ENGR 1405 Engineering Mathematics 1
2000 Fall
Course Notes
Chapter 3: Complex Numbers
Contents:
§3.1 Introduction and Definitions
§3.2 de Moivre’s Theorem
Chapter 3: Complex Numbers
§ 3.1 Introduction
and Definitions
If f (z)
= 0 where f (z) = z2 + 2z + 5, then
in order to find the roots of this equation it is necessary to find values for
“z” such that z2
+ 2z + 5 = 0. This can be
accomplished by applying the quadratic formula or by completing the
square. Here, the second approach will
be used. Hence, it is necessary to
determine values for “z” such that
|
|
z2 + 2z |
= |
-5 |
|
Þ |
z2 + 2z +1 |
= |
-5 + 1 |
|
Þ |
(z+1)2 |
= |
-4 |
Since we know that the square of any real number must be
greater than or equal to zero, it follows that the roots of the given equation
cannot, possibly, be real numbers. It
was problems such as this one that made mathematicians, of an earlier age,
realize that the real number system was not complete. A new number system, the Complex
Number System, was developed that:
(a) had the real
numbers as a subset, and
(b) allowed the
possibility of negative squares.
Complex Number:
The number z = a
+ jb = a + bj is a complex number
iff j2 = -1.
The real number “a” is called the real part of the
complex number, and
the real number “b” is called the imaginary part of
the complex number.
Since
this is a new number system, it is necessary to establish how to work with
these numbers. Hence, the following are properties are defined for any complex
numbers z1 = a1
+ jb1, and
z2 = a2
+ jb2 .
Addition:
The sum
of the complex numbers z1
and z2 is the complex
number “w” where
|
|
w |
= |
x + jy |
|
|
|
= |
z1 + z2 |
|
|
|
= |
(a1 + jb1) + (a2
+ jb2) |
|
|
|
= |
(a1 + a2) + j
(b1 + b2) |
|
Þ |
x |
= |
a1 + a2 |
|
and |
y |
= |
b1 + b2 |
Multiplication:
The
product of the complex numbers z1
and z2 is the complex
number “w” where
|
|
w |
= |
x + jy |
|
|
|
= |
z1 ´
z2 |
|
|
|
= |
(a1 + jb1) ´ (a2 + jb2) |
|
|
|
= |
(a1a2 - b1b2)
+ j (a1b2 + a2b1) |
|
Þ |
x |
= |
a1a2 - b1b2 |
|
and |
y |
= |
a1b2 + a2b1 |
Conjugate:
The
complex conjugate of the complex number
z1 = a1 + jb1 is the complex number
z2 = a2
+ jb2 iff
|
|
z1 ´
z2 |
= |
(a1 + jb1) ´ (a2 + jb2) |
|
|
|
= |
(a1a2 - b1b2)
+ j (a1b2 + a2b1) |
|
|
|
= |
(a1)2 + (b1)2 |
|
Þ |
a2 |
= |
a1 |
|
and |
b2 |
= |
- b1 |
Hence, if z2 is the complex conjugate of z1 = a1
+ jb1, then z2 = a1 - jb1 and the complex conjugate is normally
expressed as 
. Another notation
for the complex conjugate of z is z*.
Modulus:
The
square of the modulus | w | of the complex number w = x + jy is the real number
.
Division:
The
quotient of the complex numbers z1
and z2 with z2 ¹ 0 is the complex
number “w” where
|
|
w |
= |
x + jy |
|
|
|
= |
z1 ¸
z2 |
|
|
|
= |
|
|
|
|
= |
[(a1 + jb1) ´ (a2 - jb2)]
¸ [(a2 + jb2)
´ (a2 - jb2)] |
|
|
|
= |
[(a1a2 + b1b2)
+ j (a2b1 - a1b2)] ¸ [(a2)2 ´ (b2)2] |
|
Þ |
x |
= |
[a1a2 + b1b2]
¸ [(a2)2
´ (b2)2] |
|
and |
y |
= |
[a2b1 - a1b2] ¸ [(a2)2 ´ (b2)2] |
Alternate Representations:
Any
complex number w = x + jy can also be expressed in two alternate
forms.
