Complex Addition is Closed

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Theorem

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

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


Proof 1

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 Abelian Group, real addition is closed.

So:

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

Hence the result.

$\blacksquare$


Proof 2

From the formal definition of complex numbers, we have:

$z = \tuple {x_1, y_1}$
$w = \tuple {x_2, y_2}$

where $x_1, x_2, y_1, y_2 \in \R$.

Then from the definition of complex addition:

$z + w = \tuple {x_1 + x_2, y_1 + y_2}$

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

So:

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

and hence the result.

$\blacksquare$


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