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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?

Andrés E. Caicedo
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JasonW
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    There is a thread about this on MathOverflow also. There, as here, the top scorer is Mazur's $\overline{\Xi}\over\Xi$. – MJD Mar 01 '14 at 04:39
  • This is not an answer, but $dx$ and $dy$ are the best notation for confusing students. – Sawarnik Mar 01 '14 at 05:02
  • @Sawarnik do you mean contrary to using $\Delta x$ or $\Delta y$? – JasonW Mar 01 '14 at 05:05
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    @user76061 I meant that unlike $f'(x)$, the imprecise $dx$.. confuses people. Why the derivative became a quotient, what is a differential, why did $dx$ come in the integral, among other things. – Sawarnik Mar 01 '14 at 05:11
  • "A woman stepped forward and asked, / What is the strangest day? // Tuesday, the Master replied." —Kehlog Albran, The Profit –  Mar 03 '14 at 22:39
  • Not the worst notation, but still rather strange, is $n!!$ to mean $n(n-2)(n-4)\cdots3\cdot 1$ when $n$ is odd and $n(n-2)(n-4)\cdots4\cdot 2$ when $n$ is even. It should clearly mean $(n!)!$ instead, but the iterated factorial is so rare that the appropriation of $n!!$ for something else does not usually seem to cause confusion. – MJD Mar 14 '15 at 16:37
  • In his essay “How to Write Mathematics”, P.R. Halmos says “A mathematician's nightmare is a sequence $n_\epsilon$ that tends to $0$ as $\epsilon$ becomes infinite.” – MJD Dec 09 '15 at 21:34

8 Answers8

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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.

N. S.
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  • Objective achieved! – JasonW Mar 01 '14 at 03:21
  • Just curious: What did Mazur use the notation for (except for getting Lang's attention, of course)? –  Mar 01 '14 at 03:25
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    @SanathDevalapurkar Just for getting Lang's attention. I posted the link to the story, on short Lang was criticizing a lot the notations Mazur was using, so they prepared a t-shirt for Lang, and they were waiting for Lang to say the magic words... But for some reason, that particular day Lang was quiet... – N. S. Mar 01 '14 at 03:29
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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.

MJD
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The Landau big-$O$ notation is extremely strange.

  1. 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.

  2. 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.

  3. 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 $1\ne x$.

MJD
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  • It probably makes more sense if you give $O$ a variable to bind, e.g. $Ox(x^2)$ could denote the equivalence class of functions associated with the mapping $x \in \mathbb{R} \mapsto x^2 \in \mathbb{R}$. – goblin GONE Mar 01 '14 at 05:28
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    There is still no excuse for the $=$ sign. If it were instead a $\in$ sign, or some other asymmetric sign like $\prec$, instead, my objection would only be a minor quibble. – MJD Mar 03 '14 at 21:50
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    @MJD I've always treated $O$ as a function which when given a function $f$ returns the set of functions asymptotically upper-bounded by $f$, because the = in that context is absolutely appalling! Makes much more sense to say $\lambda x.4x^2-3 \in O(\lambda x. x^2)$. Regarding your point 2, I do believe $\Theta$ is gaining much more popularity in computer science circles these days, at least. – Ray Toal Mar 04 '14 at 06:08
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    On the equals sign for O notation, Knuth indicates that (at least) he thinks of (say) $O(x^2)$ as "something that is at most a constant times $x^2$"—see his letter "Teach Calculus with Big O" (blog repost, PDF of AMS) — and of the equals sign as the English "is": "Aristotle is a man, but a man isn’t necessarily Aristotle". He's channelling de Bruijn: see pp. 5–7 of Asymptotic Methods in Analysis. – ShreevatsaR Mar 04 '14 at 20:14
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    In particular, Knuth (and de Bruijn before him, and perhaps even Bachmann and Landau onwards) would write $O(x^2) = O(x^3)$ (for $x \to \infty$, not $x \to 0$ of course), with the meaning that "something that is at most a constant times $x^2$ is also at most a constant times $x^3$". E.g. de Bruijn writes, on page 6 of his book, the equation $O(x) + O(x^2) = O(x) \quad (x \to 0).$ At least de Bruijn agrees that it is abuse of notation and that the equals sign is a poor choice because it "suggests symmetry, and there is no such symmetry". But it's customary. – ShreevatsaR Mar 04 '14 at 20:25
  • @MJD, yes I agree. – goblin GONE Mar 06 '14 at 00:23
  • @RayToal, I agree that that $λx.x^2−3∈O(λx.x^2)$ is the sensible way of writing it. Or even better, $O(\lambda x.x^2-3)=O(\lambda x.x^2),$ since then we recover symmetry. Or even better, $Ox(x^2-3)=Ox(x^2)$, since this is more brief. In fact, I would suggest abbreviating this as $x^2-3 \equiv x^2 ;\mathrm{lan} ; x$, where $\mathrm{lan}$ stands for Landau. – goblin GONE Mar 06 '14 at 00:31
  • @ShreevatsaR Since the English "is" is not symmetric, I find it unfortunate that people have chosen to use $=$ in its place. When we say "Aristotle is a man" we abbreviate that in logic as $Aristotle \in SetOfAllMen$, so even in this case $\in$ is preferred to $=$ IMHO. Of course mathematical notation has numerous conventions that defy common sense. Don't even get me started on the unfortunate use of $\sin^2{x}$ to mean $(\sin{x})^2$ instead of the obvious $\sin(\sin{x})$. I believe I was 25 years old when I first discovered this was not function iteration when trig was involved! – Ray Toal Mar 06 '14 at 02:44
  • I will note that quite a few people do use the membership notation, and I feel like its usage is on the upswing for exactly these sorts of reasons. – Steven Stadnicki May 05 '14 at 21:51
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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.

