Complex Multiplication is Closed
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Theorem
The set of complex numbers $\C$ is closed under multiplication:
- $\forall z, w \in \C: z \times w \in \C$
Proof from Informal Definition
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 multiplication:
- $z w = \paren {x_1 x_2 - y_1 y_2} + i \paren {x_1 y_2 + x_2 y_1}$
From Real Numbers form Field:
- $x_1 x_2 - y_1 y_2 \in \R$
and:
- $x_1 y_2 + x_2 y_1 \in \R$
Hence the result.
$\blacksquare$
Proof from Formal Definition
From the formal definition of complex numbers, we define the following:
- $z = \tuple {x_1, y_1}$
- $w = \tuple {x_2, y_2}$
Then from the definition of complex multiplication:
- $z w = \tuple {x_1 x_2 - y_1 y_2, x_1 y_2 + x_2 y_1}$
From Real Numbers form Field:
- $x_1 x_2 - y_1 y_2 \in \R$
and:
- $x_1 y_2 + x_2 y_1 \in \R$
Hence the result.
$\blacksquare$
Sources
- 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $1$: Integral Domains: $\S 2$. Operations: Example $1$