Talk:Abuse of notation

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Contents 1 Screwdriver? 2 Very common abuse? 2.1 Misuse rather than abuse? 3 Have retreated from language criticized above 4 John Harrison 5 Infinite limits 6 Quantifiers or Definition vs Fact 7 two comments 8 Inner product vs. v^T w

[edit] Screwdriver?

I am not sure what you are trying to do with this page, in particular the opening comment is almost content-free. I always use screwdrivers to open paint tins. 8-) (and no, this is not vacuously true - I've been painting my house recently). I would think a page on abuse of notation should also describe why it is a useful thing to do, as well as describing why the examples are actual abuses of notation. (e.g. always insisting that functions and variables have distinct symbols leads to proliferation of symbols that only the most anal retentive mathematician (or painter) would delight in.) Andrew Kepert 01:45, 6 Apr 2005 (UTC)

Hi Andrew, Sounds like you could have made a better start at this than me. I used an abuse of notation recently in Combinadic so thought there should be such a page, but am still at a loss what should go into it. --J. W. McLeod 09:28, 6 Apr 2005 (UTC)

[edit] Very common abuse?

"A very common abuse of notation is using sin²(x) instead of (sin(x))²."

It's not an abuse at all according to other things I've read - fⁿ(x) = [f(x)]ⁿ for n not equal to -1 (for n=-1 it refers to the inverse function). The article Function (mathematics) seems to make no reference to this. Brianjd | Why restrict HTML? | 04:56, 2005 Apr 8 (UTC)

• Especially in the abstract, fⁿ(x) = f[f(x)] However, sin²(x) is an exception to the rule. Bluap 16:48, 3 May 2005 (UTC)
  • Seems like a double standard to me. So sin² means the square of the sine—i.e. the sine times itself—but sin^-1 is the inverse function? OneWeirdDude (talk) 23:00, 16 October 2008 (UTC)

sin²x is an unusual notation but probably not abuse. For most functions, f²(x) = f(f(x)), rather than [f(x)]². Still, all trig functions represent a common and systematic exception. This is true of all exponents, not just 2. —Preceding unsigned comment added by Eebster the Great (talk • contribs) 21:35, 2 February 2009 (UTC)

[edit] Misuse rather than abuse?

sin^k(x) doesn't simplify exposition, nor suggests any correct intuition. It's just an arbitrary exception that creates ambiguity, so I propose we designate it a misuse of notation.

< rant >

It's being widely taught to school kids, roughly together with teaching f^-1 as function inverse, and creates unjustified confusion. It's a shame!

(To make it worse, f^-1 is usually introduced without explaining the general f^k notation for composition, nor it's origin in paren-less "f f x" function application notation, and the wide abuse of exponentiation nota).

Presumed rationale for the exception:

* Trig functions are among the first functions that are commonly written without parentheses. This notation is commonly introduced without discussing order of operations w.r.t. exponentiation, making sin x² ambiguous.

* Repeated application of trig functions is nearly useless. However, sin^-1 is useful, limiting the exception to positive powers, which is ugly.

< /rant > 79.179.39.171 (talk) 22:24, 14 June 2009 (UTC)

[edit] Have retreated from language criticized above

The present version would seem to make a better stub for this topic. I don't think there is much of a controversial nature left, but there remains enough structure so that it's pretty clear what the intended topic is. --J. W. McLeod 12:48, 10 Apr 2005 (UTC)

[edit] John Harrison

Who is John Harrison? --Abdull 08:29, 30 May 2006 (UTC)

[edit] Infinite limits

I don't think qualifies as abuse of notation.

• If the domain and codomain under consideration are the extended real line, the limit may very well exist, and have the precise value of , without any notational or conceptual difficulties whatsoever.
• If the domain and codomain under consideration are , then, as described, the limit does not exist (edit: and neither does the infinity), and, therefore, the sentence is not false but meaningless when considered merely as the sum of its parts, so the idiom (essentially bringing the extended real line into a real context) gives meaning to an otherwise meaningless sentence, rather than giving an additional meaning to a meaningful one.

Dfeuer 04:36, 30 October 2007 (UTC)

Yes, has a precisely defined meaning, as our own article on limits shows. I'm getting rid of that example. -- 75.162.71.236 04:15, 8 November 2007 (UTC)

[edit] Quantifiers or Definition vs Fact

Consider the question "f(x) = 0; Is it true that f'(x) = 0?" This can either mean "at some particular point x, f(x) = 0, in which case f' evaluated at the same point need not be 0, or it can be taken as a definition, in which for all x, f(x) = 0, and f'(x) is indeed 0 at all points. Is there a standard way of disambiguating these? —Preceding unsigned comment added by 76.113.64.59 (talk) 09:25, 17 June 2008 (UTC)

The = operator means that something is always true. Without qualification, f(x) = 0 means that x is 0 for all (valid) x. So for the same domain. Xihr (talk) 10:11, 17 June 2008 (UTC)

The = operator in itself does neither imply always nor true, e.g. you can easily state "x=2" or even "2=3". Therefore, I agree with the poster of the unsigned comment above. As long as x is a free variable the question is incomplete. However, the concept of always meaning "for all x" could also be implied with the equivalence operator like this: f≡0. Schellhammer (talk) 16:38, 5 November 2008 (UTC)

[edit] two comments

1. The determinant formula for the vector product is just a mnemonic device to help in remembering the definition. Therefore I can't see how it is an abuse of notation. McKay (talk) 09:39, 3 June 2009 (UTC)

2. The section on O(.) reads like a personal essay. The way "=" is used in this context is imo an abuse of notation, but this needs to be cited from a suitable source. The other claim, that f(n) is just a value rather than a function, doesn't belong here as it is not specific to this notation. Also, ambiguity of notation is not at all the same as abuse of notation, so the example O(n^m) doesn't belong either. McKay (talk) 09:39, 3 June 2009 (UTC)

[edit] Inner product vs. v^T w

Many people seem to write the inner product <v,w> between two vectors as v^T w, although strictly speaking the result of the latter operation should be a 1 by 1 matrix rather than a scalar. This is not the same thing: consider an m by n matrix A with m and n both > 1. Then, like any matrix, A can be multiplied by a scalar, but not by a 1x1 matrix. I'm not aware of any mathematical operator that will take a one by one matrix and extract its element as a scalar or vice versa.

Can anyone knowledgable either confirm or disconfirm this as a case of abuse of notation? —Preceding unsigned comment added by 131.111.20.201 (talk) 13:28, 8 June 2009 (UTC)