Integral
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    I think this is actually meaningless. You cannot have the same variable in the integrand as in one of the limits of integration. – Steven Gubkin Mar 01 '14 at 04:21
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    I couldn't agree more. – Integral Mar 01 '14 at 04:25
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    If you are careful, you can, but there is no excuse to require the care. The $x$ in $f(x,y)$ is bound by the integral, while the $x$ upper limit is free, so that is what the derivative is taken with respect to. The result is then $f(x,y)$ with $x$ free. – Ross Millikan Mar 01 '14 at 05:07
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    The integration variable could be named anything; why on earth would you pick the one letter that would cause the most confusion. – mjqxxxx Mar 03 '14 at 22:33
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    @mjqxxxx: Because it's consistent with how one usually notates antidifferentiation; in fact, the integral above can be viewed not as intending to be a definite integral, but as intending to be an anti-derivative with a particular constant of integration determined by the lower bound $x_0$. –  May 05 '14 at 21:52
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The usage of pi:

$\pi$ is a constant. $\pi(x)$ is the prime counting function. $\prod(x)$ is a product of a sequence.

qwr
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    $\pi$ is often a homomorphism or projection map as well. – Steven Gubkin Mar 01 '14 at 04:21
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    $\phi = \frac{1 + \sqrt 5}{2}$, $\phi (n)$ is the totient function, $\Phi = \frac{1 - \sqrt5}{2}$, $\Phi_n$ is the $n$th cyclotomic polynomial, $\phi$ is often used in logic (e.g. $\psi \implies \phi$)... – MT_ Mar 01 '14 at 04:50
  • $\phi(\alpha)$ is also used in NF literature as a special operation on cardinals. If you're writing about the proof theory surrounding NF's big cardinals, it gets ugly. That's why I've pledged to overwork $\xi$ in my own writing... – Malice Vidrine Mar 01 '14 at 07:44
  • $\pi$ is also often a permutation. – ShreevatsaR Mar 04 '14 at 19:45
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    isn't used for the fundamental group as well? – MphLee Mar 06 '14 at 07:25
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    In statistics, we use $\pi$ as a variable to represent the population proportion. For example, we might be "95% confident that $\pi$ is between $0.41$ and $0.43$." Without knowing that $\pi$ is being abused, that sentence would make absolutely no sense. – wchargin Nov 17 '14 at 03:36
  • The usage of $x$ is even worse. It can represent anything. – Jackozee Hakkiuz Oct 31 '18 at 23:21
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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.

Darrin
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$$\large{\prod_{n = 1}^3 \mathbb{R} = \mathbb{R}^3}$$

Edit: Apparently this is common notation. MJD suggests a better example:

$$\large{\prod_{n = 1}^3 S \neq S^3}$$

MT_
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  • Where has that been used? I've never seen this before (I've seen $(\mathbb{R}^1)^3=\mathbb{R}^3$), though I can understand the logic behind it. –  Mar 01 '14 at 03:43
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    Why is this strange? $\mathbb{R}^3 = \mathbb{R}\times\mathbb{R}\times\mathbb{R}$. It's is a cartesian product, but still a product. Its quite natural to write like the way you showed. – Integral Mar 01 '14 at 03:47
  • Maybe I just don't work with this a lot so it seems weird to me. Usually I see it as $\mathbb{R} \times \mathbb{R} \times \mathbb{R}$ -- the use of product notation of $n = 1$ to $3$ just seems a bit unnatural to me. – MT_ Mar 01 '14 at 04:05
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    This is quite natural. The one you should have mentioned is $$\prod_{n=1}^3 S\color{red}{\ne} S^3.$$ – MJD Mar 01 '14 at 04:23
  • @MJD how does that work? – JasonW Mar 01 '14 at 04:28
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    $S$ is the circle, $S^2$ is the sphere (that is, the two-dimensional manifold that is the boundary of the ball in $\Bbb R^3$), $S^3$ is the 3-sphere, which is the 3-dimensional boundary of the ball in $\Bbb R^4$. But unfortunately $S^2 \ne S\times S $; the latter is the torus, sometimes written $T^2$, just to really confuse matters. – MJD Mar 01 '14 at 04:44
  • Here is a MathOverflow answer with a particularly horrible assertion of exactly this type: “I didn't know that, but I did know this: we cannot have $S^2=S×S$ for any topological space $S$.” Here the $S^2$ on the left is the sphere, and the other $S$es are arbitrary spaces. Terry Tao remarked “ things considered, perhaps "S" is not the best name for the topological space for this assertion.” – MJD Mar 22 '14 at 19:10
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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?

user99680
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