Recently, @solvemymaths asked the rather leading question:

As with pretty much all mathematical notation questions, I have exactly one answer: for clarity. If it’s clearer to use the obelus than a fraction - a vinculum, if that’s your cup of magic potion - then you should use it.

But where is it ever clearer?

When I saw Ed’s tweet, my first thought was “I used one in The Maths Behind ((Available wherever good books are sold.))!” In particular, I was comparing the tidal effects due to two bodies, the ratio of which normalised to $\left( \frac{m_m}{r_m^3}\right) \div \left( \frac{m_s}{r_s^3}\right).$

One could leave that as a stacked fraction: $\frac{\left( \frac{m_m}{r_m^3}\right)} {\left( \frac{m_s}{r_s^3}\right)}$ … but I think it’s hard enough to work out what that means well enough to evaluate it, let alone well enough to understand what it’s trying to get across.

Typographical neatness

There are cases where a fraction is harder to typeset than a obelus - in my blog, it’s no problem, we can handle a bit of line-wrangling (as in the paragraph above) - but in a book, typesetters get antsy about extra gaps between lines. Is $I = V \div R$ any better or worse than $I=V/R$? Probably not; although, with a more involved expression, I’d imagine an obelus is less likely to be scanned over by a casual reader.

I’ve used the obelus in places to separate pieces of an expression, especially when simplifying fractions, in the same kind of place as I’d use a $\times$. For example, using approximations justified earlier in the post, $2\cos(55º)\sin(5º) \approx 2 \times \frac{180\pi}{100} \times \frac{5\pi}{180}$, which simplified to $2 \times 5 \div 100$.

Habit

The use of $\div$ to represent the operation of division is pretty ubiquitous (at least outside of computer languages, but let’s leave that filth to one side). Instinctively, in listing the operations available, I’d have $+$, $-$, $\times$ and $\div$.

(There’s a weak argument that binary operators are The Way To Work - if we’re going to use $+$ and $-$, we should $\times$ and $\div$ and even $\hat{}$ for powers. Filth, I tell you.)

Some of Ed’s replies suggested using $\div$ to distinguish a number from a command - as in, $\frac{10}{3}$ is just a number, but $10 \div 3 = 3.\dot3$ (at least, to that kind of student). I have used it this way when reporting on the Mathematical Ninja, for example, differentiating $\frac{52}{90}$ from $5.2 \div 9 \approx 0.578$.

In summary

Written maths is about communicating with one’s audience. If I’m discussing something with peers - or with students who are good enough to know better - I’ll almost always use a fraction (and gently chide anyone doing Higher GCSE or above if they use an obelus without good reason). If I’m working with someone who struggles, and I judge $\div$ to be clearer for them, I’ll use $\div$.

Outside of that distinction, the only time I routinely use the obelus is to avoid stacking fractions.