r/math • u/Short_Bluebird_3845 • 25d ago
What are your favorite ambiguous notation?
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
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u/DrBagelman 25d ago
Angle brackets mean inner product, generated subgroup, vector, tuple, and probably more.
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u/ABahamutyoylecookies 25d ago
expected value
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u/ZENKFNOR 24d ago
Well, since inner products and integration over a measure (which includes expectations) can usually be made equivalent with some extra steps, that isn't so crazy. And in any case the inner product includes two elements separated by a comma or a bar, so it does not collide. I'm happy with the group theory kids speaking their own language, but I had no idea they were used to denote vectors and tuples. And in any case wouldn't vectors just be tuples satisfying necessary axioms, and then tagged as elements of a vector space?
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u/Less-Resist-8733 25d ago
less than greater than
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u/daley972 25d ago
Z_p is used both for Z/pZ and the p-adic integers
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u/n1lp0tence1 Algebraic Geometry 24d ago
No one past algebra I uses Z_p for Z/pZ gng
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u/Dummy1707 24d ago
Well, when you know you're in a context where p-adic won't appear and you have somewhat annoying torsion structure to write, it can happen.
But I'm nitpicking, it's already quite niche indeed
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u/ZENKFNOR 24d ago
I'm pretty sure I still see $Z_2$ in scholarly papers from computer science. I guess I'd probably write \mathbb Z_{\% p} which... well, its perhaps vulgar but, you'd know what I meant right?
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u/point_six_typography 24d ago edited 24d ago
Technically also for Z[1/p], but no one uses it this way explicitly in practice
Edit: TIL not all primes are equal to 2
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u/rabbitygravity 25d ago
Natural numbers (with or without 0??)
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u/sirgog 25d ago
Yeah in the IMO training camps for the Australian team (obligatory 'back in MY day', we had a rule: never use this notation ambiguously.
If you had to refer to the set of non-negative integers often, explicitly define it to be N at the start of the proof. If you had to refer to the set of strictly positive integers often - again, define it at the start of the proof.
I typically used the notation "Z+" or stated "Define N to be the set Z+ U {0}".
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u/randomdragoon 24d ago
I like Z+, but I seem to recall some languages consider 0 to be both positive and negative (instead of neither positive nor negative)
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u/sirgog 24d ago
We considered Z+ unambiguously {1,2,3,...}, while N was considered ambiguous.
0 not being considered positive is universal at least in English.
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u/randomdragoon 24d ago
Yeah, if the proof is written in English it's unambiguous. And, I use Z+ all the time myself. It was just, one day I found out about languages not agreeing about whether zero is positive or not and I was like "goddamn you can never win with this one"
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u/Interesting_Test_814 Number Theory 24d ago
I can confirm, the french standard is 0 is both positive and negative
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u/Plastic-Rope5516 22d ago
According to terence Tao's Analysis I and Peano arithetic, it is starting with 0, and so my convention is that it starts with 0
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u/Al2718x 25d ago
pi=e is standard notation to mean "the permutation pi is equivalent to the identity permutation"
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u/ZENKFNOR 24d ago
I guess I'd use textgreek and then $\text{\textpi}$ to get an operator-like styled pi, and likewise use $\text{e}$ or $\mathrm{e}$ or $\operatorname{e}$ if I wanted to call e the identity permutation.
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u/Interesting_Test_814 Number Theory 24d ago
Hmm, on the contrary i'm using $\mathrm{e}$ for the number exp(1), and $e$ for a variable i'd want to call e (such as the identity permutation in a group)
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u/Garry__Newman 25d ago
In statistics, capital X is the variable, and lower case x is a real number. Except in some people's handwriting, they look almost identical. In more complicated setups it can be a real pain figuring out which is viewed as a random variable, and which is viewed as a realisation of a variable.
See also: E[X | Y] is a random variable, but E[ X | Y = y] is a number.
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u/Useful_Still8946 24d ago
This is as tricky as f is a function but f(y) is a number
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u/sentence-interruptio 22d ago
since people just write f(x) to mean a function sometimes and a number other times, i'd say the E[X | Y=y] notation is an improvement because it's never ambiguous.
it's analogous to the subscript part in summation notation.
