Non-Zero Complex Numbers under Multiplication form Group

From ProofWiki
Jump to navigation Jump to search


Let $\C_{\ne 0}$ be the set of complex numbers without zero, that is:

$\C_{\ne 0} = \C \setminus \set 0$

The structure $\struct {\C_{\ne 0}, \times}$ is a group.


Taking the group axioms in turn:

Group Axiom $\text G 0$: Closure

Non-Zero Complex Numbers are Closed under Multiplication.


Group Axiom $\text G 1$: Associativity

Complex Multiplication is Associative.


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

From Complex Multiplication Identity is One, the identity element of $\struct {\C_{\ne 0}, \times}$ is the complex number $1 + 0 i$.


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

From Inverse for Complex Multiplication‎, the inverse of $x + i y \in \struct {\C_{\ne 0}, \times}$ is:

$\dfrac 1 z = \dfrac {x - i y} {x^2 + y^2} = \dfrac {\overline z} {z \overline z}$

where $\overline z$ is the complex conjugate of $z$.