare$