ThmDex – An index of mathematical definitions, results, and conjectures.
 ▼ Set of symbols ▼ Alphabet ▼ Deduction system ▼ Theory ▼ Zermelo-Fraenkel set theory ▼ Set ▼ Binary cartesian set product ▼ Binary relation ▼ Map ▼ Operation ▼ N-operation ▼ Binary operation ▼ Enclosed binary operation ▼ Groupoid ▼ Ringoid ▼ Semiring ▼ Ring ▼ Multiplicative group ▼ Multiplicative monoid ▼ Multiplicative semigroup ▼ Multiplicative groupoid ▼ Multiplicative binary operation ▼ Natural number multiplication operation ▼ Integer multiplication operation ▼ Rational multiplication operation ▼ Real multiplication operation
Definition D611
Complex multiplication operation

Let $\mathbb{C}$ be the D372: Set of complex numbers.
Let $+$ be the D606: Real addition operation.
Let $\cdot$ be the D610: Real multiplication operation.
The complex multiplication operation is the D554: Binary operation $$\mathbb{C} \times \mathbb{C} \to \mathbb{C}, \quad ((a, b), (c, d)) \mapsto (a c - b d, a d + b c)$$

Let $\mathbb{C}$ be the D372: Set of complex numbers.
Let $+$ be the D606: Real addition operation.
Let $\cdot$ be the D610: Real multiplication operation.
The complex multiplication operation is the D554: Binary operation $$\mathbb{C} \times \mathbb{C} \to \mathbb{C}, \quad (a, b) (c, d) = (a \cdot c - b \cdot d, a \cdot d + b \cdot c)$$