Appendix B: Review of Complex Numbers
The imaginary number j is defined by j 2 = -1.
Any complex number can be written as z = x + j y where x and y are real numbers. x is called the
real part of z ( symbolically, x = Re(z) ) and y is the imaginary part of z ( y = Im(z) ).
Addition and subtraction: z1 + z2 = ( x1 + j y1 ) + (x2 + j y2 ) = (x1 + x2) + j (y1 + y2)
I.e. Re (z1 + z2) = Re (z1) + Re(z2). Similarly for subtraction: z1 - z2 = (x1 - x2) + j (y1 - y2)
Multiplication: z1 z2 = ( x1 + j y1 )(x2 + j y2 )
= ( x1x2 +j x1y2 + j x2 y1 + j 2y1y2 )
= (x1x2 - y1y2) +j (x1y2 + x2 y1)
Complex conjugate: z* x - j y. That is, Re(z*) = Re(z), Im(z*) = - Im(z)
example: ( 2 + j 3)* = 2 - j 3
Properties of the conjugate:
A) (z*)* = z
B) zz* = x2 + y2 = a real number 0. Further, zz*=0 if and only if Re(z) = 0 and Im(z) = 0.
Magnitude (also called modulus) of a complex number: z =
z * z = x2 + y2
Problem: prove that, for any two complex numbers z1 and z2, z1 z2 = z1 z2
Division:
z1 z1 z2 * ( x1 x2 + y1 y2 ) + j ( y1 x2 − x1 y2 )
=
=
z2 z2 z2 *
x22 + y22
Any complex number can be represented as a point in
the complex plane, where the point (x,y) represents the
complex number z = x + j y . A vector from the origin
to (x,y) has a length equal to z by Pythagoras’s
theorem. The angle this vector makes with the +x axis
is called the phase of z. It is also called the argument
of z. The usual notation is
arg(z) = = tan-1 (y / x)
example: Find the modulus and phase of 4 + j 5
a) 4 + j 5 =
42 + 52 = 41 ≈ 6.403
b) arg(4 + j 5) = tan-1 (5 / 4) 51.34( = 0.896 rad
1B
Euler’s theorem is one of the most important formulas in mathematics:
e j θ = cos(θ ) + j sin(θ )
(Euler’s theorem)
where e is the base of natural logarithms. The result may be proven by expanding each side of
this equation in a Taylor series and comparing them term by term.
Polar representation of a complex number: If z = x + j y, then x = z cos( ) and y = z sin( ),
so that z = x + j y = z cos( ) + j z sin( ) = z (cos( ) + j sin( )) = ze j .
z = z e j arg( z )
This makes multiplying complex numbers easier:
(
z1 z2 = z1 e jθ1
)( z
2
)
e jθ2 = z1 z2 e j (θ1 +θ2 )
so that z1 z2 = z1 z2 , arg( z1 z2 ) = arg( z1 ) + arg( z2 )
2B