# Rationalizing the Denominator, and Comment Writing

We get tomorrow off of school for “Election Day.” Translated, that is the day teachers at my school write narrative comments for all their students discussing their first quarter grades. We’ll all be holed up in our apartments, trying to come up with various ways to say “this student is doing great,” “this student is doing okay,” and “this student is not doing well.” Luckily, I’m pretty fast at writing these, so I’m not concerned.

In other news, I gave my Algebra II students a quiz last week, and one of the skills covered was rationalizing the denominator where there are radicals involved. (Multiplying the top and bottom of the fraction by the conjugate.) My three musings:

(1) Why do we math teachers care so much about this? I know it’s a good skill to teach because sometimes it really does simplify expressions, but do we always want to insist that the denominator is rationalized? I always thought that it was a bit dumb — and no one really has been able to justify why teachers insist on it with such vehemence. Any ideas? [1]

(2) Ummm… in Calculus, we’re starting to work on the formal definition of the derivative and guess what? To find the derivative of $f(x)=\sqrt{x}$ using the formal definition, you have to rationalize the NUMERATOR. Harumph.

(3) For extra credit, for students who had some extra time after finishing their Algebra II quiz, I asked them if any of them could somehow rewrite the following without any radicals in the denominator: $\frac{1}{\sqrt{2}+\sqrt{3}+\sqrt{5}}$. Although no one got it, I loved watching them work on it. [2]

[1] My high school Algebra II teacher told us: “Why don’t we want radicals in the basement? BECAUSE THEY BUILD BOMBS!” I will never forget that. Love it. I totally use it. His legacy lives on.

[2] Even though I was horrified that some students’ initial step was to rewrite that as $\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\frac{1}{\sqrt{5}}$. Why is it that students NEVER understand fractions?

1. Matt E says:

My explanation for “why we have to learn this” is that it not only simplifies the expression, but it gives us a kind of standardized way to write a given number. If three students are working on a problem, one gets “root 3 over 2” for an answer, another gets “5 root 6 over 10 root 2”, and the third gets “root three fourths”, they’d have to take those extra steps anyway to verify that they actually all got the same answer. (Which, of course, doesn’t mean that it’s correct.) So rationalizing/simplifying makes it easier to verify such things a glance. I dunno, does that make sense?

2. The only convincing explanation I know goes back to The Time Before Calculators.

Many professions habitually need decimal approximations. So think of cos(pi/4). 1/sqrt(2). Now would you rather do long division to divide 1 by 1.414 (yuck!), or would you rather rationalize the denominator, and divide 1.414 by 2 (easy!)?

Of course that doesn’t explain why we still insist on it. I like Matt’s explanation, I often say something along the lines of “the state exam is more than half multiple choice… it’s important to be able to recognize equivalent expressions”.

The “radicals in the basement” line made me lol, for real. I’m using that tomorrow. I don’t have election day off (lucky!).

3. Yep, echoing Kate’s explanation, I think it goes back to the pre-calculator days.

As to why we still stress it? I’m not sure. I believe the AP Calc exams will take any equivalent form. Do we do it because that is what college professors want?

As for the fractions… if you figure out that please let me know!

4. There are a few reasons for rationalizing, some more compelling (in the calculator era) than others.

Certainly arithmetic is simpler with rationalized denominators, as the 1 / 1.414… versus 1.414… / 2 example illustrates.

Arithmetic with complex numbers comes to mind as well, where again one on occasion finds oneself rationalizing denominators (e.g. (3+2i)/(4-3i) = (3+2i)(4+3i)/25 )

In a calculus context, one often wants to simplify fractions whose denominator is a sum or difference of trig functions. Appealing to past experience with rationalizing denominators can help students understand how to be able to square the two summands individually to then be able to exploit nice identities…. ( e.g. finding the antiderivative of 1 / (1+sin(x)), say.)

Are any of these hugely compelling examples? No, not particularly. If my calc students need to remember this technique, I can help them with it fairly quickly even if they claim to have never seen it before. (Likewise we can teach polynomial long division to students who don’t know integer long division, but it probably is simpler and more effective for those who do know that arithmetic skill.)

Having said that, I suspect the most likely reason it is in the curriculum is… inertia.

5. Aaronji says:

In case anyone was wondering, the answer to how to write 1/(√2+√3+√5) without any radicals in the denominator is: (3√2+2√3-√30)/12.

6. kibrolv says: