Rationalizing the Denominator

Date: 04/09/97 at 12:35:50
From: Katrina
Subject: Fraction Denominators

Why must all fractions have denominators that are rational numbers?

Date: 07/22/97 at 00:31:57
From: Doctor Terrel
Subject: Re: Fraction Denominators

Katrina,

That's a good question!  

If we have a fraction like

            7.5
           ----- 
            9.6

there's nothing really wrong with leaving it that way. It depends 
on where the fraction came from, what it means, and where we want 
to go with it. You see, fractions (in fact, most numbers), are just 
"things" or "mathematical ideas" until we give a meaning to them. 
Often they represent a ratio of two quantities, as opposed to the 
elementary idea of "parts of a whole." So in general, I'd say: don't 
worry about any requirement that the denominator be a rational or even 
a whole number until you know the purpose for using that fraction.
 
However, I do recall from some of my earlier math courses that we 
used the phrase "rationalizing the denominator," and I think this may 
the reason you asked the question. You see, in trigonometry or algebra 
II we often had to simplify or even evaluate expressions such as:

       13                 300
    --------   and   ---------------
     sqrt 3          sqrt 5 - sqrt 2

Unless we do something tricky, we have ugly, LONG divisions on our
hands, namely,

     13                  300          300
   ------ = ?      -------------- = -------  =  ?
   1.732           2.236 - 1.414     0.822

Of course, sqrt 3, sqrt 5, and sqrt 2 were always available in 
tables. But believe me, in the old days (B.C. = before calculators), 
long division was not enjoyed by many individuals.  Instead, we were 
taught to "rationalize the denominator" like this (for the first 
example):

   13      sqrt 3      13 (sqrt 3)     13 x 1.732    
------- x -------  =  ------------  =  ----------      
sqrt 3    sqrt 3       (sqrt 3)^2          3         

Because the rest does not involve a multi-digit divisor, I leave it to 
you to finish.

The other problem is similar, but a little different.

       300             sqrt 5 + sqrt 2       300(2.236 + 1.414)
 ----------------  x  ----------------  =  -----------------------
 sqrt 5 - sqrt 2       sqrt 5 + sqrt 2    (sqrt 5)^2 - (sqrt 2)^2


     300 x 3.650       300 x 3.650
 = -------------  =  --------------  =  100 x 3.650  = 365
       5 - 2               3

I hope these two examples show that in the B.C. days we could 
calculate some pretty mean expressions, if we could make the 
denominator somehow simpler. Here the importance of the "rational 
denominator" was that the evaluation by long division was made quite a 
bit easier. Nowadays, if you need those values (as an engineer or 
architect might), a calculator does it directly, quickly, and more 
accurately.  

Here are your key sequences for a simple, non-scientific calculator:
 
 1)   13  [/]  3  [sqrt]  [=]
 
 2)   5  [sqrt}  [M+]  2  [sqrt]  [M-]  300  [/]  [MR]   [=]
 
By the way, could you reduce my original fraction to lowest terms, 
as a favor to me?
 
Thanks.

-Doctor Terrel,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/