This web page lists a number of important properties of the real numbers. Most of this is background information that you already know in some form or other and you can think of it as an information sheet setting out the basic properties of the reals that we have been using up to now. You know them all and are familiar with them and with using them, but may not have seen them listed in this way before.
A number system satisfying all the properties listed here is called an Archimedean Ordered Field. Thus the set of real numbers forms a Archimedean Ordered Field. So does the set of rational numbers. The final property describing the reals and distinguishing it from the rationals and other Archimedean ordered fields is called completeness and will be described later.
We start here by seeting out the axioms and basic properties of a field. You may have seen most of these ideas elsewhere and this might be good revision material. The material here isn't strictly speaking analysis, but it is basic and important background on what the real numbers are, which is of course needed for real analysis.
When writing these axioms, it is convenient to
introduce, as well as 0,1,
notation for the additive inverse -
A field is a set
(3) multiplication and addition satisfy the distributivity law
Note that from the way the definition is stated, a field
You already know that the identity and inverses are unique in a group. We can state this formally as a very useful proposition.
Proposition on uniqueness of identity and inverses.
Most of the usual algebraic properties of numbers
(such as (-
If -1=0 then 0+0=0=1+-1=1+0 so 1=0 by cancellation in the additive group.
Except for when
Please read the next few statements carefully. -
In particular note from the last proposition the real reason why
a negative times a negative is a positive: it is the only way
to get the distributive law to work out!
I could go on, but this is enough to get us started. Some of the propositions above are not obvious consequences of the axioms. Many other basic properties you are familiar with are now much easier to prove with these propositions done.
A further important feature of the reals that we have been
using all along is the
sequences and series syllabus, and could be
examined, but should be easy to learn and use.
I choose here to discuss
less than. We write
Definition of an ordered field.
An ordered field is a field
One is positive: 0
We give a few simple but important propositions concerning the order in a field here.
For the rest of this section fix an ordered field
For all positive
By induction on
For all positive
The last two propositions have the useful consequences that
By cases. If
Note too that the only one of the three cases above where
For all positive
Finally for this section, we note that the proof of
the triangle inequality we gave elsewhere works completely
in the setting of an arbitrary ordered field, where distance is defined by
The triangle inequality.
The next axiom concerns
integer part function.
We have seen that this function plays a useful role in our theory, but it turns
out that not all ordered fields have it defined. A field for which integer part is
defined is called an Archimedean Ordered Field.
Before stating the axiom, it is important to know how the integers
lie in an ordered field. We have seen that such a field contains elements
called 0 and 1. The additive subgroup that are generated by these
is a copy of the integers
More formally, given an
Fortunately this mapping is one-to-one (i.e., an injection). That
is because all the values (1+1+
In fact (and there is still something more to prove here)
it turns out that this mapping is an embedding
More generally a rational number
Defintion of an Archimedean ordered field.
An Archimedean ordered field is an ordered field
Examples of Archimedean ordered fields include the reals
We can add other axioms at this stage to distinguish the reals
and the rationals. For example, we could consider the square root function
(which isn't available in
Everything we have done in the course up to this point (i.e., up to but not including the monotone convergence theorem) works for any Archimedean ordered field with square roots. Actually, square roots were only required for some examples so the formal theory we have been working through works in slightly less.
However, it turns out that Archimedean ordered field with square roots
need not have such numbers as
You have seen the axioms for Archimedean ordered fields, the two key examples being the reals and the rationals, and some of the basic consequences.