# Suggestions for good notation

I occasionally come across a new piece of notation so good that it makes life easier by giving a better way to look at something. Some examples:

• Iverson introduced the notation [X] to mean 1 if X is true and 0 otherwise; so for example Σ1≤n<x [n prime] is the number of primes less than x, and the unmemorable and confusing Kronecker delta function δn becomes [n=0]. (A similar convention is used in the C programming language.)

• The function taking x to x sin(x) can be denoted by x ↦ x sin(x). This has the same meaning as the lambda calculus notation λx.x sin(x) but seems easier to understand and use, and is less confusing than the usual convention of just writing x sin(x), which is ambiguous: it could also stand for a number.

• I find calculations with Homs and ⊗ easier to follow if I write Hom(A,B) as A→B. Similarly writing BA for the set of functions from A to B is really confusing, and I find it much easier to write this set as A→B.

• Conway's notation for orbifolds almost trivializes the classification of wallpaper groups.

Has anyone come across any more similar examples of good notation that should be better known? (Excluding standard well known examples such as commutative diagrams, Hindu-Arabic numerals, etc.)

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this question is broadly useful, so perhaps better as community wiki? – Suvrit Oct 20 '10 at 20:09
In set theory we write ${}^B A$ for the set of functions from $B$ to $A$. – Andrés E. Caicedo Oct 20 '10 at 20:54
I've always assumed that the notation $A^B$ is because of the "exponential law" $(A^B)^C = A^{B\times C}$ ... – Kevin H. Lin Oct 20 '10 at 23:18
Arabic numerals ? Ah yes, they were transmitted to Europe by the Arabs. – Chandan Singh Dalawat Oct 21 '10 at 3:29
Isn't $x \mapsto f(x)$ commonplace? As for homomorphisms, they are not simply maps, and $\mathrm{Hom}(A, B)$ denotes the whole class, while $A \to B$ denotes a single mapping. – Alexei Averchenko Oct 21 '10 at 3:38

Diagrammatic notation for tensors (Penrose diagrams, birdtracks, etc.). It makes many things like the invariance of tr(A B C) under cyclic permutation into empty statements.

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Dirac's bra-ket notation

This notation is very useful when applying the Hilbert spaces in Quantum Theory. It exploits some properties of duality, eigenvalues/eigenvectors, projectors and self-adjoint operators. In mathematics, perhaps it is difficult to adopt, because mathematicians are using notations that are more general, and cannot exploit these particularities. But if you know Hilbert spaces you can learn this notation in one minute, and then it makes visible many of these nice properties.

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I have to strongly disagree with the use many physicists do of the Dirac bra-ket notation, as I'm going to explain: mathematicians would use the $\langle\, |\, \rangle$ as a function with the "blank places" to be saturated by its 2 argument (or even $\langle\, | \, | \, \rangle$ as a function to be saturated by 3 arguments, the one in the middle being an operator as in $\langle \xi | \, A | \,\eta \rangle$) [continued] – Qfwfq Oct 22 '10 at 13:33
[continued] On the contrary, I have seen many physicists use the highly "dis-functional" notation $| \, \lambda \rangle$ to denote an eigenvector with $\lambda$ as eigenvalue, and so write things like: $A | \lambda \rangle=\lambda\cdot | \lambda \rangle$ – Qfwfq Oct 22 '10 at 13:37
There are other examples of notations which don't represent the missing arguments by blanks. One is the (abstract) index notation. An index which appears repeated in covariant and contravariant positions means contraction, and if it is not repeated, it represents a free slot that can be filled by contracting with another free slot of another tensor. The blank to be filled by an argument is not visible, but we know that a free index can do this. Another example is the Polish notation, where again we don't see the missing arguments as blanks. – Cristi Stoica Oct 23 '10 at 9:46
@unknown (google). On the other hand, I like $|\lambda\rangle$ as eigenvector only if it is stated that $\lambda$ has multiplicity 1, otherwise is confusing. When multiplicity is 1, the confusion is avoided because in fact $|\lambda\rangle$ is a ray in the complex projective space. – Cristi Stoica Oct 23 '10 at 10:11
There was a long, long discussion at the n-Category Cafe one time, started by how offensive "bra" was to someone (speaking as a woman). – Todd Trimble Oct 15 '12 at 19:18

$$a^{\cdot \, n} = a\cdot a\cdots a$$ $$a^{\wedge \, n} = a\wedge a\wedge\dots\wedge a$$ $$a^{\,, \, n} = a,a,\dots,a$$ For example one could write $$\langle(x+10y-z)^{\,, \, 2}\rangle= \langle(x+10y-z),(x+10y-z)\rangle.$$ or $$\sin^{\circ(-1)}x=\arcsin x$$ or $$\sin^{\cdot(-1)}x=\frac1{\sin x}$$

