I've just rendered the following:
\documentclass[varwidth=true, border=2pt]{standalone}
\begin{document}$\textbackslash$
\end{document}
which seems to give the same as $n$. Do you know why?
As \textbackslash is for textmode and not for mathmode, I've expected TeX to fail in this situation. Instead, I get
LaTeX Warning: Command \textbackslash invalid in math mode
So why does \textasciitilde in math mode make LaTeX crash but \textbackslash not?
The definition of \textbackslash when the OT1 encoding is used is
\OMS-cmd \textbackslash \OMS\textbackslash
Both the first and the third token can't be typed by a user without some devious trick. However, the definition of \OMS-cmd is equivalent to
\def\OMS-cmd#1#2{%
\ifx\protect\@typeset@protect
\@inmathwarn #1%
\expandafter\ifx\csname\cf@encoding\string#1\endcsname\relax
...<omitted irrelevant things>...
}
The macro \@inmathwarn is responsible for the warning
Command \textbackslash invalid in math mode
After that the macro \OMS\textbackslash is expanded. If we were in text mode, the command would do a font change in a group, selecting a font in the OMS encoding (math symbols), doing
\char"6E
and a backslash would appear, because that's what an OMS encoded font has at slot "6E (hexadecimal).
If we're in math mode, the same \char"6E instruction would be performed, but here the font change has no effect. So we get the character at position "6E in the font in math family 0, where an n is found. When \char"6E is found in math mode, TeX does as if it were \mathchar"006E and in family 0, slot "6E there's an n (it's the upright text font and the ASCII code of n is exactly "6E).
Things are different in the T1 encoding, because in this case the definition of \textbackslash is
\T1-cmd \textbackslash \T1\textbackslash
and the latter macro eventually expands to \char"5C. In a text font, T1 encoding, at slot "5C there's actually a backslash.
Here's the explanation. Now, some advice:
Never underestimate warnings and
never use \text... symbol commands in math mode (don't mistake them with \textrm, \textit or similar, which are obviously legal in math mode).
As mentioned in Ulrike's answer at Teletype \textbackslash in alltt environment, when OT1 encoding is employed, the glyph for \textbackslash is obtained from the symbol font (because there is no glyph for it in the regular font). If you run my MWE and zoom in on the column depicted in the figure (the '6 or ''E column), you will see that the symbol font (cmsy10) backslash character is in slot 110, which happens to be the same slot as the letter n in the italic font (cmti10). Presumably, in math mode, the font employed is redirected to the italic font, and so \char110, which renders as the \textbackslash in the symbol font, instead renders as an italic n in the math font.
This problem can be circumvented by using T1 encoding (\usepackage[T1]{fontenc}), which provides for a text backslash in the regular font. It will render properly, even in math mode.
\documentclass{article}
\usepackage{fullpage}
%\usepackage[T1]{fontenc}
\usepackage{fonttable}
\begin{document}
\fonttable{cmti10}
\fonttable{cmsy10}
\end{document}
