# 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.)

• In set theory we write ${}^B A$ for the set of functions from $B$ to $A$. Commented Oct 20, 2010 at 20:54
• I've always assumed that the notation $A^B$ is because of the "exponential law" $(A^B)^C = A^{B\times C}$ ... Commented Oct 20, 2010 at 23:18
• Yes, among other things. Also $A^B\times A^C=A^{B+C}$, where $+$ is disjoint union. But all the great reasons for it don't help for our mind thinking that maps start with the source and end with the image, not the other way round. Commented Oct 20, 2010 at 23:20
• Arabic numerals ? Ah yes, they were transmitted to Europe by the Arabs. Commented Oct 21, 2010 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. Commented Oct 21, 2010 at 3:38

Among recent introductions, I like the notation and names (introduced by Kenneth Iverson and popularized by Donald Knuth) for the ceiling function $\lceil x\rceil$ and floor function $\lfloor x\rfloor$. Compare with the heavy "approximation by excess/defect"...

• According to Knuth, this notation and these names were introduced by Iverson in his book "A programming language" in 1962. Commented Oct 20, 2010 at 20:25
• But I have also seen some people call these "Gaussian Brackets" --- any reason why that is so? Commented Dec 15, 2010 at 13:56
• They are life-savers... Commented Jan 24, 2013 at 12:15
• To add, I also like the notation $\lfloor x\rceil$ for rounding $x$ to the nearest integer $($but mainly because I use $\left\lbrack x\right\rbrack$ to denote $\{1,2,\ldots,x\})$, but I have seen $\| x\|$, however $\|$ can mean other things $($I mean, I have used the command \| to generate it, but another command is \parallel$)$ :P Commented Oct 6, 2018 at 9:18
• While people commonly complain about running out of letters (despite having Roman, Greek, Cyrillic, and Hebrew to get started with), one much more quickly runs out of delimiters (), [], {}, <>. It is fantastic that a new pair has been invented. Commented Oct 29, 2023 at 12:38

I like notation such as $2^X$ for the set of subsets of $X$ and ${X\choose k}$ for the set of $k$-element subsets. Also $[x^n]F(x)$ for the coefficient of $x^n$ in the power series $F(x)$, and multivariate notation like $x^\alpha$ for $x_1^{\alpha_1}\cdots x_n^{\alpha_n}$, where $x=(x_1,\dots,x_n)$ and $\alpha=(\alpha_1,\dots,\alpha_n)$.

• I find the projection $[x^n]F(x)$ indeed very useful. Commented Oct 21, 2010 at 17:23
• And, in the spirit of the latter notation, the multivariate version $[x^\alpha]F(x)$ for polynomials in several variables. Commented Oct 23, 2010 at 19:14
• I particularly like $X \choose k$ when X is say an algebraic variety and so $\chi {X \choose k} = {\chi(X) \choose k}$, where $\chi$ is the Euler characteristic... Commented Oct 29, 2010 at 2:29
• I find the notation $[x^n]F(x)$ incredibly prolix, cumbersome, and difficult to understand. Far superior is inner product notation, $(x^n, F)$, which has several advantages: it is symmetric and bilinear, and is actually correct when dealing with pre-Hilbert spaces of polynomials and the natural inner product. Commented Sep 4, 2020 at 15:21
• @DavidHandelman: perhaps it is just a matter of taste or context. I think of $[x^n]$ as a linear operator on the space of power series. If $A\colon V\to W$ is a linear transformation and $v\in V$, is it always preferable to write $(v,A)$ rather than $A(v)$ or $Av$? A nitpicking further point: why is the notation prolix? $[x^n]F$ uses one less character than $(x^n,F)$. Commented Sep 5, 2020 at 18:36

I like $A \hookrightarrow B$ and $A \twoheadrightarrow B$ for "$A$ injects into $B$" and "$A$ surjects onto $B$" respectively.

• I like this but I am confused what to use for bijections. Commented Dec 18, 2010 at 12:59
• A two-headed hooked right arrow. I don't know how to do it in normal LaTeX but you can easily do it using the xymatrix package, and obviously it's easy to write on a chalkboard. Commented Dec 18, 2010 at 23:25
• These two commands work for me for the "bijetion"-arrow described by Tom: \newcommand*\longbijects{\ensuremath{\lhook\joinrel\relbar\joinrel\twoheadrightarrow}} \newcommand*\bijects{\ensuremath{\lhook\joinrel\twoheadrightarrow}} Commented Jan 21, 2011 at 21:50
• @Marcel - $\stackrel{\sim}{\to}$, a combination of the map and the isomorphism sign. Commented Nov 24, 2011 at 23:12
• Add to this the notation $\looparrowright$ for immersions. Commented Mar 25, 2014 at 16:27

Bourbaki dangerous bend symbol to mark dangerous or difficult ideas.

