In this introductory chapter some mathematical notions are presented rapidly

The set of rational numbers. A rational number is the quotient, or ratio,
of two integers, the second of which (denominator) is non-zero. Without loss of
generality one can assume that the denominator is positive, whence each rational
number, or rational for simplicity, is given by
with and
Moreover, one may also suppose the fraction is reduced, that is, z and n have no
common divisors. In this way the set is identified with the subset of rationals
whose denominator is 1. Besides sum, product and difference, the operation of
division between two rationals is defined on , so long as the second rational is
other than . This is the inverse to the product.

Rasm

1 Basic notions
A rational number admits a representation in base of the kind
corresponding to

The sequence of digits di, written after the dot satisfies one and only one of
the following properties: i) all digits are from a certain subscript onwards (in
which case one has a finite decimal expansion; usually the zeroes are not written),
or ii) starting from a certain point, a finite sequence of numbers not all zero -
called period - repeats itself over and over {infinite periodic decimal expansion;
the period is written once with a line drawn on top). For example the following
expressions are decimal expansions of rational number

The expansion of certain rationals is not unique. If a rational number has a finite
expansion in fact, then it also has a never-ending periodic one obtained from the
former by reducing the right-most non-zero decimal digit by one unit, and adding
the period 9. The expansions and define the same rational number 1;
similarly, and are equivalent representations of .
The geometric representation of a rational is obtained by subdividing
the segment in n equal parts and copying the subsegment m times in the
positive or negative direction, according to the sign of (see again Fig. 1.4),