Every high school math student has been taught how to rationalize the denominator. We tell students not to give an answer like
because it isn’t fully “simplified.” Rather, they should report it as
This is fair, even though the second answer isn’t much simpler than the first. What does it really mean to simplify an expression? It’s a pretty nebulous instruction.
We also don’t consider
to be rationalized because of the square root in the denominator, so we multiply by the conjugate to obtain
In this particular example, multiplying by the conjugate was really fruitful and the resulting expression does indeed seem much more desirable than the original expression.
But here’s where it gets a little ridiculous. Our Algebra 2 book also calls for students to rationalize the denominator when (1) a higher root is present and (2) roots containing variables are present. Let me show you an example of each situation, and explain why this is going a little too far.
Rationalizing higher roots
First, when a higher root is present like
the book would have students multiply the top and bottom of the fraction inside the radical by so as to make a perfect fifth root in the denominator. The final answer would be
Simpler? You decide.
This becomes especially problematic when we encounter sums involving higher roots. It’s certainly possible, using various tricks, to rationalize the denominator in expressions like this:
But is that really desirable? The result here is
which is, arguably, more complex than the original expression. Can anyone think of a good reason to do this, except just for fun?
Rationalizing variable expressions
Now, let’s think about variable expressions. Here is a problem, directly from our Algebra 2 book (note the directions as well):
Write the expression in simplest form. Assume all variables are positive.
The method that leads to the “correct” solution is to multiply the fraction under the radical by , and to finally write
This is problematic for two reasons. (1) This isn’t really simpler than the original expression and (2) this expression isn’t even guaranteed to have a denominator that’s rational! (Suppose or even .) Once again I ask, can anyone think of a good reason to do this, except just for fun??
So how far do we take this?
Is it reasonable to ask someone to rationalize this denominator?
You can rationalize the denominator, but I’ll leave that as an exercise for the reader. So how far do we take this? I had to craft the above expression very carefully so that it works out well, but in general, most expressions have denominators that can’t be rationalized (and I do mean “most expressions” in the technical, mathematical way–there are are an uncountable number of denominators of the unrationalizable type). All that being said, I think this would make a great t-shirt:
And I rest my case.
Glad to find that the next generation is carrying the torch, son! I’ve had this rant for decades.
Here’s my take: Ban all mention of the term simplify unless it is accompanied by the phrase for the purpose of ….
In my History of Math class today I told them about the accomplishment of which François Viète (1540-1603) was proudest: The roots of a polynomial are symmetric functions of its coefficients. Note that (x-a)(x-b)=0 is simpler for the purpose of finding the roots of the equation (that is, the zeros of the polynomial) than the equivalent x^2 -(a+b) x + ab = 0.
But for the purpose of finding where y = x^2 + 3x – 5 crosses the y-axis, I’ll take that form over the equivalent factored form y = (x – (-3 – √29)/2) ( x – (√29 – 3)/2) any day.
Fortunately, there are other words than “simplify” in this context: “find the linear factors,” or “express as a trinomial.”
Trig substitutions simplify an integrand for the purpose of recognizing something that I can integrate. I’ll give the Calculus instructors the right to leave out the obvious: “Use a trig substitution to simplify” without mentioning the obvious purpose.
Vi Hart’s video about the number Wau comes to mind. She instructs us to “complexify the following” (my term). And that’s more fun.
Yes, I totally agree about the use of the word ‘simplify’. As a teacher I always have to specify, ‘write the expression using no negative exponents,’ or ‘reduce the expression to the argument of a single log’ or whatever else. Amen to that!
I think our issue with Rationalization runs even deeper, too. Usually there’s a good purpose for the desired manipulation (like trig substitutions in calculus). But here, I don’t see any good purpose for rationalizing the denominator. Can you give a math history perspective on that? Was there a time when it was more useful, but now it’s not? I just can’t think of a very good argument why anyone would ever *need* to rationalize the denominator.
