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What is the right way to punctuate a sequence of steps that lead to a solution?

For example, consider this 3-step solution that solves the equation $ (x^2 - 4) = 0 $.

\begin{align*} x^2 - 4 & = 0 \\ (x + 2)(x - 2) & = 0 \\ x & \in \{2, -2\}. \\ \end{align*}

Is the manner in which the above steps are written correct or does it need to be punctuated more appropriately?

What bothers me in the above example is that the above steps do not read like a complete English sentence, but rather sentence fragments arranged vertically.

Discussions in the following URLs seem to make it pretty clear that mathematical text should read like complete and grammatically correct English sentences:

But my example above seems to violate this principle. Consider this, would the following form be a better style?

\begin{align*} x^2 - 4 = 0 & \implies (x + 2)(x - 2) = 0 \\ & \implies x \in \{2, -2\}. \end{align*}

What is the correct and popular way of writing such mathematical steps?

Lone Learner
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    whatever you choose, just be consistent – Dando18 Nov 07 '17 at 03:41
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    Depends totally on the intended audience. Is this a research-level thing, or something for students / potential learners - and if the latter, what level? – Aaron Montgomery Nov 07 '17 at 03:59
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    I'm with dando, just wrote normally. And stay consistent. I would think about your audience and also the calculation/steps being discussed.if it is very mechanical, strings of eq. Is fine Imo, but if it is profound or meaningful or requires thinking, words are easier to grasp. Ultimately, (Imo) you should be thinking of readability. Can an arb. researcher pick up your paper and understand the proofs after some work? (the extreme bad example being abc conj) – mdave16 Nov 07 '17 at 04:01
  • The right way is following the style guide for whatever journal you are intended to submit for publication. If it is not for publication, anything sensible is fine. – achille hui Nov 07 '17 at 04:22
  • @achillehui There is no style guide. I am writing the text for arbitrary audience. For example, I may have the text in a PDF that may be hosted in a blog or it could be in an answer I post in this website. The intention of this question is to improve my general mathematics writing style. – Lone Learner Nov 07 '17 at 04:35
  • A related question: Best practice for the notation of conditions in cases: 'if' vs 'for' vs ',' – Lone Learner Sep 11 '18 at 05:09

2 Answers2

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I'm really glad that the top example bothers you - it bothers me too. Unfortunately I think that we are in a fairly small minority. Many people seem to think it's OK (maybe, even, preferable) to write mathematics in an impenetrable way. Perhaps I am just way too cynical, but I sometimes think this is an instance of the syllogism

you don't understand what I wrote, therefore I am smarter than you

which is surely not valid. Those of us who believe that clear writing and thorough explanations are vital parts of mathematics just need to keep on setting an example against this kind of thing.

Anyway... <end of rant>... back to the point... IMHO the way you put it at the end is definitely better than the first, though possibly even better still would be to use words: $$x^2-4=0\ ,\quad\hbox{so}\quad (x+2)(x-2)=0\quad\hbox{and therefore}\quad x=2\ \hbox{or}\ x=-2\ .$$ Actually, despite what I said above, I could possibly be convinced that this is overkill - after all, it is a very simple example. On the other hand, anyone who neglects or refuses to write properly in simple cases is unlikely to explain themselves clearly in more extended contexts where the writing becomes even more important.

I would also suggest that the set notation in your final line $x\in\{2,-2\}$ is rather unnecessary, and slightly obscures the point that we are doing algebra here (though once again, this is a simple example - I might not say the same in a different case). Finally, the three parts of your statement are actually all equivalent (the logic goes both ways, not just left to right), and it might be good to make this clear, if it is something that's important for your argument.

Hang in there, keep writing as well as you can, and don't let anyone tell you that you're wasting your time! Good luck!

David
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  • When you say, "the three parts of your statement are actually all equivalent", are you suggesting that I write my example in the following manner? \begin{align} x^2 - 4 = 0 & \iff (x + 2)(x - 2) = 0 \ & \iff x = 2 \text{ or } x = -2. \end{align} – Lone Learner Nov 07 '17 at 04:29
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    This reminds me of my first Linear Algebra course. I distinctly remember the professor muttering things like "This is a sentence, so it deserves a period" when he would write full sentences interspersed with mathematics. In fact, I remember that --- that there's a style to communicating mathematics --- more than I remember learning any particular linear algebra thing! My point here is that, while students like me probably were and are rare, we do teach others extra-mathematical things, subconsciously, and it is worth paying attention to these things. – pjs36 Nov 07 '17 at 04:42
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    Yes - though only if it's important to your argument. In this case it probably is, though I could imagine an example where the right-to-left logic would be irrelevant and could therefore be omitted. – David Nov 07 '17 at 04:44
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    @pjs36 I often tell my students that every sentence must begin with a capital letter and end with a full stop. They usually laugh. I think (am I just kidding myself?) that it's not a laugh of derision or disagreement, but merely of surprise at hearing in mathematics class something that they never expected to hear again after their last English class. – David Nov 07 '17 at 04:51
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At school, mathematics is taught as "doing calculations" or "working things out". Little thought is given to style: what is jotted down in the notebook or on the blackboard is a minimal record of the steps in calculation. The logic underlying these steps is considered a matter of routine that does not need to be written out explicitly.

The situation is quite different in published mathematics—the more advanced, the more so. The basic idea to get here is that mathematical statements are essentially statements in natural language (e.g. English). Thus "$2+2=4$" is a symbolic form of the statement "The sum of two and two is four". In addition, the logical structure of the exposition is as much the real meat of the work as are the details of the calculation. Generally, mathematical statements are statements in natural language in which symbolic expressions and statements are embedded. As such, they need to be properly written according to the stylistic conventions of natural language, for clarity and ease of reading and understanding.

The case of private communication—say from you to your teacher or examiner—falls between these cases. Great clarity of exposition is not needed, because the reader already knows the material better than you do, and would not normally demand that you spend the time that such clarity would take. That said, writing mathematics well (as opposed to simply compiling a list of necessary steps) does leave a good impression and would do your reputation no harm at all. Also, if you are going to proceed far in mathematics, to the extent of writing a thesis or material for publication, you will need to learn this skill at some stage.

John Bentin
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