• Haha. I didn't know this was due to Bourbaki, I learned it from Knuth's "The TeXbook". Commented Oct 21, 2010 at 17:33
• Perhaps this should be on every line in the Bourbaki text. Commented Feb 3, 2013 at 11:28
• The signage is probably more familiar to Europeans than elsewhere. When I first read Knuth's book, I wondered what all the weird Z's were about. Commented Jun 18, 2021 at 20:48

$D_j f$ to denote the partial derivative of a function between Euclidean spaces, w.r.t. the $j$'th coordinate. For some reason Jacobi's notation $\frac{\partial f}{\partial x_j}$ has become more popular. Jacobi's notation tends to cause much ambiguity and confusion, a point which is emphasized in the book "Multidimensional Real Analysis" by Duistermaat & Kolk. For instance (this example is taken from their book), let $e_1,e_2$ be the standard basis for $\mathbb{R}^2$ and define a new basis by $e'_1 = e_1 + e_2, e'_2 = e_2$. The passage from one basis to another is as follows: If $x_1 e_1 + x_2 e_2 = y_1 e'_1 + y_2 e'_2$ then $y_1 = x_1, y_2 = x_2 -x_1$. Now the meaning of $\frac{\partial y_2}{\partial y_1}$ is ambiguous: If one interprets $y_1$ and $y_2$ as independent coordinate functions, then $\frac{\partial y_2}{\partial y_1} = 0$. On the other hand, $\frac{\partial y_2}{\partial y_1} = \frac{\partial (x_2 -x_1)}{\partial x_1} = -1$, right? This was the source of much confusion for me when I was taught multivariate calculus and the notation $D_j f$ would have eliminated this confusion entirely.

• Another reason to prefer $D_j$ or $\nabla_j$ over $\partial/\partial x_j$ is that when you're expressing tensors using the Einstein "index gymnastics notation," the derivative $\nabla_j$ has a lower index, as it should, whereas the same operator in Leibniz notation is $\partial/\partial x^j$, which looks like it transforms as an upper-index tensor.
– user21349
Commented Oct 15, 2012 at 20:39
• Regarding differential geometry: If $f:M\to\mathbb R$ is a smooth function on a manifold and $x:M\to\mathbb R^n$ is a chart, I prefer $\left(\frac{\partial f}{\partial x}\right)_j$ or $\left(\frac\partial{\partial x}\right)_j f$ (or even $\partial_j f$ if the choice of the particular chart is clear or irrelevant). Because the notation $\frac{\partial f}{\partial x^j}$ suggests that $\frac{\partial f}{\partial g}$ could be defined using only $g$, and in fact you need to know that you are restricting to the curve along which the other coordinates $x^i$ are constant. Commented Jan 24, 2013 at 21:03
• @Ben, the index in the coordinate expression $\frac{\partial f}{\partial x^j}$ for the 1-form $df$ is clearly in the low position! In fact, this is the main reason that I see for having to put the indexes of the coordinates in the high position as we do, instead of doing everything in the opposite way, which would be better in some way: we could write $f=x_1^2+x_3$ instead of $f=(x^1)^2+x^3$. Commented Jan 24, 2013 at 21:14
• Jacobi was aware of this problem when he introduced the $\partial$ notation. Unfortunately his choice of notation for avoiding this ambiguity might have led to more confusion. Commented Nov 26, 2017 at 15:41
• How would $D_j$ eliminate any confusion? You seem to assert that $D_1 y_2$ would be somehow unambiguous. But it is just as ambiguous as $\partial y_2 / \partial y_1$. If $D_1$ means differentiation with respect to $y_1$, holding $y_2$ constant, then $D_1 y_2 = 0$. If $D_1$ means differentiation with respect to $x_1$, holding $x_2$ constant, then $D_1 y_2 = -1$. More generally, there is no symbol or notation involving only one single coordinate function --- not $D_1$, not $\frac{\partial}{\partial y_1}$, nothing --- that can possibly convey full information about a coordinate chart. Commented Feb 9, 2021 at 10:31

String diagram-notation makes for example adjoint functors, monads, tensor categories,... much clearer.

• And it also connects monoidal categories to braids, knots, links, tangles... Commented Oct 20, 2010 at 20:45

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: $$a^p\!\equiv_p\!a.$$

• 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 $1 \equiv_2 5 \equiv_3 2$. Commented May 18, 2015 at 19:31
• I still like this though... :) Commented Jan 27, 2018 at 22:06
• That is indeed a cool looking Fermat's theorem. Commented Jan 28, 2018 at 4:42
• Similar notation I've used is to put the (n) over the congruence symbol. Commented May 21, 2020 at 11:16
• @JoshuaZ: Yes, that is better. A symmetric relation should be denoted with a symmetric operator. Commented Apr 11 at 13:06

I like $f\colon\thinspace M\looparrowright N$ to denote an immersion of smooth manifolds.

• I use exactly this arrow to denote an operation of a group M on a space N. Commented Oct 20, 2010 at 21:39
• I prefer the notation $\alpha: G \curvearrowright X$ for an action $\alpha$ of $G$ on $X$ (see my answer). Commented Oct 20, 2010 at 22:57
• ... And I also like the "self intersecting arrow" to denote immersions that are possibly not embeddings. Commented Oct 20, 2010 at 22:58

I am fond of subscripting asymptotic notation with the parameters that the implied constant is allowed to depend on (and on the asymptotic parameter, if needed). e.g.

• $$X = O_k(Y)$$ (or $$X \ll_k Y$$, or $$Y \gg_k X$$) means that $$|X| \leq C_k Y$$ for some $$C_k$$ depending only on $$k$$.
• $$X = o_{n \to \infty; k}(Y)$$ means that $$|X| \leq c_k(n) Y$$ for some function $$c_k(n)$$ of both $$k$$ and $$n$$, which goes to zero as $$n \to \infty$$ for fixed $$k$$.
• (Rarer) $$X = O_{n \to \infty; k}(Y)$$ means that $$|X| \leq C_k Y$$ whenever $$n \geq N_k$$, for some $$C_k$$ and $$N_k$$ depending only on $$k$$.

