Complex Numbers

What is the set of complex numbers defined as?

C = {a + bi : a ∈ R and b ∈ R} .

The complex conjugate z ̄ of a complex number z = a + bi given by?

z ̄ = a − bi.

For z ∈ C\ {0} a non-zero complex number, what is the inverse of z given by?

z⁻¹ = (1/zz^-)z^-

What is the quotient of two complex numbers α = a + bi and β = c + di, with β ̸= 0?

αβ⁻¹ = α/β = (1/ββ^-)αβ^-

What is the principal value of arg(z), denoted Arg(z)?

The member of arg(z) such that Arg(z) ∈ (−π, π].

What is De Moivre’s Theorem?

If n ∈ N and θ ∈ R, then (cos(θ) + isin(θ))ⁿ = cos(nθ) + isin(nθ).

If n ∈ N and θ ∈ R, then (cos(θ) + isin(θ))⁻ⁿ = cos(nθ) − isin(nθ) = cos(−nθ)i + sin(−nθ).

What is Euler’s formula?

e^iθ = cos(θ) + isin(θ),

What are the 3 different ways we can write complex numbers?

z = a + bi = r (cos(θ) + isin(θ)) = re^iθ

Using Euler’s formula, how can we write cosθ and sinθ?

cos(θ) = ½ (e^iθ + e^−iθ) and sin(θ) = ½ i (e^iθ − e^−iθ) ∀θ ∈ R.

Using De Moivre’s Theorem, how can we write cos(nθ) and sin(nθ)?

e^inθ = cos(nθ) + isin(nθ),

e^−inθ = cos(nθ) − isin(nθ),

What is the algebraic way of writing a complex number?

z = a + bi

What is the mod-arg way of writing a complex number?

z = r(cos(θ) + isin(θ))

What is the exponential way of writing a complex number?

z = re^iθ

What is the formula for cos(2θ)?

cos²θ - sin²θ

What is the formula for sin(2θ)?

2cosθsinθ

What is the definition of an nth root of a non-zero complex number z?

If wⁿ = z

What does it mean for a complex number, a, to be a root of the polynomial equation p(z) = 0?

Iff (z - a) is a factor of the polynomial (i.e. p(z) = (z - a)q(z) where q(z) is a polynomial of degree n - 1)

For the quadratic equation az² + bz + c = 0 where a, b, c are complex numbers, what is the formula for the solutions of a complex quadratic?

Derived from the quadratic formula, z = -b/2a + ((b² - 4ac)/4a²)^1/2

What is a real irreducible quadratic?

If a quadratic polynomial cannot be factorised into 2 real linear factors (i.e. if the roots are complex)