Complex Addition is Closed/Proof 1

Theorem

The set of complex numbers $\C$ is closed under addition:

$\forall z, w \in \C: z + w \in \C$

Proof

From the informal definition of complex numbers, we define the following:

$z = x_1 + i y_1$

$w = x_2 + i y_2$

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

Then from the definition of complex addition:

$z + w = \paren {x_1 + x_2} + i \paren {y_1 + y_2}$

From Real Numbers under Addition form Group, real addition is closed.

So:

$\paren {x_1 + x_2} \in \R$ and $\paren {y_1 + y_2} \in \R$

Hence the result.

$\blacksquare$