← Advanced Complex Numbers (HL)
Mathematics · Topic Cheatsheet

Advanced Complex Numbers (HL)

18 key results accumulated across 2 chapters.

Three equivalent forms
Ch 1
z=a+bi=r(cos⁡θ+isin⁡θ)=r eiθz = a + bi = r(\cos\theta + i\sin\theta) = r\,e^{i\theta}
Cartesian for +/−, polar for ×/÷/powers, exponential most compact.
Euler's formula
Ch 1
eiθ=cos⁡θ+isin⁡θe^{i\theta} = \cos\theta + i\sin\theta
Derived by substituting x=iθx = i\theta into the Maclaurin series of exe^x and grouping real and imaginary parts.
Euler's identity
Ch 1
eiπ+1=0e^{i\pi} + 1 = 0
Special case at θ=π\theta = \pi. Uses e,i,π,1,0e, i, \pi, 1, 0 each once.
Four quadrant values
Ch 1
ei⋅0=1,  eiπ/2=i,  eiπ=−1,  ei⋅3π/2=−ie^{i \cdot 0} = 1,\; e^{i\pi/2} = i,\; e^{i\pi} = -1,\; e^{i \cdot 3\pi/2} = -i
Multiplication rule
Ch 1
eiθ1⋅eiθ2=ei(θ1+θ2)e^{i\theta_1} \cdot e^{i\theta_2} = e^{i(\theta_1 + \theta_2)}
Imaginary exponents obey the same exponent law as real ones.
Common trap
Ch 1
Track every power of ii during Maclaurin substitution — (iθ)4=+θ4(i\theta)^4 = +\theta^4 (even power flips sign back to positive).
Locus — circle
Ch 2
∣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
Ch 2
∣z−z0∣<r|z - z_0| < r
Open disk interior. Use ≤\le for closed disk; >> for exterior.
Locus — perpendicular bisector
Ch 2
∣z−a∣=∣z−b∣|z - a| = |z - b|
All points equidistant from aa and bb form the perpendicular bisector of segment abab.
Locus — ray
Ch 2
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
Ch 2
Re(z)=c   or   Im(z)=c\text{Re}(z) = c \;\text{ or }\; \text{Im}(z) = c
De Moivre's theorem
Ch 2
(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
Ch 2
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
Ch 2
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
Ch 2
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)
Ch 2
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
Ch 2
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
Ch 2
Always list ALL qq values for zp/qz^{p/q}. Stating only one loses most of the answer.