Complex Numbers — Formula Sheet

A complex number is any number that can be expressed as z = a + bi, where a, b ∈ ℝ and i = √(-1)

The imaginary unit i is defined such that i² = -1

Complex numbers were introduced because no real number satisfies x² = -1

i³ = i² · i = (-1) · i = -i

If z is real (b = 0), then z̄ = z (conjugate of a real number is itself)

The imaginary unit i is defined as i = √(-1), which means i² = -1.

For example, if z = 3 + 4i, then z̄ = 3 - 4i.

The modulus of z = a + bi is |z| = √(a² + b²).