Complex Numbers form Ring

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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:

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


Hence the result, by definition of ring.

$\blacksquare$


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