Hello, > 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. Peter -- 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.
