The modulus of a complex number z=a+ib
(where a and b are real) is the positive real number, denoted |z| , defined by : |z|=√a2+b2.
Amplitude of a complex number
The plan has a direct orthogonal reference (O,→i,→j). Lets z a non zero
complex number and M its image.
Called the amplitude of the complex number z, any measure, expressed in radians, of the angle (→i,→OM).
Trigonometric form of a complex number
A complex number z of argument θ and modulus r, can be written in its trigonometric form z=r(cos(θ)+i⋅sin(θ)),
|z| = r,
arg(z) = θ.
Exponential notation of a complex number
For any real θ, we note ei⋅θ the complex number cos(θ)+i⋅sin(θ).
A complex number z of amplitude θ and modulus r, can be written in its exponential form z=r⋅ei⋅θ,
|z| = r,
arg(z) = θ.
Second degree equation with real coefficients
A quadratic equation with real coefficients has in ℂ:
One real solution if the discriminant Δ=0
Two real solutions if Δ>0
Two complex conjugate solutions if and only if Δ<0