Why Do We Rationalize Fractions?

Date: 11/29/2005 at 20:28:05
From: Tammy
Subject: Radicals in the denominator

Why do radicals have to be simplified from the denominator of 
fractions?  What is wrong with having a radical in the denominator?

For example, why must 1/sqrt(5) be multiplied by sqrt(5)/sqrt(5) to
obtain sqrt(5)/5?

Date: 11/29/2005 at 23:44:19
From: Doctor Peterson
Subject: Re: Radicals in the denominator

Hi, Tammy.

The main reason we tell students to do this is to have a standard form
in which certain kinds of answers can be written.  That makes it 
easier for us as teachers to check the answers, and for the students 
to check their own answers in their book.  Also, putting things in a 
standard form can make it easier to recognize similar forms in a 
complicated problem, and to combine them.  For example, if you had to 
add sqrt(5) and 1/sqrt(5), you would not see that they can be combined 
until you changed the latter to sqrt(5)/5; then you would be able to 
add them and get 6 sqrt(5)/5.

Before calculators, there was a specific reason to rationalize
denominators that went beyond this: when you actually calculate the
value of the expression by hand, it is a lot easier to divide sqrt(5)
by 5 than to divide 1 by sqrt(5).  (Try it!)  Since that was a useful
form, it became the standard form that teachers expected.  We maintain
the tradition because it's helpful to have SOME standard form, and
that one is at least as good as any.

But it's not really always clear what form is "simplest" in the first
place, and in many real situations we would not bother to rationalize.
See this page:

  Rationalizing the Denominator
    http://mathforum.org/library/drmath/view/52663.html 

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


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