Is $\infty / \infty = 1$? Lately, my friend and I were arguing about what $\infty / \infty$ equals.
My thinking was that $\infty / \infty = 1$, since no matter how high you go in the numerator, it would have to go equally as high in the denominator.
My friend pointed out that one is not the smallest it can go, and can be divided an infinite number of times. (Equaling $.\overline{0}$1)
Which is it? Or is infinity not even considered a real number and so the answer is really just undefined?
 A: Through your thought experiments, you and your friend have discovered the reason that $\infty/\infty$ cannot be assigned a consistent meaning.
The "answer" you get, if any, will depend completely on the relative rate at which the numerator and denominator grow, as you have seen.
A: In general, number systems exist because they usefully quantify things.
We invent the extended real numbers, which includes two extra numbers, $+\infty$ and $-\infty$, because they are fantastically useful for doing calculus! For example, it lets us understand
$$ \lim_{x \to +\infty} \frac{x^2 + 1}{x + 1} = +\infty $$
as saying that the function is converging to $+\infty$, as $x$ approaches $+\infty$.
It may be odd to think of this as a convergent limit, but with a little experience it will be quite natural. This lets us do other things; e.g. we can continuously extend the arctangent function by setting $\tan^{-1}(+\infty) = \pi/2$ and $\tan^{-1}(-\infty) = -\pi/2$. The arctangent is continuous at infinity, which lets us write the arctangent of a limit as the limit of the arctangent: e.g.
$$ \lim_{x \to +\infty} \tan^{-1} \left( \frac{x^2 + 1}{x + 1} \right) = \tan^{-1}(+\infty) = \frac{\pi}{2} $$
This example suggests something important: to get the most use out of our new numbers $\pm \infty$, we want the functions we use -- such as arithmetic -- to be continuous, so that we can reason as above.
In particular, we would like division to be continuous: e.g.
$$ \lim_{(x,y) \to (+\infty, +\infty)} \frac{x}{y} = \frac{+\infty}{+\infty} $$
Unfortunately, the limit on the left hand side doesn't exist, which means that we can't extend division to be continuous at $(+\infty)/(+\infty)$. Because of this, by far the most convenient thing to do is to define division so that this quotient is undefined.
If you're not familiar with multivariable calculus yet, the fact the limit above doesn't exist is a consequence of the fact that
$$ \lim_{t \to +\infty} \frac{t^2}{t} \neq \lim_{t \to +\infty} \frac{t}{t} $$
That said, if you were not considering division in general, but were instead considering the specific function defined by
$$ f(x) = \frac{x}{x} $$
Then we can continuously extend $f$ to $+\infty$ just like we did with the arctangent function, by setting $f(+\infty) = 1$. This is very analogous to the fact we like to extend $f$ so that $f(0) = 1$ as well.
A: $\infty$, as you said in the last paragraph of your question, is not a number (natural or real), so the usual arithmetical operations on numbers are not defined on it.
This is why
$\dfrac{\infty}{\infty}$
is not equal to $1$.
A: The following limits all have the indeterminate form of $\frac{\infty}{\infty}$, but they are not all $1$.
$$\lim \limits_{x \to \infty} \frac{x^2}{x}$$
$$\lim \limits_{x \to \infty} \frac{x}{x^2}$$
$$\lim \limits_{x \to \infty} \frac{x}{x}$$
However, if you are given $\frac{\infty}{\infty}$ without context, the value is indeterminate. Furthermore, note that $\infty$ is not a number, so it doesn't follow the standard rules of algebra.
We can take this one step further. $\lim \limits_{x \to \infty} \frac{x^2}{x}$ is infinite, and so is $\lim \limits_{x \to \infty} \frac{x^3}{x}$ -- their limits are the same. But doesn't that feel a bit strange? Wouldn't $x^3$ be "larger" because it's to the third power, not just the second? Well, now if we divide them, we get $\large{\lim \limits_{x \to \infty} \frac{\frac{x^3}{x}}{\frac{x^2}{x}}}$, which is $\infty$.
The conclusion overall being that, when comparing two infinite quantities, their relative growth rates -- "how fast they become infinite" -- must be considered.
