# Should one use "above" and "below" in mathematical writing?

I started thinking about this question because of this discussion:

http://sbseminar.wordpress.com/2010/08/10/negative-value-added-by-journals/

about how journals often change a paper (for the worse) after acceptance.

Here's my question: Since it doesn't make sense to number every single equation (especially if you will only refer to it once in the text), I often use the phrase "the equation above" or "the equation below" to refer to the previous or next displayed equation. I am well aware that after compiling, said equation may be on the previous or next page, respectively, and not strictly above or below. I thought this was common practice. One journal publisher changed every single one of my "the above equation" phrases to "the earlier equation". I didn't bother protesting, but in my mind it definitely made the text worse.

Do other people avoid using "above" and "below"?

• I'd say that if you refer to an equation, you should number it. Beware: a few journals insist on labelling every displayed formula. Aug 11, 2010 at 13:41
• "the earlier equation" sucks! There are lots of earlier equations. Well, there are lots of equations above, but in context it sounds a lot better to refer to the previous equation or the equation above instead of the earlier one. On the other hand, if you're going to refer to an equation why not number it? Aug 11, 2010 at 14:12
• I always thought "above" means "earlier in this text", regardless of whether that's on the same page or on an earlier page. Mirriam-Webster seems to agree with me: mw4.m-w.com/dictionary/above lists "2b : higher on the same page or on a preceding page" as one of the meanings. Aug 11, 2010 at 14:25
• I'd prefer "above", it definitely makes sense, after all we live in a digital age of scroll bars. However it may become confusing if you're writing in a two-column format. I'll propose a solution, from now on we all compile our documents in 'scroll' format, that is without page breaks, just one long page. However I do have the feeling that the university library would refuse to take my thesis in the form of an actual scroll. Aug 11, 2010 at 15:57
• I think "the equation above" is fine, even if the equation is on an earlier page. I'm not happy about "the above equation"; "above" is an adverb and a preposition, not an adjective. Years ago, I was taught that "the above equation" is as bad as the (at the time obviously barbaric) phrase "the below equation," but nowadays I occasionally see "the below equation" in print. Aug 11, 2010 at 16:06

## 10 Answers

Since it doesn't make sense to number every single equation...

I used to number only equations I referred to, but then someone pointed out the following. When you write a paper and make it public, you are de facto allowing other people to talk about your paper and simply because you in your paper see no reason to refer to some equation does not mean that other people reading your paper will not do so. Hence as a friendly gesture to your readers, you should number all your equations, and in this way allow them to refer to, say, equation (n) in your paper, instead of having to come up with some complicated reference. Since that time I have tried to number all my equations.

• Thanks, José. I actually never thought of that. It makes very good sense. In part, I was comparing equations to citations. I know it's definitely not acceptable to put a paper in your bibliography that you don't cite somewhere in the text, so I was applying the same rule to equations. But your point is going to make me think hard about what to do next time. I know that I need to get creative to describe where a mistake is in an unnumbered equation when I write a referee's report. Aug 11, 2010 at 14:10
• You're welcome. I'm just passing on the advice. I had never thought about it before either. Aug 11, 2010 at 14:40
• Excellent point. I never number all of my equations, because I want the reader to know which equations are the important ones at a glance. But now that I think about it, maybe the less important ones should not be displayed to begin with. Aug 11, 2010 at 15:02
• This is interesting advice. The reasons I might not do this seem quite subjective: 1) too much use of numeration in a paper gives it a kind of bureaucrtic feel [we could just number every line, right?], and 2) as Spiro describes, when I see a number next to an equation, that makes me think it is important and going to be used later. A numbered equation which is never cited again feels a little like an unfulfilled Chekovian promise. Aug 11, 2010 at 15:02
• Pete, your connection between the numbering of all equations in a mathematical paper and the violation of Chekhov's admonition is absolutely brilliant in its unexpectedness. Kudos! Aug 11, 2010 at 15:43

No. See above.

First of all, the question is a small matter that only a journal editor should take seriously. There are much deeper reasons that a paper might be well written or poorly written. Even among the concerns at this level, it is more important to use concise, regular, correct grammar than to follow particular journal conventions.

