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I'm not a mathematician, but this question intrigues me: Is an infinitely small percentage or part of infinity infinite? Do the two infinities "cancel out", leaving you with a real number? It seems like they would, but what number would be left? I've done some reading and it seems you can't subtract infinity from infinity, since that would leave you with an undefined number, but I'm not sure how that applies to division.

I'm not sure if this is a question that has been asked before (although something tells me it has) but I'm largely unable to find an answer to it. I don't absolutely have to know the answer, since it's a theoretical question with no real-life application, but it's been bugging me for a while, so I'd love to find out.

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  • $\begingroup$ Depends on the infinities... $\lim_{x\to \infty}\frac x{\log x}$ is a ratio of two infinities, but does not converge to a finite limit. Subtraction of infinities similarly depends on the comparison between them. $\endgroup$ – abiessu Nov 14 '13 at 15:52
  • $\begingroup$ Because the limit can be any of an infinite number of possibilities, expressions of this kind are avoided or if it is necessary to present one, labelled indeterminate forms. $\endgroup$ – hardmath Nov 14 '13 at 15:56
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Often traditional reasoning from arithmetic breaks down when you try to think about infinity, and percentages are no exception.

Examples: You agree to pay me $B(n)/n$ dollars per year, where $B(n)$ is a function giving your total wealth after $n$ years (thanks!). So as time goes by, the portion of your income you're paying me is $1/n$ (or $100/n$ percent). As the years pass, this dwindles down to an "infinitely small percentage."

  1. As time goes by, let's say you're a hard worker and immortal, and you earn more and more, say $B(n) = n$. Then every year you're paying me a dollar. The PERCENTAGE of your income that you are paying me is getting smaller and smaller each year, over time, but you're giving me a dollar a year. So in this case "an infinitely small percentage of infinity" is one.

  2. In experiment two, you're still immortal, but you don't work as hard. Now $B(n) = \sqrt{n}$. Then every year you're paying me $\frac{1}{\sqrt{n}}$ dollars. As time goes by, your income is still going to infinity, but what you're paying me is going to dwindle down to zero. So "an infinitely small percentage of infinity" is zero.

  3. In our final experiment, you're immortal and work really hard, and $B(n) = n^2$. Now you're paying me $n$ dollars a year, and an infinitely small percentage of infinity is infinity.

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An infinitely small part of infinity can indeed be a real number. For example, if $H$ is an infinite hypernatural, then $\epsilon=\frac{1}{H}$ is infinitesimal, and then an epsilonth part of $H$ will be exactly... 1.

By the way, this does have real-life applications. The infinitesimal approach is a more accessible way of teaching calculus (see Elementary Calculus), and calculus has numerous real-life applications.

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