Complex numbers as points: modulus, amplitude and trigonometric form

The modulus |z| = r = √(x² + y²) is the distance of the point from the origin, so it is never negative. An amplitude of z is any angle θ with x = r cos θ and y = r sin θ. Adding a full turn 2π gives another amplitude of the same number, so to have a single answer we use the principal amplitude, the one in ]−π, π] (in degrees ]−180°, 180°]). To find it, first find the acute reference angle α = tan⁻¹(|y|/|x|) from the lengths alone, then look at the quadrant of the point: quadrant I gives θ = α, II gives π − α, III gives α − π and IV gives −α (see the table). The book writes tan⁻¹(y/x) with the signs inside and adds or subtracts π; both ways give the same result. On the axes there is no angle to find: a positive real number has amplitude 0, a negative real number π, a positive imaginary number π/2 and a negative imaginary number −π/2. Three facts will be used all unit: |z| = |z̄| = |−z|, z · z̄ = |z|² (because (x + yi)(x − yi) = x² + y²), and if z has principal amplitude θ then z̄ has −θ (except that a negative real number keeps π), while −z has θ − π when θ > 0 and θ + π when θ ≤ 0.