Radicals and Mixed Numbers

The lesser known of two math memes currently wandering around the Internet involves an interesting equation: \[\sqrt{2\frac{2}{3}} = 2\sqrt{\frac{2}{3}}\]

This has spawned at least three discussions I’ve seen so far:

  1. What other values is this equation true for?
  2. Is this example good or bad for students?
  3. What’s with mixed numbers, anyway?

I’ll discuss each topic in turn.

When is this true?

This equation is true because \(\sqrt{2 + \frac{2}{3}} = 2 \cdot \sqrt{\frac{2}{3}} \approx 1.63\).

However, it is not true for all values. For instance, \(\sqrt{4 \frac{2}{3}} \approx  2.16\) while \(4 \sqrt{\frac{2}{3}} \approx 3.27\). So when is it true?

Let’s say we have three integer values: a, b, and c. The question is, when is \(\sqrt{a + \frac{b}{c}} = a \cdot \sqrt{\frac{b}{c}}\)? (We’ll get back to why we have to use a plus sign in the general form below.)

If we square both sides we get \[a + \frac{b}{c} = a^2 \frac{b}{c} \\ \Rightarrow \frac{ac + b}{c} = \frac{a^2b}{c} \\ \Rightarrow ac + b = a^2b \\ \Rightarrow ac = a^2b – b = b(a^2 – 1) \\ \Rightarrow c = \frac{b}{a}(a^2 – 1) \]

Since c is an integer, either \(\frac{b}{a}\) or \(\frac{a^2 – 1}{a}\) must be an integer. However, \(a^2 – 1\) and a are coprime, so \(\frac{a^2 – 1}{a}\) can’t be an integer. Therefore, \(\frac{b}{a}\) is an integer, and b is a multiple of a; \(b = ka\). Hence \(c = \frac{ka}{a}(a^2 – 1) = k(a^2 – 1)\) and \(\frac{b}{c} = \frac{ka}{k(a^2 – 1)} = \frac{a}{a^2 – 1}\).

However, when \(a = 1\), we get \(\frac{1}{1^2 – 1} = \frac{1}{0}\), which is undefined, so we have to exclude \(a = 1\). This leaves us with the following formula: \[\sqrt{a + \frac{a}{a^2 – 1}} = a \cdot \sqrt{\frac{a}{a^2 – 1}} \forall a \in N > 1\]

We can generalize further by looking at roots other than the square root. That is, if m is an integer greater than 1, \[a \sqrt[m]{\frac{b}{c}} = \sqrt[m]{a + \frac{b}{c}} \\ \Rightarrow  a + \frac{b}{c} = a^m \frac{b}{c} \\ \Rightarrow c = k(a^m – 1) \text{ and } b = ka\]

Here are some examples: \[\sqrt{2 \frac{2}{3}} = 2 \sqrt{\frac{2}{3}} \\ \sqrt{3 \frac{3}{8}} = 3 \sqrt{\frac{3}{8}} \\ \sqrt{4 \frac{4}{15}} = 4 \sqrt{\frac{4}{15}} \\ \sqrt{2 \frac{4}{6}} = 2 \sqrt{\frac{4}{6}} \\ \sqrt[3]{2 \frac{2}{7}} = 2 \sqrt[3]{\frac{2}{7}} \\ \sqrt[4]{3 \frac{3}{80}} = 3 \sqrt[4]{\frac{3}{80}}\]

Is this a good exercise for students?

On the one hand, this is an excellent demonstration of algebra. It also exploits what is a notational accident (see the next section) to create a nifty pattern. Our notation itself is arbitrary, so there’s nothing about numbers specifically that leads to this pattern; it’s completely an accident of notation, but it’s certainly a fun one.

On the other hand, some teachers have expressed concern that students may try to generalize the pattern beyond what algebra shows us. The example does run the risk of further confusing students about how notation works.

For that reason, it strikes me that this sort of exercise is best for students that are at a high level of understanding already and want to work with extensions. This is not a particularly practical exercise in learning how mathematics and numbers work: It’s a fun exercise.

What’s With Mixed Numbers Anyway?

One interesting thing I learned in discussions about this item is that what North Americans and others take for granted as “mixed fractions” are not in fact a global convention.

Mixed fraction notation has long struck me as mathematically dubious. It’s useful for teaching students number sense as you teach them fractions: \(3 \frac{3}{5}\) is easier to place quickly on a number line than \(\frac{18}{5}\).

However, once mathematics students have command of fractions in general, I think that mixed fractions do more harm than good. They’re difficult to work with. Compare \(3 \frac{3}{5} \times 2 \frac{2}{7}\) to \(\frac{18}{5} \times \frac{16}{7}\). In the first case, you either need to convert to improper fractions (the second case) or do some distributing: \(3 \frac{3}{5} \times 2 \frac{2}{7} = 6 + \frac{6}{5} + \frac{6}{7} + \frac{6}{35} \) \( = 6 + 1 + \frac{1}{5} + \frac{6}{7} + \frac{6}{35} \) \( = 7 + \frac{7}{35} + \frac{30}{35} + \frac{6}{35} \) \( = 7 + \frac{43}{35} = 7 + 1 + \frac{8}{35} \) \( = 8\frac{8}{35}\). Which is a mess.

Mixed fractions are used outside of mathematics on a regular basis. Vinyl records play at 33⅓ RPM. Before 50 Shades, we had 9½ Weeks. Freeway exits use mixed numbers for distance. The general populace use mixed fractions on a regular basis, precisely because they’re easier to understand numerically than improper fractions are.

However, most common fractions (⅓ is an exception) are just as easily written in decimals, so floppies could be either 3½” or 3.5″, or either 5¼” or 5.25″. And since 0.33 is a close enough approximation for ⅓, there’s no real need for mixed fractions. It’s a notational habit.

Meanwhile, mixed fractions are notationally inconsistent once we start working with algebra. 2⅓ means “two plus one-third”, but \(a\frac{b}{c}\) means “a times b over c”. In mathematics, juxtaposition without an operator normally implies multiplication. Even ⅓ ¼ would most likely be interpreted as multiplication. But when there is an integer juxtaposed to the left of a fraction, addition is implied.

I was surprised but not disappointed to learn that there are some countries where this is not the case. Italy in particular consistently uses a plus sign, so 4¼ = 1, not 4.25; 4.25 = 4 + ¼. France, Spain, and Portugal appear to also have largely (if not universally) adopted this convention. Where mixed numbers do appear, such as 33⅓ RPM, they’re seen as an industry-specific notation and might even be confusing. (Discussion here)

Mixed fractions are less useful when using metric (i.e., decimal) measures. Showing measurements in decimal form (5.5 km to the exit) allows easier shifting between units of the same type. Imperial units are inconsistent enough that decimal is not a particular advantage, so our mixed-numbers habit in the United States is likely to be harder to break.

All the same, though, I think that getting rid of mixed fraction notation is the right way to go. Pedagogically, it’s not that much more work to write \(\frac{17}{4} = 4 + \frac{1}{4}\) than to write \(\frac{17}{4} = 4 \frac{1}{4}\), and it arguably does an even better job of reinforcing the intended concept, i.e., that \(\frac{17}{4}\) is “a little more” than 4.

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