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Why Rationalize the Denominator?

  1. Sep 4, 2006 #1
    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!
     
  2. jcsd
  3. Sep 4, 2006 #2
    Maybe because it's easier to cancel?

    For example, if you didn't rationalize the denominator how would you reduce?
    [tex] \frac{6\sqrt{2}}{\sqrt{3}} [/tex]
     
  4. Sep 4, 2006 #3

    CRGreathouse

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    Also, [tex]\frac{a\sqrt b}c[/tex] 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.
     
  5. Sep 4, 2006 #4
    A simple argument is that it more easily gives you a feel for the size of a number. You know [itex]\sqrt{2} \approx 1.41[/itex]. But how big is [tex]\frac{1}{\sqrt{2}} \approx \frac{1}{1.41}[/tex]?

    If you rationalize the denominator, however, we have

    [tex]\frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2} \approx \frac{1.41}{2} = 0.705[/tex]

    That's a lot easier to picture.
     
  6. Sep 5, 2006 #5

    HallsofIvy

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    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.
     
  7. Sep 5, 2006 #6

    uart

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    Well for me I guess it's just an excuse to give them some practice at manipulating surds. :)
     
  8. Sep 5, 2006 #7

    arildno

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    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.
     
  9. Sep 5, 2006 #8

    shmoe

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    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 [tex]1+\sqrt{2}[/tex] in [itex]\mathbb{Z} \left[\sqrt{2}\right][/itex]? Ok, not strictly necessary to rationalize [tex]1/(1+\sqrt{2})[/tex], but it's one way to go.

    How about finding the real and imaginary parts of [tex]\frac{1+i}{2+3i}[/tex]? Though the radical is hidden, this is really the same thing.
     
  10. Sep 7, 2006 #9
    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.

    [tex]2 / \sqrt{2}+3 / \sqrt{5}[/tex] is much messier to understand as a real number than...

    [tex]\sqrt{2}+3\sqrt{5}/5 = (5\sqrt{2}+3\sqrt{5})/5[/tex]
     
  11. Sep 7, 2006 #10

    HallsofIvy

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    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.
     
  12. Sep 7, 2006 #11

    chroot

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    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
     
  13. Sep 7, 2006 #12

    HallsofIvy

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    Ahh, now you're telling !
     
  14. Mar 20, 2009 #13
    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?
     
  15. Mar 20, 2009 #14

    statdad

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    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.
     
  16. Mar 20, 2009 #15

    lurflurf

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    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.
     
  17. Mar 21, 2009 #16

    statdad

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    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.
     
  18. Mar 21, 2009 #17
    Lots of teachers do exactly this...
     
  19. Nov 24, 2010 #18
    My guess is that the original reason for rationalizing the denominator is that the long division to find the decimal answer would be easier to calculate.

    2 into 1.41...
    is easier than
    1.41... into 1.00

    I bet the tradition has persisted on its own inertia in relative ignorance of the reason for the tradition.
     
    Last edited: Nov 24, 2010
  20. Nov 25, 2010 #19

    HallsofIvy

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    Well, no one I would call a "teacher"!
     
  21. Apr 8, 2011 #20
    Why we rationalize the denominator has been a question that many students have asked. My answer is: because we told you to do it! By being able to manipulate expressions improves a student’s critical thinking skill surrounding relationships using algebraic concepts. It is critical as a student progresses to higher-level math and more in-depth critical analysis of concepts (math or non-math relationships), manipulation of expressions is critical to avoid nasty results like division by zero or taking a square root of a negative number.

    M
     
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