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 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 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: $1$