Is 10 closer to infinity than 1?

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14 points·JumpCrisscross··5 comments

I suppose it depends on one's definition of "closer".

If you define A is closer to X than B as abs(X-A) < abs(X-B), or rather abs(X-B) - abs(X-A) > 0, which is what I would be inclined to do, and take the limit as X goes to infinity, than yes, 10 is closer to +inf than 1.

Assuming that X > A and X > B (a fair assumption in this case), this becomes X-B - (X-A) > 0, or rather A - B > 0. Taking the limit as X goes to infinity, this is unchanged, and as such one could say that A is closer to infinity than B iff A>B. As 10 > 1, 10 is closer to infinity than 1.

Unfortunately, although I cannot come up with an example offhand, I am fairly sure that there is a proof the other way as well. Infinity is just weird like that.

1 and 10 are equally far from infinity. The ratio of the respective distances goes to 1 as x goes to infinity [1]. To put it in terms of time, both now and (now + 10 seconds) are equally far from forever.

[1] http://www.wolframalpha.com/input/?i=limit+as+x+goes+to+infi...

"Where the nonsense starts is with our habit of thinking of a large number as closer to infinity than a small one." Wittgenstein

Infinity its not measurable point, just like infinity/infinity is not equal to 1. So we could say that both are equally far/closer from infinity or at least impossible to state that.

infinity1 - 10 = infinity infinity2 - 5 = infinity infinity1/ infinity2 = infinity

Measure theory opened my eyes when thinking about infinities.

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