21
$\begingroup$

Can someone please explain what the symbol $\stackrel{!}{=}$, consisting of an exclamation mark (!) above an equals sign (=) means? Below is the example I'm trying to decipher:


The normalization factor is chosen such that in average, Dynamic Θ Time passes as fast as physical time. In practice it is determiend by the condition that the interval in Dynamic Θ Time corresponding to a 4-year reference period [$T_0$, $T_1$] should be of exactly the same length:

$T_1 - T_0 \stackrel{!}{=} \int_{T_0}^{T_1} a(t) dt$

$\endgroup$

5 Answers 5

12
$\begingroup$

In my classes I use it to indicate anxiety. So an equals with ! over it means "we want to show this equality is true". An equals without ! means "I am asserting this is true".

I don't know how universal this convention is, though. I do know I'm not the only person to use this convention.

$\endgroup$
7
  • 18
    $\begingroup$ I've seen ? used before in that context, but not !. $\endgroup$ Jan 8, 2010 at 0:00
  • 7
    $\begingroup$ Maybe some people use ! to indicate happiness, like "woohoo!!" $\endgroup$ Jan 8, 2010 at 0:01
  • 2
    $\begingroup$ I use ! for "this must be true! it is equivalent to the result" and ? for "is this true? it implies the result". $\endgroup$ Jan 8, 2010 at 0:02
  • 15
    $\begingroup$ I'd be wary of such things. After all, students can find themselves scared of the notation you're using already, and sometimes it could really do without the extra burden of conveying emotion. There are words which are quick to write which would do the same in a more human fashion, like "hopefully" or "woohoo!!". Besides, it turns out that if you overuse these things when the context is not clear, then people start asking MathOverflow questions wondering what they mean... $\endgroup$ Sep 6, 2013 at 1:23
  • 1
    $\begingroup$ I have encountered that notation with the interpretation "should be" or "we want it to be" e.g. in the context of optimization problems if one is more interested in the parameters that yield the maximum value than in the value of the maximum itself. $\endgroup$ Sep 6, 2013 at 6:53
21
$\begingroup$

I propose that we adopt the usage of algebraic chess annotations, where the exclamation point ! indicates a particularly good move, which is also suprising or unexpected.

In mathematics, $\stackrel{!}{=}$ should denote a useful, important, but unexpected equality.

$\endgroup$
4
  • 3
    $\begingroup$ !? (Interesting move) $\endgroup$ Sep 6, 2013 at 10:39
  • 19
    $\begingroup$ And surely I've written ?? often enough when grading exams. $\endgroup$ Sep 6, 2013 at 10:43
  • 7
    $\begingroup$ Example: $\mathord\vdash\stackrel{!}{=}\mathord\vDash$. $\endgroup$
    – Goldstern
    Sep 6, 2013 at 11:57
  • $\begingroup$ Honestly, this is exactly what I thought when this question bubbled up to the front page! $\endgroup$ Sep 6, 2013 at 15:23
8
$\begingroup$

I've seen it used as "has to be equal to"; the typical example is the point in a derivation when we use information we know by some general argument.

$\endgroup$
1
  • $\begingroup$ I learnt that way to use it in physics classes (and use it heavily myself, very useful). $\endgroup$ Jun 27, 2014 at 16:34
4
$\begingroup$

In my class I use $f'(x) \stackrel{!}{=} 0 $ to show that we look for a zero of the derivative of $f$ in order to find a local extremum or a stationary point. Other requirements like the definition of a normalization factor are possible candidates.

$\endgroup$
1
  • $\begingroup$ That is exactly what I learned, and I feel it's totally obvious: it indicates a condition that must be met. But it seems that this is not a universally agreed notation, and it has the potential to cause quite some confusion. So, perhaps I should avoid using it. $\endgroup$
    – user490981
    Sep 9, 2022 at 15:47
0
$\begingroup$

I would guess it means the same thing as the slightly more common $\stackrel{.}{=}$ or $\stackrel{\text{def}}{=}$ --- that the two sides of the equation are equal by definition.

$\endgroup$
3
  • 3
    $\begingroup$ I use Pascal notation for those conventions $x := y$. $\endgroup$ Jan 8, 2010 at 1:28
  • 3
    $\begingroup$ The $x:=y$ notation is my favorite too, but see J. S. Milne’s opinion in jmilne.org/math/words.html $\endgroup$
    – user2734
    Jan 8, 2010 at 9:36
  • $\begingroup$ Milne may have a point, but as I see it, most of his argument applies to any decoration placed over $=$ to denote definitions. In particular, if the context makes it clear we are defining something, one should use plain $=$ (or say it in words). $\endgroup$ Sep 6, 2013 at 10:47

Not the answer you're looking for? Browse other questions tagged or ask your own question.