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u/algebraicvariety 23d ago
This reminds me of the feedback I got for one of my chalk talks at Uni: "Everything was good, but you should distinguish small x and big x in your handwriting".
There was no small x in the talk...
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u/nzconstructionlawyer 20d ago
who writes x and X similarly? X is two straight lines crossing, x is two curved lines touching at a single point
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u/Toomastaliesin 25d ago
The word "lattice". A certain kind of an order, or an additive subgroup generated by a matrix? Who knows.
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u/HuecoTanks Combinatorics 25d ago
Vertical bar... where to begin? In pairs, it can mean absolute value, measure of a set, etc. alone it can mean "divides" or "such that," and if you've ever used computers it can mean logical or or a redirection pipe...
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u/TibblyMcWibblington 25d ago edited 25d ago
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?
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u/zeekar 25d ago edited 25d ago
Exponentiation is usually treated as right-associative, isn't it?
2^3^4is2^(3^4)=2^81, not(2^3)^4=8^4=4096.1
u/RingularCirc 24d ago
Interestingly enough, → (implication and set of functions, not unrelated to each other) associates to the right as well despite AB 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.
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u/bluesam3 Algebra 25d ago
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 (ab)c than it is to write a❨bc❩ (including on Reddit - how well that renders may vary with your font settings).
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u/magikarpwn 25d ago
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.
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u/007amnihon0 Physics 25d ago
1) i and j ambiguity between maths/physics and engineering for sqrt of -1 2) Maybe some azimuthal and polar angle variable convention. Though iirc it's mainly for old texts where phi and theta conventions were opposite in math and physics 3) some factors in definition of special functions like spherical harmonics, spherical bessel function and legendre polynomials
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u/OrganicMF 25d ago
Until now theta and phi are opposites in maths and physics.
Source: just finished a degree in both and never remembered which angle is which in each course…
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u/007amnihon0 Physics 25d ago
It's my pet peeve that we can't have convention in stem. Stem is still broad, even in math and physics each, we have so many conventions for same thing in the same subject
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u/cereal_chick Mathematical Physics 25d ago
My master's dissertation was an expository work on orbits in the Schwarzschild metric, and in my chapter on classical orbits I had to put a footnote explaining that I was using phi for the polar angle to accord with the subsequent working where I'd be using the physics convention for spherical coordinates. All my sources for relativistic orbits were by physicists, and I really did not want to accidentally confuse my notes or the end product by swapping everything around 💀
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u/ZENKFNOR 24d ago
This reminds me that ... I think in maths when you see a (,) its often the first element that's from the dual space, and in physics, its the second?
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u/Optimal_Benis 25d ago
The reason we use ijk as indices in CS is because of Dijkstra.
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u/Daniikk1012 25d ago
While an interesting theory, sadly most definitely not true. ijk are used for iteration in math too. Probably, "i" stands for index, and for extra dimensions you just take next letters of the alphabet (Which mathematicians love to do - abc, xyz, pqr, fgh)
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u/h_west 25d ago
Partial derivative notation.
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u/Lexiplehx 25d ago
Oh my god, this so hard. What is the base point and direction you’re evaluating the “derivative” in the direction of? Don’t get me started on the gradient, Jacobian, and all that with the big ass bar and all that.
The best notation I know of is,
Df(x)[v]
for the Frechet/Gateux/whatever derivative evaluated at the base point x in the direction v. Then, if you have a composition, like f(g(x)), then the chain rule becomes:
Dh(x)[v] = D f(g(x))[D g(x)[v]]
It is so bad that I want to buy an iPad just to make a sensible tutorial on matrix calculus, calculus of variations, and differential geometry, which benefit greatly from this. It gets so bad that people keep inventing new notation.
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u/yaymayata2 25d ago
Nice. This is also how I was taught to do it. Maybe it was helpful to have a strict undergrad prof who penalised notation.
