lightarrow, I don't understand how i represents a phase difference of 90 degrees.

Sorry not to have answered you before to this question.

I see sophiecentaur has already answered you, I add something else:

you know that you can represent any complex number z as a point on the cartesian plane, in two ways:

1. z = x + iy; x is the real part and y the imaginary part and they corresponds to the cartesian coordinates of z in the plane.

2. z = ρe

^{iθ}; ρ is the "modulus" (= |z|) and corresponds to the lenght of the arrow that goes from the origin to the point in the plane; θ is the "argument" and corresponds to the angle between the x axis and the arrow.

The second, called "exponential representation" is often more useful, for example when you have to compute the product or the division of two complex numbers z

_{1} and z

_{2}:

z

_{1}*z

_{2} = ρ

_{1}e

^{iθ1}*ρ

_{2}e

^{iθ2} = ρ

_{1}ρ

_{2}e

^{i(θ1+ θ2)}In our discussion the angles θ

_{1} and θ

_{2} can be called the phases of the two complex n.

So, if you multiply a complex number which phase is θ

_{1} by another complex number which phase is θ

_{2},

you obtain a third number which phase is the sum of the other 2 and which modulus is the product of the other two.

So, if you have a complex number z which phase is θ and you want to transform it into another complex number with the same modulus but which phase is θ + φ, you just have to multiply z by e

^{iφ}:

z*e

^{iφ} = ρe

^{iθ}*e

^{iφ} = ρe

^{i(θ+φ)}So, if you want to give it a π/2 phase difference: z*e

^{iπ/2} ecc.

i = e

^{iπ/2} since ρ = 1 and θ = π/2 in that case.