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### Rationalizing the Denominator

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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/
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Associated Topics:
Middle School Division
Middle School Fractions

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