How to punctuate a sequence of steps in mathematical text?

Question

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:

• https://tex.stackexchange.com/questions/7542/for-formal-articles-should-a-displayed-equation-be-followed-by-a-punctuation-to
• When writing in math, do you use a comma or colon preceding an equation?

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?

Answer 1

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!

Answer 2

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.