In everyday life we have this intuitive idea of two (e.g. natural) numbers being close to each other. We say things like "$365$ and $360$ are close to each other".
I was wondering if this informal notion could be made fully precise, and I came up with two not-fully-satisfactory answers.
- Two natural numbers are close to each other in proportion to how many first consequitive digits they have in common. E.g. 125800 and 125754 are close to each other, but 12580 and 12584 are even closer (in fact, they are closest they can be in this "model").
- Two natural numbers are close to each other in proportion to their ratio being close to $1$. For example, $4567$ and $4487$ are closer to each other than $4567$ and $4207$ are, because in the first case the ratio is closer to $1$.
As I see it, there are problems with both approaches.
In the first approach, firstly, the answer is base-dependent, and secondly, $12580$ and $12584$, and $12580$ and $12588$ are equally close to each other, which intuitively they're not. Thirdly, e.g. $3999$ and $3000$ are closer to each other than $3999$ and $4000$ are, which is plain crazy. So, this approch goes out the window obviously.
In the second approach, if we consider any number and the numbers one smaller and one larger than it, one of the computed ratios will be closer to 1 than the other one, even though intuitively both are as close to the "middle" number as two natural numbers can be close intuitively. For example, for $4567/4568=0.99978...$ and $4567/4566=1,00021...$ the second ratio is closer to $1$ than the first ratio.
Is there any "formalization" of this concept of two real numbers being close generally (and two natural numbers particularly) that coincides with our everyday intuition exactly?
We can think about this geometrically, actually, by thinking about closeness not as a metaphor, but literally. We already have a representation of $\mathbb{R}$ as a geometric space, namely a number line, and the distance between any two points is $$d(a,b) = \left\vert a - b \right\vert$$ Because the number line is already ordered 'algebraically' in the typical way that $\mathbb{R}$ is ordered, then closeness 'algebraically' corresponds to geometric closeness.