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Why does Zero Times Infinity not equal One ($0 \times \infty \neq 1$)?

If Infinity = $\infty$ and Zero = $\frac{1}{\infty}$

Then Zero Times Infinity = $0 \times \infty = \frac{1}{\infty} \times \infty$ which is equal to '$1$'?

What Am I doing wrong?

MJD
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zordman
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    What you're doing wrong is treating $\infty$ like a number, when it isn't. – Clive Newstead Nov 15 '14 at 15:56
  • I hate it when people treat the lemniscate as a variable to mean: $$\infty := \lim_{n \to \infty} n$$ This is just not a valid definition folks. – Nick Nov 15 '14 at 16:08
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    Why do you say $Zero = \frac1\infty$ and not $Zero = \frac2\infty$? – MJD Nov 15 '14 at 16:10

3 Answers3

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$0 \neq \dfrac{1}{ \infty}$

It is true that we have $\lim_{x\to \infty} \dfrac 1{x}= 0$, but that is not to say that $\dfrac 1{\infty} = 0$.

And if you have evaluated a limit to get the indeterminate form $0 \cdot \infty$, that is simply an indeterminate form of a limit (not a value) that tells us more work needs to be done to find the limit.

amWhy
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  • Could someone please explain the downvotes? I'd be happy to delete the post, but I'd like to know why I should do so. – amWhy Nov 15 '14 at 16:00
  • Wild guess here: when dealing with measure theory, one usually adds $\infty$ to $\Bbb R$, and defines arithmetic with it, in particular, one sees that $0\cdot \infty=0$ is quite a convenient agreement. In particular, one shouldn't reduce everything dealing with $0$ and $\infty$ to "limits" and "indeterminate forms". – Pedro Nov 15 '14 at 16:01
  • The other answers do not talk about limits and indeterminate forms. – Pedro Nov 15 '14 at 16:05
  • @PedroTamaroff Most of user's posting in this vein are misapplying what they are learning about limits. At any rate, I see little reason for downvotes. – amWhy Nov 15 '14 at 16:07
  • @Pedro Case in point, see the accepted answer of Alex Becker in the now linked to duplicate, and other answers there. – amWhy Nov 15 '14 at 16:08
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    One can say that $\frac{1}{\infty} = 0$ (meaning as a limit) without running into any contradictions, the point is that this does not imply that $0\cdot \infty = 1$! – Winther Nov 15 '14 at 16:09
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Infinity isn't explicitly a number so it doesn't follow the rules of arithmetic, you can pseudo-prove all sorts of silly things like 1=2 if you assume that infinity is a regular number, which is clearly wrong.

Rammus
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$\infty$ is not a real number. If it were then it would lead to all kinds of contradictions. The fact that it does lead to contradictions is precisely why we can't treat it the way you do here. Instead, we are much more careful and treat it differently in different parts of mathematics.

Simon S
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