# What does ! above = mean [closed]

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:

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

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## closed as off-topic by Andrés E. Caicedo, David White, Andrey Rekalo, Noah Stein, Joseph Van NameSep 6 '13 at 21:00

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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.

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I've seen ? used before in that context, but not !. – Harry Gindi Jan 8 '10 at 0:00
Maybe some people use ! to indicate happiness, like "woohoo!!" – Ryan Budney Jan 8 '10 at 0:01
I use ! for "this must be true! it is equivalent to the result" and ? for "is this true? it implies the result". – Reid Barton Jan 8 '10 at 0:02
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... – James Cranch Sep 6 '13 at 1:23
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. – Manfred Weis Sep 6 '13 at 6:53

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.

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!? (Interesting move) – Moritz Firsching Sep 6 '13 at 10:39
And surely I've written ?? often enough when grading exams. – Joel David Hamkins Sep 6 '13 at 10:43
Example: $\mathord\vdash\stackrel{!}{=}\mathord\vDash$. – Goldstern Sep 6 '13 at 11:57
Honestly, this is exactly what I thought when this question bubbled up to the front page! – Todd Eisworth Sep 6 '13 at 15:23

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.

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I learnt that way to use it in physics classes (and use it heavily myself, very useful). – kjetil b halvorsen Jun 27 '14 at 16:34

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.

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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.

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I use Pascal notation for those conventions $x := y$. – Ryan Budney Jan 8 '10 at 1:28
The $x:=y$ notation is my favorite too, but see J. S. Milne’s opinion in jmilne.org/math/words.html – user2734 Jan 8 '10 at 9:36
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). – Emil Jeřábek Sep 6 '13 at 10:47