Complex Numbers form Ring

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Theorem

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


Proof

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


We also have that:

$\forall x, y \in \C: x \times y \in \C$
$\forall x, y, z \in \C: x \times \left({y \times z}\right) = \left({x \times y}\right) \times z$

Thus $\left({\C, +}\right)$ is a semigroup.


Finally we have that Complex Multiplication Distributes over Addition:

$\forall x, y, z \in \C:$
$x \times \left({y + z}\right) = x \times y + x \times z$
$\left({y + z}\right) \times x = y \times x + z \times x$


Hence the result, by definition of ring.

$\blacksquare$


Sources