Usefulness of symbolic devices Each mathematician knows that  good notation or symbolism – which seems to be irrelevant from a purely logical point of view – makes theorems more plausible  and motivates results which would otherwise be overlooked.   Examples abound. Let me only mention such diverse things as decimal notation, $B^A$ for the set of mappings from $A$ to $B$, $\frac{{df}}{{dx}}$ for differentiation, commutative diagrams, umbral calculus, etc. 
I would be interested in a list of examples of the most useful notations or symbolic devices  together with hints for the reason of their usefulness.
 A: Group conjugation: $a^b$ means $b^{-1}ab$, so $(a^b)^c$ is $a^{bc}$ and $(a^b)^{-1}$ is $(a^{-1})^b$.
A: I really like the multi-index notation for derivatives which is quite often found in the theory of PDEs.
$D^\alpha f = \frac{\partial^{\vert\alpha\vert} f}{\partial x_1^{\alpha_1}\cdots\partial x_n^{\alpha_n}}$
Where $\alpha$ is defined as:
$\alpha = (\alpha_1,\cdots,\alpha_n),$
$\vert \alpha\vert = \alpha_1 + \cdots +\alpha_n$
It makes definining Sobolev Spaces (and their corresponding Norms) so much cleaner and straight forward:
$W^{k,p} = \lbrace f\in L^p : D^\alpha f\in L^p \text{ for all } \vert\alpha\vert \le k \rbrace$
A: For certain problems, suppression of a common entity or entry and making it understood by other means.  For a lecture involving manipulation of some algebra (ternary groups, maybe?) a form similar to matrix multiplication was used.  Instead of spelling out the matrices with all the entries, those entries that were zero were omitted, giving a more eye-friendly appearence.  I (and many others, I'm sure) use it for small incidence matrices which are not very dense; the symmetry and other relationships seem much more obvious without the clutter of zeroes.  And this is just one example of many.
Gerhard "Ask Me About System Design" Paseman, 2011.09.13
