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I'm teaching an "intermediate algebra" college course ($\approx$ junior high school or beginning high school algebra) and we have a bunch of problems on rationalizing denominators. How do I motivate this?

About the best I can think of is that it's a collection of tricks that are handy later, but I'd like to know something I can say or do that demonstrates some kind of intellectual need to my students now.

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I've always personally leaned more towards a "lowest terms" approach, for example I prefer $1/\sqrt{2}$ to $\sqrt{2}/2$. If our intention is just to teach how to use exponent laws, why isn't it done more directly? –  Alexander Gruber Apr 21 at 22:12
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IMHO rationalizing denominators is a purely aesthetic thing. It's just that e.g. every member of $\mathbb{Q}(\sqrt{2})$ can be canonically represented in the form $a+b\sqrt{2}$ with $a,b\in\mathbb{Q}$, so using this "canonical" representation is a Nice Thing To Do™. (And I strongly encourage my students not to rationalize denominators until the final solution, since it might be waste of time! –  mbork Apr 21 at 22:12
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I've once heard the justification that rationalizating denominators was useful when computations weren't so readily available (e.g. with computers). Then division using integers was way more feasible than dividing by rationals. –  Mark Fantini Apr 21 at 22:15
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One reason is that it allows you to verify quickly whether two answers are equal; if students are going to share their solutions with you and one another, or compare them to those found in a textbook, then it might be wise to streamline the form in which answers are presented. –  Benjamin Dickman Apr 21 at 23:04
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@Fantini, you should write your comment as an answer. I think it is the correct answer. In the old days (before calculators), you would have tables containing the values of, say, $\sqrt 2$, but none containing the values of, say, $1/\sqrt 2$, with the result that it became easier to multiply by $\sqrt 2$ than it is to divide by $\sqrt 2$. –  Joel Reyes Noche Apr 22 at 0:05

9 Answers 9

up vote 13 down vote accepted

My thinking is that it is just so damn useful for students to be aware of these tricks. The examples/exercise should allow them to develop a sense of when and how it is helpful to simplify an expression in this way, BUT also when it is NOT necessary.

Leading up to it by looking at fractions. Should the students write a rational number in the form $3\frac17$ or $\frac{22}7$? The answer IMHO depends on what you want to do with this number. If it is the final answer to a problem, then in this case either one is probably fine, but in some cases it will be helpful to have the integer part and fractional parts separated, because then you immediately get a good idea of the magnitude, and can answer questions like "Is the answer larger than $3$?" BUT. If this is an intermediate step, then converting $\frac{22}7$ to $3\frac17$ is usually not necessary, and may be a colossal waste of time, if for example the next step in the problem requires you to divide this intermediate result by $11$.

We surely want to teach the students the process of converting $3\frac17$ to $\frac{22}7$ and back, not as an end, but as a tool for saying something useful about the answer. We also develop their sense of how fractions and quotients work so that they have a sporting chance to learn to manipulate symbolic fractions later in their studies. But that's a bonus, and difficult to use as motivation at the point when they are learning to manipulate rational numbers.

On with the radicals. IMO the salient questions are "what are we expected to do with this result?" and "can we expect a significant simplification?" Consider the following assignments.

  • Plot the graph of the function $$ f(x)=\frac{x}{\sqrt2-1}-\frac{x}{\sqrt2+1} $$ by calculating the value of $f(x)$ at the eleven points $x=0, 0.1, 0.2,\ldots, 1.0$. The usefulness of first rationalizing the denominators becomes clear, if the student has developed a sense that there may be significant simplifications lurking in there. Calculating those fractions with a calculator requires several key strokes, properly placed parens et cetera, and is thus error-prone.
  • The same with a more symbolic expression like $$ g(x)=\frac1{\sqrt x+1}-\frac1{\sqrt x-1}. $$ This is an analogue of the grade school drill of doing a lot of manipulations with fractions, so that it is easier to later learn to manipulate rational functions, when the calculator won't do it for ya.
  • OTOH, when doing a lot of trigonometry with the usual fractional multiple of $\pi$, you can "almost" treat $1/\sqrt2$ as having a rational denominator. If we are to next square the intermediate result, I doubt anyone would first convert $1/\sqrt2$ to $\sqrt2/2$. I usually won't bother to rationalize that $\sqrt2$ in the denominator myself, so I obviously should not expect my students to do so either.

TL;DR; The projected way of using the intermediate result, or the end result, gives hints as to whether an effort in simplification is warranted. The students should learn these rationalizing tricks in order to develop a sense of whether it is worth their while to do it in a given situation.

