Why Do We Need to Study Rational Numbers?

Date: 04/22/2008 at 20:56:59
From: Jyoti
Subject: why do I need to know rational numbers?

My students want to know the connection between real life and rational 
numbers as "why do they have to study rational numbers?"

I have introduced rational numbers as the ratio of two integers where 
the denominator is not zero.  The students found the answer too 
mathematical.

Date: 04/22/2008 at 23:28:34
From: Doctor Peterson
Subject: Re: why do I need to know rational numbers?

Hi, Jyoti.

Well, why are you teaching it?  It's a good question for teachers to
ask too!

I hope they are aware that rational numbers are just a new name for
fractions, which they have presumably known about for years; all the
applications of fractions (say, in cooking or in construction) use
rational numbers.  So there are lots of practical reasons for using
them, and being able to work with them easily.

When they are called rational numbers, it is generally part of the
study of different kinds of numbers--natural, integer, real, etc.
This is mostly of theoretical interest, which (unless you are going to
be a mathematician) is mostly useful in understanding some of the
cultural significance of math.  The idea is that we can build up the
concept of numbers in several layers, starting with the most "natural"
numbers that are understood by young children, and becoming more
sophisticated and complex in order to answer new kinds of questions. 
Natural numbers are used in counting; zero is introduced because you
might also have none of something; integers allow you to subtract ANY
two numbers and get an answer that makes sense (and so model ideas
like debts or going in both directions along a line).  But integers
don't allow you to DIVIDE any two numbers; 2 divided by 3 is not an
integer, so you're stuck (or have to use remainders, an unwieldy trick
to avoid fractions)--until you invent or discover the idea of 
fractions (rational numbers), which let you divide anything (except by
0).  Each new kind of number includes the previous kind; we're
extending a concept and making it more and more useful.

The MOST important aspect of rational numbers, I think, is the
discovery that you can't stop there!  At first kids think that
fractions will cover everything they have to do; but in fact there are
real numbers (numbers that represent points on a number line) that are
NOT rational--the irrational numbers.  The ancient Greeks (such as
Pythagoras) assumed that everything could be measured using whole
numbers of small enough units, so that any two lengths would have a
rational ratio.  Then they discovered (through the Pythagorean
Theorem) that the length of the diagonal of a unit square could not be
represented by a rational number, and it messed up their whole theory.
They had to come up with a new way of explaining things that avoided
their false assumption, and in doing so, they became much stronger
mathematicians, and improved their ways of reasoning.  This event is
in a sense the reason mathematicians care so much about proofs.  And
you can't understand that without knowing what rational numbers are!

If you have any further questions, feel free to write back.


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