Guidelines for typesetting fractions

Question

I realize that this question is to some extent subjective. However, I suspect that there are some common standards regarding how fractions ought to be typeset in mathematics. The guide Mathematical Writing by Knuth, Larrabee, and Roberts states that "One of the most common errors mathematicians make when they get their own typesetting systems is to over-use the form of fractions with the horizontal bar..." (19). Therefore, I feel that it would be useful to gather a list of recommendations on how fractions ought be typeset in various contexts.

Here is a (possibly very incomplete) list of such contexts; feel free to add additional items and expand upon existing ones.

• Inline fractions. Should one use \frac, \dfrac, or something else? How about when one is typesetting an inline derivative like dy/dx? What if one wishes to typeset an inline partial derivative like d^3 f/(dz dy dx)?
• Fractions in display math. Suppose that one wishes to typeset 1/n in display math. Should one use \frac or /? If you think that / should be used, at what point is the fraction "large enough" that one should switch to using \frac?
• Fractions in subscript and superscript (e.g. exponents).
• Fractions at the top or bottom of series and integral symbols.
• Fractions nested within \frac-style fractions.
• Continued fractions.
• Fractions in plots and diagrams.

Thanks for your input!

Answer 1

I don't think it's possible to come up with strict, let alone unambiguous, answers to your questions. (And, I don't think you're asking for such guidelines...)

An important general aspect of what's considered to be good typography is the creation of an even amount of "color" -- or "average grayness", if you prefer -- across pages and across paragraphs within a page. There are obviously many factors -- so far unspecified -- that determine a page's "average grayness". E.g., are the paragraphs on average quite long or quite short? How tight is the interline spacing? Are successive paragraphs separated by extra whitespace? Are there lots of displayed equations? Taken together, these aspects should indicate fairly clearly when (and when not) to use \frac and friends in inline math situations.

One of these aspects is whether running text is set single-spaced or more loosely. Obviously, the larger the distance between successive lines of text, the less deleterious the effect of using \frac -- as well as exponents, subscripts and superscripts, and anything else (such as integral, sum, and product symbols!) that rises above or falls below the space defined by the baseline and the caps-line will be on the average color of the page. If the text is set single-spaced (and especially if the baseline skip is modest, say, 20% more than the font's nominal size), almost anything that rises above the capsline or falls noticeably below the baseline risks forcing the use of extra space between lines to avoid collisions -- and should thus be used sparingly at most in inline material.

Incidentally, having to increase the baseline skip for just one line is exactly what happens because of the (clearly deliberately undertaken) use of \frac in the first paragraph on p. 19 of the document you've cited in your posting: Observe that the space between the second and third line is quite a bit larger than the other spaces in the paragraph.

To sum up, publications (including working papers) that are typeset single-spaced should probably avoid having \frac (let alone \dfrac!) appear in running text. In contrast, if the working paper is set with one-and-one-half-spacing or (shudder) double-spacing, use of \frac is going to be much less problematic from the point of view of creating even typographic color. Some design purists might go as far as saying that double-spacing has already killed off any chance of creating good color. If color is already hopelessly compromised, any extra damage done by occurrences of \frac or \dfrac in double-spaced text may be minor, right?!

Well-designed publications should take into account whether there's a need to have significant amounts of inline math material. For instance, for the book Concrete Mathematics by Graham, Knuth, and Patashnik, a conscious decision was made to choose a baseline skip of 13pt rather than the more standard 12pt. This was done, in part, to accommodate the use of subscripts, superscripts, etc in the running text. If I remember correctly, a second reason for choosing a looser-than-normal baseline skip was that the font used for the book, "Concrete Roman", was felt to be a bit dark if employed at a more standard baseline skip of 12pt. Widening the baseline skip a touch was felt to bring about a better "average color" -- regardless of how much inline math material was present in a given paragraph. Clearly, whatever may constitute "optimal" color comes with a high degree of subjectivity. Equally clearly, I don't think there's anything wrong with having this degree of subjectivity.

Answer 2

I won't give details here, but suggest that you take a look at sections 2.4.1 and 2.52 in Ellen Swanson's Mathematics into Type, now on line. (The relevant sections start on p.17 = p.28 in the pdf file and p.27 = p.38, respectively.)

This is the guidebook used at the American Mathematical Society until author-prepared files "took over". One of the main criteria governing these practices was the goal of decreasing the number of printed pages, to minimize cost. Editorial practices have become more relaxed since then.

Answer 3

So Mico has given an excellent description of the typography, my practical take is to avoid "horizontal" fractions in running text, so here's what I do. Posting this might get my poor style and poor approach slated, but I don't mind (and in fact would like to hear what I could do better), so here we go:

I define a "running" i.e. inline fraction as follows:

\newcommand*\rfrac[2]{{}^{#1}\!/_{#2}}%running fraction with slash - requires math mode.

For example:

This is a long line of text, so we can see how the fractions look below it.

This is another line, with some fractions in it $\rfrac{1}{2}+\rfrac{A_B^C}{E_F^G}$

This is a third line, also reasonably long, but not so long it wraps.

Gives:

I felt that a simple supserscript-slash-subscript fraction was too loose, and just using 1/2 looked too big. I think it could be improved by inserting a hairspace before the slash only if there's a subscript on the numerator, as it would look more balanced.