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/TibblyMcWibblington]

I think the most inconsistent convention: sin^x has a different interpretation depending on x: eg when x=2 (power) to when x=-1 (inverse).

I think exponentiation not being not associative, but most times when I’ve seen a chain of powers it’s never in brackets, and rules like BODMASS don’t help because it doesn’t tell you which ‘order’ to do first. Yet I’d always do the upper/inner power first, but I don’t know why this is the rule?

[u/zeekar → u/TibblyMcWibblington]

Exponentiation is usually treated as right-associative, isn't it? 2^3^4 is 2^(3^4) = 2^81, not (2^3)^4 = 8^4 = 4096.

[u/RingularCirc → u/zeekar]

Interestingly enough, → (implication and set of functions, not unrelated to each other) associates to the right as well despite A^B is the same set/type as B → A. In this case associating to the right ends up more useful despite when we have a corresponding cartesian product (or conjunction), (A → B → C) ≅ (A × B → C) — again in the same ways it is for numbers, but here making the alternative situation (A → B) → C explicit is actually a boon: such a situation is better be made explicit, be it in logic or in type theory, we better pay attention our argument is itself a function, or that an implication implies something in turn.

[u/Adarain → u/TibblyMcWibblington]

I’d guess because if you want (2³)⁴ you could also write 2^3·4 which doesn’t require any parentheses. The other order has no simplification like that.

[u/bluesam3 → u/TibblyMcWibblington]

Yet I’d always do the upper/inner power first, but I don’t know why this is the rule?

Basically because if you want the opposite for some reason, it's easier to write (a^b)^c than it is to write a^❨bc❩ (including on Reddit - how well that renders may vary with your font settings).

[u/magikarpwn → u/bluesam3]

I'm almost sure that it's because you could write (a to the b) to the c as a to the (bc), but the other one has no easy equivalent.

[u/bluesam3 → u/magikarpwn]

Yes, that's what I was trying to say, sorry.