Complex Multiplication is Closed

Theorem
$$\forall z, w \in \mathbb{C}: z w \in \mathbb{C}$$.

Proof from Informal Definition
From the informal definition of complex numbers, we define the following:


 * $$z = x_1 + \imath y_1$$
 * $$w = x_2 + \imath y_2$$

where $$\imath = \sqrt {-1}$$ and $$x_1, x_2, y_1, y_2$$.

Then from the definition of complex multiplication $$z w = \left({x_1 x_2 - y_1 y_1}\right) + \imath \left({x_1 y_2 + x_2 y_1}\right)$$.

The Real Numbers form a Field, so $$x_1 x_2 - y_1 y_1 \in \mathbb{R}$$ and $$x_1 y_2 + x_2 y_1 \in \mathbb{R}$$.

Hence the result.

Proof from Formal Definition
From the formal definition of complex numbers, we define the following:


 * $$z = \left({x_1, y_1}\right)$$
 * $$w = \left({x_2, y_2}\right)$$

Then from the definition of complex multiplication $$z w = \left({x_1 x_2 - y_1 y_1, x_1 y_2 + x_2 y_1}\right)$$.

The Real Numbers form a Field, so $$x_1 x_2 - y_1 y_1 \in \mathbb{R}$$ and $$x_1 y_2 + x_2 y_1 \in \mathbb{R}$$.

Hence the result.