When should each type of vertical bar (pipe) be used?

Question

There are many ways to type a pipe. You could use \$|\$ ($|$), \$\vert\$ ($\vert$), \$\mid\$ ($\mid$), or just a plain | (not surrounded by dollar signs). You could also use vmatrix to indicate matrix determinants.

I wanted to know when is it appropriate to use each type of pipe on Mathematics Stack Exchange. For example, pipes can be used in the following cases:

• To indicate that one integer is a factor (or divisor) of another (e.g. $2|4$)
• To indicate conditions in set notation (e.g. $Dom(\sqrt{x}) = \{x \in \mathbb{R} \mid x \ge 0\}$)
• To indicate absolute value (e.g. $|-2019| = |2019| = 2019$)
• To indicate the cardinality of a set (e.g. $|\emptyset|=0$)
• To indicate the order of an element of a group (e.g. $\forall x \in K_4 ((x=e) \lor (|x|=2))$, where $K_4$ is the Klein four-group)
• To indicate the determinant of a square matrix (e.g. $\begin{vmatrix} 2 & 3\\5 & 7 \end{vmatrix}=-1$)

There is also of course the double pipe symbol ($||$), which is used for logical or in programming, concatenation, and parallel lines; and should not be confused with the number eleven.

Answer 1

Since you asked about some specific meanings when the vertical line appears:

• For divisibility, you use \mid (and \nmid) for "does not divide". For example, $a\mid b$ $a\mid b$, $4\mid8$ $4\mid8$, $4\nmid7$ $4\nmid7$. The advantage over typing just $4|8$ $4|8$ is that \mid yields extra spacing (but some authors prefer $4|8$).
• For conditions in set notation, use again \mid. (However, other symbols are also used for this purpose, not just vertical bar.)
• For absolute value, you can use simply |, for example, $|x+2|+|x-2|=4$ $|x+2|+|x-2|=4$. Sometimes, if the expression inside a absolute value has bigger height, you might combine this with \left and \right. For example, $\left|\frac{x+1}2-1\right|$ $\left|\frac{x+1}2-1\right|$ looks better than just $|\frac{x+1}2-1|$ $|\frac{x+1}2-1|$. You can also use \lvert and \rvert. $\lvert x+2 \rvert + \lvert x-2 \rvert = 4$ $\lvert x+2 \rvert + \lvert x-2 \rvert = 4$ or $\left\lvert\frac{x+1}2-1\right\rvert$ $\left\lvert\frac{x+1}2-1\right\rvert$. You can treat the order of a group or the cardinality of a set in the same way.

• For determinants, you can use vmatrix environment. For example, $$\begin{vmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{vmatrix}$$ is obtained using

  $$\begin{vmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{vmatrix}$$

Although MathJax is different from LaTeX, many things which can be used in LaTeX apply also in MathJax. So if you find some advice on math mode in LaTeX, it is reasonable to try them also here.

See also:

• The section on matrices in MathJax basic tutorial and quick reference.
• Absolute Value Symbols on TeX Stack Exchange.
• \mid, | (vertical bar), \vert, \lvert, \rvert, \divides on Tex Stack Exchange (this was already linked in Jack's answer and quid's comment).
• vertical bar for absolute value and conditional expectation on TeX Stack Exchange; Absolute value size! (\lvert doesn't scale) on TeX Stack Exchange
• a ∣ b and a ∤ b in formulas on TeX Stack Exchange
• How to automatically resize the vertical bar in a set comprehension? on TeX Stack Exchange.

Answer 2

I believe the word "pipe" is used more in computer science than in mathematics.

Unlike the strict syntax requirement in programming languages, how to type out this vertical bar in mathematical writing is mostly about typographical consideration. There are discussions regarding different commands in this question at https://tex.stackexchange.com. Mathematically, it does not really matter.