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How to get the closing single quote in math mode. The apostrophe in math mode is displayed as prime.

\documentclass{article}                                                              
\begin{document}                                                                     
Why can't I get $`a'$ in math mode.                                                  
\end{document}  
11

You should switch to text-mode, eg.

\documentclass{article}
\begin{document}
Why can't I get $`a\textrm'$ in math mode
\end{document}
9

Punctuation that is not part of a formula should go outside the formula:

When \(a>0\), we say that \(a\) is \emph{positive}.

In the interval \([0,1]\) there are infinitely many rational numbers.

It's quite rare that quotes are used as math symbols, but I've seen them in some textbook on mathematical logic. Since ' is used for denoting a prime (dash) in math, one can resort to a different trick: the lines

\DeclareMathSymbol{\mlq}{\mathord}{operators}{``}
\DeclareMathSymbol{\mrq}{\mathord}{operators}{`'}

will define two commands that will give left and right quotes in math and your input can become

Why can't I get $\mlq a\mrq$ in math mode.

but in this case `$a$' should be probably better.

I've added a companion to \mrq just for symmetry and, perhaps, greater clarity.

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