Complex Number Arithmetic & Polar Form in 2026
Complex numbers extend the one-dimensional real number line into a two-dimensional mathematical plane. Defined in rectangular form as $z = a + bi$ where $i = \sqrt{-1}$ (hence $i^2 = -1$), they are essential in electrical AC circuit analysis (impedance $Z = R + jX$), control theory, and fluid dynamics.
Fundamental Operations Reference
- Addition: Add real and imaginary parts: $(a + bi) + (c + di) = (a + c) + (b + d)i$.
- Multiplication: FOIL expansion noting $i^2 = -1$: $(a + bi)(c + di) = (ac - bd) + (ad + bc)i$.
- Division: Multiply by conjugate: $\frac{a + bi}{c + di} = \frac{(ac + bd) + (bc - ad)i}{c^2 + d^2}$.
- Polar Conversion: $r = \sqrt{a^2 + b^2}$, $\theta = \arctan(b/a)$.