The *complex plane* is a plane whose coordinates are represented by *Z*.

Figure 2.14 illustrates two circles. Consider the point *p* on both circles. In figure 2.14a, the point *p* has rectangular coordinate given by *x,y*. Where *x* is its horizontal distance along the X axis and *y* is its vertical distance along the Y axis. The *polar coordinate* of point *p* is *(r,θ)*. Where *r* is called its *position vector* and *θ* its direction. Consequently, the magnitude |**r**| of the position vector **r**, called its *absolute value* is given by:
*| r|^{2} = x^{2} + y^{2}*.

But

So,

If

In figure 2.14b, the point *p* has complex coordinate given by *Z = x + iy*. Where *x* is called the real component (Re(Z)) of Z and *iy* is called the imaginary component (Im(Z)) of Z. The entity *i* is defined as √-1. The conjugate of Z is defined as Z^{'} = x - iy. The polar form of *Z* when *r* = 1 is given by
Z = cosθ + isinθ. A plane with *Z* coordinates is a complex plane.

The mathematical entity, *e ^{iθ}*, known as

Complex numbers can be used to represent sinusoidal signals as follows:
*Acos(ωt + θ) = Re(Ae ^{i(ωt + θ)})= Re[(Ae^{iθ})(Ae^{iωt})]*.

Essentially, the equation says that

The mathematical entity *Ae ^{iθ}* is defined as the

This

that is,

The frequency ω of the sinusoidal signal is not explicitly stated in the phasor representation of the sinusoid. In general a sinusoidal signal can be mathematically represented either in its *time domain form*:
*v(t) = Acos(ωt + θ)*

Or its *phasor* or *frequency-domain form*:
*V(iω) = Ae ^{iθ} = A<θ*,
where