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In contrast, the insistence of Calculus textbooks to use $sin^{-1}$ for inverse or arc sine has baffled me. It throws even my best students off as being some kind of reciprocal, especially since we are already stuck with the idiosyncratic notation for powers like $sin^2(x)$ – Hans Schoutens Jan 24 '13 at 22:15
Sorry, should have read on and posted this under Blake's answer below. – Hans Schoutens Jan 24 '13 at 22:20
That's why you shouldn't write $\sin^2(x)$, but $\sin(x)^2$. Also, one can mix up $\sin^2(x)$ with $\sin\sin(x)$. – Turion Mar 25 '14 at 16:41
The notion $f^{\circ 2}(x)$ for $f(f(x))$ is, while not common, pretty well-established in a number of references where there might be confusion with other uses. – Steven Stadnicki Dec 16 '14 at 5:04
@Turion, I have always used $\sin(x)^2$ for the reason you say, but then a possible confusion was pointed out to me: what does $\sin(x + 1)^2$ mean? Having to write $\bigl(\sin(x + 1)\bigr)^2$ is pretty awful. (To be fair, the complainer's solution of using $(\sin x)^2$ doesn't address the ambiguity either.) – L Spice May 18 '15 at 19:25

The universal property of the univariate polynomial ring: For any commutative ring $A$, any commutative $A$-algebra $B$ and any $x\in B$, there exists one and only one $A$-algebra homomorphism from the polynomial ring $A\left[X\right]$ to $B$ which maps $X$ to $x$.

This is the so-called evaluation homomorphism at $x$. I denote this homomorphism by $\lim\limits_{X\to x}$. This has the advantage that we have $\lim\limits_{X\to 0}\dfrac{\left(X+1\right)^n-1}{X}=n$ and similar properties hold just as in classical analysis. The polynomial $\dfrac{\left(X+1\right)^n-1}{X}$ is well-defined (since $X$ is not a zero divisor in $A\left[X\right]$ and divides $\left(X+1\right)^n-1$), but if we would blindly replace $X$ by $0$ we would obtain a $\dfrac{0}{0}$ error.

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Ah, I really like calculus-inspired notation for algebraic stuff, just like the integral for ends. Looking forward to see more in this spirit! – Konrad Voelkel Dec 19 '11 at 10:54

I like $A^{\text{H}}$ for the conjugate transpose of the matrix $A$, ananlogously to how $A^{\text{T}}$ and $A^{\text{C}}$ means the transpose and the conjugate. You call it the Hermitian of the matrix for short. I learnt this notation from Rózsa Pál, but I can't tell who invented it.

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I use simply $A^\ast$ for this but I also saw $A^\dagger$ for both transpose and conjugate. – Marcel Bischoff Dec 18 '10 at 13:10
I also like $A^{-H}$ for $(A^H)^{-1}=(A^{-1})^H$. Often handy, but I suggest you not to try $A^{2H}$ or $A^{\frac12 H}$ (exercise: why is the latter ill-defined?) – Federico Poloni Dec 19 '11 at 10:49

A) Two notations I love are the rising factorial $x^\overline n$ and its falling factorial twin $x^\underline n$. They are used and advocated in the great book see http://en.wikipedia.org/wiki/Concrete_Mathematics . In passing this book uses great notations.

B) A general trick with binomials to reuse them with sets instead of numbers, here are some typical examples.

1) $\binom S k$ to denote the set of all $k$-sets of the base set $S$ .

2) $S^\underline 2$ to denote the pairs $(x,y)$ of $S$ where $x$ and $y$ are different.

3) $S^\underline k$ to denote the $k$- uplets of $S$ (each uplet has $k$ different elements).

C) Another notation I find useful when listing some (big) families of examples in a combinatorial setting. Use as variables the very numerals $1$ $2$ .. themselves instead of $x_1$ , $x_2$ ... . For example ( very untelling because too small an example) : the intersection of $123$ and $34$ is $3$.

D) I also often use {{ a,a,b,c}} for multiset. Any other standard or suggestion (or a way to avoid speaking about multiset) is welcome.

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D) Perhaps {$(a)^2, b, c$} for your multiset {{a,a,b,c}} – Qfwfq Sep 24 '11 at 21:04
for unknowngoogle : this is even better without the parenthesis around a , yet it does not work with variables ${{a1, a2}}$ where you don't know if $a1$ equals $a2$. – Jérôme JEAN-CHARLES Sep 28 '11 at 11:54
Perhaps $2a+b+c$ for your multiset: a 0-cycle. – Ben McKay Dec 1 '13 at 10:12
@BenMcKay, I agree, since a (finitary) multiset on the symbols $X$ is just an element of the free Abelian monoid generated by $X$, which thereby becomes a "module over the semiring $\mathbb{N}$" in a canonical way. – goblin May 6 '14 at 7:37

I recommend the notation $$a \equiv_n b$$ in place of $a \equiv b \pmod{n}$. It's much less verbose. The meaning is clear. And the $n$ is where it really belongs, next to the $\equiv$ it is describing.

