Rationalization Rant

Every high school math student has been taught how to rationalize the denominator. We tell students not to give an answer like

\frac{1}{\sqrt{2}}

because it isn’t fully “simplified.” Rather, they should report it as

\frac{\sqrt{2}}{2}.

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

\frac{12}{1+\sqrt{5}}

to be rationalized because of the square root in the denominator, so we multiply by the conjugate to obtain

2-2\sqrt{5}.

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

\sqrt[5]{\frac{15}{2}},

the book would have students multiply the top and bottom of the fraction inside the radical by 2^4 so as to make a perfect fifth root in the denominator. The final answer would be

\frac{\sqrt[5]{240}}{2}.

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:

\frac{1}{2-\sqrt[3]{5}}.

But is that really desirable? The result here is

\frac{1}{2-\sqrt[3]{5}}\cdot\frac{4+2\sqrt[3]{5}+\sqrt[3]{25}}{4+2\sqrt[3]{5}+\sqrt[3]{25}}=\frac{4+2\sqrt[3]{5}+\sqrt[3]{25}}{3},

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.

\sqrt[3]{\frac{x}{y^7}}

The method that leads to the “correct” solution is to multiply the fraction under the radical by \frac{y^2}{y^2}, and to finally write

\frac{\sqrt[3]{xy^2}}{y^3}.

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 y=\sqrt{2} or even y=\pi.) 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?

\frac{1}{2\sqrt{2}-\sqrt{2}\sqrt[3]{5}+2\sqrt{5}-5^{5/6}}

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.

Good discussions in the math blog world

Here are two blog posts I saw a few weeks ago. I’ve been following the comments with great interest, and the conversations have been fruitful. You should go check them out and join the conversation!

  • Critical Thinking @ dy/dan — Once again, Dan gives deserved criticism to a contrived textbook problem. Hilarious problem, and fun discussion in the comments.
  • Disagreement on operator precedence for 2^3^4 @ Walking Randomly — The title says it all, but it’s the first time I had ever thought about how 2^3^4 or expressions with carets should be evaluated. Note that it’s clear how 2^{3^4} should be evaluated. We’re just unclear on how 2^3^4 should be evaluated.

Teaching domain and range incorrectly

What’s wrong with these high-school math questions?

  1. State the domain of the function f(x)=\frac{1}{x}.
  2. Where is the function f(x)=\ln(x) undefined?
  3. State the range of the function f(x)=x^2.

As a math teacher, I’ve asked these questions before too. But I always ask them with a bit of a cringe. Do you see what’s wrong with them?

Domain

A function is only well-defined when it is defined with its domain. A function f is an association between two sets A and B that assigns only one element of B to each element of A. The set A is called the domain and the set B is sometimes called the codomain. If  f is a function mapping elements from A into B, then we often write f:A\to B. For instance, consider the function f:\mathbb{R}\to\mathbb{R} defined by

f(x)=e^x

The domain of this function is \mathbb{R}, since that’s how the function is defined. Notice I explicitly gave the domain right before defining the function rule. Technically, this must always be done when defining any function, ever.

We might ask a student, “What is the domain of f(x)=e^x?” But this is a poor question. The function rule isn’t well-defined by itself. There are many possible domains for this function, like the set of integers \mathbb{Z}, the set of positive reals \mathbb{R}^+, or rational numbers between 20 and 30. What a teacher probably means is, “What is the largest possible subset of \mathbb{R} that could be used as the domain of f(x)=e^x?” In this case, the answer is \mathbb{R}.

So I hope you see why question (1) at the beginning of this post is not a very precise question. Likewise, question (2) is not very precise either. “Where is \ln{x} undefined?” has multiple answers. The answer the teacher is looking for is (-\infty,0]. The question would be better worded, “What real values cannot be in the domain of \ln{x}?”

 

Range

What about the range? The codomain in the example f(x)=e^x is also \mathbb{R}. But the range of f is (0,\infty). The range of a function is defined to be the set of all y\in B such that there exists an x\in A with f(x)=y.

But the range depends on the choice of domain. So asking questions like (3), “State the range of the function f(x)=x^2” aren’t well defined for all the same reasons as above. The desired answer is probably [0,\infty). But the domain of f(x)=x^2 could be the integers, in which case the range is the non-negative integers. We’re not told. So in the case of (3), the more precise question would read, “State the range of the function f(x)=x^2 with x\in\mathbb{R}.”

 

Should we change our teaching?

Maybe. But maybe not. I think I’ll still ask the questions in the imprecise way I started this post. Using the more precise questions would be unnecessarily confusing for most students. But we as teachers should be aware of our slightly incorrect usage, and be ready to give a more precise and thoughtful answer to students who ask.

That being said, I think there’s room for more set theory and basic topology at the high school level. I’m a bit sad I didn’t learn the words onto, surjective, one-to-one, injective, bijective, image, and preimage until very late in my post-high-school studies. I’m not sure all students are ready for such language, but we shouldn’t ever shy away from using precise language. That’s part of what makes us mathematicians.

I ❤ precise language!