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+1 for the crucial observation that different forms (in this case of fractions) are convenient for different uses. I use this to motivate various form of e.g. a quadratic expression. –  mbork Apr 24 at 6:19
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This is excellent, although certainly nothing in the current curriculum I have uses this idea. But connecting the idea of rationalizing denominators to the general concept of "change the representation of the thing to suit the problem you're solving" is something that I could present to a class and feel like I'm being intellectually honest. –  Dan Drake Apr 24 at 14:32

Making my comment an answer: rationalizing fractions was useful when computers weren't so readily available. Division using integers is way more feasible than dividing by rationals. Even today, if you want to have a reasonable estimate by hand you don't compute $1/\sqrt{2}$ but you do quickly $\sqrt{2}/2$.

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And, by pure inertia, you'll find most results still stated in rationalized way. To check your work against published sources it is often required to rationalize. –  vonbrand Apr 22 at 12:08
    
This is a pretty reasonable explanation, but only if the curriculum somehow promotes doing reasonable estimates in your head or by hand -- ours doesn't. I'd like to change that (your example is good -- I can do "1.4/2" far better than I can do "1/.7"!), but I'm still curious to hear how someone would justify rationalizing denominators in a course in which canonical representations or mental math are not important. –  Dan Drake Apr 22 at 18:12
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I would strongly suggest encouraging mental math, and estimation (1 sig fig) of answers. This is helpful later when doing things like limits and derivatives when you use the fact that a small number squared is a much smaller number, or with graphing when you show that the cubic term dominates as x departs far from zero. –  Sherwood Botsford Sep 13 at 16:32

One idea: Calculate 1/(sqrt(3.0000001)-sqrt(3)) and compare with the answer of calculating the rationalized form (sqrt(3.0000001)+sqrt(3))/0.0000001.

Adjust the number of 0's to the precision of the calculator or software used.

In Maple with Digits:=10, the first expression gives: 3.571428571*10^7 while the second gives 3.464101644*10^7.

This would be a good occasion to show that calculators or computers don't always give the "right" answer. A discussion could also be held as to what it means to give a "right" answer.

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This is something that could conceivably work in my class, since for most (well...all) my students, if given two different expressions, one of the most effective ways to convince them that the expressions represent the same number is to find the decimal approximation for each. Agrees to 12-ish decimals == same number. But we'd need a course in which we do calculations with numbers such as yours, and mine usually sticks with at-most-two-digit integers. –  Dan Drake Apr 22 at 18:22

In the olden days before calculators and computers were commonplace, mathematicians had thick books of precomputed values, like $\sqrt{2}$, which they could reference to compute their answers to an accuracy of several decimal places.

Without having to actually purchase one of these books and try it, you can still mention that using a book to approximate the value of $\sqrt{2}/2$ was considerably easier than finding $1/\sqrt{2}$, although a calculator today would do both just as easily.

So why is it relevant today to rationalize denominators? Well, math contests (especially American ones such as the American Regions Mathematics League) want answers given in simplest form, with rational denominators. Not knowing how to do this would lose easy marks.

If you meant rationalizing denominators in expressions like $3/(\sqrt{2}+\sqrt{3})$ by multiplying top and bottom by $(\sqrt{2}-\sqrt{3})$, then it's because this technique can be used later on to greatly simplify expressions or derive proofs for higher math.

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This is a point that goes unsaid: grading mathematics homework is technically an undecidable problem. This is one of the reasons why we teach the normal forms of various objects (for example, rationalizing the denominator and writing polynomials in traditional form with descending degree terms). –  Tac-Tics Apr 22 at 3:45
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These are the worst two reasons for doing something I could possibly imagine: a) It is important to do this because some people want you to do it (the American Regions Mathematics League) even though it is pointless and have decided to punish you if you do not do it. b) It is important you do this pointless thing because it will help you in some unspecified way to do some nebulous thing when you progress to some unspecified topic ('higher math') at some unspecified time in the future. –  jwg Apr 22 at 9:43
    
@jwg I see what you mean. Well I'm a Canadian, and Canadian math contests tend to not care about rationalizing denominators, because yes we agree that it's rather irrelevant today. But if you know better reasons then by all means go ahead and give them. Otherwise I'm content with giving the honest reasons and not complaining about them being bad. –  SimonT Apr 22 at 11:58
    
