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.