On Thu, Oct 3, 2013 at 1:26 PM, Vincent Delecroix <20100.delecr...@gmail.com> wrote: >>> I guess my follow up question would be do we want infinity to be real in >>> this sense or is that just a byproduct of its implementation? I don't >>> know >>> all the uses for infinity that other sage users have, but certainly from >>> the perspective of the Riemann sphere it's a bit odd since >>> CC(infinity,0), >>> CC(0, infinity) and CC(infinity, infinity) are all distinct in sage, >>> giving >>> us 3 different complex infinities. I'm not particularly picking on CC, >>> since infinity*I and infinity are also not equal. >>> >> >> If you ask me, neither RR nor CC should contain any kind of infinity. As a >> piece of mathematical software, Sage must be mathematically correct and not >> pretend that infinity is in RR (or CC). >> >> There is a set of floating point objects and a set of real numbers; they >> have a large intersection, but both of them contain elements that the other >> >> doesn't. Plus or minus infinity and "not a number" are not real numbers; >> conversely, most real numbers, like pi, cannot be represented (only >> approximated) by floating point numbers. >> >> Of course, it is also true that floating point objects can represent >> infinity, and for a good reason. For example, if you evaluate a >> meromorphic function at a pole, then it is legitimate to say that the >> result is infinity. It is just not an element of the complex numbers, but >> of the Riemann sphere (the projective line over CC). >> >> For the real numbers, there are two separate completions: first, the >> projective line over RR, which has only one point at infinity and is a >> subset of the Riemann sphere; second, the "extended real line" containing >> plus infinity and minus infinity. It depends on the context which one is >> more useful, but both of them are definitely different from RR, because RR >> does not contain any infinite element at all. >> >> So I would say that the current behaviour of Sage (Infinity in RR giving >> True and any similar suggestion that infinity is a real number) is >> mathematically wrong and must be changed. It also contradicts the >> documentation of the infinity "ring" (in which Sage's "Infinity" object >> lives), which says that the infinity "ring" does not canonically coerce >> into any other ring. > > I do not agree. RR and CC are *badly* named in Sage. As Peter said,
RR isn't named "Real numbers"; it is named "real field with 53 bits precision" sage: RealField(53) Real Field with 53 bits of precision That said, of course, it is not a "field", so it is misleadingly named. I was just following Magma in these conventions: wstein@pixel:~/Downloads$ magma Magma V2.18-5 Thu Oct 3 2013 13:44:16 ... > RealField(); Real field of precision 30 > they are sets of floating point numbers. In particular, I found > completely valid that Infinity is an element of RR (since it is *not* > the set of real numbers). > > Vincent > > -- > 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/groups/opt_out. -- William Stein Professor of Mathematics University of Washington 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/groups/opt_out.
