4Fun:Worst/Best Notations in Mathematics

[Swapnil]

Just for fun (and for my curiosity), what notations/symbols in mathematics do you guys find really annoying or really interesting?

(Please, no whining about the fact that the Greek and the English alphabets look the same We've all been there. )

edit: let's add confusing notations to the mix too.

 

[CRGreathouse]

I was amused by a suggestion at the halfbakery to define ? as the inverse operation to the factorial function. 6?!? = 3.

 

[matt grime]

A notational problem in one sense that annoys me is the usage of p and q as summation indices in the same sum. It's ok in a book, but in a lecture it is annoying. At least p and q are pronounced differently, because even more prevelant is the usage of m and n in the same summation, and a mumbly lecturer with bad hand writing and that choice is a pain in the backside. I made a plan (which I probably forgot to do most of the time) when teaching to always use r, s and t and at most one of n and p as indices in complicated summations ona blackboard. Of course some people managae to have r's that look more like s's.

In a similar vein, and because you should commend the good as well as disdain the bad, I was always thankful to lecturers who adopted such good conventions as putting bars in their z's so they didnt' look like 2's. Very useful in a complex analysis course.

 

[CrankFan]

{ \cos^{-1} \theta } \over { \sin^{-1} \theta }

...I wouldn't say that I find it really annoying. It's just one of those notational things that could lead to confusion.

 

[Gelsamel Epsilon]

{ \sin^{2} \theta }

At one time ages ago I wasn't sure wether it was Sin of the Sin of Theta, or the whole thing squared.

Edit: Interesting to note, I've never had any trouble with greek/english letters. But I have had trouble with m/n like matt said, and also r and v.

Edit2: Also { d^{2}y } \over { dx^{2} }

 

[murshid_islam]

when i first came across f^{-1}(x), i thought it meant {1}\over{f(x)}

 

[daveb]

I always had a problem with people using bold to denote vectors, vector fields, tensors, etc since it's sometimes difficult for someone with my eyesight to tell that something is bold. When they use UPPER case to denote these and lower case to denote these, it's fine, but then you have to deal the vector space V and and an individual vector v in the space.
As for things I like, the Christoffel Symbol, Poisson bracket { } and Commutator [ ] are pretty elegant.

 

[arildno]

1/a. What the heck does the 1/ symbols do there??
Instead, a notation for the multiplicative inverse like \hat{a} is a lot better.
Similarly grumpy about the additive inverse (-1)

 

[shmoe]

Factorial always causes some problems. It's a given that a thread asking why "0!=1" someone will interpret this as "0 does not equal 1". It also makes it more difficult to express surprise and astonishment when an exclamation mark means something else.

I geatly dislike "ln" to mean natural logarithm as well, mostly because it locks people into thinking "log" means "log base 10" to everyone in the world.

 

[arildno]

Gauss detested the factorial notation, so you are not alone!

 

[matt grime]

But the only people who use != to mean not equal to are comp. sci.s, and frankly they will always be confused as long as they use = to mean assignment and == to mean equal. You can't accommodate them no matter how hard you try, which, admittedly, isn't very hard when it's me.

 

[arildno]

Comp. scis are as bad as the electrical engineers with their misuse of the letter "i".

 

[shmoe]

You mean electrjcal engjneers?


My main problem is really people who think the definition or notation they've seen in their first book/course/whatever is always the universal one used by everyone, everywhere and get confused when they find out otherwise.

 

[Daverz]

Gothic letters and script letters, particularly if you can't figure out what letter they are actually supposed to be.

 

[CRGreathouse]

{ \sin^{2} \theta }

At one time ages ago I wasn't sure wether it was Sin of the Sin of Theta, or the whole thing squared.

Edit: Interesting to note, I've never had any trouble with greek/english letters. But I have had trouble with m/n like matt said, and also r and v.

Edit2: Also { d^{2}y } \over { dx^{2} }

I'm with you on all of those. I think there needs to be some good general notation for iterated functions, distinct from the power notation. I've seen (in Dusart) the use of a subscript for repeated application of a function, but I don't think this is any better.

 

[Swapnil]

I always had a problem with people using bold to denote vectors, vector fields, tensors, etc since it's sometimes difficult for someone with my eyesight to tell that something is bold. When they use UPPER case to denote these and lower case to denote these, it's fine, but then you have to deal the vector space V and and an individual vector v in the space.

