1
Appendix SB
Complex Numbers and Algebra
SB.1
Definitions and Notation
Imaginary numbers arise when taking the square roots of negative numbers.
Thus,
3
9
±
=
, since multiplying +3 or –3 by itself gives 9. But what about
9
−
?
Whereas
9 is a real number (
3
±
),
9
−
is said to be an
imaginary
number. It is
evaluated by defining a quantity
j
as being equal to
1
−
. Then
j
2
= -1, and
9
−
=
9
2
j
. Now we have two positive quantities under the square root, so the square root
becomes
3
j
±
. This is a valid answer, because (
j
3)(
j
3) =
j
2
9 = -9 = (-
j
3)(-
j
3).
Defined in this way, imaginary numbers are a perfectly valid set of numbers, just
like real numbers, integers, or rational numbers.
j
, the basis of all imaginary numbers, is
of course itself an imaginary number, and all imaginary numbers are multiplied by
j
.
Imaginary numbers can be manipulated according to certain logical and consistent rules.
A
complex
number
x
is defined as the sum of a real number and an imaginary
number:
x = a + jb
(SB.1.1)
where
a
is referred to as the real part of
x
and
b
as the imaginary part. Complex
numbers are commonly encountered in algebra and trigonometry. For example, the
equation
x
2
+
x
+1 does not have real roots, but it does have complex roots, i.e., roots
that are complex numbers. The sine and cosine functions may be expressed in terms of
complex quantities:
2
cos
jx
jx
e
e
x
−
+
=
,
j
e
e
x
jx
jx
2
sin
−
−
=
(SB.1.2)
These relations can be readily verified using the infinite series representations of
the exponential, sine, and cosine functions.
The
conjugate
of a complex number
x
, denoted as
∗
x
, is the number that has
the same real part but a negated imaginary part. Thus,
a + jb
and
a – jb
are conjugates.
Complex quantities play a central role in electric circuits. They are the basis for
phasor notation. They are also encountered in Fourier series, Fourier and Laplace
transforms, and in the extensive applications that derive from the theory of functions of
complex variables.