I am reading Julian Havil’s book Nonplussed, and in one chapter he’s discussing hypercubes, he says that the volume of an n-dimensional cube of side length L is L^n; then he goes on to note that as n-> infinity, the volume goes to zero if L<1; volume goes to 1 if L=1, and volume goes to infinity if L > 1. Ok that makes sense to me until I ask the units of L. I mean if I tell you that the side length is one meter, then 1*1*1*…1 =1 alright. Then I say, “oops, I meant one yard, so L= 0.914 meter” so now as n goes to infinity the volume is zero (0.914 * 0.914 * ....-> 0). I can see everything is OK as long as n is some finite number, because then we can say the volume is XXX (meters^n) which is equal to YYY (yards^n) and the difference is just a units conversion (=(m/y)^n). But what happens to the conversion factor “when n goes to infinity”?(adsbygoogle = window.adsbygoogle || []).push({});

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# Volumes of Hypercubes

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