Mathematics and the abuse of notation

[K.I.L.E.R]

Got my class results back and I scored 50% exactly for maths. Chalnoth how did you go?
I didn't get one high distinction this semester.

Anyway I've been doing some equations and found that I was abusing notation when I looked at my results I thought (is this a dot product or am I just multiplying by a factor?).
One thing led to another and I found out I was using '.' in place of "*".

How often do people here abuse mathematical notation?
Does it confuse you?
How much problems has it caused? How much time does it take to get to the heart of the issue in places where you've abused notation?

 

[xatnys]

Annoying isn't it? They teach you multiplication can be indicated a number of ways, ".","*","x"; and as you get into higher level math, the one that is still "safe" to use is also the most annoying to write! Dot products, cross products, ugh

 

[Fred]

Thats not really what I would call abuse of notation. Theres nothing wrong with identifying * with . so long as you remember what you are doing (and taking a dot product of a vector and a scalar isn't exactly what I would call knowing what you are doing).

This however is the simplest abuse of notation I can think off of the top of my head, familiar to most anyone with a high school education:

(d/dx)F = 3x where F is say an analytic function defined over the real numbers, x is an element of R
==>
dF = 3x dx

The former is well defined mathematically, the latter is *not* even though since childbirth we are taught to do things that way and we use it all the time.

3x has its domain over the real numbers, dx otoh is not a real number... it is an infinitesimal. What does it mean to multiply an infinitesimal by a real number?

Of course latter on, after several grad courses in mathematics you can make this precise and 'work' in a certain way, but not without a good deal of either nonstandard analysis (or Robertson analysis) or operator analysis with differential forms, and in general great care has to be taken in what and how you define things.

The point being is you are justified in abusing this notation, b/c it works if you are careful, but keep in mind theres funny stuff going on behind the scenes.

 

[K.I.L.E.R]

I know what I'm doing. You don't know what I'm doing.

 

[epicstruggle]

I know what I'm doing. You don't know what I'm doing.

No one knows what your doing! Which is why everyone is always so scared when they first start reading your threads.