It is possible to write \frac 2 4 or \frac \pi 4 to get 2/4 and \pi/4, respectively.
However, it is not possible to write \frac \sqrt{2} 2 to get \sqrt{2}/2.
Are there consistent rules on how to write shorthand \frac?
[Tex/LaTex] About shorthand \frac
fractionsmacros
Related Solutions
It works if you switch the shorthand on before defining the command, or if you define the command after \begin{document} when shorthands are active.
As Joseph commented, it's recommendable to switch it off immediately afterwards, so that the active shorthand doesn't affect other commands.
\documentclass{article}
\usepackage[ngerman]{babel}
\shorthandon{"}
\newcommand{\foo}{"|}
\shorthandoff{"}
\begin{document}
ff
f"|f
f\foo f
\end{document}
Alternatively, you could switch it on locally within the macro definition, similar as written here by egreg, and made robust, so it can be used within moving arguments, such as \chapter and \section headings and captions.
\DeclareRobustCommand{\foo}{\shorthandon{"}\scantokens{"|\endinput}}
However, it might be useful to have a part in the preamble where shorthands are active, as in the first solution, so you can use them in your commands such as in text arguments to macros.
The author is referring to a series of commands he has defined and that are part od the file Commons.sty (the file can be found following a link in the site you linked to in your question); they basically are shortcuts allowing you to write fractions and matrices with or without delimiters; in the following document I've extracted the definitions from Commons.sty and provide an example of their use:
\documentclass{article}
\usepackage{amsmath}
\usepackage{xfrac}
\newcommand{\pa}[1]{\left(#1\right)} % encloses the argument using stretchable parentheses
\newcommand{\bra}[1]{\left[#1\right]} % encloses the argument using stretchable square brackets
% matrices for displayed expressions
\newcommand{\mat}[1]{\begin{matrix}#1\end{matrix}} % no delimiters
\newcommand{\pmat}[1]{\pa{\mat{#1}}} % parentheses as delimiters
\newcommand{\bmat}[1]{\bra{\mat{#1}}} % square brackets as delimiters
% variations of \frac and \sfrac
\newcommand{\pfrac}[2]{\pa{\frac{#1}{#2}}} % enclosed in parentheses
\newcommand{\bfrac}[2]{\bra{\frac{#1}{#2}}} % enclosed in square brackets
\newcommand{\psfrac}[2]{\pa{\sfrac{#1}{#2}}} % sfrac enclosed in parentheses
\newcommand{\bsfrac}[2]{\bra{\sfrac{#1}{#2}}} % sfrac enclosed in square brackets
% for small matrices to be used in in-line expressions
\newcommand{\sm}[1]{\begin{smallmatrix}#1\end{smallmatrix}} % no delimiters
\newcommand{\psm}[1]{\pa{\sm{#1}}} % parentheses as delimiters
\newcommand{\bsm}[1]{\bra{\sm{#1}}} % square brackets as delimiters
\begin{document}
\[
\pfrac{1}{2}\quad
\bfrac{1}{2}\quad
\psfrac{1}{2}\quad
\bsfrac{1}{2}\quad
\mat{1 & 0 & 1 \\ 0 & 0 & 1 \\ 1 & 1 & 0}\quad
\pmat{1 & 0 & 1 \\ 0 & 0 & 1 \\ 1 & 1 & 0}\quad
\bmat{1 & 0 & 1 \\ 0 & 0 & 1 \\ 1 & 1 & 0}\quad
\]
$\sm{1 & 0 \\ 1 & 0}\quad\psm{1 & 0 \\ 1 & 0}\quad\bsm{1 & 0 \\ 1 & 0}$
\end{document}
Of course, if you save the file Commons.sty in a convenient place (your local tree, for example) where TeX can find it, you can load the package and directly use the commands, as in:
\documentclass{article}
\usepackage{Commons}
\begin{document}
\[
\pfrac{1}{2}\quad
\bfrac{1}{2}\quad
\psfrac{1}{2}\quad
\bsfrac{1}{2}\quad
\mat{1 & 0 & 1 \\ 0 & 0 & 1 \\ 1 & 1 & 0}\quad
\pmat{1 & 0 & 1 \\ 0 & 0 & 1 \\ 1 & 1 & 0}\quad
\bmat{1 & 0 & 1 \\ 0 & 0 & 1 \\ 1 & 1 & 0}\quad
\]
$\sm{1 & 0 \\ 1 & 0}\quad\psm{1 & 0 \\ 1 & 0}\quad\bsm{1 & 0 \\ 1 & 0}$
\end{document}
As a side note, I would have defined \pmat and \bmat using directly bmatrix and pmatrix as provided by amsmath.
Best Answer
The command
\frachas two mandatory arguments. So you can they the first two tokens are the arguments (or simple the first two groups).\frac \sqrt{2} 2fails because\sqrtis the first token,{2}the second one.