Graphics Reference
In-Depth Information
Positive closure under addition
Thesum of two positivenumbrs is
positive.
Positive closure under multiplication
Theproduct of two positive
numbers is positive.
Definition of
'
less than
'. Wesay
a
b
if and only if
b
a
is positive.
A field of numbers with order properties has to contain the positive
numbers 1, 1
1, 1
1
1, 1
1
1
1, etc. as an increasing
sequence and so it must be infinite and contain versions of
N
,
Z
and
Q
.
A field of numbers with order properties can be modelled by an
indefinitely long line. The numbers are then matched with a dense set of
points on the line. Because squares cannot be negative (qn 2.15) the
complex numbers C do not have the properties of order.
Property of Archimedean order
-
chapter 3 onwards
Every number is exceeded by integer.
If all the numbers are marked on a line, and the integers are
marked as a special row of pegs, then the property of Archimedean
order says every number lies between two pegs. This rules out both
infinite numbers and infinitesimal numbers both of which can be
tolerated in a 'non-standard' number system. An equivalent property
which is given in many texts, is that if two positive numbers
a
and
b
are
given, then some multiple of the first will exceed the second,
b
na
.Yet
another equivalent property, due to Euclid, is that if from a quantity, a
half or more is removed, and then from what is left, half or more is
removed, and so on, then at some stage, what is left will be less than
any given quantity, however small.
Principle of completeness
-
chapter 4 onwards
Every infinite decimal sequence is convergent.
The property of Archimedean order implies that every number is
the limit of an infinite decimal sequence. The rational numbers are the
limits of terminating or recurring decimals, and the rational numbers
form an Archimedean ordered field. But there are points on a line not
corresponding to rational numbers. The completeness principle is
adopted so that the real number system matches precisely the points on
a line.
Propositions equivalent to completeness in the context of
Archimedean order are
