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