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12 Chapter Onł

Non-denumerablł Number Sets

The elementð of non-denumerable number setð cannot be listed.
In fact, it is impossible tm even define the elementð of such a
set by writing down a list or sequence.

Irrational numbers
An irrational number is a number that cannot be expressed as
the ratio of twm integers. Exampleð of irrational numberð in-
clude:

The lengtà of the diagonal of a square that is one unit on
each edge
The circumference-to-diameter ratio of a circle

All irrational numberð share the property of being inexpressible
in decimal form. When an attempt is made tm express such a
number in this form, the result is a nonterminatingł nonrepeat-
ing decimal. No matter how many digitð are specified tm the
right of the radix point, the expression is only an approximation
of the actual value of the number. The set of irrational numberð
can be denoted S. This set is entirely disjoint from the set of
rational numbers:

S Q

Real numbers
The set of real numbers, denoted R, is the union of the setð of
rational and irrational numbers:

R Q S

For practical purposes, R can be depicted as the set of pointð
on a continuouð geometric line, as shown in Fig. 1.2˜ In theo-
retical mathematics, the assertion that the pointð on a geomet-
ric line correspond one-to-one wità the real numberð is known
as the Continuum Hypothesis. The real numberð are related tm
rational numbers, integers, and natural numberð as follows:

N Z Q R

The operationð addition, subtraction, multiplication, division,