In this introductory chapter some mathematical notions are presented rapidly

1.3.1 The ordering of real numbers
Non-zero real numbers are either positive or negative. Positive reals form the
subset , negative reals the subset . We are thus in presence of a partition
The set

of non-negative reals will also be needed. Positive numbers correspond to points
on the line lying at the right - with respect to the positive direction - of the origin.
Instead of one simply writes ('x is bigger, or larger, than
0'); similarly, will be expressed by ('x is bigger or equal than 0').
Therefore an order relation is defined by

This is a total ordering, i.e., given any two distinct reals x and y^ one (and only
one) of the following holds: either or From the geometrical point of
view the relation tells that the point with coordinate is placed at the left
of the point with coordinate . Let us also define

Clearly, implies For example the relations and are true,
whereas is not.
The order relation interacts with the algebraic operations of sum and
product as follows:

(adding the same real number to both sides of an inequality leaves the latter
unchanged);

(multiplying by a non-negative number both sides of an inequality does not alter it,
while if the number is negative it inverts the inequality). Example: multiplying by
— 1 the inequahty gives . The latter property implies the well-known

1.3 Sets of numbers 13 bet

sign rule: the product of two numbers with alike signs is positive, the product of
two numbers of different sign is negative.
Absolute value. Let us introduce now a simple yet important notion. Given a
real number , one calls absolute value of the real number

Thus for any in For instance . Geometrically,
I a: I represents the distance from the origin of the point with coordinate ; thus,
is the distance between the two points of coordinates and .
The following relations, easy to prove, will be useful
1.1

(called triangle inequality) and

Throughout the text we shall solve equations and inequalities involving absolute

values. Let us see the simplest ones. According to the definition,

has the unique solution If a is any number > 0, the equation

has two solutions and , so

In order to solve

where
consider first the solutions for which , so that now the inequality
reads then consider in which case , and solve or
To summarise, the solutions are real numbers satisfying or
which may be written in a shorter way as
)

14bet 1 Basic notions

Similarly, it is easy to see that if ,

(1.3)
The slightly more general inequality
,
where is fixed and , is equivalent adding gives
. (1.4)
In all examples we can replace the symbol by and the conclusions hold.
Intervals. The previous discussion shows that Mathematical Analysis often deals
with subsets of whose elements lie between two fixed numbers. They are called
intervals.