In a mathematical document, it is common to have some English prose leading into a mathematical display and continuing after it. The display is some computation or equation, written symbolically, centered and with some space above and below. Technically the display can be part of the same paragraph as the writing before and after it. I have found this can be a bit awkward grammatically especially the frequency with which it happens. By the way I'm a PhD student in the process of writing my thesis so I'm thinking about these kinds of things a lot.
Firstly, it does not feel correct to include the display in a sentence, as in the following example (the content is arbitrary, just some minimal thing to give the idea).
Observing that $$v_1+3v_2-2v_3+8v_4=0,$$ we see that the vectors are linearly dependent.
In fact it seems better to leave out all punctuation from the display and keep the sentences before and after it self-contained. The usual approach is something like the following.
Observe the following equation. $$v_1+3v_2-2v_3+8v_4=0$$ We therefore conclude that the vectors are linearly dependent.
But after a while the expression "the following" starts to feel pretty worn out. You can mix it up by rephrasing so you say "as follows" or something, but there are not many substitutes.
I feel tempted to start writing something like this:
Observe: $$v_1+3v_2-2v_3+8v_4=0$$ We therefore conclude that the vectors are linearly dependent.
This seems very direct but grammatically dubious. Does anyone else worry about this? Any tips for a less repetitive document?
As an interesting aside, when looking for ideas on this I came across the following article on mathematical writing.
It has some good guidelines but does not address how to treat a displayed equation, or how to avoid saying "the following" over and over again. In fact if you do a text search you will see that it uses the phrase "the following" 25 times in a pretty short article.
I personally feel that form is a matter of taste, and if the taste pleases your consumer then you are successful.
Meanwhile, back in the real world, there are some really good resources defining "proper form" for technical math writing.
A well established professor once handed me a copy of "Mathematical Writing," a summary of a course by the same name at Stanford, which he called the de facto standard for form in technical math writing. I strongly recommend that anyone who is attempting to write a technical math paper read through §1 in particular for some really well thought out notes on the subject - including a few points that address your question.
There are other opinions on the matter. For instance, Dr. Kevin Lee at UC Davis teaches his students to write full punctuation in his math sentences, including sentences that have display math on a single line (the \$\$\$\$ sections in LaTeX).