One thread in history of math is an ever-changing notion of what counts as a number. For Pythagoras, √2 was a magnitude but not a number. Now it is a number.
We want a number as an answer, not an indicated operation. √25 is not a number, but 5 is. 1/√5 is not a number, but √5/5 is. I was amazed to learn that in India in the Middle Ages, things that I would regard even today as indicated operations were regarded unproblematically as numbers, such as
Math history is not as simple as having the notion of number becoming ever-more inclusive, as we were led to believe in math classes (positive integers – integers – rationals – reals – complexes – quaternions – octonions). It’s been a labyrinth of give-and-take. Rafael Bombelli used complex numbers to solve polynomial equations in the 16th C., but thought they were “sophistry,” not numbers, 350 years before David Hilbert’s “formalist” program of mathematics.
For heaven’s sake! Even 1 was not considered a number until the Middle Ages. See the bottom of page 2 of this source for a list of quotations to prove that, from an MIT course on ancient history of mathematics.
I think the purpose in general is not simplification but to express everything in a well-defined “canonical” form.
This can be very useful for example for a grader who can immediately recognise that the answer is correct.
And canonical forms have value independent of graders. Writing the equation of an ellipse in canonical form allows us to identify the lengths of semi-major and semi-minor axes and location of center in an instant.
Since the goal of mathematics is insight, not computation, canonical forms play a special role in mathematics: a diagonalized matrix, a boolean expression in disjunctive normal form, a predicate calculus proposition in prenex normal form, or invariant factor decomposition of a finitely generated abelian group. There are hundreds of examples.
I agree that canonical forms are very important.
That’s at least one important justification for rationalization.
But what’s the value of the canonical form *here*? I see GREAT value in other canonical forms, like the vertex form for a parabola, or the factored form for a polynomial, or the diagonalization of a matrix. I guess just to show that two radical expressions are equivalent? Not sure.
Second question: what IS the canonical form for the two examples I presented in the post? (higher order roots and radical expressions involving variables) Most often, like I said, such a form isn’t even possible.
I always thought that the historical reason was that it is easier to get a decimal approximation by hand with rationalized denominators. For instance, it is easier to divide 1.414. . . by 2 using long division than it is to divide 1 by 1.414. . . using long division.
With a rationalized denominator, you can easily increase your precision with long division by simply using 1.414 instead of 1.41 (say), and the reusing the exact work you did before. If you calculate 1 divided by 1.41and later decide you need the precision of 1.414, then you pretty much have to start your long division again from scratch.
So I think rationalizing was useful in the days when calculators did not exist; I do not see how the technique is useful now.
Agreed. Rationalizing denominators was truly practical back in those days. Maybe rationalization should go the way of trig tables and log tables and log trig tables, now being replaced by a calculator (that is, calculator the machine, not calculator the person). At least for students not planning to study math beyond one college course.
But see my next post for why the more theoretical you are, the more you need to rationalize denominators.
And as a mathematician I always shudder when students limit their options by predicting what math they might need. I once earned money tutoring a grad student in veterinarian school who needed Calculus to understand one of the professional papers he was assigned to read and master. When he was in high school he was like, “I’m going to be a vet. I don’t need more math.”
(Intentional youth lingo.)
You can never learn too much math! 🙂
Let Q be the rational numbers. Let Q[√5] mean the set of all numbers of the form
a + b √5 with a, b in Q. We say “Q[√5] is Q with √5 adjoined.” One can show that you can do division in Q[√5], which is to say that (a+b√5)/(c+d√5) is again of the form x + y√5 with x and y rational numbers. We can repeat this process. The field (technical term alert!) Q[√5] can again be extended to Q[√5][√3] which is all numbers of the form a + b√5 where a, b are taken from Q[√5].
That means that such things as 1/(x + y√5 + z√3 + t√5√3) can be rationalized, and furthermore that the result will be of the form a + b√5 + c√3 + d√5√3. The proof is constructive: We actually rationalize the denominator.
