>> 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, 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.
