+\newcommand*{\pair}[2]{ \left({#1},{#2}\right) }
+
+% A triple of things.
+\newcommand*{\triple}[3]{ \left({#1},{#2},{#3}\right) }
+
+% A four-tuple of things.
+\newcommand*{\quadruple}[4]{ \left({#1},{#2},{#3},{#4}\right) }
+
+% A five-tuple of things.
+\newcommand*{\quintuple}[5]{ \left({#1},{#2},{#3},{#4},{#5}\right) }
+
+% A six-tuple of things.
+\newcommand*{\sextuple}[6]{ \left({#1},{#2},{#3},{#4},{#5},{#6}\right) }
+
+% A seven-tuple of things.
+\newcommand*{\septuple}[7]{ \left({#1},{#2},{#3},{#4},{#5},{#6},{#7}\right) }
+
+% The Cartesian product of two things.
+\newcommand*{\cartprod}[2]{ {#1}\times{#2} }
+
+% The Cartesian product of three things.
+\newcommand*{\cartprodthree}[3]{ \cartprod{{#1}}{\cartprod{{#2}}{{#3}}} }
+
+% The direct sum of two things.
+\newcommand*{\directsum}[2]{ {#1}\oplus{#2} }
+
+% The direct sum of three things.
+\newcommand*{\directsumthree}[3]{ \directsum{#1}{\directsum{#2}{#3}} }
+
+% The factorial operator.
+\newcommand*{\factorial}[1]{ {#1}! }
+
+%
+% Product spaces
+%
+% All of the product spaces (for example, R^n) that follow default to
+% an exponent of ``n'', but that exponent can be changed by providing
+% it as an optional argument. If the exponent given is ``1'', then it
+% will be omitted entirely.
+%
+
+% The natural n-space, N x N x N x ... x N.
+\newcommand*{\Nn}[1][n]{
+ \mathbb{N}\if\detokenize{#1}\detokenize{1}{}\else^{#1}\fi
+}
+
+\ifdefined\newglossaryentry
+ \newglossaryentry{N}{
+ name={\ensuremath{\Nn[1]}},
+ description={the set of natural numbers},
+ sort=N
+ }
+\fi