# Complex Numbers form Ring

## Theorem

The set of complex numbers $\C$ forms a ring under addition and multiplication: $\struct {\C, +, \times}$.

## Proof

From Complex Numbers under Addition form Abelian Group, $\struct {\C, +}$ is an abelian group.

We also have that:

Complex Multiplication is Closed:
$\forall x, y \in \C: x \times y \in \C$
Complex Multiplication is Associative:
$\forall x, y, z \in \C: x \times \paren {y \times z} = \paren {x \times y} \times z$

Thus $\struct{\C, +}$ is a semigroup.

Finally we have that Complex Multiplication Distributes over Addition:

 $\, \displaystyle \forall x, y, z \in \C: \,$ $\displaystyle x \times \paren {y + z}$ $=$ $\displaystyle x \times y + x \times z$ $\displaystyle \paren {y + z} \times x$ $=$ $\displaystyle y \times x + z \times x$

Hence the result, by definition of ring.

$\blacksquare$