We're stuck with $a \equiv b \pmod{n}$ as the standard notation (for now!), because that's what Gauss came up with. I've got nothing against Gauss for not using a subscript $\equiv_n$. It seems to me that Disquisitiones Arithmeticae doesn't have subscripts anywhere. Subscripts must have been outside the graphic design space or something. So I don't blame him for resorting to $a \equiv b \pmod{n}$. Gauss did a great thing by popularizing $n \mid a-b$ as an equivalence relation of $a$ and $b$. But if we were to invent the notation today, I dare say $a \equiv_n b$ would be the modern choice. (See this post on Math.SE, where Alexander Gruber suggests the same thing in a comment.)

Of course, if there's no ambiguity, you can still just write plain $a \equiv b$. I'm talking about the cases where you need to or want to indicate the modulus $n$. It may not seem like much, but "(mod n)" is surprisingly verbose to physically write. If you're hand-writing pages or blackboards full of congruences, chances are you've already succumbed to abbreviating "(mod n)" somehow. I've seen lots of different shorthand, based on dropping the parenthesis, or some or all of the text "mod" (which is itself an abbreviation of "modulo", or if you really go by Gauss's Latin, "secundum modulum" - be thankful you're not writing that): \begin{align} a &\equiv b \quad \mathrm{mod}\ n \\ a &\equiv b \quad (\mathrm{m}\ n) \\ a &\equiv b \quad (n) \\ \end{align}

I've seen all of these used before, as well as $a \equiv_n b$. Certainly $a \equiv_n b$ is the cleanest notation.

As a free bonus, you get a cool-looking Fermat's theorem: $$a^p\!\equiv_p\!a.$$

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this notation has the advantage that it chains nicely: although we all know what it means, there is (I think) no logical sense in which $a \equiv b \equiv c \pmod n$ is the result of concatenating $a \equiv b \pmod n$ and $b \equiv c \pmod n$; whereas $a \equiv_n b \equiv_n c$ clearly is. One drawback: It may lead to useless chains like $1 \equiv_2 5 \equiv_3 2$. – L Spice May 18 '15 at 19:31

If one needs to denote the fiber (not the stalk which is standardly denoted $\mathcal{F}_{x}$) of a sheaf $\mathcal{F}$ at the closed point $x$ of the $\Bbbk$-scheme $X$, one can write

$\mathcal{F}\mid_{x}$

After all, the fiber $\mathcal{F}\otimes_{\Bbbk}\;\kappa (x)$ is the restriction (pullback) of $\mathcal{F}$ to the point $x:\rm{Spec}\;\Bbbk\rightarrow X$.

The problem is that, when you identify vector bundles with locally free sheaves, the above notation clatches with the usual notation $E_x$ for the fiber of vector bundles. On the other hand almost always the context would be sufficient to clarify which of the two notations is being used.

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In Ravi Vakil's thread on teaching schemes, there was some discussion of using the notation $\mathcal{F}(x)$, since given a section $f$ of the sheaf, its values as a function on $X$ are written $f(x)$ and lie in the fibers. – Ryan Reich Oct 21 '10 at 6:38
It's this answer, and the many, many comments below it: mathoverflow.net/questions/28496/…. Note that what I wrote is given by BCnrd about halfway down. – Ryan Reich Oct 21 '10 at 6:48
@Ryan: I appreciate the $\mathcal{F}(x)$ notation, but there's a (minor?) clutching with the widely used notation $\mathcal{O}_{X}(x)$ to denote the line bundle on the algebraic curve $X$ twisted by the divisor given by the point $x\in X$. – Qfwfq Oct 21 '10 at 11:44
It's true, this notation would lead to the unfortunate equality $\mathcal{O}_X(x) = \mathcal{O}_X/\mathcal{O}_X(-x)$. – Ryan Reich Oct 21 '10 at 16:22

The three-dot notation $f\mathrel{\scriptsize\vdots}A\to B$ to indicate that $f$ is a partial function from $A$ to $B$, meaning that $\text{dom}(f)\subseteq A$ rather than $\text{dom}(f)=A$. Partial functions are pervasive in logic, especially computability theory and set theory, and this notation is both compact and suggestive.

