Definition:Multiplication/Complex Numbers

Definition

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:

$\paren {a + i b} \times \paren {c + i d} = \paren {a c - b d} + i \paren {a d + b c}$

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 $\tuple {x_1, y_1}$ and $\tuple {x_2, y_2}$ be complex numbers.

Then $\tuple {x_1, y_1} \tuple {x_2, y_2}$ is defined as:

$\tuple {x_1, y_1} \tuple {x_2, y_2} := \tuple {x_1 x_2 - y_1 y_2, x_1 y_2 + y_1 x_2}$

Examples

Example: $\paren {1 + 2 i} \paren {3 + 4 i}$

$\paren {1 + 2 i} \paren {3 + 4 i} = -5 + 10 i$

Example: $\paren {3 + 2 i} \paren {2 - i}$

$\paren {3 + 2 i} \paren {2 - i} = 8 + i$

Example: $\paren {4 + i} \paren {3 + 2 i} \paren {1 - i}$

$\paren {4 + i} \paren {3 + 2 i} \paren {1 - i} = 21 + i$