"Intuitive" distance between two reals

Question

If we have two reals $r$ and $s$, the usual distance between them is the Euclidean distance $|r-s|$.

Are there other, "intuitive" and "natural" ways to compare how close two reals are to each other?

Well, $512$ is close to $1024$ because they are powers of $2$ that are only $1$ apart. $e$ and $\pi$ are close because they are important naturally occurring numbers relating to growth and rotations. This question needs to be more specifically worded to indicate what is being asked. (The $e$ $\pi$ example was me being facetious but the $512$ and $1024$ is actually close to a $p$-adic metric and can be used.) — fleablood, Jan 25 '24 at 18:14
.... then we can use the "trivial metric". All real numbers are close if and only if they are equal, and are far apart if they are not. In other words: the distance between $\pi$ and $27$ billion is the same as the distance between $\sqrt 2$ and $1.4142$ which is $1$ and the distance between $\pi$ and $\pi$ is $0$. — fleablood, Jan 25 '24 at 18:17
Relevant to your question: https://math.stackexchange.com/q/100977/1215020 — César VB, Jan 25 '24 at 18:21
... and there is the railroad station metric. To get from any number to another different number you must go through $0$ so if $r \ne s$ then the distance between $r$ and $s$ is $|r| +|s|$. .... all this leads to: what exactly are you asking? Are you asking if there are other metrics we can use on the reals? — fleablood, Jan 25 '24 at 18:23
There are good arguments saying metrics are neither natural nor intuitive to humans. For instance, one interpretation of your question would lead instead to quasimetrics ("it cannot be uphill both ways") — Brian Moehring, Jan 25 '24 at 18:30
There a gazillion ways to define a notion of distance between real numbers. It is a matter of opinion whether a notion of distance is intuitive or natural. Hence my vote to close this. — Rob Arthan, Jan 26 '24 at 00:54
@RobArthan well, the two votes on the current answer don't agree with you. — Alex, Jan 26 '24 at 20:42
@RobArthan if the answer gets upvotes and gives an answer my question, it automatically validates the question. — Alex, Jan 26 '24 at 21:15
Really? MSE is full of questions that have received attention but aren't really worth asking. — Rob Arthan, Jan 26 '24 at 21:18

score 2 · Answer 1 · answered Jan 25 '24 at 18:24

You could divide them by one another and see that if $|\frac{a}{b}|$ is close to 1 they are close to one another, one might however really rather want to consider $|\frac{a+1}{b+1}|$ to see that a very small or very big value means they are far apart

If the average of two numbers has a low euclidean distance to the numbers themselves they may be considered to be close to one another, one might want to consider the fraction given by taking the average of the numbers and dividing the average of the two respective distances from the average to the two numbers by it, this will be well defined unless both numbers are zero and will be close to 0 when the numbers are close to one another.

Otherwise just taking the difference between the two numbers should be closer to zero if they are close to one another

Can't think of more ways now but I'm sure there are some more

"Intuitive" distance between two reals

1 Answers1