Complex Numbers and De Moivre’s Theorem — Formula Sheet

z = a + bi

z̄ = a - bi

|z| = √(a² + b²)

arg(z) = tan⁻¹(b/a)

z = r(cosθ + i sinθ)

(cosθ + i sinθ)ⁿ = cos(nθ) + i sin(nθ)

z^(1/n) = r^(1/n)[cos((θ + 2kπ)/n) + i sin((θ + 2kπ)/n)]

z₁ + z₂ = (a + c) + (b + d)i

z₁·z₂ = (ac - bd) + (ad + bc)i

z₁/z₂ = (z₁·z̄₂)/|z₂|²

The equation \(x^2 + 1 = 0\) has no real solution because the square of any real number is non-negative.

A complex number is any number of the form \(z = a + bi\), where \(a\) and \(b\) are real numbers and \(i = \sqrt{-1}\).

Modulus of \(z = a + bi\) is \(|z| = \sqrt{a^2 + b^2}\).

\(n^{\text{th}}\) roots of unity are solutions to \(x^n = 1\), given by \(\cos(2k\pi/n) + i\sin(2k\pi/n)\), \(k = 0,1,\dots,n-1\).

Defined as \(i = \sqrt{-1}\), so \(i^2 = -1\).