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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.

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A related MO question – Andrey Rekalo Sep 13 '11 at 12:43
Community Wiki? – Igor Rivin Sep 13 '11 at 13:28
Is the other MO question essentially a duplicate or is there a different focus here? – Qiaochu Yuan Sep 13 '11 at 16:15

Group conjugation: $a^b$ means $b^{-1}ab$, so $(a^b)^c$ is $a^{bc}$ and $(a^b)^{-1}$ is $(a^{-1})^b$.

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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:

$W^{k,p} = \lbrace f\in L^p : D^\alpha f\in L^p \text{ for all } \vert\alpha\vert \le k \rbrace$

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I was going to mention the differential operator D used in ODEs. So, for example, $Dy=\frac{d}{dx}y$. I originally thought it was a useless notation, until I realized that it greatly simplified things. For example, $(D^4+4D^3+6D^2+4D+5)y=0$ is difficult to solve until you express it as $(D^2+1)(D^2+4D+5)y=0$. – Joel Reyes Noche Sep 14 '11 at 2:39

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

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