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

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 $x_1, x_2, y_1, y_2$.

Then from the definition of complex multiplication $z w = \left({x_1 x_2 - y_1 y_1}\right) + i \left({x_1 y_2 + x_2 y_1}\right)$.

The Real Numbers form a Field, so $x_1 x_2 - y_1 y_1 \in \R$ 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:

$z = \left({x_1, y_1}\right)$

$w = \left({x_2, y_2}\right)$

Then from the definition of complex multiplication $z w = \left({x_1 x_2 - y_1 y_1, x_1 y_2 + x_2 y_1}\right)$.

The Real Numbers form a Field, so $x_1 x_2 - y_1 y_1 \in \R$ and $x_1 y_2 + x_2 y_1 \in \R$.

Hence the result.

$\blacksquare$

Sources

• C.R.J. Clapham: Introduction to Abstract Algebra (1969)... (previous)... (next): $\S 1.2$: Example $1$