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
Formulation 0
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 \begin{equation} \mathbb{C} \times \mathbb{C} \to \mathbb{C}, \quad (a, b) (c, d) : = (a c - b d, a d + b c) \end{equation}
Formulation 1
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 \begin{equation} \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) \end{equation}