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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.

  1. 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").
  2. 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?

alex811
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  • If $d(a,b)$ is the "distance" between reals $a,b$, we can define it to be $$d(a,b)=\frac{|a-b|}{a^2+b^2}$$ – Rushabh Mehta Nov 02 '18 at 05:22
  • What is the denominator for? – alex811 Nov 02 '18 at 05:24
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    Depends on whether mean absolute closeness. In which case 32 and 543 are closer than 234,567,765 and 234,567,045. Or preportionally close, or some other close. Intuition without specifics in math, is pointless and I don't think there is any reason to pursue this line of thought to be honest. – fleablood Nov 02 '18 at 05:25
  • @fleablood In full agreement. This is just for developing intuition – Rushabh Mehta Nov 02 '18 at 05:29
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    There are many different metrics that are in frequent use over the reals. One of the other popular metrics to be used is the one for the Riemann Sphere (or riemann circle in this case). Draw the real line. Draw a unit circle with center on the origin. Map a point from the real line to a point on the circle by drawing a line between the point on the real line and the top of the circle and record the point where it intersects the circle. Now, find the usual geometric distance between points on the circle. – JMoravitz Nov 02 '18 at 05:51
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    "Intuition exactly" is something of an oxymoron for the purposes of a mathematical definition. That said, if you're restricting to positive numbers, one could define the so-called logarithmic distance $\delta(a, b)$ between $a, b$ to be $\log \left\vert\frac{a}{b}\right\vert = | \log a - \log b|$. This has the advantages of (A) capturing proportionality and (B) symmetry in $a, b$ (resolving the issue raised in method (2)). So, for example, $\delta(4567, 4487) = 0.017672\ldots$, whereas $\delta(4567, 4207) = 0.082106\ldots$. – Travis Willse Nov 02 '18 at 05:52
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    More precisely, this notion "captures proportionality" in the sense that for any $\lambda, a, b > 0$ we have $\delta(\lambda a, \lambda b) = \delta(a, b)$. – Travis Willse Nov 02 '18 at 05:54

1 Answers1

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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.

Juan Sebastian Lozano
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  • Yup, this is essentially a metric space. – Rushabh Mehta Nov 02 '18 at 05:21
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    Actually I think you were thinking about it exactly like one might if you think of $\mathbb{R}$ as a purely algebraic object. Like, you were trying to come up with a measure of how similar two numbers were, but this isn't really a measure of how similar they are, it is just a measure of how far apart they are geometrically that matches their ordering. It is only when you have that, for example, $\mathbb{Z}$ is a finitely generated group that this notion of distance becomes a notion of similarity in the truest sense, I guess. – Juan Sebastian Lozano Nov 02 '18 at 05:27
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    Ah, the comment I was replying to was deleted :P – Juan Sebastian Lozano Nov 02 '18 at 05:28
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    My comment was "Boy, was I thinking too much into the matter... :)". Here you go :P – alex811 Nov 02 '18 at 05:30