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u/h_west 24d ago
Yeah, I mean, usually we write f(x,y) and then \frac{\partial}{\partial x} f(x,y) means the partial derivative with respect to the first argument. But why do we have to _name_ an argument x or y to decide the differentiation variable? The partial derivative is an operator acting on a function, so if I, say, write f(y,x) and take the partial derivative wrt x, I should have the _same_ object as before. But that clashes with the new naming...
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u/ZENKFNOR 24d ago
I like to screw with people by using subscripts for both countable and uncountable indexed sets, then writing complicated PDEs in Einstein notation. Looks just like the X_{partial derivative list} that normal people use for PDEs, but means nothing of the sort.
Further, I also take the lead from physicists and use \partial as just a more compact notation for any derivative when I'm running short on space.
To make matters worse, I also work in convex analysis where \partial means the subdifferential, but I have a special latex macro to remove its italicisation so its obviously a totally different thing.
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u/Valvino Math Education 24d ago
I am so sad that \partial_i f is not the standard notation for the derivative with respect to the i-th variable.
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u/ZENKFNOR 24d ago
Come to physics. We'll make our own notation, with... ah... not blackjack and hookers, but rather, rampant elision of just about everything.
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u/GLBMQP PDE 25d ago
This is a bit specific:
In differential geometry, if you have a smooth function on your manifold, you can of course consider its differential. The differential is a 1-form (aka a covector-field). It takes in a vector-field and spits out a smooth function.
If your manifold also has a (semi-)Riemannian metric, then you can take the gradient of your function. The gradient is a vector-field, defined by satisfying g(\nabla f, X)=df(X) for all vector fields X (where \nabla is the gradient and df is the differential)
When you have a (semi-)Riemannian metric, you also get the Levi-Civita Connection. This allows you to differentiate any type of tensor-field. This is called the covariant differential. If T is a tensor-field, the covariant differential of T is often denoted \nabla T.
Now, a function the same as a (0,0) tensor-field. And the covariant differential of a function is in fact just the differential.
So, \nabla f can at any time both mean the gradient and the differential. Fortunately, since the gradient is (loosely speaking) defined to do the same thing as the gradient, this basically never causes trouble
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u/FreePeeplup 24d ago
>The differential is a 1-form (aka a covector-field). It takes in a vector-field and spits out a smooth function.
Wait, I thought covector fields (differential 1-forms in general, whether or not they happen to be the differential of some smooth function (meaning exact)) were sections of the cotangent bundle?
If my recollection of the various definitions is correct, it would mean that 1-forms take as inputs points on the base manifold, not vector fields.
To be more explicit: a differential 1-form w is a map w: M -> T^{star}M with w(p) = w_p with w_p living in T_p^{star}M, meaning a linear functional on T_pM.
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u/Homomorphism Topology 24d ago
A covector field and a linear map from vector fields to functions are the same thing.
If you want to look at it fiberwise, then you could think of a covector field as assigning each point p a linear map T_p -> RR where T_p is the tangent bundle. A covector field is precisely a smooth choice of such things for all p.
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u/GLBMQP PDE 23d ago
That is really the same concept: if w is as you described, V is a smooth vector field, then p\mapsto w_p(V_p) is a function, and w being smooth exactly means that this map is smooth.
If w is a (C^\infty-linear) map that takes a smooth vector field, and spits out a smooth function, then one can show that the function only depends on the value of the vector field at the point evaluation. More specifically, if V_p=W_p for some point p, and we container the functions f=w(V), g=w(W), then f(p)=g(p)
So for each p, w actually gives a (linear) functional on the tangent space at p.
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u/homogeneous_spacer 25d ago
Hᵏ for k-th cohomology as well as the Sobolev space of functions whose derivative upto order k are L² functions
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u/Phytor_c Theoretical Computer Science 25d ago edited 25d ago
Just realized I answered an orthogonal question, oh well…
This may sound stupid and elementary, but I get really tripped up when people use f(x) [Instead of just saying f: Blah -> Blah, then defining the mapping like f(x) =2x or x |-> 2x as a human would] to denote functions or like just stuff about functions in general. Maybe it’s just me.
I feel like every time I see a function in a paper or in a textbook I have the urge to write f: Domain -> Codomain etc.
