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\$
–user13618
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\$
–user14972
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.”

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answered Aug 22 "15 in ~ 15:50 \$endgroup\$
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