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|=sqrt(a^2+b^2)`.

Amplitude of a complex number

The plan has a direct orthogonal reference `(O,vec(i),vec(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 `(vec(i),vec(OM))`.

Trigonometric form of a complex number

A complex number z of argument `theta` and modulus r, can be written in its trigonometric form `z=r(cos(theta)+i*sin(theta))`,
|z| = r,
arg(z) = `theta`.

Exponential notation of a complex number

For any real `theta`, we note `e^(i*theta)` the complex number `cos(theta)+i*sin(theta)`.

A complex number z of amplitude `theta` and modulus r, can be written in its exponential form `z=r*e^(i*theta)`,
|z| = r,
arg(z) = `theta`.

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