A complex number is written as z = a + jb
where a is the real part of z and b is the
imaginary part of z. j is the square root of -1 so
j2 = - 1. A complex number can also be seen as a vector
in a two dimensional space with axes Re z and Im z.
In this space the vector will extend from the origin to the point
(a, b). Every complex number a + jb
has a complex conjugate given by a - jb.
For 2 complex numbers y = a + jb and
z = c + jd the following holds.
y = z only if a = c and b = d.
If a complex number is zero, then both its real and imaginary parts are
Addition:- y + z = (a + c) + j(b
Subtraction:- y - z = (a - c) + j(b
Multiplication:- yz = (a + jb) (c + jd)
= ac + jad + jbc + j2bd =
(ac - bd) + j(ad + bc)
Reciprocal:- 1/(a + jb) = (a - jb)/
[(a + jb)(a - jb)] =
(a - jb)/(a2 + b2)
Division:- y/z = (a + jb)/(c + jd) =
[(a + jb)(c - jd)]/[(c + jd)(c - jd)] =
[(ac + bd) + j(bc - ad)]/(c2 +
This can also be written :-
y/z = (ac + bd)/(c2 +
d2) + j(bc - ad)/(c2 +
Polar Form of Complex Numbers
A complex number z = a + jb can be written is polar form
as z = r e jq where
r2 = a2 + b2. r is called
the magnitude or modulus or absolute value of z. and
q is the phase or argument of z.
De Moivre's equation
( a + jb )n =
(r ejq )n =
rn (ejq )n =
rn (ejnq ) =
rn ( cos nq
+ j sin nq)
where r2 = a2 + b2
and q = tan-1(b/a)
Operations on complex numbers in polar form
Operations like multiplication, division,
powers and roots are very easy if complex numbers
are in polar form. Consider 3 complex numbers:-
z = r e jq
z1 = r1
z2 = r2
Multiplication:- z1z2 =
e j(q1 +
Division:- z1/z2 =
e j(q1 -
Powers:- zn = rn e
Roots:- z1/n = r1/n e
Logs:- ln z = ln r +
nth roots of unity
11/n = [e j 2pm]1/n =
cos(2pm/n) + j sin(2pm/n)
The nth roots of unity are obtained for values of m from 0 to n - 1.
(-1)1/n = [e j (2m + 1)p]1/n =
cos(p(2m + 1)/n) + j sin(p(2m + 1)/n)
The nth roots of -1 are obtained for values of m from 0 to n - 1.
Complex Numbers in Oscillations and Waves
For oscillations, the following relation is useful.
e jw t =
cos w t + j
sin w t
cos w t =
Re [e jw t]
sin w t =
Im [e jw t]
For waves, the following is useful.
e j(kx - w t) =
cos(kx - w t) + j
sin(kx - w t)
Functions of a Complex Variable
A function f(s) = 1/(s + a), where s is a
complex number, is a function of a complex variable s or a complex function.
Of particular interest in complex functions is the location of the poles(points at
which the function goes to infinity) of the
function. In f(s) there is only one pole at s = - a.
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Copyright 1996 © Cuthbert A. Nyack.