I have to agree with @jwg here: this isn't the olden days; my students all have smartphones and laptops, and dividing by a radical to get a decimal approximation is easy. My students are college students taking a high school math course; none of them are ever going to do a math competition. My course uses an online homework system, so whoever designed it couldn't have been thinking about computational difficulty, and was not trying to prepare anyone for a contest. –  Dan Drake Apr 22 at 18:17
    
I think that b) is not an honest answer, and a) should not be given as an answer without complaining about it, unless you want your students to learn that 'math is pointless'. Fair enough if you really believe that rationalizing denominators is crucial, but you shouldn't pass on other people's meaningless conventions without passing comment on them and allowing your students to understand that they are norms not laws (the actually honest approach). –  jwg Apr 22 at 20:06

(Hi, Dan!) Although there are various not-unreasonable motivations to "rationalize denominators", I think mostly it is mostly just another not-very-well-motivated rule to test students' compliance. It is true that polynomials in radicals, with rational coefficients, are simpler than rational expressions in radicals, unless something simplifies surprisingly, it really makes little difference. Perhaps an example of one of the few times this might really matter is testing exact equality, where the "reduced" polynomial expression is canonical and unique, while rational expressions are not unique. Nevertheless, when looking at polynomials in several algebraic numbers, it quickly becomes non-trivial to decide on nice minimal/canonical polynomial expressions, also. This already happens with $n$th roots of unity with highly composite $n$.

For that matter, what about denominators involving several square roots? Or, worse, several roots of higher degrees? These are tractable, but the usual curriculum stops surprisingly soon... giving more credence to the suspicion that it's just a thing to browbeat students about.

In short, a larger version of this reduction/simplification is useful, but the version almost universally taught stops far too soon, and thus (at least passively) has the effect merely of a test of obedience to rules. Not my favorite thing to test.

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(Hi Paul!) Sadly, I think your answer is the best; when I look at my course and curriculum, the bits with rationalizing denominators do seem to be little more than a test of obedience or compliance. :( –  Dan Drake Apr 22 at 18:32
    
If we're looking for base motives, then in addition to testing compliance, we also have the fact that teachers are lazy and want papers to be easier to grade. If you require students to state all answers in a canonical form, it's quicker and easier to check if their answers are right. –  Ben Crowell Sep 14 at 16:58
    
@BenCrowell, indeed, and, after all, it is highly non-trivial to specify a range of "simple enough" as a requirement for "answers", while giving artificial (if defensible) specifications may be easier, and easier to communicate. This ease does give an erroneous impression to students, I think, in terms of what is or isn't legitimate mathematics. Easier to discuss/pick-on such issues than more substantive ones. –  paul garrett Sep 14 at 17:06

What follows is an edited and expanded version of comments, and a list of examples, that I posted 12 June 2001 (and later in 26 September 2007, in a more abbreviated form) in the Math Forum discussion group AP-calculus.

I believe rationalizing the denominator was originally positioned so early in the curriculum --- algebra 1 and geometry for division by $\sqrt{n},$ and algebra 2 for division by something like $m + \sqrt{n}$ --- was partly for reasons having to do with numerical calculation, and partly for reasons having to do with algebraic combination and simplification of exact numerical values. Incidentally, if you look at textbooks written 50 to 150 years ago, you don't really see much of an expectation that radicals were numerically approximated (this view being based on worked examples in the text and answers to exercises), except for trigonometry texts. However, the numerical aspect becomes much more important in applications that occurred outside of mathematics (mainly in science courses), so I suspect what happened is that the training in appropriately rewriting radical expressions so that square root tables and such could be easily used was left to the math courses.

I personally think there has been too much emphasis on rationalizing the denominator in the past 40 years (perhaps in the past 20 years the emphasis has been more appropriate), especially in classes below the precalculus level, but I also think it's easy to forget just how often the technique of rationalization shows up in math, even if we restrict ourselves to the lower undergraduate level. As for me, when departmental and/or course supervisor constraints allowed me to do so, I DID NOT REQUIRE answers to be in denominator-rationalized form in high school or college algebra classes, or in precalculus classes. However, I felt it was an important skill for anyone getting at least as far as calculus. Thus, in calculus courses, I tried to make up for this inattention to rationalization (both by me and by other teachers) by working the topic in at a number of places. I did this mainly by working examples in class and by assigning problems (with an appropriate hint) like #1-6 below.

