TITLE: What is infinity divided by infinity?
QUESTION [25 upvotes]: This should be a simple question but I just want to make sure.
I know $\infty/\infty$ is undefined. However, if we have 2 equal infinities divided by each other, would it be 1?
And if we have an infinity divided by another half-as-big infinity, would we get 2? For example $\frac{1+1+1+\ldots}{2+2+2+\ldots}=\frac12$?

REPLY [18 votes]: I will quote the following from Prime obsession by John Derbyshire, to answer your question.

Nonmathematical people
  sometimes ask me, “You know math, huh? Tell me something I’ve
  always wondered, What is infinity divided by infinity?” I can only
  reply, “The words you just uttered do not make sense. That was not a
  mathematical sentence. You spoke of ‘infinity’ as if it were a number.
  It’s not. You may as well ask, ‘What is truth divided by beauty?’ I have
  no clue. I only know how to divide numbers. ‘Infinity,’ ‘truth,’
  ‘beauty’—those are not numbers.”