So true. I also hate the fat that they use bold letters to denote vectors which is impossible to do when you are writing on a piece of paper. I prefer arrows but those are overused as well.

 

[Swapnil]

You know, I really hate the fact that testbooks usually leave out the \hat{ } (hat) symbol on unit vectors. Now I am getting used to it because usually the only unit vectors we usually work with are \hat{n} and \hat{t}, the unit normal and the unit tangent vector, respectively.

 

[Swapnil]

{ \cos^{-1} \theta } \over { \sin^{-1} \theta }
...I wouldn't say that I find it really annoying. It's just one of those notational things that could lead to confusion.

That's why I always use \arcsin{\theta} and \arccos{\theta}

 

[Swapnil]

Another notation I hate is the use of \cdot (small dot) to represent dot product. Its looks so much like multiplication! I know, I know ... you are never going to multiply vectors because there is no such thing. But still, I think its a bad notation. I personally prefer making a small circle instead of a dot like this: \vec{A}\circ\vec{B}.

 

[CRGreathouse]

Another notation I hate is the use of \cdot (small dot) to represent dot product. Its looks so much like multiplication! I know, I know ... you are never going to multiply vectors because there is no such thing. But still, I think its a bad notation. I personally prefer making a small circle instead of a dot like this: \vec{A}\circ\vec{B}.

Oh good, that way you can cnfuse it with composition instead. Yeah, they aren't functions you can compose... but they aren't numbers yo ucan multiply, either.

 

[matt grime]

Another notation I hate is the use of \cdot (small dot) to represent dot product.

You dislike dots to represent dot products? What do you think we should use instead of a DOT for a DOT product. Since vectors are not, in general, numbers how can there be any notion that you are 'multiplying numbers'. Further note that in the 1-d case, when they are just numbers it *is* just mulitplication of numbers.

 

[Swapnil]

Further note that in the 1-d case, when they are just numbers it *is* just mulitplication of numbers.

What are you talking about? How can vectors exist in 1-dimension?

 

[matt grime]

Well, modulo the hazy notion of 'exist' (vectors surely exist in a vector space, if anything, not a dimension?) you do know what the dimension of a vector space is? You do know there are 1 dimensional vector spaces that are canonically isomorphic to the underlying fields? \mathbb{R} is a vector space, the dot product on this vector space is just mulitpliction.

 

[chroot]

I have to agree that Swapnil's use of a small circle to represent the dot product is an example of a notation that would drive me nuts.

My favorite notation is undoubtedly abstract index notation. It's so powerful, so streamlined, and so immediately useful.

<br /> R_{ab} - \frac{1}<br /> {2}Rg_{ab} = \frac{{8\pi G}}<br /> {{c^4 }}T_{ab} <br />

- Warren

 

[chroot]

What are you talking about? How can vectors exist in 1-dimension?

An example of a one-dimensional vector space is the real line, which is spanned by the single basis vector (1).

- Warren

 

[Tomsk]

I think the tex dots should be bigger though, I usually do dots slightly bigger than normal for dot products.

I read an article on wikipedia the other day- this one: http://en.wikipedia.org/wiki/Curl

which says that \nabla\times\vec{A} is an abuse of notation, but I don't see why, del is (d/dx,d/dy,d/dz), right? So surely curl is del cross something?

 

[chroot]

Well, technically del is an operator, not a vector, but it behaves like one.

- Warren

 

[matt grime]

I fail to see how anyone can find dots in dot products confusing, I mean u.v has a clear meaning dependent on what u and v are, further, if you object so much why bastardize another notation when there is the perfectly acceptable inner product (u,v) or <u,v> notation at hand. Heck you can even use u*(v) using the dual space. And if u and v are 1-d vectors, so elements of the basefield, then u.v is u times v, so there is no contradiction at all in the usage of the symbol.

I mean, it is reasonable to note that the f^2(x) and sin^2(x) have contradictory meanings, sin is after all a function and f^2 (x) often means f(f(x)).

 

[Tomsk]

Also, with factorial notation, I always find myself saying the number in an excited way. 3! becomes THREE!

 

[Swapnil]

I think the tex dots should be bigger though, I usually do dots slightly bigger than normal for dot products.

I read an article on wikipedia the other day- this one: http://en.wikipedia.org/wiki/Curl

which says that \nabla\times\vec{A} is an abuse of notation, but I don't see why, del is (d/dx,d/dy,d/dz), right? So surely curl is del cross something?

