Confusing Notations

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  • #26
Zurtex
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Hurkyl said:
I've heard an interesting story related to this. :smile:

A professor wanted to emphasize to his calculus students that using "f" for a function and "x" for real number was merely conventional, and to shock them, he proceeded to write down this definition:

x(f) := f^2

And then work out its derivative, dx/df.

As it so happened, the professor was so confused that he had difficulty working out this simple derivative!!!
Haha that's a good one, I'll need to remember that one. By brain had to do a lap before I could do that one.
 
  • #27
mathwonk
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this fixation on the "usual meaning": of a symbol causes me great difficulty in teaching my class. I.e. I often start over with what I want a letter to mean, but students often want it to continue to have the same meaning it had earlier.


For example suppose you prove the mean value theorem for a function f on an interval [a,b], i.e. if f is continuous on [a,b] and differentiable on (a,b), then at some popint c in (a,b) we must have f'(c)[b-a] = f(b)-f(a).

OK now you want to rpove the corollary that a function f continuosu on [a,b] with derivative zero on (a,b) is constant.

So you say well let c,d be any two points of [a,b], and I will prove that f(c) = f(d). Well you are already in trouble. If you try to apply the theorem to the interval [c,d], some people will not be able to let the symbols c,d polay the roles that were played before by a and b. And they will think c is the point where the dertivative was evaluated.

So you have to stop and go back and change the names of the new interval to some other letters, and you do not have any more available since people today where i live dislike greek letters,.....
 
  • #28
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Off topic a bit: we're running out of variables.
Have you seen Omniglot?

http://www.omniglot.com/

We're set for a couple centuries, I think.
 
  • #29
arildno
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Hurkyl said:
I've heard an interesting story related to this. :smile:

A professor wanted to emphasize to his calculus students that using "f" for a function and "x" for real number was merely conventional, and to shock them, he proceeded to write down this definition:

x(f) := f^2

And then work out its derivative, dx/df.

As it so happened, the professor was so confused that he had difficulty working out this simple derivative!!!
A great story, Hurkyl!
That reminds me of an occasion when I was a student, and took a course in real analysis :

Our lecturer had use for the constant function equal to 1, and brazenly defined it as 1(x).
A collective shudder went through the classroom, and one of the brightest gasped:
"You CAN'T call a function 1(x)!!"

The lecturer then threw his hands up in mock resignation, and exclaimed:
"So, what should I call my function then; 2, perhaps?"
I thought I learnt a lot that day; in particular in how we ought to choose notations intelligently, rather than to slavishly follow conventions..
 
  • #30
matt grime
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mathwonk said:
this fixation on the "usual meaning": of a symbol causes me great difficulty in teaching my class. I.e. I often start over with what I want a letter to mean, but students often want it to continue to have the same meaning it had earlier.


For example suppose you prove the mean value theorem for a function f on an interval [a,b], i.e. if f is continuous on [a,b] and differentiable on (a,b), then at some popint c in (a,b) we must have f'(c)[b-a] = f(b)-f(a).

OK now you want to rpove the corollary that a function f continuosu on [a,b] with derivative zero on (a,b) is constant.

So you say well let c,d be any two points of [a,b], and I will prove that f(c) = f(d). Well you are already in trouble. If you try to apply the theorem to the interval [c,d], some people will not be able to let the symbols c,d polay the roles that were played before by a and b. And they will think c is the point where the dertivative was evaluated.

So you have to stop and go back and change the names of the new interval to some other letters, and you do not have any more available since people today where i live dislike greek letters,.....

Using the same letter in different ways can cause confusion, but spare a thought for the student who got onto a degree program in maths and when faced with

f(x)=x

asked me if the x on the left hand side was the same x as the one on the right.

And, for, Telos, there are far more things in mathematics than we could ever write down, even if we agreed a notation for them.

And, back to one of mathwonk's posts, how about a little Wittgenstein:

When a student sees on the board 2,4,6,8... and is asked to continue the sequence, what does it mean when he says "i've got it" and writes down 10,12,14,.. what is "got it", why does the thing he "gets" agree with what you or I get?

So, why when we see f(p) do we indeed "get" that f is a funtion of p.
 
  • #31
Like Masta Coda said, trying to get everyone to use the same notation is like trying to get everyone to use the same language. Plus some notations are better than others for only certain situations. I had the same problem as you with f(x) not being f multiplied by x. f and x are never constantly a function and a variable of a real number. What I find is best is to never consider a(b) as a being multiplied by b but always when only the second variable is in parenthesis to understand the first as a function of the second. Represent multiplication with both parenthesis, a dot, or no parenthesis in the case of variables.
(f)(x)=fx=f*x= 5x
f=5
Part of notation is the ease of writing and communicating it. I still find it confusing how ordered pairs and interval notation can look exactly alike with parenthesis. (a,b) could mean the point or the interval. The keyboards inability to neatly express many things in math also annoys me.
 

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