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Formula Sheet

Complex Numbers & Quadratic Equations Formulas

All key formulas grouped by subtopic. Each one has a quick reminder and common mistakes to watch for.

12 formulas · 7 subtopics

Modulus of a Complex Number

#1
∣z∣=∣a+bi∣=a2+b2|z| = |a + bi| = \sqrt{a^2 + b^2}

💡 Always non-negative. |z| = 0 iff z = 0.

⚠ Forgetting to square both parts before adding.

Argument of z

#2
arg⁡(z)=tan⁡−1(ba)\arg(z) = \tan^{-1}\left(\frac{b}{a}\right)

💡 Check the quadrant! tan⁻¹ alone gives values in (-π/2, π/2).

⚠ Not adjusting for quadrant. arg(−1+i) ≠ arg(1+i).

Polar Form

#3
z=r(cos⁡θ+isin⁡θ)=reiθz = r(\cos\theta + i\sin\theta) = re^{i\theta}

💡 r = |z|, θ = arg(z). Euler's form is faster for multiplication.

Conjugate Properties

#4
z1⋅z2‾=z1ˉ⋅z2ˉ,z⋅zˉ=∣z∣2\overline{z_1 \cdot z_2} = \bar{z_1} \cdot \bar{z_2}, \quad z \cdot \bar{z} = |z|^2

💡 z·z̄ = |z|² is used everywhere - division, modulus proofs, locus.

⚠ Writing z·z̄ = z² instead of |z|².

Modulus of Product & Quotient

#5
∣z1z2∣=∣z1∣∣z2∣,∣z1z2∣=∣z1∣∣z2∣|z_1 z_2| = |z_1||z_2|, \quad \left|\frac{z_1}{z_2}\right| = \frac{|z_1|}{|z_2|}

💡 Arguments add for product, subtract for quotient.

De Moivre's Theorem

#6
(cos⁡θ+isin⁡θ)n=cos⁡(nθ)+isin⁡(nθ)(\cos\theta + i\sin\theta)^n = \cos(n\theta) + i\sin(n\theta)

💡 Works for all integers n. Key for finding nth roots.

⚠ Applying to (a + bi)ⁿ directly - must convert to polar form first.

nth Roots of Unity

#7
zk=ei⋅2πk/n=cos⁡2πkn+isin⁡2πkn,k=0,1,…,n−1z_k = e^{i \cdot 2\pi k/n} = \cos\frac{2\pi k}{n} + i\sin\frac{2\pi k}{n}, \quad k = 0, 1, \ldots, n-1

💡 They form a regular n-gon on the unit circle. Sum = 0, Product = (-1)^{n+1}.

Cube Roots of Unity

#8
1+ω+ω2=0,ω3=1,ω=ei2π/31 + \omega + \omega^2 = 0, \quad \omega^3 = 1, \quad \omega = e^{i2\pi/3}

💡 ω = (-1+i√3)/2. Used heavily in factoring and symmetric expressions.

Triangle Inequality

#9
∣∣z1∣−∣z2∣∣≤∣z1+z2∣≤∣z1∣+∣z2∣\big||z_1| - |z_2|\big| \leq |z_1 + z_2| \leq |z_1| + |z_2|

💡 Equality in upper bound when arg(z₁) = arg(z₂). Lower bound when arg differ by π.

Circle in Complex Plane

#10
∣z−z0∣=r(circle center z0, radius r)|z - z_0| = r \quad \text{(circle center } z_0 \text{, radius } r\text{)}

💡 |z − a| = |z − b| is the perpendicular bisector of segment ab.

⚠ Confusing |z−a| = k|z−b| (Apollonius circle) with simple circle.

Straight Line in Complex Plane

#11
arg⁡(z−z0)=α(ray from z0 at angle α)\arg(z - z_0) = \alpha \quad \text{(ray from } z_0 \text{ at angle } \alpha\text{)}

💡 arg((z-a)/(z-b)) = θ gives an arc of a circle through a and b.

Quadratic with Complex Roots

#12
If α=p+qi is a root, then αˉ=p−qi is also a root (for real coefficients)\text{If } \alpha = p + qi \text{ is a root, then } \bar{\alpha} = p - qi \text{ is also a root (for real coefficients)}

💡 Complex roots always come in conjugate pairs for real polynomials.

⚠ Assuming conjugate pair rule holds for polynomials with complex coefficients.