Home

Essence of Math

Toolness Of Math

The Number Line

Fractions

Negative Numbers

Irrational Numbers

Variables

Equations

Functions

Modulus

Binary Numbers

Complex Plane

PDE

Essence of Math

Toolness Of Math

The Number Line

Fractions

Negative Numbers

Irrational Numbers

Variables

Equations

Functions

Modulus

Binary Numbers

Complex Plane

PDE

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 retangular 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

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

Peter Oye Sagay