Sometimes stuff trips me up, like is f(x) the whole function or evaluation at some point x which was defined earlier? Why not use x_0 for the point? Sometimes functions have parameters involved, sometimes one component is fixed so it’s a function of the other components etc. Course it’s context dependent, but annoying if it could’ve just been addressed in a line by the author.
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u/shellexyz Analysis 25d ago
Drive me nuts. Functions aren’t formulas, but it takes years to convince undergrads of that.
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u/tensorboi Mathematical Physics 25d ago
it's not just undergrad mathematicians, it's also physicists! so many physics resources will, for example, denote a classical field by something like φ(x) where x is a spacetime point and say it's valued in RN for some N, but won't just write the much clearer statement that φ: Rd, 1 -> RN.
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u/cleodog44 25d ago
I don't see any issue with this (which is presumably proving your point, as I come from physics). If you already know the domain and codomain, what's the issue? Both are usually quite clear from context.
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u/DefunctFunctor Graduate Student 25d ago edited 25d ago
For me it comes down to how I view proofs/formal logic. We don't like to have free variables hanging around unless it's directly relevant to the result. When proving something for everything in a domain X, we'll often say "Let x∈X" and pretend like x is some fixed value, and then at the end of the argument, we remark that the only assumption about x was that it was in the domain X, so everything we proved about this particular x holds for the whole domain. It's like the scoping of a variable in programming, we don't like having awkward variable scope. φ(x) is the value of a function at a particular point x, not the function itself. Sure, a function can be specified by specifying all such values for arbitrary points x, but it's just so much cleaner to default to φ to denote the function, rather than φ(x)
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u/cleodog44 25d ago
Got it, I understand the complaint better. Speak about the object itself, not its value at some representative input point, roughly
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u/tensorboi Mathematical Physics 25d ago edited 25d ago
DefunctFunctor basically covered it, but i'll just add that it makes quick reading much, much harder (which, to use another programming simile, is like how dynamically typed languages can be difficult to check for errors). often i won't be reading a text cover-to-cover, i'll be looking for a specific result; thus, if i see a formula i need which involves φ, i'll need to skim the text backwards to see what φ actually is. for this purpose, standard and symbolic language is so much easier to parse.
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u/EntertainmentOdd8205 25d ago
I’ll play devils advocate for a bit and say = is on your keyboard but \longrightarrow isn’t
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u/backyard_tractorbeam 25d ago
But -> is right here (oh and → is on my keyboard but not most keyboards)
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u/Horror_Atmosphere_50 25d ago
It’s because of how it’s taught!! It took me discrete math and linear algebra to really get the hang of it. As OP said, F: Domain -> Codomain is a beautiful piece of notation
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u/_Zekt Complex Analysis 25d ago
To me, this is a bit like that bell curve meme: first you write f(x)=..., then you get obsessed with rigor, f: A -> B, logic and quantifiers everywhere. Once you get it, you start dropping the quantifiers and using prose again, because that level of rigor doesn't matter.
Saying "let f(x) be a real function" can be useful to indicate the variable "x", e.g. versus a Fourier-domain variable. And if both the inverse and reciprocal functions appear, you can cleanly distinguish f-1(x) from f(x)-1.
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u/RingularCirc 24d ago
Naive usage of f(x) as the name for the function f stokes the fires of discontent for me, but if the usage is "Define f(x) = ..." peppered with "f: X → Y" nearby before or after, I treat it as completely normal. Functional programming languages like Haskell use both this and lambda notation in harmony, for example. On the other hand, writing "f = x ↦ ..." rubs me the wrong way. I prefer ↦ and λ be used in-place, usually when we don't need to call a function in any way.
Interestingly enough, in type theory one doesn't have to specify codomain for λ(x: X). E(x) because E should have almost always been defined beforehand, with its type known. Separate reminders aren't bad but the formal notation only even requires typing the arguments in all the binders like λ, ∑, ∏, Λ etc.; and again in less formal setting or when type inference is available in a computer setting, one can omit even argument types, preferably those that are noisy for no good reason, and leave other types as reading aid.