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Joel, do you know who introduced it? I've heard the rumor it was Mathias, but I'm not sure. – Andrés E. Caicedo Oct 20 '10 at 20:52
I use $\rightharpoonup$ for partial functions. – Andreas Blass Oct 20 '10 at 22:23
Use a broken arrow instead: f : A - - - > B. I have no idea how three dots is suggestive of the domain being something smaller than A. – KConrad Oct 21 '10 at 18:15
I just meant that it suggests that $f$ is something like a function from A to B, without being intrusive. This notation is really useful in situations where you have numerous partial functions of different arities running around. – Joel David Hamkins Oct 21 '10 at 18:22
Never saw the three-dot notation. It looks like a smudge or tiny dead gnat to me. I use the notation mentioned by Andreas. – Todd Trimble Oct 30 '10 at 18:28

The ever-controversial reverse Polish notation for functions: $f(x) = xf$. Thus in composition, the order makes sense: $(g \circ f)(x) = x f g$ (this point is moot for the fortunate Hebrew- and Arabic-speaking mathematicians). I hate this notation in practice but I can't deny that it is objectively right and "just makes sense" in more or less the same way that the original post discusses writing $B^A = A \to B$. Please no one vote this up.

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One partial compromise is to subscript a function $f: X \to Y$ as $f_{Y \leftarrow X}$. For instance, the composition of two inclusion maps $\iota_{Z \leftarrow Y} \circ \iota_{Y \leftarrow X}$ becomes $\iota_{Z \leftarrow X}$. – Terry Tao Oct 21 '10 at 17:11
Hehe, `objectively right'. – Greg Muller Oct 21 '10 at 18:42
@Terry Tao: I've seen that one from time to time, and it is pretty nice when you are dealing with different spaces (especially when you have a category of them and, as you write, a functorial assignment of maps). For self-maps of X, it leaves...something to be desired. – Ryan Reich Oct 21 '10 at 20:36
Abstractly an arrow from $X$ to $Y$ has and an orientation but no direction. It could be drawn either up/down/right/left/slant 30°/slant-45°/... . But we are heavy victims of the typographical habits, and this in spite of modern computer possibilities (not means :Tex being a terrible tool!). The best thing to do is to think of an arrow as slanted +200°or in three dimensions, then the typographical induction tends to disappear completely (at least for me and I guess for Hebrew/Arabic writers too). So the motto (not necessarily a categorist's one) is "let's do multidimensional algebra!". – Jérôme JEAN-CHARLES Nov 23 '10 at 23:50
• I also like the notation $x \prec y$ to denote majorization of a vector $x$ by a vector $y$; once defined, this notation relieves quite lot of burden.

• On a related note, I also prefer the notation $A \succeq 0$ to signify that $A$ is a positive semidefinite matrix (some prefer to use the perhaps "more natural" $A \ge 0$, but since I frequently deal with nonnegative matrices, the $\ge$ is out)

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Unfortunately, "majorization" has at least two different meanings... – darij grinberg Oct 22 '10 at 10:45
Majorization theory involves also the useful notation $x_\uparrow$ and $x_\downarrow$ for vectors with real entries. – Denis Serre May 18 '11 at 15:14

All of the notations created to simplify writing category theory. For instance, the idea of drawing a circular arrow inside of a diagram to indicate that that diagram is commutative. As well as the idea of putting an angle in the top left or bottom right of a square diagram to indicate that it is a pushout or pullback. And finally, the notation of augmenting any of these notations with $\simeq$ to indicate that the diagram is only "up to homotopy".

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If $\mathcal{C}$ is a category and $X,Y\in\mathrm{obj}(\mathcal{C})$, I like the notation $\mathcal{C}(X,Y)$ to denote $\mathrm{Hom}_{\mathcal{C}}(X,Y)$.

So, $\mathcal{C}(X,X)=\mathrm{End}_{\mathcal{C}}(X)$.

What do you think of the notation $\mathcal{C}(X):=\mathrm{Aut}_{\mathcal{C}}(X)$ ?

This would be consistent with the notation (or similar notations) $\mathsf{DIFF}(S^1)$ (resp. $\mathsf{TOP}(S^1)$ ) for diffeomorphisms (resp. homeomorphisms) of the circle, i.e. the $\mathrm{Aut}$ in the category $\mathsf{DIFF}$ of smooth manifolds (resp. $\mathsf{TOP}$ of topological manifolds), sometimes used in topology (see e.g. here and here. And (see e.g. here) $\mathsf{TOP}(n)=\mathrm{Aut}_{\mathsf{TOP}}(\mathbb{R}^n)$.