Of course, if there is a parameter that influences all the constants (e.g. the ambient dimension) then it is better to explicitly state at the beginning that all constants will depend on this parameter so that one does not have to put in the explicit subscripts in all the time.

It can be instructive to rewrite some basic notions in analysis in this sort of notation, just to get a slightly different perspective. For instance, if $$f: {\bf R} \to {\bf R}$$ is a function, then:

• $$f$$ is continuous iff one has $$f(y) = f(x) + o_{y \to x; f,x}(1)$$ for all $$x,y \in {\bf R}$$
• $$f$$ is uniformly continuous iff one has $$f(y) = f(x) + o_{|y-x| \to 0; f}(1)$$ for all $$x,y \in {\bf R}$$
• A sequence $$F = (f_n)_{n \in {\bf N}}$$ of functions is equicontinuous if one has $$f_n(y) = f_n(x) + o_{y \to x; F,x}(1)$$ for all $$x,y \in {\bf R}$$ and $$n \in {\bf N}$$ (note that the implied constant depends on the family $$F$$, but not on the specific function $$f_n$$ or on the index $$n$$)
• A sequence $$F = (f_n)_{n \in {\bf N}}$$ of functions is uniformly equicontinuous if one has $$f_n(y) = f_n(x) + o_{|y-x| \to 0; F}(1)$$ for all $$x,y \in {\bf R}$$ and $$n \in {\bf N}$$
• $$f$$ is differentiable iff one has $$f(y) = f(x) + (y-x) f'(x) + o_{y \to x; f,x}(|y-x|)$$ for all $$x,y \in {\bf R}$$;
• (similarly for uniformly differentiable, equidifferentiable, etc.)

(These formulations are close to the nonstandard analysis formulations of these concepts, which uses similar but not quite identical asymptotic notation, but that is another story.)

• The subscripts seem a little bulky. Couldn't you just write, e.g. $f(y) = f(x) + o_{y \to x; f,x}(1)$? I guess it's not really changing much. Commented Oct 21, 2010 at 17:18
• Maybe it's time for you to go to really big-O notation: $X=\underset{n\to\infty}{\overset{k}{\LARGE\mathcal O}} Y$ for $O(\cdot)$ and then $f_n(y)=f_n(x)+\underset{|y-x|\to 0}{\overset{F}{\LARGE\mathrm O}} 1$. for $o(\cdot)$. :D Commented Oct 21, 2010 at 17:34
• The "really" big-O notation is a little bit confusing; since normally we write summation like this with $Y$ depends on the parameter $k$, but here we have the constant $C_k$ depends on it instead. Commented Oct 22, 2010 at 1:07
• Chang: definite sums can depend on their upper index. Commented Oct 22, 2010 at 6:08
• I think O-o notation is one of the worse. It does not worth to improve --- better to start from scratch. Commented Jan 30, 2013 at 19:18

The notation $M^{\oplus n}$ and $M^{\otimes n}$ to denote, respectively, nth direct sum and nth tensor product.

The notation $X \mathbin{\pi} Y$ to denote product of objects in an abstract category, and the analogous with the "upside down $\pi$" for coproduct. I once have seen this being used by B.Keller in a talk. It'd be nice to have a smaller $\Pi$ (resp. $\amalg$) symbol instead.

• $\sqcap$ resp. $\sqcup$?
– Max
Commented Nov 8, 2010 at 18:39
• You can get your smaller Π sign with a bit of a hack. This isn't exactly robust, though, since the size is constant relative to the surrounding size (but can be fixed if one looks up the right command): $X \mathbin{\scriptsize{\Pi}} Y$ and $X \mathbin{\scriptsize{\amalg}} Y$ Commented Nov 24, 2011 at 21:08
• I prefer to use the same notation as in arithmetic: $\times$ for binary product, $\prod$ for product of an arbitrary family, $+$ for binary coproduct, $\sum$ of coproduct of an arbitrary family. A small pi-like symbol for binary product seems to me to be a step in the wrong direction. Commented Dec 10, 2012 at 0:28
• Yeah I agree; I much prefer $\times$ and $+$ for categorical product and coproduct.
– ಠ_ಠ
Commented Jul 15, 2017 at 2:53
• I also use $×$ and $\prod$ and $+$ and $\sum$, but whenever a category has a sufficiently nice functor to $\mathrm{Set}$ preserving / reflecting / creating colimits, I switch to $\coprod$. For example I don’t use $+$ in $\mathrm{Top}$ or $\mathrm{Meas}$. (And I’d probably also use $\coprod$ for toposes …) Commented Jul 15, 2017 at 4:26

$a \vee b$ and $a \wedge b$ to denote the maximum and minimum of the numbers $a$ and $b$. (This seems to be well-known only among probabilists.)

