Complex Numbers

Complex numbers make oscillations and AC circuits algebraic. Learn Cartesian, polar and exponential forms, modulus, argument and conjugate, the Argand plane, Euler's formula e^(jθ) = cos θ + j sin θ, De Moivre's theorem, n-th roots, and phasor/impedance applications.

Electrical engineers write j for √−1 because i is current. With complex numbers, capacitors and inductors become impedances 1/(jωC) and jωL, and sinusoidal steady-state analysis reduces to algebra with phasors.

An analogy

A complex number is a point (or arrow) on a plane: the real part is east–west, the imaginary part north–south. Multiplying by j rotates the arrow by 90°.

Arithmetic

Add/subtract in Cartesian form; multiply/divide in polar form (or multiply by the conjugate to divide in Cartesian). z z̄ = |z|².

Euler's formula and phasors

A sinusoid A cos(ωt + φ) = Re{A e^(jφ) e^(jωt)}; the phasor A∠φ carries amplitude and phase. Differentiation becomes multiplication by jω.

Roots and powers

De Moivre gives powers and n-th roots, which are equally spaced on a circle. Roots of unity underpin the FFT.

Engineering use

AC circuit analysis, power factor (S = P + jQ), transfer functions H(jω), control-system poles, signal processing (I/Q), and quantum mechanics.

Formulas

• Polar / exponential: z = r(cos θ + j sin θ) = r e^(jθ), r = √(x² + y²), θ = atan2(y, x)
• Euler: e^(jθ) = cos θ + j sin θ; e^(jπ) + 1 = 0
• Multiply / divide: z₁z₂ = r₁r₂ e^(j(θ₁ + θ₂)); z₁/z₂ = (r₁/r₂) e^(j(θ₁ − θ₂))
• De Moivre: zⁿ = rⁿ e^(jnθ)
• n-th roots: z^(1/n) = r^(1/n) e^(j(θ + 2πk)/n), k = 0 … n − 1
• Impedances: Z_R = R, Z_L = jωL, Z_C = 1/(jωC)

Worked example: Forms and powers

Given: z = 3 + 4j; w = 1 + j

Find: |z|, arg z, w⁸.

• |z| = 5, arg = 53.13°
• w = √2 e^(jπ/4) → w⁸ = 16 e^(j2π) = 16

Answer: 5∠53.13°; (1 + j)⁸ = 16

Worked example: Series RLC impedance

Given: R = 10 Ω, L = 50 mH, C = 200 µF at 50 Hz

Find: Z.

• X_L = 15.71 Ω, X_C = 15.92 Ω
• Z = 10 + j(15.71 − 15.92) = 10 − j0.21 Ω

Answer: |Z| ≈ 10.0 Ω — almost at resonance

Common mistakes

• Wrong: arg z = atan(y/x) in all quadrants. — Right: Use atan2(y, x).
• Wrong: Adding complex numbers in polar form. — Right: Convert to Cartesian to add.

Applications

• AC circuits: Impedance and phasor analysis.
• Signal processing: I/Q signals, FFT roots of unity.
• Control: Poles and frequency response H(jω).

Key takeaways

• z = x + jy = r e^(jθ).
• Multiply/divide in polar, add in Cartesian.
• De Moivre for powers and roots.
• Phasors turn AC analysis into algebra.

Going further

• Riemann sphere: Complex infinity and mappings.
• Phasor power: S = V I* = P + jQ.
• Quaternions: Extending complex numbers to 3D rotations.