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Is there any particular reason you've chosen to make your links single letters, instead of pointing them out more explicitly? – S. Carnahan Dec 10 '12 at 3:25
@S. Carnahan: not really... I just wanted the links to be kind of parenthetic references, because those MO pages weren't "standard" definitory places for the notations. If it's considered weird of annoying according to MO netiquette, I'll just edit :) – Qfwfq Dec 11 '12 at 14:09
I'm using these notations privately already, especially the $\mathcal{C}(X)$ one and I didn't know there was anyone else doing it! I'd upvote twice if I could! – Turion Mar 25 '14 at 16:44

In commutative algebra with many variables, repeating lists of variables in polynomial arguments and various rings gets very tedious. I suggest using $X_{1..n}$ instead of $X_1,\ldots,X_n$.

Here's an excerpt from Bourbaki's Commutative Algebra, page 222:

For every formal power series $f\in A[[X_1,\ldots,X_n]]$, $$f(X_1,\ldots,X_n)-f(Y_1,\ldots,Y_n)=\sum_{i=1}^n (X_i-Y_i)h_i(X_1,\ldots,X_n,Y_1,\ldots,Y_n)$$ where the $h_i$ belong to $A[[X_1,\ldots,X_n,Y_1,\ldots,Y_n]]$.

And here's how it looks with my suggested notation:

For every formal power series $f \in A[[X_{1..n}]]$, $$f(X_{1..n})-f(Y_{1..n})=\sum_{i=1}^n (x_i-Y_i)h_i(X_{1..n},Y_{1..n})$$ where the $h_i$ belong to $A[[X_{1..n},Y_{1..n}]]$.

I think this notation is maximally succinct, and helps a reader from getting lost in long lists of variables.

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I can see that being handy when one needs to call out some of the indices. For your example from Bourbaki, I would sooner use X decorated with a hat or overbar to indicate a tuple. Gerhard "That May Just Be Me" Paseman, 2013.02.03 – Gerhard Paseman Feb 3 '13 at 8:26

For quite some time, Giovanni Sambin has been advocating the notation $A\between B$ for "overlapping" sets, that is, for inhabited intersections, where $\exists_{x} \ x \in A \cap B$. He makes his case in a constructivist frame, but I think that a lot of ink, chalk---and keyboard pushing...---would be spared if one wrote $A\between B$ instead of $A\cap B \neq \emptyset$, even in non-constructive mathematics.

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Doesn't extend nicely to $A\cap B\cap C\ne\varnothing$. – Akiva Weinberger Sep 1 '15 at 23:57
Well, one can just overload the overlapping symbol, $\between (A, B, C)$; it would again be more economical than the traditional exploitation of the associativity of $\cap$. – Basil Sep 2 '15 at 7:37

To say that $u$ and $v$ are orthogonal you can spell out "The scalar product of $u$ and $v$ is equal to zero", i.e.:

$\langle u,v \rangle=0$

but you can also use the binary symbol $\perp$ to write the sentence "$u$ orthogonal to $v$" more directly, i.e. $u\perp v$.

Analogously, to say that sets $A$ and $B$ have empty intersection, of course you can spell out "$A$ intersection $B$ equals the empty set", i.e.:

$A \cap B = \emptyset$

But it would be nice if there was a binary symbol (like a barred $\cap$ symbol, not to be confused with the $\pitchfork$ symbol for transversality) to say directly "$A$ does not intersect $B$ (nontrivially)".

I don't think this symbol already exists in LaTeX.

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You could use $\perp$ for that as well... – Harry Altman Oct 22 '10 at 13:28
I write $A\thinspace )\thinspace(\thinspace B$ for $A\cap B=\emptyset$ and similarly $A\supset\hskip-5pt\subset B$ for $A\cap B\ne\emptyset$. – Christian Blatter Nov 9 '10 at 9:26
Surely you should write $A \supset \subset B$ and not $A)(B$. – Ryan Reich Nov 16 '10 at 10:43
You can use $A \parallel B$ for $A\cap B=\varnothing$, since parallel lines don't meet, and so do disjoint sets. – Asaf Karagila Oct 13 '11 at 17:31
@Asaf that metaphor for || doesn't work if A and B are subsets of projective space... SCNR. – Konrad Voelkel Dec 19 '11 at 10:59

(This would be a comment on notation for partial functions, but I don't have the reputation points, as I just joined MO.) Though this is by no means standard, for personal use I've adopted the following system of arrow decorations that captures many standard types of binary relations. For a relation f from A to B, use $\rightharpoonup$ to indicate $\forall x\in A ~\exists y \in B~~xfy$, $\rightharpoondown$ to indicate $\forall x\in A~\exists^{\leq 1} y\in B~~xfy$, $\leftharpoondown$ to indicate $\forall y\in B~\exists x\in A~~xfy$, and $\leftharpoonup$ to indicate $\forall y\in B~\exists^{\leq 1}x\in A~~xfy$. So, $\rightarrow$ is for functions, $\leftrightarrow$ is for bijections, $\leftharpoonup\hspace{-1em}\to$ is for injections, $\leftharpoondown\hspace{-1em}\to$ is for surjections, $\rightharpoondown$ is for partial functions, $\rightharpoonup$ is for serial relations, and so on.