• This notation is also used in a lattice or Boolean algebra for least-upper-bound and greatest-lower-bound, which agrees with your meaning in a total order, such as the reals. Commented Oct 20, 2010 at 21:01
• Sometimes I have seen $a \cup b$ and $a \cap b$ for the greatest common divisor and least common multiple of positive integers $a, b$. Much better than $(a, b)$. Commented Oct 21, 2010 at 3:59
• AByer, it's by analogy with $\cup$ and $\cap$ that I remember that $\vee$ is maximum and $\wedge$ is minimum. (Otherwise I tend to think that since $\vee$ points down, it must be the minimum.) Commented Oct 21, 2010 at 4:20
• Actually, I find this notation incredibly frustrating for the reason Michael mentions in his comment; $a\vee b$ is "clearly" a cartoon of a point lying below both $a$ and $b$.
– JBL
Commented Oct 22, 2010 at 13:19
• $[a \vee b] = [a] \vee [b]$ Commented Oct 22, 2010 at 20:03

Instead of $[X]$ one often sees $\mathbf 1_{X}$ (especially in probability work?). This is neat because it literally is 1 on $X$. Also it has the advantage over $[X]$ that you can write things like $(2+\mathbf 1_{X})^2$ for the function that is $9$ when $X$ occurs and $4$ otherwise; $(2+[X])^2$ would be less appealing here. On the other hand, if there is a lot of notation replacing "$X$" this is not so good: $$\mathbf 1_{n_k\in \{n: n\text{ prime}\}}.$$

• so how about $1[X]$? Commented Oct 21, 2010 at 2:25
• $\mathbf{1}_X$ is conventionally used for characteristic function of a set. Commented Oct 21, 2010 at 3:42
• The problem with 1_{X}, as you observed, is that it pushes the main stuff to the small font of the subscript. I don't understand the advantage you wrote of: what's wrong with (2+[X])²? IIRC, expressions of this sort are used without hesitation in Graham-Knuth-Patashnik (Concrete Mathematics). Commented Oct 28, 2010 at 6:51

I once came across the notation $\underline{n}$ for the set $\lbrace 1,2,\dots,n\rbrace$. It came in very handy to write $i \in \underline{n}$ instead of $1\leq i \leq n$ or $i \in \lbrace 1,2,\dots,n\rbrace$.

• Sometimes people write ${}[n]$ for this. I agree, a shorthand of this kind is quite useful. Commented Oct 20, 2010 at 20:51
• In set theory, every ordinal number is equal to the set of smaller ordinal numbers (a definition due to von Neumann), and this makes for infinitely many conveniences similar to the ones you mention. Thus, n={0,1,...,n-1} and in general $\alpha=\{\beta | \beta\lt\alpha\}$. Commented Oct 20, 2010 at 21:04
• In numerical linear algebra MATLAB notation is sometimes (ab)used, e.g. 1:n. Commented Oct 21, 2010 at 2:42
• This can be confusing though, since the topologist's simplex $[n]$ is the ordered set $\{0<...<n\}$. The rationale is that $n$ counts the number of arrows. Commented Oct 21, 2010 at 4:52
• @Andres Caicedo, see also Richard Stanley's answer: $\binom{X}{k}$ is a nice notation for the $k$-subsets of a set $X$, mirroring the binomial coefficient $\binom{x}{k}$.
– JBL
Commented Oct 22, 2010 at 13:22

The notation for transversality:

$M \pitchfork N$

• I like to use an analogous symbol for tangency: an intersection "cap" with a bar over it, tangential to it. Commented Jan 17, 2021 at 14:05
1. For rising and falling factorials: $x^{\overline{n}}$ and $x^{\underline{n}}$ à la Knuth. Much better than the traditional way to write the Pochhammer symbol: $(x)_n := x^{\overline{n}}$. In a book I'm writing, I use the notation $x^{\uparrow n}$ and $x^{\downarrow n}$, which I find much less clumsy (consider $(2x+1)^{\overline{6k-2}}$ vs $(2x+1)^{\uparrow6k-2}$). Anyway, the utility in either of these notations is seen in the umbral calculus; it makes the connection to "ordinary" calculus much more apparent, such as with $$\Delta x^{\uparrow n} = n x^{\uparrow n-1}\qquad\text{compared to}\qquad D x^n = nx^{n-1}.$$

2. The simple idea of omitting parentheses for function application: $f\,x$ as opposed to $f(x)$. I think this often makes some mathematics look cleaner, especially when the argument isn't especially complex. It also allows for some nice (= convenient) abuse of notation, such as in $$\left[ (-1)^{p - m - n} z \prod_{j = 1}^p \left( z D_z - a_j + 1 \right) - \prod_{j = 1}^q \left( z D_z - b_j \right) \right] G(z) = 0,$$ where $D_z:=d/dz$. Note this equation isn't a product (entirely); upon expansion, we'd have $D_z G(z)$ terms.

3. Do fractions count? Imagine having to write $$\sqrt{(x^2 + 2x + 1)\div (5x^3 - 3x^2 + 2x - 7)}$$ instead of $$\sqrt{\frac{x^2 + 2x + 1}{5x^3 - 3x^2 + 2x - 7}}.$$

4. Big-O notation. Though often abused, this is a much less clumsy way to express boundedness and asymptotics and errors and even lets you begin to do some algebra with them (provided you're careful). I don't think doing such is as obvious when you write it all out manually.

5. $\square(x)$ for the square wave, $\triangle(x)$ for the triangle wave, $Ш(x)$ for the Dirac comb (seriously, see Appel's "Mathematics for Physics and Physicists"). These are more cute than explicitly useful.

6. Notation used with musical isomorphisms as a way to do raising and lowering of indices. We have $X^\sharp$ which raises the index (in the context of Einstein summation) and $X^\flat$ which lowers the index. Here, $\flat$ and $\sharp$ are isomorphisms between tangent $TM$ and cotangent bundles $T^*M$: $\flat:TM\to T^*M$ and $\sharp:T^*M\to TM$.

