An equation or other mathematical construct is separated from a qualifying expression by a \quad. Examples:

When the qualifying expression is a prepositional phrase it is given standard sentence spacing. Examples:

The first example was typed as (using the equation* environment provided by the amsmath package)

\begin{equation*}
\min_x c^Tx \quad \text{subject to $Ax=b$, $x\ge 0$}.
\end{equation*}

Here, the qualifying phrase is placed inside a \text command, which jumps out of math mode and formats its argument as regular text, with the usual interword spacing in effect, and we re-enter math mode for the conditions. This is better than writing

\min_x c^Tx \quad \text{subject to} ~Ax=b, ~x\ge 0.

with hard spaces. Note that \text is a command from the amsmath package, and it is similar to the LaTeX command \mbox and the TeX command \hbox, both of which work equally well here.

When an equation contains a conjunction such as and or or, the conjunction has a \quad on each side. Examples:

In the second example, one might argue for a \quad before the qualifying “for”, on the basis of case 1, but it I prefer the word spacing. This example was typed as

\begin{equation*}
     a = \sum_{j=1}^n c_j v_j \quad \text{where} \quad
     \text{$c_j = \langle a, u_j\rangle$ for $j=1,2,\dots,n$}.
\end{equation*}

Two or more equations are separated by a \qquad. Examples:

It is important to emphasize that one might diverge from following these (or any other) guidelines, for a variety of reasons. With a complicated display, or if a narrow text width is in use (as with a two-column format), horizontal space may be at a premium so one may need to reduce the spacing. And the guidelines do not cover every possible situation.

My guidelines are the same ones that were used in typesetting the Princeton Companion to Applied Mathematics, and I am grateful to Sam Clark (T&T Productions), copy editor and typesetter of the Companion, for discussions about them. Cases 1 and 3 are recommended in my Handbook of Writing for the Mathematical Sciences (2017).

The SIAM Style Guide (link to PDF) prefers a \qquad in Case 1 and \quad in Case 3 with three or more equations. The AMS Style Guide (link to PDF) has the same guidelines as SIAM. Both SIAM and the AMS allow an author to use just a \quad between an equation an a qualifying expression.

In the TeXbook (1986, p. 166), Knuth advocates using a \qquad between an equation and a qualifying expression.

When we write down mathematical working, perhaps in the course of doing research, we are likely to need to go back and make changes to what we have just written. A fast drying ink helps avoid smudges, and it also means we can start writing on the reverse side of a page without pausing to let the ink dry or using blotting paper. Left-handed writers may always need such an ink, depending on their particular style of writing. My favourite fast-drying ink is Noodler’s Bernanke Blue, which is named after Ben Bernanke, former chairman of the Federal Reserve. Downsides are feathering (where the ink spreads out to create rough edges) and bleeding (where ink soaks through on the other side of the page) on lesser grades of paper.

As mathematicians are major consumers of coffee, what we write risks being spilled on. (Indeed this is so normal an occurrence that Hanno Rein has written a LaTeX package to simulate a coffee stain.) An ink should be reasonably water resistant if spills are not to blur what we have written. Many popular inks have poor water resistance. One of the most water resistant is Noodler’s Bulletproof Black, which also “resists all the known tools of a forger, UV light, UV light wands, bleaches, alcohols, solvent…”. It’s great for writing cheques, as well as mathematics. (But it may not be the blackest black ink, if that matters to you.) Another water resistant ink is Noodler’s Baystate Blue (see below).

Sometimes, writing with an interestingly named pen, paper or ink can provide inspiration. While many Coloverse inks have a space or physics theme, I am not aware of any mathematically named inks.

Vibrant inks are great for annotating a printout or writing on paper with heavy lines, squares or dots. An outstanding ink in this respect is Noodler’s Baystate Blue, an incredibly intense, bright blue that jumps off the page. The downsides are bleeding and staining. An ink to be used with caution! I also like the much better behaved Pilot Iroshizuku Fuyu-gaki (orange) and Pilot Iroshizuku Kon-peki (blue).

For marking scripts one obviously needs to use a different colour than the student has used. Traditionally, this is red. A couple of my favourites for marking are Waterman Audacious Red and Cult Pens Deep Dark Orange.

Many inks have one or more of the properties of shading, sheen and shimmer. Shading is where an ink appears darker or lighter in different parts of a stroke; sheen is where an ink shines with a different colour; and shimmer is where particles have been added to the ink that cause it to shimmer or glisten when it catches the light. The strength of all these effects is strongly dependent on the pen, the nib size and the paper, and for shimmer the pen needs to be moved around to distribute the particles before writing. (Sheen and shimmer do not show up well on the scans shown in this post.) Among my favourites are Diamine Majestic Blue (blue with purple sheen), Robert Oster Fire and Ice (teal with shading and pink sheen) and Diamine Cocoa Shimmer (brown with gold shimmer). For “monster sheen”, try Organic Studios Nitrogen Royal Blue, but note that it is hard to clean out of a pen.

Much mathematics is written on bar mats, napkins, paper towels, backs of envelopes, newspapers, tablecloths, and anything that is to hand, especially when mathematicians work together away from their desks. These surfaces are not fountain-pen friendly. One promising ink for such situations is Noodler’s X-Feather, which was created to combat feathering on cheap paper. (I have not tried this ink.)

Finally, three good all-round inks that I’ve used a lot when working on my books and papers are Diamine Asa Blue, Diamine Oxford Blue, and Diamine Red Dragon.

The first example is from proofs of one of my recent papers:

The article had been submitted as LaTeX source and it was reasonable to assume that the only differences between the proofs and what we submitted would be in places where a copy editor had imposed the journal style or had spotted a grammatical error. Fortunately, I know from experience not to make that assumption. These two sentences contain two errors introduced during copy editing: the term “Anderson acceleration” has been deleted after “To apply”, and “We denote by unvec” has been changed to “We denote by vec” (making the sentence nonsensical). The moral is never to assume that egregious errors have not been introduced: check everything in journal proofs.

In a similar vein, consider this extract from another set of proofs:

There is nothing wrong with the words or equations. The problem is that an unwanted paragraph break has been inserted after equation (2.6), and indeed also before “Only”. This set of proofs contained numerous unwanted added new paragraphs.

Here is an extract from the proofs of my recent SIAM Review paper (with Natasa Strabic and Vedran Sego) Restoring Definiteness via Shrinking, with an Application to Correlation Matrices with a Fixed Block:

We noticed that the word “how” appears at the end of a line four times within seven lines—an unfortunate coincidence. We suggested that the production editor insert a hard space in the LaTeX source between one or more of the hows and the following word in order to force different line breaks. Here is the result as published:

What’s wrong with this example, from a paper in the 1980s?

The phrase “best unknown” should be “best known”!

The next example is from a book:

At first sight there is nothing wrong. But the is suspicious: why , and why is this term that depends only on inside the integral? It turns out that the equation should read

When you realize that the left parenthesis and the digit share the same key on the keyboard you can start to see how the error might have been made at the typing stage.

The final example (from a 2013 issue of Private Eye) is completely different and illustrates a rare phenomenon:

If you cannot see anything wrong after a minute or so, click here. This phenomenon, whereby white spaces in successive lines join up to make a snake, is known as rivers of white. The fix, as in Example 2, is to force different line breaks.