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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?

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    $\begingroup$ There is a thread about this on MathOverflow also. There, as here, the top scorer is Mazur's $\overline{\Xi}\over\Xi$. $\endgroup$ – MJD Mar 1 '14 at 4:39
  • $\begingroup$ This is not an answer, but $dx$ and $dy$ are the best notation for confusing students. $\endgroup$ – Sawarnik Mar 1 '14 at 5:02
  • $\begingroup$ @Sawarnik do you mean contrary to using $\Delta x$ or $\Delta y$? $\endgroup$ – user76061 Mar 1 '14 at 5:05
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    $\begingroup$ @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. $\endgroup$ – Sawarnik Mar 1 '14 at 5:11
  • $\begingroup$ "A woman stepped forward and asked, / What is the strangest day? // Tuesday, the Master replied." —Kehlog Albran, The Profit $\endgroup$ – user856 Mar 3 '14 at 22:39
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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.

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  • $\begingroup$ Objective achieved! $\endgroup$ – user76061 Mar 1 '14 at 3:21
  • $\begingroup$ Just curious: What did Mazur use the notation for (except for getting Lang's attention, of course)? $\endgroup$ – user122283 Mar 1 '14 at 3:25
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    $\begingroup$ @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... $\endgroup$ – N. S. Mar 1 '14 at 3:29
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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.

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  • $\begingroup$ Could this actually be considered correct? $\endgroup$ – user76061 Mar 1 '14 at 4:31
  • $\begingroup$ I believe it is correct. $\endgroup$ – MJD Mar 1 '14 at 4:31
  • $\begingroup$ So it treats the equivalence like a variable? $\endgroup$ – user76061 Mar 1 '14 at 4:32
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    $\begingroup$ I'm not sure what you mean. An equivalence relation, as any relation, is a set. The $=$ signs assert that the sets are equal. $\endgroup$ – MJD Mar 1 '14 at 4:33
  • $\begingroup$ Wait. It just clicked, I misread. Sorry $\endgroup$ – user76061 Mar 1 '14 at 4:34
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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 $1\ne x$.

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  • $\begingroup$ 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 $x \in \mathbb{R} \mapsto x^2 \in \mathbb{R}$. $\endgroup$ – goblin GONE Mar 1 '14 at 5:28
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    $\begingroup$ 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. $\endgroup$ – MJD Mar 3 '14 at 21:50
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    $\begingroup$ @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. $\endgroup$ – Ray Toal Mar 4 '14 at 6:08
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    $\begingroup$ 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. $\endgroup$ – ShreevatsaR Mar 4 '14 at 20:14
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    $\begingroup$ 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. $\endgroup$ – ShreevatsaR Mar 4 '14 at 20:25
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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.

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    $\begingroup$ I think this is actually meaningless. You cannot have the same variable in the integrand as in one of the limits of integration. $\endgroup$ – Steven Gubkin Mar 1 '14 at 4:21
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    $\begingroup$ I couldn't agree more. $\endgroup$ – Integral Mar 1 '14 at 4:25
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    $\begingroup$ 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. $\endgroup$ – Ross Millikan Mar 1 '14 at 5:07
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    $\begingroup$ The integration variable could be named anything; why on earth would you pick the one letter that would cause the most confusion. $\endgroup$ – mjqxxxx Mar 3 '14 at 22:33
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    $\begingroup$ @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 $x_0$. $\endgroup$ – user14972 May 5 '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.

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    $\begingroup$ $\pi$ is often a homomorphism or projection map as well. $\endgroup$ – Steven Gubkin Mar 1 '14 at 4:21
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    $\begingroup$ $\phi = \frac{1 + \sqrt 5}{2}$, $\phi (n)$ 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. $\psi \implies \phi$)... $\endgroup$ – MCT Mar 1 '14 at 4:50
  • $\begingroup$ $\phi(\alpha)$ 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... $\endgroup$ – Malice Vidrine Mar 1 '14 at 7:44
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    $\begingroup$ isn't used for the fundamental group as well? $\endgroup$ – MphLee Mar 6 '14 at 7:25
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    $\begingroup$ 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. $\endgroup$ – wchargin Nov 17 '14 at 3:36
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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.

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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:

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

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  • $\begingroup$ 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. $\endgroup$ – user122283 Mar 1 '14 at 3:43
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    $\begingroup$ 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. $\endgroup$ – Integral Mar 1 '14 at 3:47
  • $\begingroup$ 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. $\endgroup$ – MCT Mar 1 '14 at 4:05
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    $\begingroup$ This is quite natural. The one you should have mentioned is $$\prod_{n=1}^3 S\color{red}{\ne} S^3.$$ $\endgroup$ – MJD Mar 1 '14 at 4:23
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    $\begingroup$ $S$ 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. $\endgroup$ – MJD Mar 1 '14 at 4:44
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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?

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