Why start at x, y, z - User contributions [en-gb]

Parentheses are overused

2021-07-02T19:28:28Z

David Eppstein: nvm already above

[[Category:Ambiguities]]

Parentheses are used to represent all sorts of operations and objects, many of which conflict with each other.

Grouping parts of an expression: \( (x+1)(x+2) \)

Argument of a function: \(f(x)\) is "\(f\) applied to \(x\)". ([[There is no function application symbol]])

Greatest common divisor: \((a,b) = \gcd(a,b)\)

Counting combinations: \( {n \choose k} = \frac{n!}{k!(n-k)!} \)

Repeated differentiation : \( f^{(n)}(x) = \frac{\mathrm{d}^nf}{\mathrm{d}x^n} \)

Vectors or one-column matrices: \( \begin{pmatrix} a \\ b \end{pmatrix} \)

Ideals: \((2)\) is the ideal generated by 2, \((a,b,c)\) is the ideal generated by \(\{a,b,c\}\).

Parentheses are overused

2021-07-02T19:28:12Z

David Eppstein: or binomial coefficients

[[Category:Ambiguities]]

Parentheses are used to represent all sorts of operations and objects, many of which conflict with each other.

Grouping parts of an expression: \( (x+1)(x+2) \)

Argument of a function: \(f(x)\) is "\(f\) applied to \(x\)". ([[There is no function application symbol]])

Greatest common divisor: \((a,b) = \gcd(a,b)\)

Counting combinations: \( {n \choose k} = \frac{n!}{k!(n-k)!} \)

Repeated differentiation : \( f^{(n)}(x) = \frac{\mathrm{d}^nf}{\mathrm{d}x^n} \)

Vectors or one-column matrices or binomial coefficients: \( \begin{pmatrix} a \\ b \end{pmatrix} \)

Ideals: \((2)\) is the ideal generated by 2, \((a,b,c)\) is the ideal generated by \(\{a,b,c\}\).

Big O notation

2021-06-30T18:02:25Z

David Eppstein: New article

[[Category:Unpleasantness]]

Big \(O\) notation is commonly written as if it were an equality:
\[ x^2+7x+53=O(x^2). \]
It is not an equality: the left hand side describes a function of \(x\), but the right hand side is...something else. In an attempt to make this more meaningful, some authors have resorted to treating \(O\)-notation as defining a set of functions, and using set membership instead of equality:
\[ x^2+7x+53\in O(x^2). \]
This does not work either, for expressions like
\[ x^2+7x+53=x^2+O(x). \]

The real meaning of this type of expression appears to be as a proxy for wrapping the entire expression in quantifiers and replacing the equality by an inequality:
\[ \exists X\in\mathbb{R}^+\ \exists C\in\mathbb{R}^+\ \forall x\in\mathbb{R}^+\ (x>X \Rightarrow x^2+7x+53\le Cx^2). \]

But even with this interpretation you still have to know, from information beyond the expression, what kind of limiting behavior you are studying: the quantification above describes the limiting behavior as \(x\to\infty\), not as \(x\to 0\).

Maybe the simplest solution is to tell students that "=O" is a special combination of symbols, meaningful only as a combination.

Powers of trigonometric functions

2021-06-30T17:44:06Z

David Eppstein: or log

[[Category:Ambiguities]]
[[Category:Inconsistencies]]
[[Category:Unpleasantness]]

When writing a power of a trigonometric function or logarithm, it's common to write the power as a superscript before the brackets:

\[ \sin^2(x) = (\sin(x))^2 \]

This conflicts with the conventional notation for inverses and repeated application of functions:

\[ f^2(x) = f(f(x)) \]

\[ y = f^{-1}(x) \implies x = f(y) \]

In his entry on Notation in the Edinburgh Encyclopedia <ref>The Edinburgh Encyclopedia, 1830, pp. 398-399. [https://babel.hathitrust.org/cgi/pt?id=chi.21708663&view=1up&seq=431 Copy on HathiTrust]</ref>, Charles Babbage is scathing about this convention.

<blockquote>Although a definition can not be false, it may be improper.</blockquote>

==References==

<references/>