With which notation do you feel uncomfortable ?

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There is a famous anecdote about Barry Mazur coming up with the worst notation possible at a seminar talk in order to annoy Serge Lang. Mazur defined $\Xi$ to be a complex number and considered the quotient of the conjugate of $\Xi$ and $\Xi$: $$\frac{\overline{\Xi}}{\Xi}.$$ This looks even better on a blackboard since $\Xi$ is drawn as three horizonal lines. 


My favorite example of bad notation is using $\textrm{sin}^2(x)$ for $(\textrm{sin}(x))^2$ and $\textrm{sin}^{1}(x)$ for $\textrm{arcsin}(x)$, since this is basically the same notation used for two different things ($\textrm{sin}^2(x)$ should mean $\textrm{sin}(\textrm{sin}(x))$ if $\textrm{sin}^{1}(x)$ means $\textrm{arcsin}(x)$). It might not be horrible, since it rarely leads to confusion, but it is inconsistent notation, which should be avoided in general. 


I never liked the notation ${\mathbb Z}_p$ for the ring of residue classes modulo p. At one point, it confused the hell out of me, and this confusion is easily avoided by writing $C_p$, $C(p)$ or ${\mathbb Z}/p$. 


I personally hate the notation $x \vert y$, for "$x$ divides $y$". Of course, I'm used to reading it by now, but a general principle I follow and recommend is: Never use a symmetric symbol to denote an asymmetric relation! 


Physicist will hate me for this, but I never liked Einstein's summation convention, nor the famous bra ($\langle\phi$) and ket ($\psi\rangle$) notation. Both notations make easy things look unnecessarily complicated, and especially the braket notation is no fun to use in LaTeX. 


Mathematicians are really quite bad when it comes to notation. They should learn from programming langauges people. Bad notation actually makes it difficult for students to understand the concepts. Here are some really bad ones:



Writing a finite field of size $q$ as $\mathrm{GF}(q)$ instead of as $\mathbf{F}_q$ always rubbed me the wrong way. I know where it comes from (Galois Field), and I think it is still widely used in computer science, and maybe in some allied areas of discrete math, but I still dislike it. 


The notation ]a,b[ for open intervals and its ilk. Sorry, Bourbaki. 


My candidate would be the (internal) direct sum of subspaces $U \oplus V$ in linear algebra. As an operator it is equivalent to sum but with the side effect of implying that $U \cap V = \lbrace 0\rbrace$. Whenever I had a chance to teach linear algebra I found this terribly confusing for students. 


I think composition of arrows $f:X\to Y$ and $g:Y\to Z$ should be written $fg$ not $gf$. First of all it would make the notation $\hom(X,Y)\to\hom(Y,Z)\to \hom(X,Z)$ much more natural: $\hom(E,X)$ should be a left $\hom(E,E)$ module because $E$ is on the left :) Secondly, diagrams are written from left to right (even stronger: Almost anything in the western world is written left to right). And i think the strange (1) needed when shifting complexes is an effect of this twisted notation. 


I get very frustrated when an author or speaker writes "Let $X\colon= A\sqcup B$..." to mean:
If they just meant "form the disjoint union of $A$ and $B$" this would be fine. But I've seen speakers later use the fact that $A$ and $B$ are disjoint, which was never stated anywhere except as above. You should never hide an assumption implicitly in your notation. 


As Trevor Wooley used to always say in class, ``Vinogradov's notation sucks....the constants away." For those who don't know, Vinogradov's notation in this context is $f(x)\ll g(x)$ meaning $f(x) = O(g(x)).$ (if you prefer bigO notation, that is). 


I rather dislike the notation $$\int_{\Omega}f(x)\,\mu(dx)$$ myself. I realize that just as the integral sign is a generalized summation sign, the $dx$ in $\mu(dx)$ would stand for some small measurable set of which you take the measure, but it still rubs me the wrong way. Is it only because I was brought up with the $\int\cdots\,d\mu(x)$ notation? The latter nicely generalizes the notation for the Stieltjes integral at least. 


I don't like (but maybe for a bad reason) the notation $F\vdash G$ for $F$ is left adjoint to $G$. Any comment ? 





The term "symplectic group" used to mean the group $U(n,{\mathbb H})$. It's as if people called $U(n)$ and $GL(n,{\mathbb R})$ by some single name. 


My personal pet peeve of notation HAS to be algebraists writing functions on the right a la Herstein's "Topics In Algebra". I don't know why they do it when everyone else doesn't. I think one of them got up one day and decided they wanted to be cooler then everyone else, seriously... 


A cute idea but for which I have yet to find supporters is D. G. Northcott's notation (used at least in [Northcott, D. G. A first course of homological algebra. Cambridge University Press, London, 1973. xi+206 pp. MR0323867) for maps in a commutative diagram, which consists in enumerating the names of the objects placed vertices along the way of the composition. Thus, if there only one map in sight from $M$ to $N$, he writes it simply $MN$, so we has formulas looking like $$A'A(ABB") = A'ABB" = A'B'BB" = 0.$$ He also writes maps on the right, so his $$xMN=0$$ means that the image of $x$ under the map from $M$ to $N$ is zero. I would not say this is among the worst notations ever, though. 


Students have big difficulties when first confronted with the $o(\cdot)$ and $O(\cdot)$ notation. The term $o(x^3)$, e.g., does not denote a certain function evaluated at $x^3$, but a function of $x$, defined by the context, that converges to zero when divided by $x^3$. 


I have struggled with 'dx'. I've spent years trying to study every different approach to calculus that I could find to try and make sense of it. I read about the limit definitions in my first book, vector calculus with them as pullbacks of linear transformations or flows/flux, differential forms from the bridge project, kforms, nonstandard analysis which enlarges $\mathbb{R}$ to give you infinitesimals (and unbounded numbers) but the same first order properties and lets integral be defined as a sum, constructive analysis using a monad to take the closure of the rationals to give reals... but I am still just as confused as ever, I understand that the mathematical notation doesn't have a compositional semantics but still don't really get it  one of the problems is despite not really understanding it, or having any abstract definition of it.. I can still get correct answers and I really hope this doesn't become a theme as I study more topics in mathematics. 


p < q as in "the forcing condition p is stronger than q". 


I hate the short cut $ab$ for $a\cdot b$. Everyone get used to it, BUT it creates very deep problem with all other notation; say you never can be sure what $f(x+y)$ or $2\!\tfrac23$ might be... Also in modern mathematics people do not multiply things too often, so it does not have sense to make such a short cut. Yet the shortcut $x^n$ is really bad one. One can not use upper indexes after this. It would be easy to write $x^{\cdot n}$ instead. 

