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MathGroup Archive 2007

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Re: is 3?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg73669] Re: is 3?
  • From: "David W.Cantrell" <DWCantrell at sigmaxi.net>
  • Date: Sat, 24 Feb 2007 02:17:09 -0500 (EST)
  • References: <ermdak$hnh$1@smc.vnet.net>

"Tony Harker" <a.harker at ucl.ac.uk> wrote:
> Dear Oleksandr,
>
>               That's interesting: but looking at what is produced by
>   ArcSinh[2]/ArcCsch[2] == 3 // FullSimplify
>  makes me wonder just how Mathematica achieved this.

Spec., before saying True, it gives the warning

N::meprec : Internal precision limit $MaxExtraPrecision =
49.99999999999999` reached while evaluating -3 + ArcSinh[2]/ArcCsch[2].

> In other words, what
> does the message that Bob Hanlon so carefully suppressed mean? Similar
> messages sometimes occur when checking the results of simple equations by
> back-substituting, especially when the solutions contain surds, and
> appear to show that Mathematica is diving off into real numbers to prove
> results involving integers and powers of integers.

Yes, I find such things disconcerting. Would someone please explain what's
going on? It's probably been explained here before; if so, maybe just a
reference to an earlier thread could be given.

BTW, another example, which was never explained, appeared here at the end
of last Nov. in the thread "Unexpected Warning with ArcTan" by Andrew
Moylan.

Simplify[ArcTan[1/(1/2 + (1/2)*(-1 + 1/E^40))]]  yields  ArcTan[2*E^40].
But that correct result is preceeded by the warnings

Power::infy : Infinite expression 1/0. encountered.
Power::infy : Infinite expression 1/0.^1. encountered.

which make it seem as though approximate arithmetic were involved in some
intermediate steps.

David W. Cantrell


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