That said, in my opinion it's okay to say "above" and "below" to refer either to an equation, or more often to a paragraph of the paper. E.g., you might say "By the above discussion...". But it may be clearer to do something else for equations. If you just say "above", it may not be clear if it means immediately above (or previous) or something else; but if you say "immediately above", that's a bit strained. My rule is to say "this" equation if it is immediately previous, and otherwise to number the equation even if I only explicitly cite it once. If the equation is not explicitly cited, but is just part of the narrative, then like you I wouldn't number it. But there are authors who number every equation.

For paragraphs, I sometimes say "above" and "below", if the cited paragraph is in the same section and subsection.

• Thanks. I know it's a small matter. I also agree with you about citing paragraphs in the same section using above and below. My point about numbering equations is I don't like having (for example) 200+ numbered equations when only 30 or so of them are going to be referred to more than once. It seems like unnecessary clutter to me. (Especially if some of these one-time-only equations are already long and scary. Adding a number makes them more frightening...) Aug 11, 2010 at 14:06
• This is more or less how I go about things too. Somewhere along the line I started writing "the last display" when citing the previous displayed equation. I mostly reserve this for the case where there is some smaller in-line equation (or even other remarks with symbols) between the display and my citation of it. Aug 11, 2010 at 16:25

I think, for the reasons Jose points out, that you should err on the side of numbering more. In particular, an equation which you refer to more than a line or two after it appears should probably be numbered. I've also found this makes editing easier: If you insert a new paragraph, with several equations in it, you don't need to look for all of the references you've broken.

I agree with you, though, that it never would have occurred to me that "above" couldn't refer to a line on a previous page. Perhaps this is a difference between those of us who read on screens, with our scroll bars, and those who page through print outs?

Of course, different areas tend to be more or less equations-heavy, but an author's individual style matters too. Two people could write up the exact same result with dramatically different number of equations. So if you observe that your papers tend to be dense with equations, it makes sense to me to try to differentiate by numbering only the ones that will be referred to regularly (my previous comment notwithstanding, because I shy away from using too many equations).

That being said, I would avoid the "above" and "below" as not artful. "Previous" is a good word, or making use of \pageref would allow you to be more precise. It's difficult to be more precise without a concrete example, but in most cases you should be able to refer to the equation by describing it. E.g. "in the long exact sequence we just established", "the previous norm inequality implies that", ...

Like Greg Kuperberg, I think of this as a small stylistic point compared to many others. For me it's more important to choose numbering of sections, subsections, equations, definitions, and such to maximize the reader's ease of understanding what is going on. Some built-in LaTeX styles used in books and journal articles work against readability. Typical is a reference to Proposition 7 or 3.7, which may be hard to go back and find quickly (especially if Proposition 3.7 occurs in Section 3.4). Similarly, referring back to equation (38) may send the reader on a lengthy hunt through the previous 40 pages of a long article. Also of dubious help to the reader is consecutive default numbering of the sort: Definition 1, Theorem 2, Remark 3, Lemma 4, Definition 5, etc.

• "Also of dubious help to the reader is consecutive default numbering of the sort: Definition 1, Theorem 2, Remark 3, Lemma 4, Definition 5, etc. " This is worse than numbering with section numbers (Definition 1.1, Theorem 1.2, Definition 1.3, Definition 2.1, Definition 2.2 ...), but infinitely better than numbering each type of object separately (Definition 1, Theorem 1, Definition 2, Definition 3, Theorem 2, ...). Aug 11, 2010 at 16:24
• Jim, this is interesting. Could you given an example of what you think might be a friendly referencing scheme for equations in a lengthy document? Thanks. Aug 11, 2010 at 16:24
• David, I could not agree with you more!!! Aug 11, 2010 at 16:25
• Jose: It's easier to give bad examples from the papers I've looked at recently. But my own preference is for documents with moderate-sized sections numbered 2.4 or such, along with numbered equations or displays within that section numbered (3) or such. Then refer back to 2.4(3) in later sections. But I do tend to see "equation (38)" references that go back many pages and thus require a tedious search. Whatever system is used, it should always aim to facilitate reading. Stylistic choices do of course depend on the type of material and length of article or book. Aug 11, 2010 at 17:36
• When I hear "references that go back many pages and thus require a tedious search" I reach for my hyperref manual. Oct 12, 2010 at 13:05

Practically, numbering equations works, sequenced in the article from beginning to end. While it is a courtesy to readers and arbiters, later, who discuss lines that the author did not, it most certainly adds specificity, without tortuous sentence structure or word choice. "Equation 24" will always be just that, regardless of its placement on the page, and regardless of reprint formulation or quality.