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I was sold when I saw the notation for injections, but I'm not so happy with the notation for surjections, as compared to the usual symbol $\twoheadrightarrow$. There is nothing in that arrow to suggest "surjection" to me. – Ryan Reich Nov 16 '10 at 10:46

This is probably the exact opposite of what the thread starter intended, but here's an instance where it might have been useful to have overloaded notation! I recently found that in propositional logic, $p \to q$ obeys much the same rules as exponentiation $q^p$: for instance, we have $(r^q)^p = r^{q \times p}$, and similarly, $p \to (q \to r)$ iff $p \land q \to r$. This is apparently due to the universal property for exponential objects, as applied to a Boolean algebra viewed as a poset category. I suppose it's also an instance of the Curry—Howard correspondence.

More generally, it seems like it isn't such a bad idea to conflate exponentiation and arrows - it looks nicer, to me at least, to write that a function of the type $A \to (B \to C)$ is naturally isomorphic to a function of the type $A \times B \to C$, than to write about $A \to C^B$. Even, as some have suggested, $A \to {}^B C$ or $C^B \leftarrow A$ would look nicer. On the other hand we'd lose the association with cardinal arithmetic if we do this...

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The lack of a nice obviously symmetric notation for $\binom{a+b}{b}$ has bothered me; Dijkstra suggested in EWD 782 the notation $P(a,b)$, generalizing it also to $P(a_1,\ldots,a_k)$ for $\binom{a_1+\ldots+a_k}{a_1,\ldots,a_k}$. (Though I certainly disagree with him about $\binom{n}{k}$ being useless - you certainly do want to think about it that way a lot of the time.) I haven't actually had any reason to use this since I saw it but I can certainly think of times I would have.

Also the double-parentheses multichoose notation $\left(\!\binom{n}{k}\!\right)$ is nice because it lets you say "...and this is n multichoose k (which is equal to this binomial coefficient)" instead of just jumping directly to a binomial coefficient whose relevance may not be immediately obvious. But I suppose that's not really on the level of giving you a better way to look at things.

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Isn't there a nice obviously symmetric notation for ${a+b} \choose b$, namely ${a+b} \choose {a,b}$? – Rasmus Bentmann Oct 21 '10 at 18:34
Oh, true. That is a good point. It looks a bit clunky but it works. – Harry Altman Oct 21 '10 at 20:28
$\left(\!\binom{n}{k}\!\right)$ – JBL Oct 22 '10 at 13:26
is given by \left(\!\binom{n}{k}\!\right) – JBL Oct 22 '10 at 13:27
$\binom{a+b}{a,b}$ often means something else when $a=b$, namely the number of ways of partitioning $2a$ into two groups of size $a$. Thus $\binom{4}{2,2}$ is $3$ not $6$. – David MJC Nov 10 '10 at 19:40

Multi-factorials are handy. Sometimes results can be expressed compactly by introducing a double factorial or possibly higher factorial. For example

$$\int_0^{\pi/2} \sin^{2n+1} \theta \:\: d\theta = \frac{(2n)!! }{ (2n+1)!!}$$

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I wish there were a notation that didn't scream "iterated factorial", though (not that one sees this very much). I forget: does $n?$ mean anything? The question mark is handy because it suggests having to make a choice, as in "even or odd?" – Ryan Reich Nov 16 '10 at 10:48
The question mark has a meaning in C and programming languages derived from C. The notation a ? b : c; means to do b if a is true, otherwise do c. The question mark also means "optional" in regular expressions. For example, the regular expression ab?c matches abc or ac. I don't know whether either of these notations would make sense imported into math. On a related note, sometimes I would like to import C's % operator into math notation. – John D. Cook Nov 16 '10 at 23:30
True, but C also doesn't have a factorial operator, and ! means something entirely different again. There's not much reason to make mathematical notation agree with programming design choices. As for %, we always have "mod". – Ryan Reich Nov 29 '10 at 9:08
The problem with "mod" is that it is usually an equivalence relation and not a function. That is, you see "a equiv b mod m" more than "a mod m". I'm not sure the latter is common notation or that people agree in detail what it means. – John D. Cook Nov 29 '10 at 15:22

I think that Inuit numerals are cool. (http://en.wikipedia.org/wiki/Inuit_numerals) They are useful for vigesimal type things.

