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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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    There is no well defined arithmetic with $\infty$, and in particular $x*\infty=\infty$ is certainly not a generally valid rule.2013-05-31

2 Answers 2

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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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    After taking calculus, it seems like the best answer to this question is "Take calculus." How foolish I was...2013-12-15
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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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    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: http://math.stackexchange.com/questions/405497/i-would-like-to-know-an-intuitive-way-to-understand-a-cauchy-sequence-and-the-ca/405515#comment867848_4055152013-06-02