I recognize $infty/infty$ is undefined. However, if we have actually 2 equal infinities separated by each other, would it be 1?
And if we have an infinity split by another half-as-big infinity, would we gain 2? For instance $frac1+1+1+ldots2+2+2+ldots=frac12$?
sos440: In NSA, boundless numbers don't have actually specifiable sizes, and also you can't uniquely determine a sum choose $1+1+1+ldots$ v a specific hyperreal. Hyperreals have the right to be defined as equivalence classes of order under one ultrafilter. Due to the fact that ultrafilters can't be explicitly constructed, you can't, in general, take limitless sums $sum a_i$ and $sum b_i$ and also say even if it is they refer to the very same hyperreal. An ext correct if you offered Conway's surreal numbers. In the surreals, it would be organic to combine $1+1+ldots$ through $omega$, although there is quiet an pass out as discussed by Karolis. $endgroup$
Aug 11 "12 at 14:50
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Essentially, you offered the price yourself: "infinity over infinity" is not identified just because it should be the result of limiting processes of various nature. I.e., since such a meaning would be given for the benefits of completeness and also coherence through the truth "the limiting ratio is the ratio of the limits", your
$$ frac1 + 1 + cdots2 + 2 + cdots = lim_n o infty fracn2n = frac12 $$
and, speak (this is my choice)
$$ frac1 + 1 + 1 + cdots1 + 2 + 3 + cdots = lim_n o infty fracnn(n+1)/2 = 0 $$
would have to be equal (as they typically define $infty/infty$), which does not happen.
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answer Aug 11 "12 at 12:06
Filip ChindeaFilip Chindea
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$egingroup$ this equations, that course, assumes the you actually median a limiting procedure of the sort, and that the number of terms ~ above the top and the bottom accrue at the same rate (more or less). $endgroup$
Aug 11 "12 in ~ 14:20
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I will quote the complying with from Prime obsession by john Derbyshire, to answer her question.
Nonrebab.netematical civilization sometimes questioning me, “You recognize rebab.net, huh? call me miscellaneous I’ve constantly wondered, What is infinity split by infinity?” I deserve to only reply, “The native you just uttered carry out not do sense. That was not a rebab.netematics sentence. You spoke of ‘infinity’ together if it to be a number. That not. Girlfriend may as well ask, ‘what is truth separated by beauty?’ I have actually no clue. I just know just how to divide numbers. ‘Infinity,’ ‘truth,’ ‘beauty’—those room not numbers.”
answered Aug 22 "15 in ~ 15:50
Bhaskar VashishthBhaskar Vashishth
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To sophisticated a little bit on the comment by sos440, there room at the very least two viewpoints to the issue of infinity/infinity in calculus:
(1) $frac inftyinfty$ as an indeterminate form. In this approach, one is interested in the asymptotic actions of the proportion of 2 expressions, which room both "increasing there is no bound" as their usual parameter "tends" come its limiting values;
(2) in one enriched number mechanism containing both boundless numbers and also infinitesimals, such together the hyperreals, one have the right to avoid stating things prefer indeterminate forms and also tending, and also treat the concern purely algebraically: for example, if $H$ and also $K$ are both unlimited numbers, then the ratio $frac H K$ have the right to be infinitesimal, infinite, or finite appreciable, relying on the family member size the $H$ and also $K$.
One benefit of approach (2) is the it permits one to talk about indeterminate forms in concrete fashion and distinguish several cases depending top top the nature that numerator and also denominator: infinitesimal, infinite, or appreciable finite, before discussing the technical concept of limit which has tendency to it is in confusing come beginners.
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Note 1 (in an answer to user Xitcod13): below an infinitesimal number, in a number system $E$ prolonging $ ebab.netbbR$, is a number smaller sized than every hopeful real $rin ebab.netbbR$. One appreciable number is a number enlarge in absolute worth than some optimistic real. A number is finite if it is smaller in absolute worth than some optimistic real.