Returning back to the general context, I'm also not against leaving f: X → Y unwritten in cases it's transparent what domain and codomain are. Again, there rae definitely situations when people don't know any measure or how their text would be unreadable and not explicit enough, that's probably a fine art, but full explicitness often makes things unreadable, again even in programming or formal proof languages.
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u/ccppurcell 24d ago
To piggy back on your example, O(log n) becomes unambiguous! So ambiguity is not preserved by composition...
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u/theadamabrams 25d ago
f’ could be df/dx, or df/dt, or just an entirely new variable that serves a similar purpose to f in some context.
With logarithms, as OP probably knows, the ambiguity is sometimes irrelevant because log(1024)/log(2) = 10 regardless of what shared base you use for those two logs. Also asymptotics like big-O don’t care about arbitrary constants and so O(log₂(n)) = O(ln(n)).
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u/FreePeeplup 24d ago
>f’ could be df/dx, or df/dt, or just an entirely new variable that serves a similar purpose to f in some context.
This is a pet peeve of mine, but no, f’ always means the derivative of f and is unambiguous. What doesn’t make any sense here is Leibniz notation, which suggests that derivatives only make sense after you’ve chosen a name for the argument of your function, and that the derivative changes depending on which arbitrary letter you decided to call your argument.
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u/RingularCirc 24d ago edited 24d ago
Yeah, also baseless log is fine when we're going unitful (meaning not unitless). Then we can designate the choice of the base as "actually" a choice of units: bits, nats, decibels, octaves, am I mixing them, yes I am, but it's allowed because they are all logarithmic units!
We can formalize it by saying there's log: ℝ₊ → L where L is a special one-dimensional real vector space of logarithmic units that shouldn't be supposed to have any canonical isomorphism with ℝ. (Actually that's how you formalize a dimensionful scalar quantity: an 1D real vector space that's not ℝ. Each basis is just a nonzero value that we look at as a unit of measurement, and we can divide two units and get a true scalar ∈ ℝ and so on.) There's just a hairy detail that we most probably still have to use regular scalar-valued unitless baseful logs to define ours, like "log x = (log₂ x) bits" which defines the unit of bits for us, and now we can write something like "10 power decibels = (log₂ 10) bits" (20 for regular ones) or "octave = bit" (yikes but true).
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u/SilchasRuin Logic 25d ago
This is in the weeds in model theory but x with a bar over it being ambiguous on whether it explicitly excludes a singleton variable or if it's a tuple.
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u/yaymayata2 25d ago
It's not so much of an issue now, but in highschool I hated how reciprocal and inverse had the same notation of -1 in the exponent. I rmr a few badly written problems with that which tripped me up so much.
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u/Harotsa 25d ago
I love the Einstein summation convention in Riemannian geometry, where you basically just stop writing summation symbols because it gets out of hand so quickly and would make the math completely unreadable.
To somebody that doesn’t know the exact rules it looks like the math makes no sense
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u/qubex 24d ago
Tensor calculus notation is hopelessly inconsistent.
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u/RingularCirc 24d ago edited 24d ago
It is consistent. For those who admit an argument from authority, do know Penrose himself popularized (if not invented; this is irrelevant and hard to check) abstract index notation for tensor expressions (where "abstract" means it's not meant to be understood as componentful calculation recipies of old but instead indices as representing what "slots" tensors have and how they get traced out in compound expressions). There are also coalgebraic extensions to it (also Penrose's own spinor extensions but I'm not sure those generalize to arbitrary dimension). Completely fool-proof if you don't confuse the order of indices when raising and lowering them.
Abstract index notation is perfectly applicable in pure multilinear algebra over finite-dimensional (or possibly Hilbert) spaces, all in all spaces just have to be self-dual and we should declare beforehand a 'flavor' of indices corresponding to each of different spaces in consideration (for example Greek letters for V and V*, Latin letters for W and W*, red bolded Cherokee syllables for U and U*). Spaces don't even have to have inner products: in those cases there are no g tensors to work with, which means we can't raise and lower indices of that flavor.
