Jonas -- the implementation of QQbar elements is precisely via real intervals and a polynomial satisfied by the number (not always the min poly unless that is forced, since that can be expensive).
This I think that the only issue is in how real interval field elements are represented. No-one (or almost) has ever liked the current form ending in a ? which does of course have a logic behind it but is *very* hard to explain (and to remember). The sqrt() - style representation is only really practical for number of degree 2 -- in principle one could do more general radical expressions, and sqrt(2)+sqrt(3) looks rather nicer than sage: QQbar(2).sqrt()+QQbar(3).sqrt() 3.146264369941973? but that is (a) expensive and (b) only applies to rather special algebraic numbers. John On 13 July 2015 at 18:49, William Stein <wst...@gmail.com> wrote: > On Mon, Jul 13, 2015 at 10:15 AM, Simon King <simon.k...@uni-jena.de> > wrote: > > Hi! > > > > On 2015-07-13, Nathann Cohen <nathann.co...@gmail.com> wrote: > >> sage: sqrt(2) # a symbolic ring element > >> sqrt(2) > >> sage: QQbar(sqrt(2)) # an algebraic value > >> 1.414213562373095? > >> > >> It is true that this final '?' sounds more like a '...', as if some > additional > >> digits were hidden in a value stored as a float/double. Yet it is exact. > >> > >> How could we replace it? Ideally, that would be a 'sqrt(2)' but can we > always > >> provide such a representation cheaply? Could we display it as 'sqrt(2)' > at least > >> when it is free to do so? > > > > The elements of QQbar are the solutions of algebraic equations. As you > > probably know, the solutions of algebraic equations of degree > 4 can, in > > general, not be expressed that nicely. But it seems like an appealing > > idea to show a nice expression for algebraic numbers of degree up to 4. > > > >> If we cannot get rid totally of this numerical representation, what > would you > >> think of replacing this '?' by a 'alg', which would be (slightly) more > >> informative, e.g.: > > > > 1.4142134... looks exact to me: "..." seems to suggest that Sage knows > all > > (potentially infinitely many) digits but can't show them all, whereas > "?" seems > > to suggest that the last shown digit is questionable (i.e., subject to > rounding > > errors), i.e., "?" seems to suggest that Sage doesn't know the exact > value. > > Yes. Also, with interval arithmetic, that is precisely what it means: > > sage: RealIntervalField(53)(sqrt(2)) > 1.414213562373095? > sage: QQbar(sqrt(2)) > 1.414213562373095? > > It's certainly not good that the above two print in the same way. > The (mysterious [1]) person who made those design choices -- Carl > Witty -- isn't contributing anymore, or I'd ask his opinion. > > > [1] http://stackoverflow.com/users/684532/carl-witty > > > > > So, I'd prefer to display elements of QQbar as floating point numbers > > (what default precision?), always rounded DOWN to the last digit that is > > displayed, and followed by "..." (not "?") unless the displayed value is > > exact. So, what is displayed is an initial part of the potentially > > infinite sequence of digits. > > > > Best regards, > > Simon > > > > > > -- > > You received this message because you are subscribed to the Google > Groups "sage-devel" group. > > To unsubscribe from this group and stop receiving emails from it, send > an email to sage-devel+unsubscr...@googlegroups.com. > > To post to this group, send email to sage-devel@googlegroups.com. > > Visit this group at http://groups.google.com/group/sage-devel. > > For more options, visit https://groups.google.com/d/optout. > > > > -- > William (http://wstein.org) > > -- > You received this message because you are subscribed to the Google Groups > "sage-devel" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to sage-devel+unsubscr...@googlegroups.com. > To post to this group, send email to sage-devel@googlegroups.com. > Visit this group at http://groups.google.com/group/sage-devel. > For more options, visit https://groups.google.com/d/optout. > -- You received this message because you are subscribed to the Google Groups "sage-devel" group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-devel+unsubscr...@googlegroups.com. To post to this group, send email to sage-devel@googlegroups.com. Visit this group at http://groups.google.com/group/sage-devel. For more options, visit https://groups.google.com/d/optout.