See the following for a complete discussion:
https://www.physicsforums.com/showthread.php?t=131416

 

[jtbell]

I was amused by a suggestion at the halfbakery to define ? as the inverse operation to the factorial function. 6?!? = 3.

I'd prefer to use the inverted exclamation point from Spanish: 6¡!¡

 

[CRGreathouse]

Also, with factorial notation, I always find myself saying the number in an excited way. 3! becomes THREE!

3! = 2601218943565795100204903227081043611191521875016945785727541837850835
15694738224067857795813045708261992057589224725953664156516205201587379198458
08325291052446903888118841237643411919510455053466586162432719401971139098455
27278537099345629855586719369774070003700430783758997420676784016967207846280
22903210716166986726054898844551425719398549944893959449606404513236214026598
30732493697704776060676806701764916694030348199618814556251955925669188308255
42947596537274845624628824234526597789737740896466553992435928786212515967483
97602950569669992728467056374713753301924831358707612541268341586012944756601
54207495899525635430682886346310849656506827715529962567908452357025521862223
30016700834523443236821935793184701956510729781804354173890560727428048583995
72902172661229129842051606757903623233769945396419147517556755769539223380305
53085999774416757843528159134613403946049012695420288383471013637338244845066
93348484440711931292537694657354337375724772230181534032647177531984537341478
32704845798378661870325740593892421570969599463055752106320326349320922073832
33563099232675044017017605720260108292880423356066430898887102973807975780130
49576342838683057190662205291174822510536697756603029574043387983471518552602
33386635713910104633641976909739743228599421983704697910995630338960467588986
57111765666700391567481531159439800436253993997312030664906013253113047190288
91856203766669164468791125249193754425845895000311561682974304641142538074897
72337595538066171980140467793561479363526626568333950976000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000

 

[Swapnil]

I also dislike the fact the a lot of texbooks don't make a distinction between definitions and equality. They use the symbol = for stating a definition intead of other good symbols like \equiv, \mathbf{:=}, \mathbf{::}, \triangleq, \circeq.

 

[matt grime]

The second of those, := does mean, 'by definition', so if they are just using =, they are doing it wrong.

 

[Robokapp]

how do you 3!?

((3!)!)! ?

 

[0rthodontist]

You know, I really hate the fact that testbooks usually leave out the \hat{ } (hat) symbol on unit vectors. Now I am getting used to it because usually the only unit vectors we usually work with are \hat{n} and \hat{t}, the unit normal and the unit tangent vector, respectively.

You know, special notation for vectors as opposed to other variables actually bothers me a little. Not just on unit vectors which is OK (but not what I'm used to) but on any vectors at all, I don't like being told I have to put a little arrow over it or make it bold or it's not a vector.

 

[StatusX]

I like the set builder notation, it's so powerful. My only problem is with things like: {x_n | U_n open}. It seems like this should be properly written as {x_n | n \in A} where A={n | U_n open}, but I usually just write it the first way.

 

[arildno]

You know, special notation for vectors as opposed to other variables actually bothers me a little. Not just on unit vectors which is OK (but not what I'm used to) but on any vectors at all, I don't like being told I have to put a little arrow over it or make it bold or it's not a vector.

That's why mathematicians doin't bother about the arrows over vectors (elements in vectyor spaces).

 

[George Jones]

I was amused by a suggestion at the halfbakery to define ? as the inverse operation to the factorial function. 6?!? = 3.

Yikes - this would mean that a rather forceful interrogbang would be rendered impotent?!

 

[Swapnil]

You know, special notation for vectors as opposed to other variables actually bothers me a little. Not just on unit vectors which is OK (but not what I'm used to) but on any vectors at all, I don't like being told I have to put a little arrow over it or make it bold or it's not a vector.

I guess its just a matter of perspective. You see the special notation(s) for vectors as a constrainst while I see them as a freedom to precisely express myself.

 

[matt grime]

But you have complete freedom to express yourself because the context makes it clear what the notation means. Indeed that is most of your bete noires put in one summary: you dislike that the meaning of notation is left to context and wish for special notation. Hence you wish that we shouldn't use dots when dotting vectors becuase in a different context dots mean something else (something that is actually a special case of the dot product), and that vectors must have special trimmings so we know they're vectors.

If I say

"let V be a vector space over F and suppose if I pick v in V and f in F with fv=0 then either f=0 or v=0"

then I know precisely what is a vector in there, what isn't, and it is clear that the two uses of 0 refer to completely different objects. What is imprecise about that?

