This web page discusses perhaps the most obviously basic part of analysis, or of mathematics, that is the numbers we work with. The numbers are usually classified as being whole numbers (integers), fractions (rationals), real numbers, or complex numbers. However there is more to this than just saying what each kind of number is. We should also describe the fundamental properties (or axioms) satisfied by these numbers so that we are sure our work relies on only these and no other implicit or hidden property.

You should be reasonably familiar with the usual number systems of mathematics, the natural numbers, the integers, the rationals, the reals, and the complex numbers. This web page introduces these in a fairly informal way to agree some notation and terminology for the way these main number systems used in these pages. These systems will be subject to a lot more discussion as our work gets more advanced and the methods we have of describing them get more detailed.

The natural numbers are the usual counting numbers,
0,1,2,3,4,next

number

One complication is that there
are two competing conventions for

Apart from these arithmetic operations, the most important feature
of the natural numbers is induction. This is the main method of
proof for anything to do with natural numbers and is implicit in
any mathematics where the elipsis

Principle of Induction.

Suppose a statement ~~(~~ is
true for some number
~~(~~ is true
then ~~(~~ also holds. Then ~~(~~ holds for
all

You will have seen many examples of this principle being used
to good effect, and will see many more examples in these notes.
Usually (but not always) the base case

~~(~~ is the
case of the smallest natural number

An important and useful variation of induction is what I like to
call total induction which says that in proving
the induction step, ~~(~~, it is permissible to assume
that ~~(~~ folds for all

Theorem.

Suppose a statement ~~(~~ is
true for some number
~~(~~ is true
for all ~~(~~ also holds.
Then ~~(~~ holds for all

A variation of this is the least number principle (also
called smallest counterexample

) as follows.

Theorem.

Suppose a statement ~~(~~ is
false for some number ~~(~~ is false,
i.e., a number ~~(~~ is false
and ~~(~~ is true for

The next important ingredent is subtraction. By adding to
the set of natural numbers the negatives of these numbers we get
the integers. An integer is a whole number, which may be
positive or negative. The set of integers is denoted

The set of integers has a very close connection with the natural
numbers. Indeed the set of natural numbers is the set of integers
that are greater than or equal to zero. The negative numbers makes
arithmetic in the integers much more convenient, however, and
(

The next stage is to extend our number system still further to add
multiplicative inverses

. Note that this is true for all rational numbers, and (unlike in primary school) we do not care whether the fraction is "proper" or not. So0

The main complication is that a rational number will not have a unique representation as

The operations of addition and multiplication have to be extended to the new system. That's OK, and it is done as you learnt in primary school, essentially by putting the result over a common denominator. big advantage of the rationals over the integers is that in the rationals you can also divide by any nonzero number since (

.~~
~~

.

The disadvantage of the rationals is that there are still lots of
operations one cannot do. The most obvious is taking square roots.
You may have come across the well known theorem attributed to Euclid that
says that *algebraic* numbers. However, studying this is best done as part of
a course in algebra, rather than analysis.

Analysis takes a more radical and much more powerful approach of
closing the field of rationals under the process of *taking limits*.
This is what the field of real numbers