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Now I wonder where in mathematics you would like to do actual vigesimal calculations? – Konrad Voelkel Dec 19 '11 at 10:44

In the notation of Time scale calculus, the ordinary calculus derivative df/dt and the forward difference operator $\Delta f$ are both written as $f^\Delta$. Indefinite sums and indefinite integrals are both written as $\int{f(t)\Delta t}$ and called indefinite integrals. The context would say $\mathbb{T}=\mathbb{Z}, \mathbb{T}=\mathbb{R}$ or other $\mathbb{T}\subset\mathbb{R}$.

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What do you mean by "indefinite integral"? I used to hear this term referred to the "family of primitives" of a given function, like in: $\int f(x) dx=F(x)+C$ – Qfwfq Oct 22 '10 at 13:27
(btw, it wasn't me to downvote) – Qfwfq Oct 23 '10 at 18:17
Yes, when the time-scale is the real numbers, the indefinite integral $\int f(t)\Delta t=\int{f(x)dx}=F(x)+C$ and when time=integers, $\int f(t)\Delta t=\Delta^{-1}f(x)=F(x)+C$ (not the same F in each case though of course). – Roy Maclean Oct 27 '10 at 12:04

Cauchy-Binet as a generalized Pythagoras theorem.

Let $X$ be an $n \times k$ matrix with $n \ge k$. For any $k$-index $I=i_1...i_k, \; 1 \le i_1 < ... < i_k \le n$, there is some advantage to denote by $X_I$, the determinant of the $k \times k$ submatrix of $X$ with rows indexed by $I$. For any two such $X,Y$, we can state the Cauchy-Binet formula as a pairing $$\det (X^TY)= \sum_{I} X_I Y_I$$ where the sum is over all $n \choose k$ $k$-indices. This is a Pythagoras theorem for $X=Y$ since it says that the the volume-squared of the parallelepiped spanned by the $k$ columns of $X$ in $\mathbb{R}^n$ is the sum of squares of the volume of the projections on the $n \choose k$ $k$-dimensional coordinates.

For any $n \times m$ matrix $A$ with $m,n \ge k$ and $k$ indices $I,J$, we also denote by $A_{IJ}$ the determinant of the $k \times k$ submatrix of $A$ with rows indexed by $I$ and column indexed by $J$. Then for $X(m \times k)$ and $Y(n \times k)$, we have by Cauchy-Binet twice, $$\det(X^TAY)=\det(X^T(AY))=\sum_{I}X_I(AY)_I =\sum_I X_I \det(A^IY)=\sum_I X_I \sum_J A_{IJ} Y_J,$$ where $A^I$ is the $k \times n$ matrix given by the rows of $A$ indexed by $I$ and we note that $(AY)_I= \det(A^IY)$ and $(A^I)^T_J=A_{IJ}$. This notation thus allows us to view Cauchy-Binet (usually stated with $m=n,A=I$) as an extension of the usual $x^TAy=\sum_{ij}A_{ij}x_iy_j$ for $k=1$.

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This is not really about notation, but it's a very good point. – darij grinberg Nov 2 '10 at 0:09

Instead of writing $$|x-y|\le \varepsilon,$$ I used to write $$x\lessgtr y\pm \varepsilon.$$ You may read it as $x$ is more-or-less $y$ plus-minus $\varepsilon$.

One may also write something like $$x\lessgtr e^{\pm\varepsilon}\cdot y$$ which is much better than $$|\ln(y/x)|\le\varepsilon$$

It is easier to read, especially if instead of $x$ and $y$ you have long expressions.

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What would you write for $|x-y|\geq \epsilon$? – Nate Ackerman Jan 24 '13 at 23:24
You often need to say that two values are close. I do not see a place where you need values which are far, but in principle you could write $x\lessgtr y\mp \varepsilon$ – Anton Petrunin Jan 25 '13 at 19:56
I'm not happy about the "principle" in Anton's reply to Nate's comment. In the body of the question, the upper inequality (with $<$ and $+\varepsilon$) and the lower inequality (with $>$ and $-\varepsilon$) are to be understood as combined by "and", whereas in the comment, the upper and lower inequalities are intended to be combined by "or". Allowing both uses of the notation seems to be inviting confusion. – Andreas Blass Jan 26 '13 at 16:22
Andreas, you are right, I made this reply without much thinking :) – Anton Petrunin Jan 30 '13 at 19:10
Actually, I like $x =^\epsilon y$ for this concept. – Lee Mosher Feb 2 '13 at 23:43

Using $(a, b, ... )$ is handy to denote a column vector, which is the transpose of the row vector $[a, b, ... ]$, especially in linear text. Correspondingly, all displayed matrices should be written with brackets, not parentheses. This notation agrees with the usual identification of coordinates with column vectors.