7. Using $\operatorname{cis}\theta = \cos\theta + \mathrm{i}\sin\theta$ (cosine i sine), which is nice for obvious reasons (yes, $\omega = e^{\mathrm{i}\theta}$ is nice too) and $\operatorname{cas}\theta = \cos\theta + \sin\theta$ (cosine and sine), which is used in e.g., the Hartley transform.

8. Notations for hypergeometric functions $${}_pF_q \!\left( \left. \begin{matrix} a_1, \dots, a_p \\\\ b_1, \dots, b_q \end{matrix} \; \right| \, z \right) = {}_pF_q(\mathbf{a},\mathbf{b};z)$$ and Meijer-$G$ functions: $$G_{p,q}^{m,n} \!\left( \left. \begin{matrix} a_1, \dots, a_p \\\\ b_1, \dots, b_q \end{matrix} \; \right| \, z \right)=G_{p,q}^{m,n} \!\left( \left. \begin{matrix} \mathbf{a} \\\\ \mathbf{b} \end{matrix} \; \right| \, z \right)$$

9. Notation for general continued fractions: $$\underset{j=1}{\overset{\infty}{\LARGE\mathrm K}}\frac{a_j}{b_j}=\cfrac{a_1}{b_1+\cfrac{a_2}{b_2+\cfrac{a_3}{b_3+\ddots}}}.$$ The $\mathrm{K}$ comes from German's "Kettenbruch", which is "continued fraction."

I think that's good for now. There are probably lots more. :)

To end, I'll say one notation I do not like: the use of fraktur. Most of the time it just looks ugly and no one can actually write fraktur letters.

• 9. I am a fan of this. :D Commented Oct 21, 2010 at 4:53
• You're wrong that nobody can write Fraktur letters. If you put some effort into it then you absolutely can draw them. Commented Oct 21, 2010 at 18:13
• @KConrad: Sure you can, but you shouldn't. As with other scripts (say Latin or Cyrillic), the old script for writing German has a version for typesetting and a version for handwriting: Sütterlin (en.wikipedia.org/wiki/Sütterlin). If you like Fraktur, you will love it. Commented Oct 22, 2010 at 15:00
• @Bourrigan: I whole-heartedly disagree. Fraktur letters aren't that hard to write, and are easy to read. Sütterlin is illegible spaghetti, and even worse, the letters more often than not aren't what you expect them to be (the 'h' looks like an 'f', the 'e' like an 'n', etc.). It would be severely unpedagogical to use the script for anything other than to demonstrate how not to make a legible script. Commented Nov 16, 2010 at 9:26
• "Bruch" is also the standard word for "fraction" in German, so the literal translation would be "chain fraction". Commented Apr 17, 2011 at 15:41

Writing $\int_{x=0}^{2 \pi} \sin x dx$ rather than $\int_0^{2 \pi} \sin x dx$ can be very useful when there are integrals stacked several layers deep. EG

$$\int_{x=-\infty}^{\infty} \int_{y=-\infty}^{\infty} e^{-(x^2+y^2)/(2 \sigma)} dx dy = \int_{r=0}^{\infty} \int_{\theta=0}^{2 \pi} e^{-r^2/(2 \sigma)} r dr d\theta.$$

• David, you forgot the $dydx$ on the left. Commented Oct 16, 2012 at 3:30
• This can also be done by writing $\int_x^{2\pi} dx\, \sin x$ etc., which is a little bit shorter. I have often seen this in physics.
– user22882
Commented Jan 24, 2013 at 18:32
• Do you still need the $dx dy$ on the left if you specify the integration variable in the left part of the symbol? Commented Jul 9, 2013 at 6:42
• @Federico: the "$dx$" is very expressive in integrals for a variety of reasons. It makes clear that what you are integrating is actually a differential form, which is useful even in one dimension: for example $\int f(u)du=\int f(u(t))u'(t)dt$, and $\int_{t=a}^b udv=uv|_a^b-\int_{t=a}^b vdu$. Also, in Physics it makes physical quantities add up correctly ($\int F\cdot ds$ for the work done by a force makes it clear that work is measured in units of force times units of length). Commented Dec 25, 2013 at 0:41
• It's also useful when there's variable changes flying around in the background. I teach my calculus students this notation at the same time we go over $u$-substitution in an attempt to cut down on the number of students plugging in their $x$-boundaries for $u$. Commented May 13, 2019 at 18:44

The notation $\perp$ to denote either orthogonality, or to indicate independent random variables, or perhaps even to indicate relatively prime numbers.

• Wait, if $n \perp m$ means that n and m are relatively prime, then does $(n,m)$ represent their inner product? Far out... Commented Oct 20, 2010 at 20:19
• With Knuth's proposal to write $k \perp n$ for k relatively prime to n' we can write by the Moebius inversion formula $\sum_{1 \le k \le n} \left[ k \perp n \right] = \sum_{1 \le k \le n} \left[ k \mid n \right] \mu\left(\frac{n}{k}\right).$ Note that by the conventions of the Iverson bracket when the Iverson-bracketed statement is false, it annihilates anything it is multiplied by - even if that other factor is undefined'. Commented Oct 22, 2010 at 15:46
• @David: I think it's appropriate for the "inner product" to be a (bi-)multiplicative function rather than an additive one, since "orthogonality" is a multiplicative concept. Note: "far out" was a term of appreciation! Commented Oct 22, 2010 at 22:03
• There is also a nice use of $\bot$ in logic to mean absurdity, i.e. the negation of $\top$. Commented Oct 15, 2012 at 19:28
• The mobius formula suggestion of Bruce’s comment. I needed to see it: $\sum_{1 \le k \le n} \left[ k \perp n \right] = \sum_{1 \le k \le n} \left[ k \mid n \right] \mu\left(\frac{n}{k}\right).$ Commented Nov 23, 2014 at 15:07

The original problem with the indices was that they were used to label coordinates, so mathematicians preferred more and more coordinate independent operators, while physicists continued to use indices. Then, Penrose realized that it has to be something beyond the indices that makes them useful - mainly the Einstein summation convention - and proposed the abstract index notation. This notation is almost identical in form with that of coordinate indices, but it is invariant, like the notation used by mathematicians, and maintains the simplifications due to the use of indices. The indices are not interpreted as labeling coordinates, but as representing the type of vectors and tensors and how they act on each other.

