infinity is not a number
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The concept of "number" is pretty flexible. Humans started with whole numbers, then included fractions, then irrational numbers, then zero, then negative numbers, then imaginary and complex numbers. The "extended reals" include +infinity and -infinity: https://en.wikipedia.org/wiki/Extended_real_number_line …
I'm a big fan of the extended reals. Perhaps i should have said "naive manipulations treated as a number are invalid", but that didn't have the same zing to it
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are the extended reals anything more than a convenient shorthand for limits? I was never really able to see a clear distinction
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Are real numbers anything more than a convenient shorthand for limits of rationals? Some people think that too. It's a matter of opinion. Luckily, you don't need to have an opinion about these things! The math works fine even if you don't have an opinion.
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Even that is questionable. If you want 1/infinity = 0 then you have problems, but if you think of it as infinitesimally small but non-zero, then you can generate an ordered field that extends R. In that field you're treating infinity as a number and using naive manipulations.
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Interesting, and a good point. I haven’t internalized the hyperreals fully (which is what I think you’re referring to). They’ve always been of recreational interest to me, but I never gained an intuition of the notion of “standard part” or nilpotent differentials.
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Teaching math: First we scold the students and tell them not to do all sorts of stuff. Then we take the good students into the back room and tell them how to do this stuff without getting in trouble.
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Actually, the best students are those who, after we've taken them to the back room and told how to stay out of trouble, promptly ignore all our secret handshakes and mystic nostrums, and continue to swashbuckle with all the vigorous aplomb of the invincible beginner.
End of conversation
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number
systems including infinity may fail to have the properties you expect and be an unsuitable domain for the problem at handThanks. Twitter will use this to make your timeline better. UndoUndo
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Another cool extension (of Q) is p-adic numbers
Thanks. Twitter will use this to make your timeline better. UndoUndo
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