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I take it you also hate having to write the transpose when you're constrained to writing on one line? :) – J. M. Oct 21 '10 at 11:45
Yes, particularly when stacking vectors $a, b, ...$, in which case you have to write the artificial $[a^T, b^T, ... ]^T$ without this convention. – John Bentin Oct 21 '10 at 13:52
It is, of course, not $[a, b, \dots]^T$ but $[a\;b\;\dots]^T$. This is an awful convention because even once you remind yourself why the $T$ is there you (or I at least) am not convinced that it has any meaning other than to satisfy a badly chosen precedent. I don't know why anyone would vote this down. – Ryan Reich Oct 21 '10 at 16:27
Don't forget the invariant literature which uses $\left[v_1,...,v_n\right]$ not for the matrix formed by the columns $v_1$, ..., $v_n$, but for its determiannt... – darij grinberg Oct 21 '10 at 17:04

I recently saw the following notation in the context of divisors on algebraic varieties, and I liked it very much.

Suppose that $D$ and $E$ are reduced divisors on a normal algebraic variety $X$. One can use $D \vee E$ to denote the reduced divisor with support equal to $D + E$ and $D \wedge E$ to denote $(D + E) - (D \vee E)$. I could imagine variants on this if $D$ and $E$ are non-reduced (involving taking max's, respecitvely mins, of the divisors component-wise).

EDIT: I'm slightly curious as to why this was downvoted. I guess it's too common to be interesting?

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It is not me , I do not know algebraic variety, but as a guess may be there is some order there and so you have an inf and sup or even a lattice that would fully justify these notation and be rather common at that. – Jérôme JEAN-CHARLES Nov 23 '10 at 23:54

I use the notation

$V \oplus^{\perp}W$

to denote orthogonal direct sum [Edit: direct sum of, say, subspaces of a given inner-product space].

Or

$(M,g) \times^{\perp} (N,g')$, or simply $M \times^{\perp} N$, to denote (orthogonal) cartesian product of Riemannian manifolds.

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I'd like to know why this was downvoted: you simply don't like the symbol, or there's some deeper reason? – Qfwfq Oct 23 '10 at 22:13
I didn't downvote it, but I don't like the notation because it is redundant and emphasizes the wrong thing. For vector spaces, there is no intrinsic inner product with respect to which a direct sum can be either orthogonal or not. If you have two inner product spaces, then the direct sum is always orthogonal unless specified otherwise, because there is no one way to do it otherwise. Likewise for the direct product of two manifolds with a Riemannian metric. It is better to have a notation for when the sum or product is not orthogonal, and to specify how. – Ryan Reich Nov 16 '10 at 11:00
I have seen that some authors use the notation $V\obot W$ for this purpose (provided by mathabx package in latex), see tex.stackexchange.com/a/61882/39306. – Name May 27 '15 at 17:50

Has this already been mentioned? If a group $G$ acts on a commutative group $A$ by homomorphisms, $G \to Aut(A)$, then use $a^g$ to denote the action. Especially if the group multiplication on $A$ is written multiplicatively, where we can say things like $(ab)^g = a^g b^g$. This can come up especially in Galois theory; I remember Lang using this notation in his Algebra to prove Hilbert's Theorem 90, and I thought it was very neat, and enhanced the readability of notation as well.

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I like this notation for actions in general, but for the particular case of actions on groups written in multiplicative notation, this is quite confusing, as superscripts already denote too many other things in this context. You ruled out conjugation by making the group commutative, but it’s still ambiguous which superscripts in a thing like $(a^kb)^g$ denote action by $G$, and which denote powers with integer exponents. – Emil Jeřábek Dec 7 '13 at 13:09
@ToddTrimble, I agree with Serre (Cohomologie Galoisienne, p. 42): "Si $s \in G$ et $x \in E$, le transformé $s(x)$ de $x$ par $s$ sera souvent noté ${}^sx$ [mais jamais $x^s$, pour éviter l'horrible formule $x^{(st)} = (x^t)^s$]." (Nonetheless, Harish Chandra does use your suggestion for the conjugation of a group on itself or its Lie algebra.) – L Spice May 18 '15 at 19:36

I like the notation $A\mathrel{\in\in}\mathcal C$ for objects (rather than morphisms) in a category (not my own invention).

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Has anyone come across any more similar examples of good notation that should be better known?

Some interesting glyphs:

1. Combinatorial Principles in Set Theory:

2. Bisimulation:

• Given two states p and q in S, p is bisimilar to q, written p ~ q, if there is a bisimulation R such that (p,q) is in R.
3. Boxplus operator in Coding Theory

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I always considered 1. to be a prototypical example of bad notation. – Emil Jeřábek Dec 19 '11 at 13:27

## protected by François G. Dorais♦Jul 9 '13 at 16:41

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