There is no such thing as ∞, there is only [∞]. x/∞ = NaN. It's exactly as meaningful as x/Tuesday or x/your_face.

To define [∞] you need two things. A specific number in question, and a precision. Let's pick 333. Precision say ±0.01%. [∞] is then any number such that 333/[∞] is 0 to within 0.01%, that is, smaller than 0.00005, meaning for our purposes [∞] is 6,660,100 and all larger numbers. Note the placement of the one is at five significant figures. To be precise [∞] >= 6.6601*10^6.

Let's recurse. We can easily see that [∞] exists for any integer or real number, because division works. That's the [∞] mathematicians like to talk about - the [[∞]], basically.

We can also see [∞] does not exist for [∞]. If we divide [∞] by 5*10^-5, we don't get a number, we get nonsense. [∞] is not an integer or a real number, it is a set of numbers. You can multiply any particular entry by 20,000 if you want, but that doesn't change which members belong in the set.

You can see that no matter how precise you need to be, there will be at least one member for [∞]. The set is never empty. However, statements such as 'infinitely precise' are meaningless. You can't have a number ±[4, 12, 0.09, 333]. At best that's four numbers, not 'a' number. What about [∞] defined on 333 ± 1/[∞]? Recursive, doesn't converge. NaN.

Numbers with finite precision do not approximate nature. Nature is inherently finite in precision, because infinity is not a number, and precision is a number.

Fun fact: 0 is also not a number. You can get arbitrarily close to zero, but you can't actually reach it. [0], not 0.

## 2 comments:

What is ∞/2? For every integer n, there is exactly one even number 2n, so ∞/2 = ∞. What is ∞/∞?

It could be anything, even infinity! There are ∞ × ∞ complex integers, but the complex integers can be serialized by spiraling out from the origin, so ∞ × ∞ = ∞.

Infinity behaves in so many ways unbecoming of a number that no sane person would ever call it a number. OTOH, zero meets all conditions of being a number except that you can't divide by it.

I think it's safe to say that any finite number divided by infinity is zero. The one thing you can't do with zero is the only thing you can do with infinity.

For every integer n, there is exactly one even number 2n, so ∞/2 = ∞∞/2 = ∞? Divide by ∞, 1/2 = 1. Nope. You can't divide by 0, and you can't divide by [∞].

What is ∞/∞?It could be anything, even infinity!

Only if ∞ != ∞. This is an error known as equivocation. Maybe ask about Ω/∞ or something, because it's not surprising that 1 != 2.

Mathematicians have been extremely dumb about 0 and ∞ for centuries. You can't divide by [∞], and in exactly the same way 0/0 is undefined.

What is 2 divided by [6, 4, 7, 333, 0.0009]? Not each individually - all at the same time. In other words, in what way 1/3=1/2? Infinity is a set, not a particular member of the set.

Fun fact: 0 does not behave like a number. You can figure it out for yourself. Maybe I'll explain later.

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