I think that there are advantages and disadvantages in both notations. Though, many tensor operations, especially contraction and type change, are easier to define and perform by using indices.

The following fields can benefit of this notation: Linear Algebra, Representation Theory, Group Theory, Differential Geometry.

This notation can naturally be related to Penrose's diagrammatic notation.

• I agree that abstract index notation was a brilliant observation. However, the problem with tensor calculus in general and (abstract) indices in particular, is that it does not conceptualize the calculations being performed. This can be very pragmatic, as a tensor expression admits multiple interpretations, but it can also be confusing. For instance a contraction could be indicating composition, an inner product, or a trace, while skew symmetrization could be indicating a wedge product or a Lie bracket. Commented Oct 22, 2010 at 21:07
• @David MJC: That's one reason because I like it. Some operations like those you mentioned are, from a viewpoint, indeed "equivalent". We can see this notation like a forgetful functor, or as a polymorphism in object oriented programming. It introduces a higher layer of abstraction, which allows us to see relations between apparently unrelated things. You are right that it hides some concepts, but it reveals others. At a much smaller scale it is like Category Theory, exposing hidden connection and general patterns. But I do not prefer it over using operators, I view them somehow complementary. Commented Oct 23, 2010 at 11:02

I like to interpret $f(x)$ as meaning $f\circ x$, otherwise known as the pullback $x^*f$. For instance $x$ could be the standard real valued coordinate on a line. This makes rigorous sense of the concept of a "variable" and hence also dependent and independent variables ($y=f(x)$). In the example of functions on a line, $f'=dy/dx$ is simply a ratio of 1-forms.

Such an interpretation also answers the common complaint that $f=f(x)$ confuses a function with its values. Instead it represents the very common shorthand of omitting pullbacks!

• Did you invent it your-self? If not did you see this notation in some books? Commented Nov 8, 2010 at 1:00
• Any book which describes a function f as a relation y=f(x) between dependent and independent variables is using this notation, so in that sense I didn't invent it. Also for mathematicians such as E.Cartan, points were always variable, i.e., functions on some unspecified parameter space: $x\in X$ means "x is a function on an unspecified domain with values in X", which is Grothendieck's "functor of points". However, I don't have a reference for my interpretation, and it does generate a laugh (e.g. in a colloquium) to say that confusing a function and its values amounts to omitting pullbacks. Commented Nov 8, 2010 at 22:00
• This one is nice both because: a) it makes sense of the reverse-Polish notation for functions, $f(x) = xf$, and b) it makes sense of the term "random variable" in probability, which it took me a long time to understand the meaning of. Commented Nov 25, 2011 at 1:54
• This point of view is espoused in Munroe's 1958 AMM article "Bringing calculus up to date" (jstor.org/stable/2308879). He gives a very little bit of history, mostly without references. Commented Jun 28, 2012 at 22:01
• @AntonPetrunin have you never seen people talk about "the function f(x)" ? Commented Oct 13, 2021 at 0:59

A good notation and a bad notation (in my opinion).

Good: $p' = (1 - \frac1p)^{-1}$. It is commonly enough used in analysis (Holder inequality) that it is good to have a shorthand, and it makes clear that the conjugate exponents are dual pairs: $(p')' = p$.

Bad: $p^* = \frac{np}{n-p}$ the Sobolev conjugate in Sobolev inequalities. It hides the dependence on the spatial dimension $n$, and overloads the $*$ for something that does not have a duality: $(p^* )^* = \frac{(2p)^*}{2} \neq p$.

• Having shorthand for the dual to a norm is indeed convenient $\left|\mathbf{A}\right|_{p^{\prime}}$. :D But maybe there is a better superscript than the already heavily used prime? Commented Oct 22, 2010 at 1:22
• @J.M.: both of these are historical (so don't blame me for them); also, when else do you see a prime applied to a number? Commented Oct 22, 2010 at 10:03
• @Willie, it is a convenient way to denote zero! Commented Oct 22, 2010 at 16:43
• Ah, so we could write Euler's formula as $\sum_{n \geq 0} n^{-s} = \prod_p (p')^s$! Commented Oct 30, 2010 at 18:22
• I like Todd's pun very much, although it should be $\sum_{n\geq 0} n^{-s} = \prod_p (p^s)'$, shouldn't it? Anyway "Zeta(s) is the product over p prime of p to the power s prime" reads nicely! Commented Nov 8, 2010 at 22:09

I really like the arrow notation for limits: $$\frac{\sin(x)}{x} \xrightarrow{x\rightarrow 0} 1.$$ I've seen people use this on the blackboard, but I don't think I've seen it in print. The right-hand side of an arrow expression can be decorated with a "+" or "-": $$\frac{\sin(x)}{x} \xrightarrow{x\rightarrow 0^+} 1^-.$$ Arrow expressions can be treated as propositions (e. g., $x\rightarrow 0$ implies $\frac{\sin(x)}{x}\rightarrow 1$), but this is usually less succinct than the stacked arrows. However, it's easier to chain limits this way:

If $f$ and $g$ are continuous [in the sense of elementary calculus], then so is $f\circ g$: if $a$ is fixed and $x\rightarrow a$, then $g(x)\rightarrow g(a)$ (since $g$ is continuous), so $f(g(x)) \rightarrow f(g(a))$ (since $f$ is continuous), QED.

