Complex multiplication

Complex multiplication combines complex numbers using distributivity and the rule i² = −1. In polar form, it multiplies magnitudes and adds angles.

Complex number: A number expressible as a real part plus an imaginary part, often written a + bi. Multiplication acts on these numbers and preserves their algebraic structure.
Imaginary unit: The number i satisfying i² = −1. Its defining square determines the sign of the real term when products are expanded.
Rotation matrix: A matrix that rotates vectors in a plane while preserving their lengths and orientation. Multiplication by a unit complex number represents the same planar rotation.
Complex addition: The operation that adds complex numbers by adding their real and imaginary components separately. Unlike multiplication, addition shifts points rather than combining rotations and scales.
Complex conjugate: The complex conjugate of a + bi is a − bi, obtained by reversing the sign of its imaginary part. Conjugation interacts with multiplication by conjugating each factor and their product.
Real number: A number on the continuous number line, including rational and irrational numbers. Real numbers form the scalar components used to build complex numbers.
Phasor: A complex number encoding the amplitude and phase of a sinusoidal quantity. Multiplying phasors combines amplitudes and adds phase angles.
Quaternion multiplication: Multiplication of quaternions, a four-dimensional number system with noncommutative products. It extends rotation-related multiplication but loses the commutativity of complex products.
Polar form: A representation of a complex number by its magnitude and angle from the positive real axis. In this representation, multiplication becomes magnitude multiplication and angle addition.
Addition of complex numbers: The operation that adds real parts and imaginary parts separately. Distributive expansion of a product relies on adding its component terms.

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