Significance of Rational Numbers

Date: 01/11/2003 at 11:24:12
From: Chirag
Subject: Significance of rational numbers 

We know that any rational number can be constructed using integers, 
and that an irrational number can be constructed using a sequence of 
rational numbers, but isn't that entirely due to the way rational 
numbers are defined? 

Why are rational numbers defined the way they are?

Date: 01/12/2003 at 12:05:01
From: Doctor Kastner
Subject: Re: Significance of rational numbers 

Hi Chirag -

Why are the rationals defined the way they are? That's an interesting
question. Let me address it from a formal point of view, and to do
that we need to talk about abstract algebra a little. Abstract algebra
is the branch of mathematics that studies number systems and their
associated operations. An example will help make this clearer. 

A group (G) is a set of elements, together with a binary operator (.)
that satisfies the following properties:

1) Closure. If a and b are in G, a.b is in G
2) Associativity: If a,b,c are in G, then a.(b.c) = (a.b).c
3) Identity. There is an identity element (I) in G such that a.I = a
   for every element in G
4) Inverse. For each element a in G, there is an inverse b = a^(-1)
   such that a.(a^(-1)) = I

I should make it clear that the a^(-1) does not necessarily mean 1/a.
We are dealing with generic inverses here. 

Groups are all around us in math. The classic example is the set of
integers together with the binary operator +. It is clear that it is
closed and associative, and the identity is 0, while the inverse 
of a is -a. 

But notice that the integers with the operator * do NOT form a group.
Multiplication of the integers satisfies the first three properties
(the identity is 1), but given an element a, the inverse element 1/a
is not always an integer. Even worse, what if the number you want to
find an inverse of is 0?

Other topics of study in abstract algebra include (in order of
increasing number of properties) rings and fields. In addition, rings
and fields require two operations (+ and * if we are talking about the
rationals), not just one. Groups, rings and fields are all connected,
and it is common for mathematicians to start with a group and try to
expand it into a ring or a field. 

The group of integers under + forms a group, but suppose I want to 
make a field using that as my base. First, I know that I need another
operation, in this case *. Both operators in a field must also satisfy
the group properties (except for a multiplicative inverse for the 0
element), and we also need commutivity (a+b = b+a, a*b= b*a) and the
distributive property (a*(b+c) = a*b+a*c)). 

Those extra ones aren't bad, but I still don't have the right set of
numbers for the multiplicative inverse. If I consider the integer a,
my field must also contain the inverse, 1/a. So let's put those in the
set. But since this should be closed, I need to include the numbers
b*1/a in my field too. Now put those in. You can see where this is
going. The result is the rational numbers. It turns out that the
rationals are the simplest field that can be constructed using the
integers as a starting point.

Finally, I should mention that the set of irrational numbers does not
form a field (or even a group for that matter) since it isn't closed
under +. Pi and its additive inverse -pi are both in the collection,
but their sum is 0 which is not an irrational number.

I hope this helps.  Write back if you're still stuck, or if you have
other questions.

- Doctor Kastner, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 01/16/2003 at 12:23:50
From: Chirag
Subject: Thank you (significance of rational numbers )

Thanks so much for answering my question so promptly. It was really 
very satisfying to get such a comprehensive answer.