This can be made rigorous, say, with nonstandard analysis, although there are probably more elementary ways.

Sometimes, we need to use a limit as a subexpression in a formula, rather than just stating that the limit equals something. For this, I like the notation $f(x)|_{x\rightarrow a}$ in favor of $\lim_{x\rightarrow a}f(x)$. To me, it's an obvious and intuitive extension of the notation $A(x)|_{x=a}$, which is commonly used to denote the expression that results when $x$ is replaced by $a$ in the expression $A(x)$ (in which $x$ occurs free).

• Interesting... could you tell where you saw this first? Commented Oct 27, 2010 at 8:44
• @J. M. - actually, I can't recall. :-( It's possible that I use it more than others, but I'm certainly not the inventor. I probably haven't seen "x -> a" treated as a proposition exactly as I did above, but it's basically equivalent to "x~a" (in nonstandard analysis, to mean that the difference between x and a is smaller than all standard positive reals). Commented Oct 28, 2010 at 5:35
• As an undergraduate in Cape Town I saw and used "$\frac{\sin x}{x} \rightarrow 0$ as $x \rightarrow 0$" almost as often as the $\lim$ notation.
– Max
Commented Nov 14, 2010 at 11:33
• This notation was always present in my education (in Bremen) and is especially popular in functional analysis where there is $\to$ for strong convergence and $\rightharpoonup$ for weak convergence.
– Dirk
Commented Nov 25, 2011 at 7:18
• @Max, I hope your post-undergraduate education has taught you a different value for that limit! Commented Jun 14, 2012 at 19:23

As a freshman, I "invented" the notation

$H \lhd ! \; G$

to say that $H$ is a characteristic subgroup of $G$, i.e. a subgroup invariant under any automorphism of $G$ (whereas a normal subgroup $N\lhd G$ is only invariant under the inner automorphisms).

• I love this! :) Commented Jan 27, 2018 at 21:25

I found the notation $K_\bullet$ for a complex (in with objects an abelian category or as an objects of the derived category) is very helpful. Otherwise people have to write something like $\cdots \to K_{n}\to \cdots \to K_{2}\rightarrow K_{1} \to K_{0}$ which just contains exactly the same amount of information.

Since the standard notation for open interval $(a,b)$ can be confused with the coordinates, gcd, and other stuffs (open brackets have been used A LOT!), I've seem notations like

$]a,b[$

occurred in the book "Elementary Classical Analysis" by Marsden, and we can denote half-open half-closed interval like this:

$]a,b]$ or $[a,b[$.

• I have to -1 this one. Nothing annoys me more than seeing $[x,y[ \cup ]a,b[$ and such. Commented Oct 21, 2010 at 4:36
• I was a fan of this notation until I read Quadrescence's comment. Commented Oct 21, 2010 at 6:30
• The point is that $]a,b[$ is the French way to write the open interval (this explains Harry's comment). It is still taught in high school, and students never learn about $(a,b)$, even at university. Only researchers adapt to this notation once they write in English. I agree that $]a,b[$ is clearer. Commented Oct 21, 2010 at 6:48
• @Ryan Reich: Quadrescence's example is only annoying when improperly typeset. Look at the difference in LaTeX between $[x,y[\cup]a,b[$ [x,y[\cup]a,b[ versus $\left[x,y\right[\cup\left]a,b\right[$ \left[x,y\right[\cup\left]a,b\right[. Commented Oct 21, 2010 at 17:22
• I prefer Knuth's notation, which uses $(a.\,.b)$ for the open interval and $[a.\,.b]$ for the closed one. Commented Oct 22, 2010 at 19:38

One can decorate a subscript or superscript by additional symbols to indicate what the subscript or superscript is doing. For instance, consider a truncation $f 1_{|f| \leq N}$ of a function to its values whose magnitude is at most $N$. One could of course call such a function something like $f_N$, but why not call it $f_{\leq N}$ instead? Then one can do things like "Decompose $f = f_{\leq N} + f_{>N}$, where $f_{\leq N} := f 1_{|f| \leq N}$ and $f_{>N} := f 1_{|f| > N}$." Notation of this type is sometimes used in PDE, particularly with regard to Littlewood-Paley frequency projections.

Similarly, one could imagine the operation of shifting $f$ by $N$ to be denoted something like $f_{+N}$ rather than $f_N$, etc..