To quote one lecturer I had (for optimization) "we write vectors in ordinary roman script because I assume you're smart enough by now to know what is a vector and what isn't".

 

[Swapnil]

But you have complete freedom to express yourself because the context makes it clear what the notation means. Indeed that is most of your bete noires put in one summary: you dislike that the meaning of notation is left to context and wish for special notation. Hence you wish that we shouldn't use dots when dotting vectors becuase in a different context dots mean something else (something that is actually a special case of the dot product), and that vectors must have special trimmings so we know they're vectors.

True.

If I say
"let V be a vector space over F and suppose if I pick v in V and f in F with fv=0 then either f=0 or v=0"

then I know precisely what is a vector in there, what isn't, and it is clear that the two uses of 0 refer to completely different objects. What is imprecise about that?

Yeah, I have to agree that a mathematical statement can be precise without having any special notations PROVIDED that a proper context is given. But I would still prefer notations over no notaions just as a matter of elegence rather than a matter of precision.

To quote one lecturer I had (for optimization) "we write vectors in ordinary roman script because I assume you're smart enough by now to know what is a vector and what isn't".

But such assumptions can be dagerous sometimes. You can't just assume that your audience is homogoneous. There bound to be some people who "aren't smart enough."

 

[arildno]

But those not smart enough has been weeded out way before mathematics lectures deal with abstract vector spaces..(or so we should hope!)

 

[matt grime]

But you're arguing for notational complexity when there is absolutely no need for any. How can that possibly be remotely elegant?

 

[Swapnil]

I am not specifically talking about vector spaces (but even if I was, I wouldn't be able to argue either way because I don't have much experience with vector spaces).

 

[Son Goku]

One notation that tripped me up, is the use of the standard way of writing a multiple a.b being used to denote the gorup operation in group theory, when the group is an additive group.
Like a fool, it would take a few seconds for me to realize a.b would actually be a + b, in the case of the integers under addition.

It similarly caused some confusion in the case a.0 = a. It took me a while to see that dot as denoting the group operation and not necessarily multiplication.

Of course the notation is completely sensical.

 

[Swapnil]

I love the \cup and the \cap notaions for sets and the \wedge and the \vee notations for logical statement. They are so intuitive.

 

[mathwonk]

i find my time wasted most by notation in which a lot of emaning is contained in tiny symbols that are easily changed in transcription. the secretary always messes them up, or the type setter, and one cannot easily recover the correct meaning.


i do not know abetter ,one, but the notation that speaks least clearly to students is integral notatuions. they never believe that the integral of f(t)dt from t = a to t=x is a function, with proeprties like continutiy, differentiability.


people who write capital f for the antidertivative of lower case f, and then write them in identical size, and shaoe, are probably trying to confuse me as to their accuracy on a test.

similarly a's and alpha's in a book look identical say in courant;s calculus, perhaps the, only flaw that book has for me.


tensor product notation is also confusing to many people as it suggests that all elements are decomposable.

actually summation notation, i.e. dreaded sigma notation, is a bugbear for almost all my calculus students.

not to mention simple functional notation. how isa student suppsoed to tell the difference between f(x) measning f evaluated at x, and a(x) meaning a multipleid by x, when they ahve the same notation.

even Mathematica cannot tell the difference, and requires the functional evaluation be written with square brackets. it is always entertaining and challenging when a student computes that 3cos(x/3) = cos(x).

 

[CRGreathouse]

I love the \cup and the \cap notaions for sets and the \wedge and the \vee notations for logical statement. They are so intuitive.

The logical ones trip me up. I first learned Boolean algebra (when very young) from a book that used + for or, \cdot for and, \oplus for xor, and an overbar for not. The set notation is intuitive for me, though.

 

[Swapnil]

not to mention simple functional notation. how isa student suppsoed to tell the difference between f(x) measning f evaluated at x, and a(x) meaning a multipleid by x, when they ahve the same notation.

even Mathematica cannot tell the difference, and requires the functional evaluation be written with square brackets. it is always entertaining and challenging when a student computes that 3cos(x/3) = cos(x).

True. I agree.

actually summation notation, i.e. dreaded sigma notation, is a bugbear for almost all my calculus students.

I don't know. How can the sigma notation be initimidating? Its just a nice shorthand to write the sum of a bunch of terms that have a common pattern.