Mathematical Fundamentals
115
orthogonal coordinates
polar coordinates
magnitude
absolute value
phase
argument
complex conjugate
variable
domain
In several branches of engineering the symbol j is preferred to i, because it is
more easily distinguished from the symbol of current. In this book, the symbol
i is used exclusively.
Given the preliminary definition of i, the complex numbers are defined as
the couples
x + iy
(2)
where x and y are real numbers called, respectively, real and imaginary part of
the complex number.
Given two complex numbers c
1
= x
1
+ iy
1
and c
2
= x
2
+ iy
2
the four
operations are defined as follows
Sum : c
1
+ c
2
= (x
1
+ x
2
) + i(y
1
+ y
2
)
Difference : c
1
- c
2
= (x
1
- x
2
) + i(y
1
- y
2
)
Product : c
1
c
2
= (x
1
x
2
- y
1
y
2
) + i(x
1
y
2
+ x
2
y
1
)
Quotient :
c
1
c
2
=
(x
1
x
2
+ y
1
y
2
) + i(y
1
x
2
- x
1
y
2
)
x
2
2
+ y
2
2
.
If the introduction of complex numbers dates back to the XVI century, their
geometric interpretation, that gave an intuitive framework for widespread use,
was introduced in the XVIII century. The geometric interpretation is simply
obtained by considering the geometric number c = x + iy as a point of the plane
having coordinates x and y. This interpretation, depicted in fig. 1, allows to
switch from the orthogonal coordinates x and y to the polar coordinates and
, called magnitude (or absolute value) and phase (or argument), respectively.
The x and y axes are called, respectively, the real and imaginary axes. The
magnitude of a complex number is calculated by application of the Theorem of
Pythagoras:
2
= x
2
+ y
2
= (x + iy)(x - iy) = cc
(3)
where c is the complex conjugate of c, also depicted in fig 1
a complex number is the angle formed by the positive horizontal semi-axis with
the line conducted from the geometric point to the origin of the complex plane.
The argument is signed, and the sign is positive for anti-clockwise angles (see
fig. 1).
A.2
Variables and Functions
In mathematics, the entities that one works with are often arbitrary elements of
a class of numbers. In these cases, the entities can be represented by a variable
x defined in a domain D. In this appendix, we have already used some variables
implicitly, for instance, to state the properties of a field.
When the domain is an interval of the field of real numbers having extremes
a and b, we can say that x is a continuous variable of the interval [a, b] and we
write a x b.
1
The expressions can be derived by application of the usual algebraic operations on real
numbers and by substituting i
2
with -1. In order to derive the quotient, it is useful to multiply
and divide by x
2
- iy
2
.
2
It is easy to show that the magnitude of the product is equal to the product of the
magnitudes. Vice versa, the magnitude of the sum is not equal to the sum of the magnitudes