• Hum, I would prefer if $f 1_{|f| \leq N}$ be written as $f|_{\leq N}$ to make clear that it is a spatial cutoff. $f_{\leq N}$ suggest so me a vertical cut-off: $=f$ if $|f| \leq N$ and $=N$ otherwise. This could be a nice notation when doing Calderon-Zygmund... Commented Oct 21, 2010 at 17:41
• I'm just making this up, but it seems you could be more consistent with the projection $[x^n]$ as in $[x^n]f(x)$ by doing something like $[\le N]f$: "Decompose $f=[\le N]f + [>N]f$", or if you're daring, "decompose $f=([\le N] + [>N])f$." Commented Oct 21, 2010 at 17:42

$$a^{\cdot \, n} = a\cdot a\cdots a$$ $$a^{\wedge \, n} = a\wedge a\wedge\dots\wedge a$$ $$a^{\,, \, n} = a,a,\dots,a$$ 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 $$\sin^{\cdot(-1)}x=\frac1{\sin x}$$

• 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 $sin^2(x)$ Commented Jan 24, 2013 at 22:15
• Sorry, should have read on and posted this under Blake's answer below. Commented Jan 24, 2013 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 $\sin\sin(x)$. Commented Mar 25, 2014 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. Commented Dec 16, 2014 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.) Commented May 18, 2015 at 19:25

To denote an action $\alpha: G\times X \rightarrow X$ of a group $G$ on a space $X$, there is the nice piece of notation:

$\alpha: G \curvearrowright X$

or simply

$G \curvearrowright X$ (the latter when the action is understood from the context).

E.g. you can say something like: $\rm{GL}(V) \curvearrowright V$ linearly. Or, to say that $W$ is an invariant subspace for $G \curvearrowright V$, you just write: $G \curvearrowright W$.

Another example: $\rm{Ad}:G \curvearrowright \mathfrak{g}$, and so on.

• I prefer for the arrow to be a nearly-full circle, coming from X back to X. Warning: if you put the arrow on the lower part instead of upper, it looks like you've written G twice but in two different fonts. Commented Oct 21, 2010 at 2:31

I know some people absolutely DESPISE using coordinates and components to do "tensor analysis", but sometimes there is no recourse, and then Einstein's summation convention is a big help.

• Have you ever seen Penrose's notation? It seems crazy at first, and I've never used it but it actually seems like a pretty good compromise between "abstract" notation and co-ordinates. Commented Oct 21, 2010 at 1:55
• I love both Penrose's notation (which is to a first approximation the same as the string diagrams mentioned in another answer) and the Einstein summation convention. Neither of them solves all problems; each is sometimes more efficient than the other. (In particular, I find sums of several terms get ugly quickly in string diagrams. So for proving an associativity, I'll typically try string diagrams first, but for proving a Jacobi identity, I'll go for Einstein summation.) Commented Oct 21, 2010 at 3:15

$G \circlearrowleft X$ (or $G \circlearrowright X$) to denote that $G$ acts on $X$.

Edit by A.H. :
Here are some latex definitions that produce the symbol that David Speyer describes in his comment:

\def\acts{
\hspace{.1cm}
{
\setlength{\unitlength}{.30mm}
\linethickness{.09mm}
\begin{picture}(8,8)(0,0)
\qbezier(7,6)(4.5,8.3)(2,7)
\qbezier(2,7)(-1.5,4)(2,1)
\qbezier(2,1)(4.5,-.3)(7,2)
\qbezier(7,6)(6.1,7.5)(6.8,9)
\qbezier(7,6)(5,6.1)(4.2,4.4)
\end{picture}
\hspace{.1cm}
}}


and

\def\acted{
\hspace{.1cm}
{
\setlength{\unitlength}{.30mm}
\linethickness{.09mm}
\begin{picture}(8,8)(0,0)
\qbezier(1,6)(3.5,8.3)(6,7)
\qbezier(6,7)(9.5,4)(6,1)
\qbezier(6,1)(3.5,-.3)(1,2)
\qbezier(1,6)(1.9,7.5)(1.2,9)
\qbezier(1,6)(3,6.1)(3.8,4.4)
\end{picture}
\hspace{.1cm}
}}

• I prefer $G \curvearrowright X$.
– Mark
Commented Oct 20, 2010 at 20:34
• I use a notation like that, but I have not found it in latex: the closest I can find is $G\subset X$ but with the $\subset$ turned into an arrow (so that $G$ appears as the label on the arrow, you could say) I distinctly dislike using \circlearrowleft for this! :) Commented Oct 20, 2010 at 20:35
• In hand writing, I prefer a roughly 3/4 circle which starts at about 4 oclock and ends at about 2 oclock. That way the arrow starts at X and returns there. Important note: if draw the arrow going the other way, people may mistake it for a capital G. Commented Oct 20, 2010 at 22:36
• I agree with David Speyer. You want the arrow to begin and end at $X$ and go by way of $G$. (Not that this answers the original question, which was to come up with notation that substantially improves understanding...) Commented Oct 21, 2010 at 1:39
• \lcirclearrowright in MnSymbol looks ok. Commented Oct 22, 2010 at 1:22

As mentioned in a comment, $\lfloor x\rfloor$ is much better notation than $[x]$ for denoting the greatest-integer function. Most especially since it doesn't collide with the $10^6$ other things that $[]$ is used for, e.g. the $0,1$ function Richard Borcherds mentioned.

I very much like, though haven't had much use for, the notation $n{q\atop \cdot}$ for $|GL_n(q)/B|$, pronounced "$n$ $q$-torial". Famously, it extends to a polynomial function of $q$, and when $q=1$ we have $n{1\atop \cdot} = n!$

(Oops: I left out the $/B$ the first time, thanks Jim and David.)

• Allen, that's not quite right about the cardinality of $\mathrm{GL}_n(q)$. For instance, when $q=1$ you get $0$. (Maybe you want the number of points on $G/T$?) Commented Oct 21, 2010 at 6:44
• I'm pretty sure G/B is what Allen wants. Commented Oct 22, 2010 at 13:40