Theorems well-known to those who study enough abstract algebra show that one can generalize this with square roots of any rationals adjoined to the rationals. Since Euclidean (straightedge and compass) constructions can do iterated square roots, all of this has a nice geometric interpretation too.
I won’t speak today to the usefulness of rationalizing more general expressions, but it seems that a student of abstract algebra shouldn’t be handicapped by having never practiced rationalizing fractions with square roots in the denominator.
Exercise: To Q adjoin c = the cube root of 5. Can you do division in the resulting set of numbers, Q[c], which is to say numbers of the form x + yc ?
Exercise: generalize the above exercise.
I meant “a + b√3 where a, b are taken from Q[[√5]” at the end of paragraph 1.
I agree 100% with everything here. In fact, the very examples you quoted came up in my Algebra class last semester.
And I admitted in my original post that rationalization in those particular cases is useful. But what about ANY expression involving radicals in the denominator? And what about variable expressions? I’m not sure I see the value as much there.
I also agree 100% that students should not limit their mathematical options. But in this case, I felt like the book and curriculum were presenting contrived problems without providing a justification. “For fun” is even a valid justification as far as I’m concerned, but it should be acknowledged.
I prefer problems that are not contrived. I prefer telling students why what they’re doing will be worth their time, or letting them discover it. It doesn’t have to be a technical explanation. “Fun” is a good motive. Another good motivator is to suggest a short reading. An often missing motivator is to ask a student to reflect on her work to ask what future direction it could go in.
I did that with my comment above, where I say “generalize,” and I did it with my Pi R Squared post where I end with, “Can you think of anything that involves π but doesn’t involve a circle?”
This is George Polya’s Step 4 in his famous book How To Solve It.
But I admit that sometimes I am more concerned with how much I cover in a lesson than with how much I uncover.
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I teach high school math. Today I taught rationalizing the denominator. I fully understand the concept and I agree with the above comments. For extra credit, I asked the students to find who came up with the idea of rationalizing the denominator and why. I understand the why. What I haven’t been able to find is the “who.” Is there one person credited with the concept?
I’m not sure. In the 12th century, Islamic commentators on Euclid’s Elements, Book X, rationalized denominators, but I’m not sure which one would have been first. Here are some possibilities (from Victor J. Katz, op. cit., pp. 332-333, http://www.amazon.com/History-Mathematics-3rd-Victor-Katz/dp/0321387007 ): Al-Khwarizmi, al-Khayyami, Sharaf al-Din a-Tusi, Ibn Mun’im, al-Samaw’al, al-Karkhi (i.e. al-Karaji), al-Samaw’al, or Ibn al-Baghdadi.
But even if you found the first person to mention it, that wouldn’t necessarily be the one who should get the credit. L’Hopital gets credit for a Calculus rule because he was the first to mention it in print, but in fact John (Johann) Bernoulli is more likely to be the creator of L’Hopital’s rule, at least according to Ron Larson (at http://www.amazon.com/exec/obidos/ASIN/039593320X/ ).
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My take on this for years has been that there are a lot of things that people are asked to do at one point in their mathematical careers that really aren’t that important later. Rationalizing the denominator is one: the only good reason I have ever been able to come up with for that is that it makes it easier on the grader. I don’t think anyone in a calculus course would get upset with 1/sqrt(3) as an answer.
Similarly, all of the hysteria about improper fractions disappears at some point, and if you wrote 2 (and) 1/75 rather than 151/75 in a calculus course you would be the odd duck.
The big problem with all of these “rules” about how the answers are presented is that they play into the whole “mathematics as algorithms” point of view rather than “mathematics as thinking.” You give good reasons for why someone might want to rationalize, or not rationalize, but the algorithm people want there to be a rule that is blindly followed without people thinking or questioning it, IMO.