Definition:Multiplication/Complex Numbers

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The multiplication operation in the domain of complex numbers $\C$ is written $\times$.

Let $z = a + i b, w = c + i d$ where $a, b, c, d \in \R$.

Then $z \times w$ is defined as:

$\left({a + i b}\right) \times \left({c + i d}\right) = \left({ac - bd}\right) + i \left({ad + bc}\right)$

This follows by the facts that:

Real Numbers form Field and thus real multiplication is distributive over real addition
the entity $i$ is such that $i^2 = -1$.

Complex Multiplication

When the formal definition of complex numbers is used, complex multiplication is defined thus:

Let $\left({x_1, y_1}\right)$ and $\left({x_2, y_2}\right)$ be complex numbers.

Then $\left({x_1, y_1}\right) \left({x_2, y_2}\right)$ is defined as:

$\left({x_1, y_1}\right) \left({x_2, y_2}\right) := \left({x_1 x_2 - y_1 y_2, x_1 y_2 + y_1 x_2}\right)$


Example: $\left({1 + 2 i}\right) \left({3 + 4 i}\right)$

$\left({1 + 2 i}\right) \left({3 + 4 i}\right) = -5 + 10 i$

Also see