(∞) is essentially the opposite
of dividing by zero
, in two ways. Firstly, it actually gives you an answer, instead of blowing up the Universe
; and secondly, that answer is vanishingly small -- technically "approaching zero
" -- instead of mind-destroyingly large
. Part of the problem is (as Dr. Math explains here
), infinity is not actually a number; it is a concept, like happiness
, only one which stands in for idea of the endless set, but not an endless set of any particular number. Consider the simple expediency of the infinite line starting from a set point. The line begins at that point and proceeds infinitely along the direction of the line. If we mark the line once every centimeter, we've got an infinite number of notches, so dividing a finite number by the number of notches on the line will get you that "approaching zero" answer. But suppose we add notches at every half centimeter mark? By so doing, we've unequivocally doubled the number of notches, but our notch count is still the same, "infinity"; and our divided number is still the same "approaching zero." And this can be repeated again by simply returning to our point of origin and making the line run in both directions from the point. Still the same infinity of notches, even if it is now 4×∞, and the same "approaching zero" result from the division.
And the same principle applies to the numerator
as to the denominator
. One divided by infinity is "approaching zero." One billion divided by infinity is "approaching zero." An inconceivably large but finite number representing the total number of bit-states in a digital Universe
divided by infinity is "approaching zero." This concept becomes theologically significant when contemplating whether we, with out finite minds, can have Knowledge of God
, thought to have either an infinite mind or at the very least an inexpressibly large one.
But, given all of the above, there's an exception
. Well, two exceptions. Firstly, zero divided by infinity yields the same result as zero divided by any other number, which is.... zero. And so, 0/2 is simply another way of denoting, "sorry, but I have no halves." And in just the same way, 0/∞ is simliarly a way of denoting, "so sorry, but I have no infinitely small divisions" or "were you looking for a unit approaching zero? Regretfully, I've none to offer you." Fortunately, dividing the number zero poses none of the potentially mind-blowing effects of dividing by
zero. The second exception is the really fun one, which is dividing infinity by infinity. ∞/∞ = 1. Oh, well sort of. Actually, it depends, now doesn't it? Remember everything I was telling you a moment ago about how you could double and quadruple and otherwise parse the divisions of your infinite line and still reach the same answer? Well, with an infinity on top of the equation that is no longer so, because (2∞)/∞ = 2 and (42∞)/∞ = 42
. One can have a negative infinity as well, so that ∞/-∞ = -1.
So, if you've got something you'd really, really like to just diminish to the point of effective nonexistence
, go ahead and divide it by infinity. It's safe. Unless, naturally, it's an infinity itself, in which case you're stuck with it. Except that, just maybe, you can divide it by an infinity of infinities -- ∞/(∞×∞) = 1/∞!!