Complex Numbers under Addition form Group

Theorem

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

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

Proof

Taking the group axioms in turn:

Group Axiom $\text G 0$: Closure

Complex Addition is Closed.

$\Box$

Group Axiom $\text G 1$: Associativity

Complex Addition is Associative.

$\Box$

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

$\Box$

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

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

$\blacksquare$

Sources

• 1968: Ian D. Macdonald: The Theory of Groups ... (previous) ... (next): $\S 1$: Some examples of groups: Example $1.04$
• 1974: Robert Gilmore: Lie Groups, Lie Algebras and Some of their Applications ... (previous) ... (next): Chapter $1$: Introductory Concepts: $1$. Basic Building Blocks: $2$. GROUP: Example $5$