The Unclear Impact

A common name for the inverse sine function is

arcsin 𝑥

The usual name for inverse hyperbolic sine function is

arcsinh 𝑥

Now some Wikipedians have decided to start calling it

arsinh 𝑥

I've never seen this before and it makes me think "arse". I don't like it!

A more common alternative is

sinh⁻¹ 𝑥

Some computer scientists use

asinh 𝑥

Some pedants, claiming that hyperbolic trig functions aren't connected to arcs, argue for

argsinh 𝑥

Who came up with "arsinh" and why? Is it too late to kill off this notation? It seems like a case of Wikipedia editors run amok. It reminds me of how some of them made up their own very precise definition of "order of magnitude" - and now run around correcting people who say "to within an order of magnitude" if it doesn't match their made-up definition.

When I want to be understood I'll use "arcsinh". When I want to act mathematical I'll use "sinh⁻¹". If I wanted to save space I'd write "asinh". I might use "argsinh" if I were pretending to be a pirate. But I'd only use "arsinh" if I wanted to be an arse.

By the way, I don't like debates about notation. So I don't know why I'm talking about this when I have more interesting things to talk about. Just blowing off steam, I guess.


I think nish would be nice.

@johncarlosbaez I don't like arsinh either. I prefer arcsinh or asinh...

@johncarlosbaez In Hungary, I learned the inverse functions of hyperbolic functions as “area sine hyperbolic” and “area cosine hyperbolic” (I guess it’s alluding to the relation with the enclosed area of the hyperbole instead of the arc length). We used the notation arsh and arch for them (but arcsin and arccos for the trigonometric inverse function). Maybe Wikipedia is trying to meet halfway between the English and the Central/Eastern European notation in this case?


@johncarlosbaez Perhaps it's not fair to blame Wikipedians because "arsinh" is mentioned in Abramowitz & Stegun (just above 4.6.4).

The start of Section 4.6 in Abramowitz & Stegun.

@johncarlosbaez Wikipedia claims this is from ISO standards ( I guess these people who have nothing to do with maths decided to make their own standards to milk some cash.

@taxyovio @johncarlosbaez From the same standardization people who so helpfully brought us "lb" for the binary logarithm and "lg" for the decimal logarithm.

@11011110 @taxyovio - ah yes, the same sort of people who introduced mebibytes so we wouldn't get confused by megabytes:

@taxyovio From the version of this standard that I saw on the Internet (they were draft version: maybe it changed in-between?), this standard uses “arcsin” and “arcsinh”.

@MartinShadok @taxyovio - if you want more people to understand you, you'll say "arcsinh".

I think arsinh is short for Arsenio Hall, best known for his role in 'Amazon Women on the Moon'.

@FritzAdalis - I'm glad there's some rational explanation for it.

@johncarlosbaez The talk page for that one is a master class in petty obstinacy. Someone tries to change the page every couple years, but a couple dedicated people keep it as is.

@johncarlosbaez argsinh x!

@tca -arrghsinh 𝑥!

@jitseniesen - okay, very interesting. Of course, Abraham and Stegun are strongly indicating what they consider the *standard* notation, and that's arcsinh.

@0x10f - when archangels sin it's an archsin.

@johncarlosbaez Also, I'd like to just point this out (I don't have an opinion on this subject) -- you said that a notation makes you think "arse", and that Wikipedia editors seem to have "run amok" (I happen to edit Wikipedia), but then you say that you don't like debates about notation. This seems self-contradictory to me -- doesn't publicly making a statement with prominently stated contentious claims constitute an initiation of such a debate?

@mikolaj - yes, it's self-contradictory. As I said, "So I don't know why I'm talking about this...."

@johncarlosbaez in "French Mathematics" I've never seen the arcsinh but only the argsinh notation.

@uxor - very interesting! And someone said the Germans are to blame for arsinh. So I guess different conventions are common in different languages.

@johncarlosbaez I somewhat dislike sinh⁻¹ as that might be confused with 1/sinh.

And we should not forget the proper pronunciation of these functions: cosh sounds like koʃ, sinh is Portuguese and sounds like siñ, the same is valid for tanh.

@ralf_muschall - Indeed! When teaching calculus I avoided expressions like sinh⁻¹, or more commonly sin⁻¹, precisely because students can easily confuse them with reciprocals.

Pronouncing sinh correctly is a cinch.

@3j0hn - aha! I haven't dared look at the talk page, but that sounds like the sort of thing I feared.

@johncarlosbaez Aren't these names basically just a convenient mnemonic in any case? I mean, if you so desperately want to break notational compatibility, why not invent something completely new instead of sin/cos/tan...?

That "ar for area" stuff just feels like a fraudulent retcon, and anyway it IS an arc length, if using Minkowski metric 😀

@TorbjornBjorkman - names are just conveniences, which is why I don't like arguing about them, but also why I get annoyed when people try to reform the world by making up new names - it rarely winds up reforming the world, usually it just adds confusion.

And yeah, I agree about the Minkowski metric.

@johncarlosbaez @3j0hn David Jeffrey has a better and more general proposal which I wish would catch on: Inv sin(x), Inv tanh(x), Inv f(x) with a space for subscripts indicating branches.

In math displays inline one could instead use a hacek (looks like a v, for inVerse) atop the f.

But arc really only works for trig functions.

@johncarlosbaez but also consider