As an editor, my choice is to number all equations, and bold those that are referred to by the text of the article.

Equation identification, when there is textual sectioning, by chapter or other means, can include the section as predicate: e.g.: 5-24 refers to equation 24 in chapter 5. Texts that use this system (or any system) wisely place an explanation at the forward.

Obviously, longer papers/articles, or complex sectioning, both complicate the system for the reader, the ultimate consumer for the publication (e.g.: 5.3-24). Personally I would avoid it; I can't imagine the necessity of referring to large counts of equations in other chapters or sections, and in those rare cases, for a reference such as "see equation 5-24", I would consider adding a postscript: "in Chapter 5.3" or "at page 137".

Every equation ought to be numbered in print publications or fixed-format electronic publications; if an equation is not important enough to be included as a numbered equation in the article, it ought not be included at all. As for "above" and "below", I've learned them contextually as meaning "prior" and "later" in the current article. I've never understood it to mean exactly one equation above or one equation down. In fact, I've even seen absolute and relative references together, as in "see equation 12 above." If an equation reference goes too far forwards or backwards, it makes sense to repeat the equation renumbered with a new number in this location. It's much easier to look at it on the same page rather than have to flip back and forth.

Relative references make some sense for fixed publication media such as printed copies of journals. Absolute references, such as pointers and index numbers and URLs, make more sense for variable view-model media such as electronic publications (HTML particularly).

I agree with the other answers (above, and below) for mathematical writings for print publications such as journals. However, there is an extra consideration for electronically published items in electronic journals or particularly in forums like this web-site, Mathoverflow.

Users have the option of controlling their viewing model on electronic publication systems and changing what appears at the "top" of their electronic page. They may choose chronological order in order to view comments in the same order they were submitted, allowing ease in understanding the flow of commentary. They may choose reverse chronological order, for example when they are revisiting a question just to see what the latest entries have been. They may also choose to order the results by popularity or relevance (with popularity of votes being an electronic self-selected polling of relevance by other readers).

On forums like Mathoverflow, references to other writer's contributions as "the answer above" or the "answer below" are rendered meaningless and confusing by the fact that the physical ordering of the answers is different for different readers and at different times. Reader preference can re-order the answers according to time submitted (oldest first, newest first) or by popularity (votes thus far); the popularity is evanescent as the number of votes will also change over time.

Certain options lead to difficulty in following threads. For example, comments tend to be initally shown in descending order of votes, destroying the temporal ordering of conversations or the ordering of comments spread out amongst multiple entry boxes. I would have expected that long comments spread out amongst multiple boxes would be discouraged; but they seem to be rather prevalent among mathoverflow. I find that I always have to click on the "show additional comments" button in order to be able to follow the unfolding of the comments and understand the conversation in the commentary.

• I agree with you, but my question was (implicitly) specifically about printed journal articles. But your point is well taken. Oct 12, 2010 at 12:31
• +1 to Spiro. Your answer is off-topic; it is very interesting but it belongs in meta. Oct 12, 2010 at 13:08
• Spiro, my response was too long to include as a comment. I've edited my answer to include an answer to your implicit question. Oct 12, 2010 at 13:24

I think it is a matter of style whether to keep numbered formulas to a minimum or to a maximum, or to opt to something in between. I usually try to have as few, as possible. "The above equation" seems to me like a slight abuse of both grammar and common sense: one usually has lots of equations above the point of reference! I may occasionally write this myself, but normally I prefer something like "the last equation".

• But wouldn't "the last equation" be at the end of the paper? :)
– JRN
Apr 5, 2011 at 8:08
• @Joel: depends on what you consider the default "anchor point". Should I have to make the reference you mention, I'd write the last equation of the paper. (A different viewpoint may eventually lead to the conclusion that "the last equation" refers to the very last equation of all the math! :-)
– Seva
Apr 5, 2011 at 8:48

I would use the following equation in a sentence immediately preceding the equation. I would use the equation below if there were more text in between.

I would tend to avoid these formulation to be frank, I would re-organise the text to avoid forward references to an equation in general, unless I speak about a celebrated equation. For example The law or large numbers below has first appeared in... , Einstein's equations below describe.... Compare with The following equation is the strong law of large numbers: [Displayed Eq.].

If I refer to an equation above, I number it and refer to the number: In Eq. (xx), $a$ is the acceleration, ...