MISCELLANEOUS LIST OF EXAMPLES FOR RATIONALIZING

1. These limits can be evaluated without taking derivatives if you first apply a binomial rationalization step: $$\lim_{x \rightarrow 1}\frac{x^2 - 1}{\sqrt{2x+2\,} \; - \; 2} \;\;\; \text{and} \;\;\; \lim_{x \rightarrow 0}\frac{1 - \cos x}{x} $$

2. To rewrite $$ \ln \left( \frac{x + \sqrt{x^2 - 1}}{x - \sqrt{x^2 - 1}} \right) \;\;\; \text{as} \;\;\; 2\ln\left (x + \sqrt{x^2 - 1}\right),$$ it helps if you first rationalize the numerator. Note: Putting $x = \sec \theta$ gives an identity that is sometimes useful.

3. To differentiate $x^{\frac{1}{2}},$ $x^{\frac{1}{3}},$ etc. using the limit definition of the derivative, you'll want to rationalize numerators.

4. The derivative of $$ \frac{\sqrt{a-x} \; + \; \sqrt{a+x}}{\sqrt{a-x} \; - \; \sqrt{a+x}} $$ is much easier to put into the more useful form $$ \frac{a^2 \; + \; a\sqrt{a^2 - x^2}}{x^2\sqrt{a^2 - x^2}}$$

if you rationalize the denominator BEFORE differentiating.

5. Let $a \neq 0,$ $b,$ and $c$ be real number constants. To verify that $$ \lim_{n \rightarrow \infty} \left( \sqrt{an^2 + bn} \; - \; \sqrt{an^2 + cn} \right) \;\;\; = \;\;\; \frac{1}{2\sqrt{a}}(b-c),$$ it helps to rationalize the numerator first.

6. The linearization of $$\frac{1+x}{1-x} \;\; \text{at} \;\; x=0$$ is easy if you begin by multiplying both the numerator and denominator by $1+x.$ After doing this, you get to ignore the $x^2$ terms that appear additively with constants or with multiples of $x.$ The result will be $1 + 2x.$ More generally, rationalization ideas can be used to obtain the quotient rule for derivatives by multiplying/dividing by an appropriate conjugate and ignoring all but first order terms, and similar methods can be used to approximate $f(x+h,\,y+k)$ for rational functions $f(x,y)$ when $h$ and $k$ are close to $0.$

In the same way, one can show by rationalization methods that for $\delta$ and $\epsilon$ near $0,$ we have $$\frac{1}{1 + \delta} \; \approx \; 1 - \delta \;\;\; \text{and} \;\;\; \frac{1}{1-\delta} \; \approx \; 1 + \delta \;\;\; \text{and} \;\;\; \frac{1+\epsilon}{1+\delta} \; \approx \; 1 + \epsilon - \delta $$ These and other approximations are discussed in Philip L. Alger's 1957 text Mathematics for Science and Engineering (see pp. 145-155 of Chapter 6: Numerical Calculations) and in William Charles Brenke's 1917 text Advanced Algebra (see Chapter IX, Section 146: Useful Approximations, pp. 126-127). These approximations are often more important for giving approximations that are valid over a range of variable values than for giving individual and isolated numerical approximations. This is especially useful when an exact algebraic form is difficult to work with, such as in a differential equation (recall the pendulum equation). For instance, $$\tanh M \;\; = \;\; \frac{e^{M} \; - \; e^{-M}}{e^{M} \; + \; e^{-M}} \;\; = \;\; \frac{1 \; - \; e^{-2M}}{1 \; + \; e^{-2M}} \;\; \approx \;\; 1 - 2e^{-2M}$$ is an approximation that is correct to $16$ decimal places when $M = 10.$ This particular approximation for $\tanh M$ is obtained in the same manner I've just shown, and then used to find the lowest eigenvalue in the high barrier limit for a quantum mechanical particle confined to a double potential well, in Charles S. Johnson and Lee G. Pedersen's 1974 Problems and Solutions in Quantum Chemistry and Physics (see Problem 4.8(b) on pp. 105-106).

Another example can be found in Jerry B. Marion's 1970 text Classical Dynamics of Particles and Systems (see p. 270). Marion uses the approximation $$\theta \;\; = \;\; \frac{2\pi}{1 - \frac{\delta}{\alpha}} \;\; \approx \;\; 2\pi\left(1 + \frac{\delta}{\alpha}\right) \;\; = \;\; 2\pi + \frac{2\pi\delta}{\alpha}$$ near the end of a derivation of the precession of Mercury's orbit as predicted by Einstein's Theory of Relativity. The term $\frac{2\pi\delta}{\alpha}$ represents the approximate precession per orbit, which in Mercury's case works out to approximately $43$ seconds (angle measure) per century.

