Generations of math teachers have drilled into their students that they must reduce fractions. That serves some purpose in the early years, but somewhere along the way students need to learn reducing fractions is not only unnecessary, but can be bad for communication. For example, if the fraction 45/365 comes up in the discussion of something that happened 45 days in a year, the fraction 45/365 is clearer than 9/73. The fraction 45/365 is not simpler in a number theoretic sense, but it is psychologically simpler since it’s obvious where the denominator came from. In this context, writing 9/73 is not a simplification but an obfuscation.
Simplifying fractions sometimes makes things clearer, but not always. It depends on context, and context is something students don’t understand at first. So it makes sense to be pedantic at some stage, but then students need to learn that clear communication trumps pedantic conventions.
Along these lines, there is a old taboo against having radicals in the denominator of a fraction. For example, 3/√5 is not allowed and should be rewritten as 3√5/5. This is an arbitrary convention now, though there once was a practical reason for it, namely that in hand calculations it’s easier to multiply by a long decimal number than to divide by it. So, for example, if you had to reduce 3/√5 to a decimal in the old days, you’d look up √5 in a table to find it equals 2.2360679775. It would be easier to compute 0.6*2.2360679775 by hand than to compute 3/2.2360679775.
As with unreduced fractions, radicals in the denominator might be not only mathematically equivalent but psychologically preferable. If there’s a 3 in some context, and a √5, then it may be clear that 3/√5 is their ratio. In that same context someone may look at 3√5/5 and ask “Where did that factor of 5 in the denominator come from?”
A possible justification for rules above is that they provide standard forms that make grading easier. But this is only true for the simplest exercises. With moderately complicated exercises, following a student’s work is harder than determining whether two expressions represent the same number.
One final note on pedantic arithmetic rules: If the order of operations isn’t clear, make it clear. Add a pair of parentheses if you need to. Or write division operations as one thing above a horizontal bar and another below, not using the division symbol. Then you (and your reader) don’t have to worry whether, for example, multiplication has higher precedence than division or whether both have equal precedence and are carried out left to right.
15 thoughts on “Pedantic arithmetic rules”
In his book, Basic Arithmetic, Serge Lang said the same thing.
Is there anything Serge Lang hasn’t written a book on? :)
Don’t you think that the need for manual computation underlies a lot of practices that produce less-than-optimal results? For example, statistical hypothesis tests that needed to be calculated by hand or quite unpowerful machines in their time, and more principled inference was intractable. And now we are stuck with it.
Luca: Yes. Statistics enshrines several practices that were once a practical necessity.
Honestly, I think that myths like this one endure because it provides an excuse for people to not put in the effort and time. If people believe that success comes by flashes of intellectual lightning, why go through the arduous effort of learning and trial and error?
As a high school teacher that is aware of the arbitrariness of these “rules” I always stress to my learners that rationalizing the denominator is not a rule, but a social convention brought on by history.
My problem / difficulty is why are we still required to teach it any more? Does it have value beyond the history of the need? It is still tested on ACT/SAT/Accuplacer type college entrance exams, so clearly the exam writers value it.
I’ve wondered about this from the calculation side over the years. When approximations to radicals were given in tabular form, wouldn’t it make sense to rationalize denominators and then perform calculations with the approximations. To me, that would mean that you would be dividing the decimal result by an integer, not the other way around which is much more efficient.
Simplified fractions are simpler, so it’s worthwhile to teach. I’d rather think about 2/7 than 4282/14987. But sometimes other factors trump this form of simplicity. Sometimes, as above, psychological simplicity is more important than mathematical simplicity.
Was this comment meant for my Mozart post?
There is one place where “clearing the denominator” has theoretical significance. When you re-write
(a-b√2)/(a² – 2b²)
you’re showing that ℚ[√2] is a field, not just a ring. This has nothing to do with arithmetic or calculations, but matters to Galois theory.
@Jonathan, That makes a lot of sense. So teaching rationalizing the denominator in high school becomes a precursor skill to the higher math possible in future classes. I can live with that, because there are many skills that need to be taught where the payoff is not known until later classes. Thank you.
@Glenn, as a student I hardly could live with that. I really wanted to understand what we were taught and know why we did things. Those times when a teacher said something along the lines of “just copy from the blackboard what I write there, it’ll make sense to you next week” or “let’s just call it a convention for now” were frustrating for me.
But I assume that providing a far outlook on what some techniques would later be used for might have intimidated students who were not as interested and quick in maths as I.
I believe a teacher should always honestly address honest questions. If other students aren’t interested in or prepared for the answer, no problem. They’ll just tune it out.
Sometimes teachers aren’t able to a question and give an evasive answer so they can save face. Better teachers would model the way they’d hope their students would respond: “I don’t know, but I’ll find out.”
Some questions aren’t honest. They’re just whining. Teachers aren’t obligated to answer those, though sometimes a really good teacher will be a little sly and answer an insincere question as if it were sincere.
@John and Martin, My question and concern is absolutely genuine, and when my learners asked why I pulled out a slide rule and explained the history of math. We I explained that we still rationalize numeric denominators because of the ease of division without a calculator. We still do it because of tradition and entrenched expectations.
That is a reasonable explanation as far as that goes. Knowing there is a higher mathematics allows me to connect the history and tradition within a larger context as well, and that is important to me too. I tell my learners I will never lie to them or make stuff up. I completely agree with you both on those counts.
It is so frustrating when my learners ask, “Why did they tell me there was no solution to these quadratics. Yes there is, it is just a complex solution.” Their next question is always, “Why did our teachers lie to us?”
As far as real solutions and complex solutions to quadratics, that’s a great opportunity to talk about the importance of context. Whether there is or isn’t a solution depends on the universe of discourse. I write a little more about this here: What do you mean by can’t?.