"I don't get why we teach rationalizing the denominator. It was originally introduced as a way to better estimate square roots without a calculator (Ex: it's easier to divide by 2 than divide by sqrt2). It doesn't seem like these estimations are necessary anymore, now that calculators will do this." --@veganmathbeagle
I goes back to the days before scientific calculators, when we had tables of roots to work with. These were generally rounded to the ten- or hundred-thousandth place, and no one wanted to deal with a divisor having that many digits. I think we still teach them because they allow students working with trig ratios of "special triangles" to simplify the quotients to recognizable ratios. Is that a good enough reason? I doubt it, but until we design courses and assessments with less emphasis on computation and more emphasis on interpretation, application, and validation, teachers will continue to value these symbolic gymnastics.
Agreeing with Jen: I think we rationalize denominators because it gives a more 'exact' answer (since you can't divide by an irrational number). However, students have pointed out that sqrt 2 is not a meaningful number in many real contexts, whereas 1.414 is.
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Chris Brownell
3/17/2014 06:07:55 am
I agree completely with this question. This is a task that is fraught with fruitless frustration in thousands of students. Don't get me wrong, I am in favor of some frustration for my students, lots in fact. But this topic goes beyond the pale to the point of fruitlessness.
The only time rationalized denominators come in handy is in the values of the special trig functions as has already been mentioned and in a few albeit cool pattern problems.
There is the connection to complex numbers in the same vein. You rationalize a denominator by multiplying by the radical, or by a radical conjugate -- well, to remove an imaginary denominator component, you multiply by the complex conjugate.
That said, I do not know whether I'm arguing in favour of keeping "rationalizing", or whether I'm arguing in favour of ALSO removing it in the case of complex numbers... mainly because I don't teach complex numbers. Anyone else know if it has applications there?
I agree with Jen and Chris about the apparent futility of this activity, as it is usually taught.
However, this is one of the first time students abstract manipulate expressions to view them in different forms, and is an extension of the idea that numbers can have different equivalent representations, which is a key idea to understand later.
As for complex numbers, they are critical to being able to solve differential equations, and complex analysis is a critical field in physics. That being said, I had minimal exposure to complex numbers in high school and I did fine at university which is when they are introduced.
So thumbs up to rationalizing the denominator if the purpose is to look at different representations of numbers and not follow a mindless routine, and thumbs down to early complex numbers (there are other pre-calculus ideas which I prioritize over complex numbers. Eg. understanding how to subdivide objects into an infinite number of pieces).
Rationalizing complex numbers allows it to be put into a + bi form, that is, turn it into an actual coordinate point so you can then do stuff like turn into polar form and do the rules for roots and so forth.
This conversation has gone through most of it's path. One note I would like to add is that we lose the idea of equivalent forms of a value that has the denominator already rationalized. I think of the standard unit circle and trigonometry chart values that students are often encouraged to memorize. Values like sqrt(wholenumber)/integer are common. I would hope that many expose this ratio as equal to some other integer/sqrt(wholenumber) that more directly relates to the ratio of 2 sides using a standard SOHCAHTOA relationship (or the reciprocals). I make more comments on this Rationalizing Denominators in a post (http://mathbutler.wordpress.com/2014/06/24/2-is-better-than-1/).
I also agree that conjugate multiplication and rationalizing a denominator helps with complex number manipulations, however the abstractness may be more appropriate for higher-ed as David Wees suggests. I too didn't have exposure to complex number like math until college and did fine.
If one is concerned with using conjugates to multiply denominators for rationalization, I've had more success with relating to a difference of squares factorization fill in the blank exercise. Knowing that squares turn square roots and imaginary expressions into real number values leads well into "How do we make the expression have squared terms only? Where have we seen this before?"