← Advanced Complex Numbers (HL)
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Advanced Complex Numbers (HL)

Chapter 2 · Loci, Multi-angle, Rational Powers

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Locus — circle
∣z−z0∣=r|z - z_0| = r
All complex numbers at distance rr from z0z_0 form a circle of radius rr centred at z0z_0.
Locus — disk
∣z−z0∣<r|z - z_0| < r
Open disk interior. Use ≤\le for closed disk; >> for exterior.
Locus — perpendicular bisector
∣z−a∣=∣z−b∣|z - a| = |z - b|
All points equidistant from aa and bb form the perpendicular bisector of segment abab.
Locus — ray
arg⁡(z−z0)=α\arg(z - z_0) = \alpha
Half-line from z0z_0 in direction α\alpha (not the full line).
Locus — vertical / horizontal line
Re(z)=c   or   Im(z)=c\text{Re}(z) = c \;\text{ or }\; \text{Im}(z) = c
De Moivre's theorem
(cos⁡θ+isin⁡θ)n=cos⁡nθ+isin⁡nθ(\cos\theta + i\sin\theta)^n = \cos n\theta + i\sin n\theta
Source of multi-angle identities — expand binomially and equate real/imaginary parts.
Double-angle from De Moivre
cos⁡2θ=cos⁡2θ−sin⁡2θ,sin⁡2θ=2cos⁡θsin⁡θ\cos 2\theta = \cos^2\theta - \sin^2\theta, \quad \sin 2\theta = 2\cos\theta \sin\theta
From (c+is)2=c2−s2+2ics(c + is)^2 = c^2 - s^2 + 2ics.
Triple-angle from De Moivre
cos⁡3θ=4cos⁡3θ−3cos⁡θ,sin⁡3θ=3sin⁡θ−4sin⁡3θ\cos 3\theta = 4\cos^3\theta - 3\cos\theta, \quad \sin 3\theta = 3\sin\theta - 4\sin^3\theta
Rational power — number of values
zp/q   has   q   distinct valuesz^{p/q}\;\text{ has }\;q\;\text{ distinct values}
When p/qp/q is in lowest terms. They sit equally spaced around a circle of radius rp/qr^{p/q}.
Phasor (real-world bridge)
Asin⁡(ωt+φ)  ⟺  AeiφA \sin(\omega t + \varphi) \;\Longleftrightarrow\; A e^{i\varphi}
Encode amplitude (modulus) and phase (argument) of an oscillation as one complex number.
Common trap — locus rewriting
Rewrite the equation as ∣z−z0∣|z - z_0| BEFORE reading off the centre. ∣z+2∣=3|z + 2| = 3 means z0=−2z_0 = -2, not +2+2.
Common trap — multi-valued powers
Always list ALL qq values for zp/qz^{p/q}. Stating only one loses most of the answer.