Why Rationalize the Denominator?

**BSMSMSTMSPHD** · Sep4-06, 05:07 PM

I'm wondering why we teach algebra students that they MUST rationalize the denominator of a fraction containing a radical. Many books that I have seen state "it is often desireable to rationalize the denominator..." but I can't think of an obvious reason why.

I did a cursory Google search and came up empty, so I came back here. So, any good reasons out there?

As usual, Thanks!

**FrogPad** · Sep4-06, 05:51 PM

Maybe because it's easier to cancel?

For example, if you didn't rationalize the denominator how would you reduce?
$LaTeX Code: \\frac{6\\sqrt{2}}{\\sqrt{3}}$

**CRGreathouse** · Sep4-06, 05:56 PM

Also, $LaTeX Code: \\frac{a\\sqrt b}c$ with gcd(a,c) = 1 and b squarefree is a unique representaton of a number, so it's easy to see if two numbers are the same or not.

**Moo Of Doom** · Sep4-06, 10:07 PM

A simple argument is that it more easily gives you a feel for the size of a number. You know $LaTeX Code: \\sqrt{2} \\approx 1.41$ . But how big is $LaTeX Code: \\frac{1}{\\sqrt{2}} \\approx \\frac{1}{1.41}$ ?

If you rationalize the denominator, however, we have

$LaTeX Code: \\frac{1}{\\sqrt{2}} = \\frac{\\sqrt{2}}{2} \\approx \\frac{1.41}{2} = 0.705$

That's a lot easier to picture.

**HallsofIvy** · Sep5-06, 05:56 AM

Originally Posted by BSMSMSTMSPHD

I'm wondering why we teach algebra students that they MUST rationalize the denominator of a fraction containing a radical. Many books that I have seen state "it is often desireable to rationalize the denominator..." but I can't think of an obvious reason why.

I did a cursory Google search and came up empty, so I came back here. So, any good reasons out there?

As usual, Thanks!

I'm wondering why you think we teach students that they MUST do any such thing. Certainly, since adding or subtracting fractions involves getting a "common denominator" which involves factoring, it is often simpler to have a rational denominator but seldom necessary. In fact, a text I recently used has a section on "rationalizing the numerator" which, while less common, is useful in some problems.

**uart** · Sep5-06, 09:09 AM

Originally Posted by BSMSMSTMSPHD

I'm wondering why we teach algebra students that they MUST rationalize the denominator of a fraction containing a radical.
As usual, Thanks!

Well for me I guess it's just an excuse to give them some practice at manipulating surds. :)

**arildno** · Sep5-06, 12:41 PM

Hmm..never knew I "had to" do this, but upon reflection, I think I've mostly followed this rule.
I think it is a type of aesthetic:
While we readily can envisage that we have an ugly amount of some specified part, we do not like the specified part itself to be ugly.

I.e, while I unproblematically accept that I have a square-root of 2-amount of one-half, I dislike to have one square-root-of-2'th.

**shmoe** · Sep5-06, 02:36 PM

Having a more 'standard' form is desirable when comparing things, especailly when it comes time to grade the students work.

Ok, maybe a better reason, is the multiplicative inverse of $LaTeX Code: 1+\\sqrt{2}$ in $LaTeX Code: \\mathbb{Z} \\left[\\sqrt{2}\\right]$ ? Ok, not strictly necessary to rationalize $LaTeX Code: 1/(1+\\sqrt{2})$ , but it's one way to go.

How about finding the real and imaginary parts of $LaTeX Code: \\frac{1+i}{2+3i}$ ? Though the radical is hidden, this is really the same thing.

**Robokapp** · Sep7-06, 06:49 AM

I think the reason is that if the denominator is a rational number, it is in a simpler form and fractions can then combine, making operations more clear.

$LaTeX Code: 2 / \\sqrt{2}+3 / \\sqrt{5}$ is much messier to understand as a real number than...

$LaTeX Code: \\sqrt{2}+3\\sqrt{5}/5 = (5\\sqrt{2}+3\\sqrt{5})/5$

**HallsofIvy** · Sep7-06, 11:27 AM

Let me point out one more time that no one teaches students that they must rationalize denominators! As many people have pointed out, there are often good reasons for wanting to do that. There are also sometimes reasons for wanting to rationalize the numerator of a fraction instead.

**chroot** · Sep7-06, 04:11 PM

There's no mathematical reason why anything ever needs to be simplified into any conventional form. 1/sqrt(2) is just as valid a fraction as any other.

The only reason people are taught to simplify things in specific ways (like reducing all fractions, rationalizing denominators, etc.) is because it makes it easier for the teacher to quickly grade papers.

- Warren

**HallsofIvy** · Sep7-06, 05:16 PM

Ahh, now you're telling !

**calculift** · Mar20-09, 12:28 PM

I understand that the tradition has to do with pre-calculator days when division was done with logarithms and a radical denominator complicated the process. Any old-school slide rule folks that can shed some light on the situation?

**statdad** · Mar20-09, 01:34 PM

Originally Posted by calculift

I understand that the tradition has to do with pre-calculator days when division was done with logarithms and a radical denominator complicated the process. Any old-school slide rule folks that can shed some light on the situation?

That would be my experience, sans the slide rule - when calculations were done by hand, or with simple calculation aids, division with an integer always beat the prospect of division by a decimal.

**lurflurf** · Mar20-09, 05:39 PM

Originally Posted by chroot

There's no mathematical reason why anything ever needs to be simplified into any conventional form. 1/sqrt(2) is just as valid a fraction as any other.

The only reason people are taught to simplify things in specific ways (like reducing all fractions, rationalizing denominators, etc.) is because it makes it easier for the teacher to quickly grade papers.

- Warren

It is a little more general than grading. Most simplifying is done because it is more simple, and thus more useful for some purpose. It is also true that sometimes different forms are prefered for different purposes and some different forms are of comparable simplicity for some purposes. Even in those cases it is helpful to standardize the result to make it easier to compare to other results. That comparison might be done for grading, but it is also uses in error checking and equality checking.

**statdad** · Mar21-09, 12:36 PM

Originally Posted by lurflurf

It is a little more general than grading. Most simplifying is done because it is more simple, and thus more useful for some purpose. It is also true that sometimes different forms are prefered for different purposes and some different forms are of comparable simplicity for some purposes. Even in those cases it is helpful to standardize the result to make it easier to compare to other results. That comparison might be done for grading, but it is also uses in error checking and equality checking.

I agree. If I wanted to make grading easier I would tell students not to show work, only answers, and do away with partial credit.