# How many times more than $0$?

If I have $10$ apples, but you have $5$ apples, then I have $2$ times more apples than you. But what if I have $10$ apples, but you don't have any apples?

If you look at the graph $f(x)=\frac{10}{x}$, it shows that when $x$ approaches $0$, $f(x)$ approaches infinity. So it means I have infinity times more apples than you?

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Cobold, I presume you meant $y = \frac{10}{x}$ instead of $y = \frac{10}{0}$, and I edited appropriately. Hope it's ok. –  Srivatsan Nov 28 '11 at 21:43
Re "If I have 10 apples, but you have 5 apples, then I have 2 times more apples than you.": English-usage sticklers, including yours truly, will disagree: you have two times as many, or one time more. –  msh210 Nov 28 '11 at 21:47
@msh210: "two times as many", or "twice as many"? –  Arturo Magidin Nov 28 '11 at 21:55
That is to say, nothing wrong with "two times", even if there's a shorter alternative (and even if I personally happen to be one of those people who also like using "thrice" instead of "three times"). –  Ilmari Karonen Nov 28 '11 at 21:59
@msh210, so, if I have 10 apples, and you don't have any, then I have infinity-minus-one times more apples than you. Good, I'm glad we got that settled. –  Gerry Myerson Nov 28 '11 at 22:10

It would be more accurate simply to say that the question has no answer -- your ten apples cannot be described as a multiple of my zero apples.

You can choose to call that "infinity times", but for most purposes that is just a way to fool yourself into thinking that you have answered the problem when in fact you haven't. More specifically, if you know you have "two times as many as 5" apples, you can use this to compute the exact number of apples -- but knowing that you have "infinity times as many as 0 apples" tells you nothing useful.

It's better just to leave the result of the division-by-zero undefined. You don't have any times zero apples.

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You have to be careful because infinity ($\infty$) is not a number, so saying "I have infinity apples" doesn't really make sense.

It would be better to say "I have impossibly-many apples" perhaps. And that should match your intuition from the example you gave: it is impossible to multiply zero by any number that would be large enough to make it equal to 10.

Zero is unique in this respect: any other number, no matter how big or small, can be multiplied by some other number to get 10. But not zero.

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OP never used the phrase "infinitely-many apples", but rather was trying to make a case for the phrase "infinity times as many apples" as a person with zero apples. –  Austin Mohr Nov 28 '11 at 21:53
I wasn't claiming OP used this phrase. I was giving an example of a sentence that made no sense, then one that perhaps did. I then attempted to segue back to OP's example. Sorry if it was confusing. –  Fixee Nov 28 '11 at 22:29
You have indeed given an example of a sentence that makes no sense, though maybe not the one you intended! –  The Chaz 2.0 Nov 28 '11 at 22:49
@TheChaz Which sentence did I write that makes no sense to you? My intention was to highlight "infinitely many apples" as nonsensical, but you are confused by something else? Or you believe my intention is otherwise? –  Fixee Nov 28 '11 at 23:04