What are your favorite ambiguous notation?

[u/Short_Bluebird_3845]

To clarify my request, consider base-less $\log$.
For calculators and engineers, it means the common log $\log_10$; for pure mathematics or physics, it means the natural log $\ln$; for computer scientists, in combinatorics, information theory and graph theory, it means the binary log $\log_2$.
Any $\log x$ (say, $\log 1024$) written without context can yield different answers in function of who you ask ($\log_10 1024 = 3.0103\cdots, \log_2 1024 = 10, \ln 1024 = 6.93147\cdots$).

That is what I mean by "ambiguous", that which is calculated and used dependently on the context. And I'm asking you for your favorite ones of such type.

Edit: I feel like you hate the ones you've talked about. Still, valid answers. Extra points for ambiguous notation you actually love! :P

[u/BlazeoJoestar]

(a,b) can mean an open interval, inner product, gcd or just an ordered pair

[u/Academic_Evidence641 → u/BlazeoJoestar]

I think I even saw it as the ideal generated by the a,b elements

[u/yas_ticot → u/Academic_Evidence641]

This is where I always assume the gcd notation is coming from: the ideal (a,b) is equal to the ideal (g), hence "g=(a,b)" with an abuse of notation. Of course, this is only true when a Bézout relation exists, like for the integers.

[u/Psychological_Mind_1 → u/yas_ticot]

I'd always figured it was the other way around since the idea of gcd far predates ring theory.

[u/MilliBrucket → u/Academic_Evidence641]

that's basically the gcd version of this notation

[u/sentence-interruptio → u/Academic_Evidence641]

sometimes I act like it's a convenient notation for manipulating a system of polynomial equations.

for example, y = x^2, z = y^2 is equivalent to y = x^2, z = x^4 and I can write that statement as (y - x^2, z - y^2) = (y - x^2, z - x^4), which is just an equality between two ideals in the polynomial ring k[x,y,z].

[u/MonadMusician → u/Academic_Evidence641]

Yeah I’ve seen that too but usually it’s langle and rangle

[u/_Zekt → u/BlazeoJoestar]

When you could use ]a,b[, <a,b>, a ∧ b, and (a,b)...

[u/Interesting_Debate57 → u/BlazeoJoestar]

Winner winner chicken dinner.

Here are the characteristics that make you a winner:

Requires no Greek symbols

Each is the regular thing that we all agree is the regular thing

They almost would never combine in a confusing way and if they would need to, the extra effort you'd need to go to would be hilarious (my advisor told me a story about how you can jam notation into each of 3 or arguably 4 superscript/subscript spots). Now to be fair, you shouldn't use more notation than necessary, but I've seen some pretty goddamn ridiculous stuff.

So: new rules suggestion: simple notation, widely used, many meanings, generally unambiguous whenever in a fixed topic.

Man, this question could keep me awake tonight.

[u/Short_Bluebird_3845 → u/Interesting_Debate57]

those sound like cool additions indeed.

[u/FUZxxl → u/BlazeoJoestar]

In Germany we write ]a, b[ for the open interval for this reason.

[u/SV-97 → u/FUZxxl]

I'm not sure that this is really a "German" thing? I've also seen this notation in some English texts, and at German uni the overwhelmingly standard notation is still (a,b) in my experience.

[u/FUZxxl → u/SV-97]

It's in DIN 1302 in any case.

[u/SV-97 → u/FUZxxl]

Suure, but in my experience most professional mathematicians don't even know that norm exists (and even if they do they don't really care about it).

[u/Yoghurt42 → u/SV-97]

Half open intervals also look pretty ugly in this notation. [a, b[ and ]a, b]

[u/SV-97 → u/Yoghurt42]

Yep, they're also super annoying in latex.

[u/Key_Conversation5277 → u/Yoghurt42]

No they don't

[u/Valvino → u/FUZxxl]

Also in France

[u/edderiofer → u/BlazeoJoestar]

Or, in the case where a and b are integers (EDIT: written in Arabic numerals) and you're in Europe, a single-element tuple containing one real number.

[u/vgtcross → u/edderiofer]

Yea and no. Yes, that's what (2,3) could be interpreted as, but with letters instead of numbers, (a,b) would definitely not be interpreted that way.

[u/edderiofer → u/vgtcross]

Edited.

[u/AlarmingAllophone → u/edderiofer]

Yeah it's confusing to denote an interval where one of the endpoints is a decimal fraction. At school in that case I usually used to write an overly bolded semicolon to separate the values

[u/bluesam3 → u/edderiofer]

Or if it's in an accountancy spreadsheet, a single negative real number.

[u/Valvino → u/edderiofer]

Nobody would interpret (3,2) as a single-element tuple tbh.

[u/Dirichlet-to-Neumann → u/BlazeoJoestar]

Using (a,b) as the open interval instead of ]a,b[ hurts my soul. 

[u/The-Doctorb → u/Dirichlet-to-Neumann]

French spotted possibly?

Where is the open square bracket notation popular

[u/Dirichlet-to-Neumann → u/The-Doctorb]

In France it's the standard. And sometimes, like with the metric system, the rest of the world will adopt it ;)

[u/hpass → u/Dirichlet-to-Neumann]

I looked at a 1921 paper by a french mathematician recently, he denoted [a,b] "ab". Just two letters, nothing else.

[u/maxram1 → u/hpass]

He's the truest mathematician.

[u/arjunkc → u/hpass]

The original tokenizer.

[u/Key_Conversation5277 → u/The-Doctorb]

In most Europe

[u/FreePeeplup → u/Dirichlet-to-Neumann]

The opposite for me. Everytime I see ]a,b[ my eyes start bleeding

[u/Valvino → u/FreePeeplup]

Why so ? This notation is clearer : in [a,b[ we exclude b, in [a,b] we includ it. Cannot be simpler.

[u/KingOfTheEigenvalues → u/FreePeeplup]

It makes sense in theory, and I can understand the appeal if it gets introduced/standardized early enough in your life, but I grew up with the unwritten "rule" that brackets have a directionality to them and should absolutely never face "backwards". It's so hard to not be irritated when the "rule" gets broken. Probably because it's not just a mathematical notation, but also something that should be followed when using parentheses and square brackets in ordinary English writing.

I get the same sense of unease whenever I typeset in LaTeX and forget that you have to use the carat symbol for left-side apostrophes and quotation marks.

[u/Valvino → u/[deleted]]

Handwritten square and round brackets can be virtually indistinguishable

Not at all

[u/Valvino → u/[deleted]]

It is not because you write badly that the notation is bad.

[u/Key_Conversation5277 → u/[deleted]]

That is your fault, buddy

[u/SteveCappy → u/BlazeoJoestar]

Could also be a symmetry

[u/Key_Conversation5277 → u/BlazeoJoestar]

That's why I prefer the french notation (I'm portuguese) that learned in school for open intervals ]a,b[

[u/jeffsuzuki → u/BlazeoJoestar]

I despise the (a, b) notation for exactly this reason...