# Complex Addition is Closed

## 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.

$\blacksquare$

## Sources

- 1969: C.R.J. Clapham:
*Introduction to Abstract Algebra*... (previous) ... (next): Chapter $1$: Integral Domains: $\S 2$. Operations: Example $1$ - 1981: Murray R. Spiegel:
*Theory and Problems of Complex Variables*(SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Axiomatic Foundations of the Complex Number System: $1$