Complex Numbers under Addition form Abelian Group

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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

Complex Addition is Closed.

$\Box$


G1: Associativity

Complex Addition is Associative.

$\Box$


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

Complex Addition is Commutative.

$\Box$


Infinite

Complex Numbers are Uncountable.

$\blacksquare$


Sources