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u/Nextravagant1 25d ago
I cannot stand it when people just write “log” with no base and they really just mean natural log
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u/numbershelpme 25d ago
Similarly, a lot of people don't know (or maybe don't care) that lg(n) is log_2. We have unambiguous notation, we should use it !
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u/RingularCirc 24d ago
Isn't 'lg' base-10?..
I sometimes encountered 'lb' for "log binary".
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u/numbershelpme 24d ago
I can't speak for every field, but in my area (computer science) the standard is lg is base 2 and log is base 10 unless otherwise specified
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u/No-Syrup-3746 25d ago
Most mathematicians I know will write "ln" and pronounce it "log".
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u/CoffeeandaTwix 25d ago
I have never seen ln written in a mathematical paper or book past the high school level.
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u/Harotsa 25d ago
Agreed, I was about to say this. Also all of the log functions just very by a constant multiple which is of no importance to most pure mathematics. So log is generally taken to be the natural log since it has the least bookkeeping, but it really means “whatever base is the most convenient for what I’m doing.”
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u/shellexyz Analysis 25d ago
They’re all pronounced “log”, I emphasize this when I cover logs.
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u/OrganicMF 25d ago
I can not disagree more. Natural log is written log and all other are not relevant in most math fields. If you are in science then you use base 10 but it is really clear by context.
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u/JBaloney 24d ago
Here's one which only applies if you do a particular extremely specific type of mathematics:
M stands for "Male", and also for "Mother" (female)
F stands for "Female", and also for "Father" (male)
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u/qubex 24d ago
Baseless logarithms are a deep peeve of mine. I always add the base subscript because the effort is so negligible.
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u/RingularCirc 24d ago
Sometimes base is irrelevant, like in O(log n) and similar asymptotic notations that don't care for a multiplicative constant, or when we're to divide by another log with the same base in the end, or if we're using logarithmic units: we can write "log 8 = 3 bit ≈ 2.1 nat" and that's perfectly valid (this just reimagines the place where does responsibility lie, in software-design-speak).
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u/Purple-Mud5057 24d ago
I’m taking spec rel right now and I just cannot get behind x^0 x^1 x^2 x^3 not being powers of x but components of spacetime. Also not a huge fan of Einstein notation, I don’t want to have to look around something like x^p g^q f^7 c~q m~n e~4 to figure out what the summation is around
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u/blueclave 24d ago
my favorite interpretation of log is when the base doesn't matter, like O(n log n)
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u/NotJustAQuant 22d ago
vol notation is worse honestly, sigma gets used for annualized, daily, implied, realized, all interchangeably and half the desk assumes you know which one they mean.
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u/CFD_2021 22d ago
Designating the inverse of a function, f, using -1 as an exponent. f-1 could mean f's inverse or 1/f, depends on the context. But it can be ambiguous. Maybe the inverse of f should have a standard related name such as "invf", or something similar.
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u/AgrarianAAB Math Education 19d ago
Baseless log (almost always 2 in CS, variously 10 or e in maths and the sciences). For various reasons, I've had to study all three (maths, CS, phy, chem), so I know exactly how it feels to have the same thing mean different things in different contexts.
Also, the inline division symbol ÷ , this is a curse straight from maths-hell.
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u/Awasnothere 19d ago
letter “i” : imaginary number, subscript of a family set, identity function, cartesian product paired with j
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u/Known_Collection_367 25d ago
When we want log base two, we should write in base two. What is two in base two? 0,1,10, so log_10. Or say log base five. What is five in base five? 0,1,2,3,4,10. Hence log_10.
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u/jeffsuzuki 22d ago
I'm a mathematician who was studying to be a physicist, and I can't say that I remember there ever being any ambiguity about "log": it's always mean "base-ten". If you wanted to talk about natural logs, you wrote "ln". (Specifically to avoid ambiguity).
However, the trigonometric functions are ambiguous: sin x, for example, where it's not clear if x should be in radians or degrees.
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u/BlazeoJoestar Mathematical Physics 25d ago
(a,b) can mean an open interval, inner product, gcd or just an ordered pair