infinity is not a number
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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
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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good point
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It's kind of funny that most people have an easier time accepting and understanding extended reals rather than limits. I know it has several downsides, but I can't help but wonder whether teaching non-standard calculus in hs would help the general public in learning the subject.
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Being able to rigorously extend sets of numbers to include infinite and infinitesimal quantities (i.e. extending an Archimedean number system into a non-Archimedian one) in a way that their arithmetic makes sense is pretty commonplace in abstract algebra, e.g. for mappings.
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But they are limitless
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