# Deciding on which equations to number

I’m writing a master’s thesis in theoretical and mathematical physics, specifically in general relativity. To better organize the thesis, I’m using the well-known division in works in mathematics to organize important results in lemmas, propositions, theorems and corollaries. Outside those blocks there are also equations relating to the overall discussion connecting the results.

I’m quite unsure about the equation numbering. I see several options:

• I number all equations.
• I number only the important ones which I’ll need later.
• I number just the equations on the discussions and not those inside the proofs of the lemmas, propositions, theorems, corollaries, etc.

What are the pros and cons of these approaches?

• Refer to Math StackExchange numbered question (2362196). Jan 21, 2018 at 21:17
• There is a nice functionality in TeX where you number all equations, but only those that you call back in a label are displayed. In this way, you don't have to worry about choosing which ones are important and need to be tagged: this is decided when you refer to them, which tautologically makes them relevant. Jan 22, 2018 at 0:34
• My two cents on the matter: only number the equations to which you refer later on (either by using the \equation environment or by using \tag on a normal equation). Don't refer to equations by writing (...) equation (n), but give your equations a \label and refer to that instead. Also, use the package hyperref (even better, cleverref), so that when you refer to equations you get clickable links in the compiled pdf. Jan 22, 2018 at 7:58

(Maths/Engineering Copy Editor speaking.) Well, first, I advocate against these two solutions (see below for some reasoning):

• Number all displayed equations.
• Number only referenced equations.

So, the solution lies somewhere in between. The basic idea is that you do number:

• All referenced equations (obviously).
• All important equations. To measure this, I would say that an equation is important if the reason why you made it a displayed equation is to emphasize it.
• Unreferenced equations "parallel" to a referenced ones. Imagine you have two similar equations in similar contexts in your work; then you should either number both or none.
• Equations you "feel" (whatever it means) someone else might want to reference.

What you in general do not number:

• Equations put on display for that they are too large.
• Intermediate steps of computations.
• Equations in abstracts.
• You number a multiline equation only with one number.

Hope this helps. Almost-final words: Use your common sense. Final words: Be self-consistent.

Why I don't like all displays numbered:

• It clutters the page if there are many equations. You also might number things that are basically one equation (think LaTeX's \intertext).
• The numbering ceases to function as an emphasis.
• If the equations are long or the columns narrow, it can eat up precious space.

Why I don't like only referenced equations numbered:

• You don't allow anything else referenced than what you yourself decided to reference.
• May I know why do you think the first two options are bad? Jan 21, 2018 at 20:57
• Abstract equations are rarely if ever "displayed," except possibly in mathematics. Jan 21, 2018 at 21:43
• I don't understand your reasons for thinking that numbering everything is bad. What does it mean for an equation to "deserve" a number? Are equations intrinsically "better" if they have one? You could argue that numbering everything removes the numbering as a signal for emphasis (I'm unclear whether that's what you're really saying), but frankly, it's not great at that job (and there's plenty of other, better, tools for that) and the loss of future referenceability can be more than enough to make up for that if you cared about the emphasis to begin with.
– E.P.
Jan 22, 2018 at 1:05
• @MassimoOrtolano But a reviewer can have to refer to anything in a paper, not just displayed equations. This is normally done using line numbers - many journals helpfully send out review versions with line numbers down the side. Jan 22, 2018 at 9:20
• @EspeciallyLime Yes, but in a thesis you usually don't put line numbers. And even when you can refer to equations with line numbers, as a reviewer, I really prefer to have the possibility to refer to equation numbers. Jan 22, 2018 at 9:23

For a master's thesis, and particularly if it's a field that you will continue to work in at PhD level or elsewhere, I would make the case that you should number all the equations.

For my MRes dissertation, I attempted to take the middle road described in yo's answer, and I only numbered the equations that I thought at the time were more important. (It's available here if you want to see the balance for yourself.) As I moved into my PhD in the same field, though, and as I used that dissertation as a reference work for both journal publications and my later PhD thesis, but most often in my personal notes as I developed those ideas further, I very often found that I wanted to reference equations that I hadn't thought important enough to number at the time, so my notes are full of references to "the second equation between 3.18 and 3.19" in the dissertation.

Frankly, I just don't see any reason to think that numbering too many equations is a bad thing. What are the actual downsides?

• That the page looks cluttered? Then it's most likely a problem with intrusive typesetting.
• That it looks like you're emphasizing equations that are unimportant? That just stems from the misconception that numbering must always convey emphasis; it can do that if you so choose, but it doesn't need to.
• That the numbers get too big? That's not actually a problem.
• That it's slightly inconsistent in which equations get numbers (as opposed to shorter calculations that can be done inline)? It is indeed inconsistent, which comes from the fact that some equations are shorter and some are longer, and if you remove the association between numbering and emphasis by being consistent with the former, the claimed "inconsistency" ceases to be a problem.
• That you still don't have ways to reference the inline calculations? You don't, but your chances of being able to pinpoint a paragraph as the one above eq. N.nn still increase.

Now, I do see the case that if you're encapsulating material inside theorems or, particularly, their proofs, they do become isolated to a higher degree, and it need not make a lot of sense to number everything in the proof of a minor lemma if it's never going to be referenced in print from outside that proof. However, depending on the ways you're going to use your thesis, you might e.g. want to improve that lemma, in which case your notes will thank you for the ability to reference the proof they're improving on.

This is obviously a matter of taste, though, and it's all subjective, but do give a thought to your future self when taking that decision.

