Complex Addition is Closed

Theorem

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

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

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:

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

So:

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

Hence the result.

$\blacksquare$

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:

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

So:

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

and hence the result.

$\blacksquare$

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: $1$
2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): complex number