While this may be a fruitless pursuit of anecdotes, I still ask: what is the strangest (or most blatantly wrong (at least in the eyes of common notation)) mathematical notation you have ever seen?
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$\begingroup$There is an old story about Lang and Mazur, Mazur tried to get Lang attention by using the worst notation possible. He wrote Xi conjugated over Xi, which looks like:
$$\frac{\overline{\Xi}}{\Xi}$$
P.S. You can read the story, narrated by Paul Vojta, in the AMS Notices issue dedicated to Lang:AMS Nottices Lang
It is on pages 546-547.
$\endgroup$ 3 $\begingroup$The single worst use of mathematical notation I have ever seen was in a set of lecture notes in which the author wanted to construct a sequence of equivalence relations, each one ($\equiv_n$) derived from the previous one ($\equiv_{n-1}$). After $i_0$ iterations of this procedure, the construction has no more work to do, and the sequence has converged to a certain equivalence relation $\equiv$ with desirable properties. The notes contained this formula: $$\equiv_{i_0+1}=\equiv_{i_0}=\equiv$$
I regret that I did not make a note of the source.
$\endgroup$ 7 $\begingroup$The Landau big-$O$ notation is extremely strange.
One writes $$f(x) = O(g(x))$$ which looks like $f$ is the composition of $O$ and $g$, but it is nothing of the sort. Is $O()$ an operator that can be applied to any term? Can I write $$O(x^2) = O(x^3)$$ or $O(x^2) = 2x^2$? Not normally.
It is easily confused with a whole family of similar notations for similar notions; computer programmers regularly talk about $O(n)$ algorithms when they mean $\Omega(n)$ algorithms, for example. This is exacerbated because someone decided that instead of using mnemonic abbreviations, it would be a good idea arbitrarily assign every possible variant of the letter ‘o’ in naming them. Then when they ran out of letter O’s they used $\Theta$, seemingly because it looks enough like an O that you might confuse it with one.
It is written with an $=$ even though the relation is asymmetric! We have both $x=O(x^2)$ and $x=O(x^3)$ although $O(x^2)$ and $O(x^3)$ are not the same, and we have both $1 = O(x)$ and $x = O(x)$ even though $1\ne x$.
I took a long time to get used to derivative of integrals like this $$\frac{\partial}{\partial x}\int_{x_0}^x f(x,y) \ dx$$
It's just too much $x'$s in the same formula, and each one has a different meaning. Nevertheless, its common to see people writing down this way.
$\endgroup$ 5 $\begingroup$The usage of pi:
$\pi$ is a constant. $\pi(x)$ is the prime counting function. $\prod(x)$ is a product of a sequence.
$\endgroup$ 7 $\begingroup$From a proof that convergence a.e. implies convergence in measure for $\mu(\Omega)<\infty$:$$\bigcup_{r\geq 1}\bigcap_{n\geq 1}\bigcup_{j\geq n}\{|{f_j-f}|>\frac{1}{r}\}=\{\omega:f_j(\omega) \not \to f(\omega)\}$$
Also, labeling graphs of functions as $f(x)$ (which I end up still doing to my undergraduates, who are bored when I mention my reservations about it), $\coprod$, "Random Variable," calling a domain the preimage but switching it to a connected open set in complex talk, etc. etc. etc.
$\endgroup$ $\begingroup$$$\large{\prod_{n = 1}^3 \mathbb{R} = \mathbb{R}^3}$$
Edit: Apparently this is common notation. MJD suggests a better example:
$$\large{\prod_{n = 1}^3 S \neq S^3}$$
$\endgroup$ 7 $\begingroup$How about using pairs of letters like $r,s$ or $u,v$ , or $m,n$ when writing on a blackboard? Unless you're extremely careful, the two in any pair get very easily confused with each other. Or, when you're told you have two collections of objects ( with maybe some additional propreties ) , say $S,X$ , and then you have that $a$, or worse $x$ is an element in $S$. Isn't it so much better to just say $s$ is in $S$, and $x$ is in $X$ ; isn't an element $s$ in $S$ better than any other letter?
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