7. To express the quotient of two complex numbers in rectangular form, when each of the complex numbers is given in rectangular form, you'll want to use a "rationalization of the denominator" technique. Related to this is finding the real and imaginary parts of a rational function of a complex variable (e.g. verifying the Cauchy-Riemann equations, finding a harmonic conjugate of a rational function, investigating certain orthogonal families of curves, etc.).

8. For numerical purposes (e.g. reducing round-off errors during a computer computation), the quadratic formula $$ x \;\; = \;\; \frac{-b \; \pm \; \sqrt{b^2 - 4ac}}{2a}$$ is in some cases more usefully expressed as $$ x \;\; = \;\; \frac{2c}{-b \; \pm \; \sqrt{b^2 - 4ac}}$$

9. To show that ${\mathbb Q}[\sqrt{2}]$ (i.e. real numbers of the form $r + s\sqrt{2}$ where $r$ and $s$ are rational numbers) is a field, a "rationalization of the denominator" technique is useful when verifying the multiplicative inverse part of the definition of a field.

10. Rationalizing techniques are useful to obtain non-radical forms for the general equation of a hyperbola and an ellipse directly from their geometric definitions. Related to this is the general idea of rationalizing an algebraic equation (say, for an algebraic curve or an algebraic surface -- see Cayley's 1868 paper On Polyzomal Curves, otherwise the Curves $\sqrt{U} + \sqrt{V} +$ &c. $=0,$ which begins on p. 470 here, for some eye-opening stuff) and of solving radical equations.

11. It is easy to find a simple expression for the following sum if each denominator is rationalized: $$\frac{1}{\sqrt{1} + \sqrt{2}} \; + \; \frac{1}{\sqrt{2} + \sqrt{3}} \; + \;\frac{1}{\sqrt{3} + \sqrt{4}} \; + \; \cdots \; + \; \frac{n}{\sqrt{n} + \sqrt{n+1}}$$

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Why assume that your students should learn to rationalize the denominator? It's true that there are instances when rationalizing a denominator is helpful, but my guess is that in your college algebra context you're probably trying to have your students write $\sqrt{2}/2$ instead of $1/\sqrt{2}$ for their final answers. As others have pointed out, rationalizing the denominator was important back before calculators, but nowadays both answers can be estimated as decimals just as quickly using a calculator. In other words, $\sqrt{2}/2$ isn't really a "simplified" form of $1/\sqrt{2}$ like it used to be.

My suggestion: Let your students answer their questions with or without rationalizing the denominator. It may cost you a little extra effort when you grade answers, but also free up more of your class time to teach more important concepts.

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Having an irrational number in the denominator presents problems in computation, whether you are doing it by hand or with a calculator. If an irrational number is non-terminating, using a rounded or terminated approximation always introduces a computational error. It's harder to estimate or control that error when it appears in the denominator of a fraction than in the numerator, and especially if the denominator is anywhere close to zero.

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Could you expand on why it's harder to control the error with the irrational number in the denominator? My intuition agrees with you, and a quick check shows my windows calculator is more accurate with (1/89)*0.5 than (1/89)/2, but I have no idea why. –  aknauft Apr 22 at 6:58
    
My windows calculator claims that both (1/89)*0.5 and (1/89)/2 are equal to 0.00561797752808988764044943820225 (and neither involves any irrational numbers in either numerator or denominator)?? –  jwg Apr 22 at 9:48
    
If the computational error, call it delta, is in the numerator, you will have (a + delta)/b, so the error is simply delta/b. But if it's in the denominator, you have a / (b+delta), which is harder to simplify. The magnitude of the error in the division then does not depend on delta only, but on how close (b + delta) is to zero: division by a small number has a larger possible error in the final result than division by a large one. –  Confutus Apr 22 at 16:41

I like the historical example. Computationally division is lots slower than multiplication somewhat depending on the numbers. (A quick check shows 8-20 times slower) A good compiler would optimise dividing by 2 with a shift operation. I don't know if generally dividing by a long decimal is either more time consuming or less accurate than dividing by a small integer. These obviously don't matter to someone using a calculator, but could well matter when doing the billions of operations needed to update a screen.

In addition I think there is merit in having canonical forms. Why do you ask students to reduce 3/6 to 1/2. Both are valid. Why not leave an an answer as 31+12-16 instead of 27?

Certainly checking agreement between two polynomials starts getting messy if they are in arbitrary order. (and if there are multiple variables I'm not aware of any canonical form for the answer)

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