Complex Numbers under Addition form Group

From ProofWiki
Jump to navigation Jump to search


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

The structure $\struct {\C, +}$ is a group.


Taking the group axioms in turn:

Group Axiom $\text G 0$: Closure

Complex Addition is Closed.


Group Axiom $\text G 1$: Associativity

Complex Addition is Associative.


Group Axiom $\text G 2$: Existence of Identity Element

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


Group Axiom $\text G 3$: Existence of Inverse Element

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