Compound words are common in mathematical writing and it can be hard to remember how to hyphenate them. Unfortunately, there are no hard and fast rules. In this article I give some guidance and illustrative examples. The principle to keep in mind is that hyphenation should help to avoid ambiguity.
In phrases of the form “adjective noun noun” or “noun adjective/participle noun” a hyphen is usually used: closed-form solution, nineteenth-century mathematics, error-correcting code. But if the adjective follows the noun then no hyphen is needed: solution in closed form, mathematics of the nineteenth century, code that is error correcting. Here are some other examples:
- nearest-neighbor interpolation,
- higher-dimensional discrete Fourier transforms,
- large-scale optimization problem,
- minimum-norm solution but solution of minimum norm,
- first-order differential equation but differential equation of first order,
- the parameter-dependent ODE but the ODE is parameter dependent,
- rank-1 matrix but the matrix has rank 1.
In examples such as finite-difference method and finite-element method it is a matter of convention and taste whether to hyphenate. Some authors do and some don’t. Most authors do not hyphenate singular value decomposition.
Compounds beginning with adverbs ending in ly are not hyphenated, since they are usually unambiguous. Examples: slowly converging sequence, highly oscillatory integrand, continuously differentiable function, numerically oriented examples.
An important special case is compounds beginning with ill, well, little, much, and best, the first two of which are particularly common in mathematical writing. Here, a hyphen is used for a compound of two words used adjectivally, but if the compound itself is modified then no hyphen is used. Examples (these also apply with ill replaced by well):
- This is an ill-conditioned problem.
- This is a very ill conditioned problem.
- The problem is ill conditioned.
- This problem is very ill conditioned.
If the first example were to be written as This is an ill conditioned problem then it could be read as if ill were an adjective modifying the compound conditioned problem. Confusion is unlikely in this instance, but in ill-prepared contestant the hyphen is needed unless we are talking about a contestant who is prepared but not well.
Here are two further examples that are complete sentences.
- MATLAB allows a two-dimensional array to be subscripted as though it were one dimensional.
- This approach is particularly well-suited to high-precision computation.
The hyphen in well-suited in the last example is not essential, but is rather a matter of taste.
I know from personal experience that it is hard to achieve good, consistent hyphenation when you are concentrating on all the other aspects of writing. This is where having the services of a copy editor is extremely valuable. To benefit, you need to publish with a journal or book publisher that takes copy editing seriously (SIAM, PUP, CUP, OUP, …).
I give the final word to an Oxford University Press style manual, as quoted in the Economist Style Guide:
If you take hyphens seriously, you will surely go mad.
I am indebted to Sam Clark of T&T Productions for checking this post (and for saving me from many hyphenation blunders in my last two books).