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.

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

- 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