What is infinity times the reciprocal of infinity?

I was talking with a friend about interesting properties of numbers and their theoretical contradictions and solutions when we came up with this. What is the answer?

So...
$x * ∞ = ∞$
and...
$\frac{1}{x}*x=1$

So what do you get when you do...
$\frac{1}{∞}*∞$? $∞$ or $1$?

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But infinity isn't a number... –  Guess who it is. Dec 10 '11 at 5:38
@J.M. But it can still be used in equations. –  JShoe Dec 10 '11 at 5:42
@JShoe: No, it can't. It isn't well defined. –  mixedmath Dec 10 '11 at 5:43
Usually in measure theory, Lebesgue integrals and things like that, we avoid "to divide" by 0 and by $\infty$. To prove things, these cases are considered apart. I'm with J.M. infinite isn't a number. –  leo Dec 10 '11 at 5:44
There is no well defined arithmetic with $\infty$, and in particular $x*\infty=\infty$ is certainly not a generally valid rule. –  Marc van Leeuwen May 31 '13 at 9:31

Unfortunately, infinity is not a number, and cannot be manipulated as such.

For example, $\lim_{n \to \infty} x = \infty$, and $\lim_{n \to \infty} x^2 = \infty$, but

$\lim_{n \to \infty} \frac{x}{x} = \frac{\infty}{\infty} = 1$,

$\lim_{n \to \infty} \frac{x}{x^2} = \frac{\infty}{\infty} = 0$,

$\lim_{n \to \infty} \frac{x^2}{x} = \frac{\infty}{\infty} = \infty$,

In fact, we can make $\frac{\infty}{\infty}$ converge to anything we want.

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...Huh? That seems wrong, but hey, everything does until you understand it. Thanks! –  JShoe Dec 10 '11 at 5:57
Infinity isn't something that easily succumbs to intuition, @JShoe. :) –  Guess who it is. Dec 10 '11 at 6:01
Those $=$ signs above need to be taken very loosely... –  LarsH Apr 13 '12 at 16:03
isn't it a little misleading to say "cannot be manipulated as such" without mentioning the partial manipulations you can do on the extended real line? I think the OP deserves to hear he isn't totally insane for thinking sometimes operations with infinity make sense. –  symplectomorphic Jun 2 '13 at 7:37
After taking calculus, it seems like the best answer to this question is "Take calculus." How foolish I was... –  JShoe Dec 15 '13 at 23:45

John Wallis, who originally introduced the symbol "$\infty$" in the 17th century, used it to denote a specific infinite number, and furthermore exploited an infinitesimal of the form $1/\infty$ in area calculations. Extended number systems that include infinitesimals were used by such giants as Leibniz, Euler, and Cauchy, and are in active use today.

Today the symbol $\infty$ is not generally used in this sense. However, the wording of the question by the OP suggests that this is the sense he may have in mind, rather than the traditional meaning in the context of "indeterminate forms". The OP should keep in mind that the modern use of the symbol is different from its historical use.

If $H$ is an infinite number (meaning that it is greater than every real number), then $H \times \frac{1}{H}=1$ as for ordinary real numbers. See for example http://www.math.wisc.edu/~keisler/calc.html

The answer given by user mixedmath assumes that you are dealing with "indefinite forms", and is also correct.

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But $\infty$ cannot be used to denote such infinite numbers (for the simple reason that they are not unique), so this does not answer the question that was posed. (This is not meant to imply the question was well formulated, just that this is not a plausible interpretation of what might have been meant.) –  Marc van Leeuwen May 31 '13 at 9:27
The OP specifically uses the term "numbers" in his question, suggesting that his use of the symbol is not the traditional one we are all used to. –  user72694 Jun 2 '13 at 7:30
Similarly, the OP's insistence that "$\infty$ can be used in equations" suggests that he does view it as a number. Of course, he needs to realize that this use is different from the traditional one. –  user72694 Jun 2 '13 at 7:32
I think you greatly overestimate the mathematical maturity of OP. Seeing something like $\lim_{x\to0}\frac1x=\infty$ would lead to conclude that "$\infty$ is used in equations" (not to mention the cruder $\frac10=\infty$), but the conclusion that $\infty$ is a number does not follow. I would think that the expression I gave is an example of the traditional use of $\infty$. –  Marc van Leeuwen Jun 2 '13 at 8:03
Marc, I think we are in full agreement. The traditional use of the symbol is decidedly NOT a "number". This should be made clear to the OP, and I was careful to do so. Yet the intuition that "infinity times 1/infinity equals 1" is a valid one, and helps intuition in many situations. See, for example, the student's response in the context of Cauchy sequences here: math.stackexchange.com/questions/405497/… –  user72694 Jun 2 '13 at 8:09