• That the page looks cluttered? Then it's most likely a problem with intrusive typesetting. – You cannot really blame this on typesetting. As the entire point of equation numbers is being referenced, you have to be able to quickly browse through them. This is only possible if are somewhat optically prominent. Being optically prominent on the other hand means that they are disturbing when you do not care about them right now (e.g., for normal reading). Jan 22, 2018 at 15:03
• @Wrzlprmft Sure, and there's some give-and-take there. Numbering everything can look (a bit) cluttered and that can be something that ought to be avoided. My core point is that a blanket ban on "clutter" as brought on by numbering everything isn't really warranted.
– E.P.
Jan 22, 2018 at 18:29

As you can see by the other answers, this is one of those topics, where opinions differ (and do so sometimes quite loudly). As you mention that you are writing your master thesis, you should also ask your advisor about personal preferences or which style is the norm in the sub-field. This might save you from having to redo everything in the end.

However instead of adding another personal opinion (which naturally differs from the others posted), a general remark:

In general, referencing formulas by number is distracting. These references tend to break the flow, as I need to stop reading and find the right formula, somewhere far away, probably even on a different page. (If it is directly above, do not reference it by number, write something like "the preceding formula". Even a short look back, just to see that it is right there can be annoying.)

I agree that those references are sometimes a necessary evil. Still, whenever you use one, try to think about if it is really needed. Sometimes it is just a symptom of a structural problem. If you need a lot of single references, your arguments might simply be badly ordered, as you seem to be not proving things at the point where they are needed. If you are referencing a single formula a lot, why is it just a formula and not a lemma or a definition?

If you still need to reference something, try to help the reader by making it easier to grasp. Some examples that come to mind:

• Naming formulas: Often formulas are not only arrangements of symbols but have some underlying meaning or idea, which you can use as a short colloquial name. It does not have to be a global name that is established in literature, just something that helps to understand, remember and identify it. This does not replace the equation number but makes it more bearable. An example would be something like:

[...] From this we have now shown monotony:

(1.23) a < b

[...]

Using the monotony (1.23) we have...

• Warning the reader: If you are going to use a formula later on, tell the reader about it. If you tell me that you will need this estimate again in step five of the proof, I can be prepared and will be far more likely to remember it.

• Referencing with location: If you really need a formula from three proofs ago, then don't just tell me its number, but that its from the proof of theorem so and so and ideally even from which part of the proof.

• Repeating things: It is rarely done, however there is no law against repeating a formula. If you need to re-discus a formula from three chapters ago in detail, write it down again completely. Keep the number from before and tell me, where it is from, but if the complete next paragraph is about it, it really helps to have it written down in front of me.

Of course all of those can and sometimes should be combined as necessary.

• While I agree that "referencing formulas by number is distracting", I think that kind of misses the point of the question. It's a good piece of advice for how to write text that may have to reference equations, but the question is more about how to write the equations themselves. And if you want to make the connection by arguing that one should leave equations unnumbered to block others from referencing them by number, I don't agree with that. Jan 22, 2018 at 7:07
• You might disagree, but as mentioned opinions differ widely on this. However if others need to reference one of my equations in another work (I'm excluding editing/refereeing work which needs to reference arbitrary lines of text anyway), the problems are exactly those I already mentioned. Either my work is structured badly, or their work is not really up to par. The former case indeed is my fault, but at least the latter case I can try to prevent by not giving them a point of attack. A parallel concept might be the idea of encapsulation in software, which has similar motivations.
– mlk
Jan 22, 2018 at 13:12
• Although this post does not really provide a clear answer to the question (and I agree with the first paragraph that this is field-dependent and should be agreed upon with the supervisor/advisor), it contains some really good advice on writing texts with formulas! Jan 22, 2018 at 13:19

Number all the equations, important intermediate steps and final step.

That way it is easier for anyone to refer to exactly the one they want to discuss, including you.

So eq 1, then equ 1.1 as intermediate step etc.

• Numbering all equations just in case it may happen that someone wants to discuss that equation is tantamount to people that prepare for the apocalypse or something of the sort. Jan 22, 2018 at 0:47
• @PedroTamaroff What's the actual downside here? Does the consistent numbering harm the text in some way? Or is the effort in using \begin{equation}\end{equation} consistently really that monumental?
– E.P.
Jan 22, 2018 at 1:01

After reading many printed-out papers, I can only recommend: number every single equation. When a paper tells me to look for "Equation (2.3)" and I have to sift through dozens of pages because the author only numbered important/referenced equations and I could find (2.2) on page 6 and (2.5) on page 28, it's extremely annoying. Whereas if every equation is numbered, since there is at least one or two big equations on each pages, following the reference becomes very quick.

The downsides of numbering everything (the only real one I see is that if the equation is too large, things may get ugly, but this is usually fixable) are insignificant compared to this. Besides, what if you forego the numbering because your equation is large, but later on you find yourself wanting to refer to the big equation...? The only equations I wouldn't (and don't) number are single equations in a proposition/lemma/theorem/... statement, as you can refer to the proposition/lemma/... by number instead.

(My experience is in pure mathematics.)

I'm going to take a different view to most: number only the equations you refer to.

The rationale for numbering anything else (other people, or you yourself in subsequent work, might want to refer to other equations) doesn't make sense to me. In the (IMO highly unlikely) case that you want to refer to a specific equation in another work, it is a terrible idea to make the reader go away and look up what it says; you should always write out the equation in addition to giving the reference. Now it doesn't really matter whether the equation has a number; you can just write "we will use the equation [whatever it says], from [theorem, page or section reference] of [citation]" and readers will be able to find it easily if they want to check it.

• I've written something like the following: "This lemma has almost been proven in [1]. Looking at page 13, the third equation, we can already see that $w < \infty$. Since the problem only appears when $w = \infty$, ...". Taking an formula in the middle of a technical proof and inserting it into another paper does not make much sense, either, and repeating the entire proof would be even more pointless. Jan 23, 2018 at 15:46