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I hedged with "usually". Feel out your audience when you teach. Plus, honestly, one-point compactification of C is less intuitive than two point of R (where
by jbaber 3y ago
I hedged with "usually". Feel out your audience when you teach.
Plus, honestly, one-point compactification of C is less intuitive than two point of R (where you'd have + and - infinity), so I stand by it being not great to bring up. There are many models here ("infinity's not a number", "there is one, unsigned infinity", "there are two infinities", "there are transfinite numbers", ...) and picking the right one to bring up at the right time, I argue, is part of good teaching. Not just "time to expose this person to the definition that's the "fanciest" I've personally got experience with".
- frognumber 3y agoI usually go with "all of the above" rather than "the fanciest." With my six-year-old, I did: - Countable / uncountable infinity (but I didn't go beyond that) - Positive / negative infinity, optionally with NaN - One-point compactification of the real numbers (which is pretty simple -- draw a circle, and map points on the line to the circle) I don't recall if I ever did the one-point of C, but likely I did. The major point, actually, is to understand that there are many ways one can extend numbers, not all compatible with each other, to give bigger sets of numbers. The specifics weren't too important.