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This is just a question that popped into my head which I lack the knowledge to answer (or even to know whether there is an answer, honestly). Does the idea of an infinitesimal unit of measurement even make sense theoretically? If I had a line segment, could it be of infinite length when measured in such units?

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    $\begingroup$ See non-standard analysis. In short, yes, there are rigorous developments of analysis that include infinitesimals. Depends on what you mean by "unit," of course. $\endgroup$ – Thomas Andrews Jul 22 '14 at 4:51
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The answer is affirmative. Thus, the unit interval $[0,1]$ can be partitioned into $H$ subintervals of infinitesimal length. These are called hyperfinite partitions. Wallis in fact used the symbol $\infty$ in the sense of $H$ above but today it is not customary to use it this way so I used $H$ instead (Wallis was the one who introduced the infinity symbol in the first place).

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What mathematicians mean by 'infinitesimal' and how the word is commonly used are sometimes different. The common definition is that it is a quantity too small to be measured so the simple answer to your question is No. However, we could still ascribe values to sub-measureable quantities, see Could we assign a numerical value to an infinitesimal? And of course it's normal to leave infinitesimals as abstract values with certain properties resulting from their small magnitude. I can't help with your second question though.

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For an infinite line segment, its length is infinite inches, infinite kilometers, infinite parsecs, or infinite angstroms - all are same. Angstroms and light-years does not make any difference as infinity + infinity = infinity * infinity = infinity

As infinity is not a number but a concept, it cannot be compared with any real number. A crude definition of infinite is 'a number larger than the largest real number' is also mathematically senseless.

Einstein said nothing is infinite except the universe and human stupidity.

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  • $\begingroup$ I'm not talking about an infinite line segment (does such a thing exist? I think by definition line segments are closed intervals on both sides, i.e. finite, but someone can correct me if I am wrong). My question was whether it is mathematically sensible to define some sort of infinitesimal standard of measurement such that these closed intervals would be of infinite length when measured in those units. Also, be careful with $\infty+\infty=\infty*\infty=\infty$, because those statements are only sensible for particular types of infinity (e.g. $\aleph_0$). $\endgroup$ – hexaflexagonal Jul 22 '14 at 19:17

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