4Fun:Worst/Best Notations in Mathematics

[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.

 

[CRGreathouse]

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.

Yeah, beats me. It was my favorite notation up until midway through my high school years.

 

[FrogPad]

I hate anytime that I have to use "v" and "V". I can never size them properly, so I have to write my v's with horizontal "wings" on each side, so that it looks something like a $\nu$. Then after awhile they start to look like check marks.

I also hate when I need to write, "0", "o", or "O".

For some reason, I really dislike the $\hat i , \, \hat j , \, \hat k$ notation. I prefer, $\hat x$ or $\vec a_x$

Oh, and for some reason I love to use $\lambda$ anytime I need a temporary variable of some sort.

My biggest pet peeve is when professors are inconsistent with their notation (*cough* physics)

 

[mathwonk]

we once complained a professor was using confusing notation in class and he replied "yes, I intend to exploit the confusion in the proof."

 

[mathwonk]

once you begin to teach you view all notation in a new light.

it is amazing how many students fail to give any meaning at all to the simpelst looking notation. In 30 years I have never had a class in which several did not say that a derivative was something like
lim h-->0 f'(x) = f(x+h)-f(x)/h. they seem not tor ead these sequences of symbols like words in a senrtnence at all.

and sigma notation is never understood even by a fraction of my students.

these students cannot comprehend mentally that there are more than one term there just ebcause the sigma notation says so. . i.e. summation as i=1,...n, of f(n) does not speak to them at all. they have to see a strong of f(1) + f(2) +...+ f(n) wriiten out to get it.

now lately the svcholarship rpogram is bringing stronger stduents and perhaps i am hamopered by my old assumptions, but we shall see.

the following question always mows them down, even if announced in advance:

define carefully the riemann integral as a limit, explaining the meaning of any special symbols you use, such as "delta x"

 

[mathwonk]

the thing that freeaked me out as a student was when aprofessor tried to get us to understand a bit of duality by writing f(p) as p(f) and pointing out that the point could be viewed as acting on the function. i thought I was going to have an anxiety attack.

 

[Swapnil]

One more thing that I hate is that there is no good notation to denote that something is a statement other that the = sign. This causes a lot of problems. For exapmle, in induction you have to prove that $$P(n)\Rightarrow P(n+1)$$ but most students forget that P(n) is a statement like "something = something" which is either true or false. It does not equal one side or the other, which is a common misconception.

 

[FrogPad]

once you begin to teach you view all notation in a new light.

it is amazing how many students fail to give any meaning at all to the simpelst looking notation. In 30 years I have never had a class in which several did not say that a derivative was something like
lim h-->0 f'(x) = f(x+h)-f(x)/h. they seem not tor ead these sequences of symbols like words in a senrtnence at all.

I think a lot of the time, it is taught this way.

If asked what a derivative is, it seems that the correct response is to regurgitate the expression you showed above.

Calc I-III was like this for me. It was a continual process of recalling a grouping of symbols to put down on the paper. It wasn't until a later math course where we used, Strang's "Introduction to Applied Mathematics" did I see that math can be an expression of the author. I really enjoyed the book, because it was less about regurgitating answers, and more about understanding the underlying idea of the topic at hand.

I've had a few professors, who when asked a question would stop, explain what needs to be done, and then write the required symbols to justify themselves. Other professors would slap symbols on the board, then explain what they are doing. The latter professors are the ones I would end up having trouble learning from them. I usually feel one step behind them, because I'm continually asking myself in the back of my mind, "why?".

 

[mathwonk]

just keep asking,
'huh?"

thats what you are paying for.

 

[Swapnil]

One of the other things that really bugs me is when they use famous notations as constants or famous constants as variables. For example, its really confusing when in some texts they use $\Sigma$ and $\Pi$ to stand for constants or when they use the letter $\pi$ as a variable. I always go "Huh?," but then I realize they are just trying to play mind games.

 

[radou]

One of the other things that really bugs me is when they use famous notations as constants or famous constants as variables. For example, its really confusing when in some texts they use $\Sigma$ and $\Pi$ to stand for constants or when they use the letter $\pi$ as a variable. I always go "Huh?," but then I realize they are just trying to play mind games.

I only saw $$\Sigma$$ and $$\Pi$$ stand for planes, while attending descriptive geometry, which I didn't find specially annoying.

 

[FrogPad]

just keep asking,
'huh?"

thats what you are paying for.

hehe.
true that

 

[cliowa]

There's another incredibly annoying thing some mathematicians do (one can see this very often in books):
They take two different things, but give them the exactly same notation (because the word begins with the same letter or because traditionally it always is n, so the author can't brake with tradition), stating that it will be clear in the context which one is meant. I hate that!

 

[mtanti]

Once I was explaining factorials to a friend in a chat room and I said 'do you know what is 3x2x1?' I said '3!'.

The reply was 'why are you shouting?'

It also gets pretty confusing when you put the factorial in a question: 'what is 3!?'

You end up having to use brackets around the number (3!)

 

[Swapnil]

Personally, I don't like the $$\sqrt$$ sign, because it makes it seem that exponents are something magical. Think of the things that might go through a middle school student's mind when he is asked to take raise $$\sqrt 3$$ to the power of 3! ( BTW, I mean just the number 3, not 3 factorial!).

Plus, the squareroot sign is also long and ugly.

 

[Swapnil]

Another fact that I dislike is the use of the term "function." Sometimes, a function simply means a correspondence (a rule by which we assign each object in a set some other object(s) in some other set).

Other times it means a special TYPE of correspondence (the one which takes an object in a set and maps it to a single UNIQUE object in another set).

 

[CRGreathouse]

Another fact that I dislike is the use of the term "function." Sometimes, a function simply means a correspondence (a rule by which we assign each object in a set some other object(s) in some other set).

Other times it means a special TYPE of correspondence (the one which takes an object in a set and maps it to a single UNIQUE object in another set).

I never use it in that first sense. I use the term "relation" instead.

 

[chroot]

As CRGreathouse mentioned, the use of the word "function" is not appropriate for a relation which is multi-valued. If your teachers are using the term this way, I pity them.

- Warren

 

[0rthodontist]

Another fact that I dislike is the use of the term "function." Sometimes, a function simply means a correspondence (a rule by which we assign each object in a set some other object(s) in some other set).

Other times it means a special TYPE of correspondence (the one which takes an object in a set and maps it to a single UNIQUE object in another set).

Are you talking about the use of "function" to denote partial functions?

 

[Swapnil]

What do you mean by "partial functions"?

 

[CRGreathouse]

What do you mean by "partial functions"?

Let $f:A\rightarrow B$ be a function. Then $f':A'\rightarrow B$, where $A\subsetneq A'$ and f' is equal to f on A and undefined otherwise, is a partial function from A' to B.

Essentially, it's a function that isn't defined everywhere. Division on the integers is an example, since division by 0 is undefined.