AcademyComplex Arithmetic
Academy
Complex Multiplication
Level 1 - Math I (Physics) topic page in Complex Arithmetic.
Multiplying Complex Numbers
Complex multiplication uses the distributive property and the key identity \(i^2 = -1\).
Multiplication Formula
For \(z_1 = a + bi\) and \(z_2 = c + di\):
Multiplication
\[z_1 \cdot z_2 = (ac - bd) + (ad + bc)i\]
Derivation
FOIL
\[(a + bi)(c + di) = ac + adi + bci + bdi^2\]
Simplify
\[= ac + adi + bci - bd\]
Combine
\[= (ac - bd) + (ad + bc)i\]
Geometric Meaning
Multiplication by a complex number causes both rotation and scaling:
- The modulus multiplies: \(|z_1 z_2| = |z_1| \cdot |z_2|\)
- The argument adds: \(\arg(z_1 z_2) = \arg(z_1) + \arg(z_2)\)
PolarMult
\[(r_1 e^{i\theta_1})(r_2 e^{i\theta_2}) = r_1 r_2 e^{i(\theta_1 + \theta_2)}\]
This makes multiplication by \(e^{i\theta}\) a pure rotation by angle \(\theta\).
Example
Example
\[(1 + i)(2 + 3i) = 2 + 3i + 2i + 3i^2 = -1 + 5i\]