Rationalizing the Denominator

Date: 05/03/99 at 20:31:23
From: Karen
Subject: Fractions and square roots

Why can't you have a square root in the denominator of a fraction?

Date: 05/04/99 at 12:35:30
From: Doctor Peterson
Subject: Re: Fractions and square roots

Hi, Karen.

Technically, it's not a fraction unless the numerator and denominator 
are integers, but I think you're probably asking why we have to 
simplify an algebraic fraction by rationalizing the denominator. Why 
aren't you allowed to leave a square root in the basement if it isn't 
doing any harm?

It's not really that you CAN'T, but that you don't want to, or rather 
that it's traditional not to. In part, the concept of rationalizing 
the denominator is left over from the days when all arithmetic was 
done by hand. Which would you rather calculate:

       1            1
    ------- = --------------
    sqrt(2)   1.414213562373

or

    sqrt(2)   1.414213562373
    ------- = --------------
       2            2

I think you'll find the latter is much easier to do! Even with 
computers, I suspect the latter is likely to be more accurate.

Secondly, though, it's a good idea to have a standard form in which to 
write expressions like this, even if you aren't going to evaluate 
them, because it lets you compare them more easily. Think of it as a 
"mug shot" at the police station. They always line you up the same 
way and take pictures of the front and side, so different people can 
be compared easily. Here, if I asked you to tell me whether 1/sqrt(2) 
and sqrt(2)/2 are the same, they hardly look like the same "person" 
- but when you make them both "turn the same way" by rationalizing the 
denominator, the fact that they are the same becomes obvious.

The same two reasons explain why we reduce fractions to lowest terms: 
ease of calculation and standardization of form.

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