# Core As Flashcards

If z = a + bi, what is the complex conjugate?

z* = a - bi

What is the modulus argument form of the complex number z?

z = r(cosθ + isinθ)

where |z| = r and arg z = θ

|z₁z₂| =

|z₁| * |z₂|

arg(z₁z₂) =

arg(z₁) + arg(z₂)

If z₁ = x₁ + iy₁, what does |z - z₁| = r represent on an argand diagram?

A circle with centre (x₁, y₁) and radius r

If = x₁ + iy₁ and z₂ = x₂ + iy, what does |z - z₁| = |z - z₂| represent on an argand diagram?

A perpendicular bisector of the line segment joining z₁ and z₂

If z₁ = x₁ + iy₁, what does arg (z - z₁) = θ represent on an argand diagram?

Half line from, but not including, the fixed point z₁ making an angle θ with a line from the fixed point z₁ parallel to the real axis

What does 1/α + 1/β equal?

(α + β) / αβ

What does 1/α + 1/β + 1/γ equal?

(αγ + βγ + αβ) / αβγ

What does 1/α + 1/β + 1/γ + 1/δ equal?

(αγδ + βγδ + αβδ + αβγ) / αβγδ

What does α² + β² equal?

(α + β)² -2αβ

What does α² + β² + γ² equal?

(α + β+ γ)² - 2(αγ + βγ + αβ)

What does α² + β² + γ² + δ² equal?

(α + β+ γ+ δ)² - 2(αγ + βγ + αβ + γδ + βδ + αδ)

What does α³ + β³ equal?

(α + β)³ -3αβ(α + β)

What does α³ + β³ + γ³ equal?

(α + β+ γ)³ - 3(α + β+ γ)(αγ + βγ + αβ) + 3αβγ