What is a fraction?

The other day, I saw a tweet joking that while calculus teachers insist that \(\frac{dy}{dx}\) is not a fraction, the LaTeX is \frac{dy}{dx}.

That reminded me of my longtime complaint that “fraction” as a mathematical term is vaguely defined, so I asked my Twitter followers a simple question: “Which is a fraction?” I provided these options: 0.7, π/e, both, neither.

The motivations for those options (Twitter limits polls to four) were based on the two elements of the most common definition of “fraction”, which I’ll take from MathWorld: “A rational number expressed in the form a/b.”

So I provided a rational number and a number expressed in the form a/b. If the allegedly standard definition, requiring both form and value, was the prevailing one, the most common answer to my poll should be “neither”.

Here are the results:

My poll was shared. I do not have 1000+ followers.

These results surprised me. I was expecting some level of spread, and I was also expecting (and got) comments about, “Well, the technical definition is….” I also added an addendum to assume that π/e is irrational after someone pointed out that it hasn’t been strictly proven to be, but I think most people did make that assumption.

What I wasn’t expecting was a lack of a majority opinion, and an even split over whether “fraction” is about value or about form.

What does “both”, the plurality winner, mean here? Does it mean that all rational numbers and all numbers of the form a/b are fractions? How, then, is the word “fraction” a meaningful term?

Worse, does it mean that any number that can be written in fractional form is a fraction? Or, even farther, that any expression that can be so written is a fraction? Are all real numbers fractions? Are all expressions fractions?

Hic sunt dracones.

And yet, despite the dangers, that was the most common answer.

Beyond that, those people taking a clear stand were evenly split on value and form. None other than Ben Orlin (I’m not worthy!) pointed out that, if “fraction” is simply a synonym for “rational number”, then what’s the point in having two terms? Just call it a rational number. (A rebuttal that was not offered: There are other places where we have a friendly term and a technical term, such as “counting number” and “positive integer”. Still, a valid point.)

Personally, I was expecting the people favoring form to dominate the people favoring value, and that didn’t happen. At all. Not even slightly. (272 to 271, so I suppose “very slightly”.)

And what of the “technically correct” answer that requires both form and value, i.e., that a fraction is a rational number expressed in a fractional form?

Last place.

In a quote tweet, someone asked me to define “fraction”. Because I try not to provide my own opinion during these sort of polls, I answered honestly: “Words mean whatever communities agree they mean.”

What’s interesting here is that there is a distinct lack of agreement within the community here about what ought to be a fairly fundamental concept.

Which on the one hand is fine. It’s absolutely the nature of the beast. But many people who are reading this are mathematics teachers, and many mathematics teachers act as if terms are set in stone, with clear definitions that everyone agrees on.

So therein lies my caution: If we don’t agree on this, what else do we not agree on? And how many of us, prior to a poll such as this, would have absolutely sworn that there was agreement on this? As mathematics teachers, how often do we communicate to our students that things are concrete truths when they are more flexible than that?

Things for each of us to think about.

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