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Rational and Irrational Numbers

Date: 2/2/96 at 0:4:38
From: Anonymous
Subject: Rational and Irrational numbers

What is the difference between rational and irrational numbers?

Date: 2/2/96 at 14:34:37
From: Doctor Syd
Subject: Re: Rational and Irrational numbers


Before we talk about rational and irrational numbers, let's make clear 
one other definition.  An INTEGER is in the set: 

{...-3, -2, -1, 0, 1, 2, 3, ...}

It is just a positive or negative whole number.  Thus 454564 is an 
integer, but 1/2 isn't.

Now, a rational number is any number that can be written as a ratio of 
two integers (hence the name!).  In other words, a number is rational if
we can write it as a fraction where the numerator and denominator are 
both integers.  Now then, every integer is a rational number, since 
each integer n can be written in the form n/1.  For example 5 = 5/1 - 
thus 5 is a rational number.  However, numbers like 1/2, 
45454737/2424242, and -3/7 are also rational since they are fractions 
where the numerator and denominator are integers.  

An irrational number is any real number that is not rational.  By "real" 
number I mean, loosely, a number that we can conceive of in this world,
one with no square roots of negative numbers (numbers where square roots 
of negative numbers are involved are called complex, and there is lots 
of neat stuff there, if you are curious).  A real number is a number 
that is somewhere on your number line.  So, any number on the number 
line that isn't a rational number is irrational.  For example, the 
square root of 2 is an irrational number because it can't be written as 
a ratio of two integers.  

How would you imagine we would show something like that?  The proof 
is a proof by contradiction.  We assume that the square root of 2 CAN 
be written as p/q for some integers, p and q, and we get a contradiction.  
The proof has a little trick to it, but if you're curious about it, write back 
and I can tell you more!  

Other irrational numbers include:
square root of 3, the square root of 5, pi, e, ....

I hope this answers your question.  There are lots of neat properties of 
rational numbers, irrational numbers and real numbers.  For instance, it
turns out that if you were to try to gauge how many rational numbers, 
irrational numbers, and real numbers there are between 0 and 1, you 
would find that while there are infinitely many of each kind of number, 
there are many, many more irrational numbers than rational numbers.  The 
sizes of the infinities involved are somehow a little different.  Another 
property is that between any two rational numbers on the number line 
there is an irrational number; also, between any two irrational numbers 
there is a rational number.  

So, these are some things for you to ponder!  Write back if you have any 
more questions!

-Doctor Syd,  The Math Forum

Associated Topics:
Elementary Number Sense/About Numbers

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