Graphics Reference
In-Depth Information
Theorem
qn 9, 14, 15
An infinite decimal is equal to a rational
number if and only if it is either terminating or
recurring.
Theorem
qn 16
If two distinct infinite decimals are equal, then
one is terminating and the other has recurring
9s.
Theorem
qn 18
There is no rational number whose square is 2.
Theorem
qn 20
There is an irrational number between any two
rationals.
Theorem
qn 20
The irrational numbers are dense on the line.
Countability
Definition
If there is a sequence of distinct terms from a set
which contains all the terms of the set, then that
set is said to be
countably infinite
.
Theorem
qn 25
The set of rational numbers is countably
infinite.
Theorem
qn 23
The set of infinite decimals between 0 and 1 is
not countably infinite.
The completeness principle: infinite decimals are convergent
The properties of numbers which we have worked with so far
consist of
(1) algebraic properties listed in appendix 1,
(2) properties of less than, based on the notion of positiveness, in
chapter 2,
(3) the Archimedean Principle in chapter 3.
We have been able to show that in a number system satisfying these
properties, every number is the limit of an infinite decimal sequence (qn
3.51).
Now the rational numbers, by themselves, satisfy (1), (2) and (3), so
these number properties do not ensure the existence of square roots or
of limits for non-recurring infinite decimals.
To have a number system with all the properties we expect, namely, the
real number system, R, we adopt one more principle, the completeness
principle.