Polar form: w
= x + jy = r
[cos(q ) + j sin(q )]
Euler form: w
= x + jy = r
ejq
Here 
is the modulus
of the complex number
and q = arg(w) is called the argument of the
complex number where
|
|
q |
= |
Tan-1(y/x) |
if x > 0 |
|
or |
q |
= |
Tan-1(y/x)
- p |
if x < 0 and y £ 0 |
|
or |
q |
= |
Tan-1(y/x)
+ p |
if x < 0 and y > 0 |
|
or |
q |
= |
-p / 2 |
if x = 0 and y < 0 |
|
or |
q |
= |
+p / 2 |
if x = 0 and y > 0 |
[The
argument of zero is undefined.]
If -p £
arg(w) < +p, then q = arg(w) is the principal argument
of w.
Argand Diagrams:
Complex numbers can be represented as points on a grid that
appears identical to the Cartesian co-ordinate grid (the xy-plane)
except that the horizontal axis is called the real axis, and the vertical axis
is interpreted as the imaginary axis.
The axes
are labeled respectively as Re{z}
and Im{z}.
From the
Argand diagram one can see that the argument of a complex number is not unique.
If q = arg(w) is the principal argument of w, then q + 2np (for all integers n)
are also arguments of w, [because
cos(q + 2np ) º
cos q and sin(q + 2np ) º
sin q for all integers n].
Sample Problems
Sample
Problem 1: Determine the value of z = (-2 + j5)
+ (7 - 6j)
Sample
Problem 2: Determine the value of z = 7
(3 + j4) - (2 + j) (5 - 6j)
Sample
Problem 3: Determine the value of 
Sample
Problem 4: Evaluate 
Alternative solution:
The coefficients in the binomial expansion of (a + b)4 are the entries in the n = 4 row of Pascal’s triangle:
1 4 6 4 1 .
Also note that
j 2 = -1, j 3 = -j, j 4
= 1 and j(n+4) = jn for all n.
Then
§ 3.2
de Moivre’s Theorem
The last sample problem of the previous section was rather long and tedious. It would be nice if some method existed by which the amount of work needed for this and other similar problems could be reduced. It turns out that it is indeed possible by using some elementary real number properties.
The
tools needed are:
(a) the Pythagorean
theorem
(b) elementary
geometry
(c) some basic
series expansions already studied
(d) the Binomial
theorem for arbitrary real exponents
Given
any complex number z = x + jy,
it is possible to represent it on an Argand diagram
By means
of the Pythagorean theorem and elementary geometry, we know that
In our
study of series it was found that the Maclaurin series expansion for f (x) = e x was
If we
set x = j q, and use the fact that
j n+4 = j n for n ³
1 and
j 2 = -1, j 3 = -j, j 4
= 1 then
Hence, by combining the three representations for complex numbers, the following is easily shown to be true
This
result is called de Moivre’s theorem, and is used primarily for the following
situations
(a) determining
integer powers of complex numbers,
(b) finding rational
powers of complex numbers (especially roots)
(c) establishing the
various trigonometric identities.
When
using de Moivre’s theorem to find a non-integer power of a complex number z, express z in its polar or
Euler form using the complete set of arguments (q + 2np ).
Sample
Problem 5:
Find all
three distinct cube roots of 1.
We need
to solve w3 = z for w,
where z = 1.
| z
| = | 1 + 0j | = 1 and arg(z) = arg(1 + 0j) = 0
(principal value).
In polar
form, w3 = z
= 1 cos(0 + 2np ) + j 1 sin(0 + 2np ), where n = any integer.
By de
Moivre’s theorem,
w = [1 cos(2np ) + j 1 sin(2np )](1/3)
= 1(1/3) [cos(2np / 3) + j sin(2np / 3)]
=
cos(2np / 3) + j sin(2np / 3)
This
generates only three distinct complex numbers, which can be evaluated by
setting n to any three consecutive values:
n = -1 Þ w1
= (-1 - jÖ3) / 2 , n = 0 Þ w1 = 1 , n = +1 Þ w1
= (-1 + jÖ3) / 2 .
Further examples of each of these applications will be done in class.
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