Complex Multiplication is Closed

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