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Firstly, yes, you can divide any number and get even smaller one but hear me out.

My logic goes as follows:

Let there be a number X. This number behaves similiarly to infinity, except it is not infinitely large, it is infinitely small. You could define this number as 0.000...1, where ... is an infinite number of zeroes. In the same way you can multiply infinity to get infinity, you can divide X, although you'd still get just X. So X is not really a number, more like an idea.

This brings up some interesting possibilites, like:

For an open interval (a, b), a + X would be the first number in that interval and b - X would be the last number of that interval. Therefore you could convert it to a closed interval as [a + X, b - X]

And you can divide with this number as well, you would get either inifnity or minus infinity.

A friend claims that such notion exists, and is called 'Zero plus'. I have failed trying to search for it, so is anything like this established?

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  • $\begingroup$ Not sure there could exist anything behaving exactly as you describe. Also, you have to decide what sort of numbers and norm/measure you are using when you say small. When you say $a+X$ would be the first number in the interval $(a,b)$, that doesn't work (or even make sense) for open intervals of real numbers, by the nature of $\mathbb{R}$. But, at first glance, you should definitely at least look up and read about infinitesimals and hyperreal numbers. $\endgroup$ – Christopher.L Apr 18 at 21:58
  • $\begingroup$ Thanks, I'll look it up! $\endgroup$ – Meowxiik Apr 18 at 22:03
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    $\begingroup$ You may be interested in my answer to "Is there a symbol for the idea of the smallest value greater than zero?" $\endgroup$ – Mark S. Apr 18 at 22:08
  • $\begingroup$ Yup, it would seem that my question is a duplicate for that one. Thanks for the link! $\endgroup$ – Meowxiik Apr 18 at 22:10
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In standard reals there is no such number (and there is no infinity either). There is a so called "non-standard analysis" which uses hyperreal field instead of usual reals. However, even in hyperreals there is no such things as "smallest positive number" or "last point of an open interval": for every number $x > 0$ we have $\frac{x}{2} < x$, and this property is too useful to give up.

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  • $\begingroup$ Yes, I think the "$x/2<x$" property is key to why any such notion must be of limited use. $\endgroup$ – Mark S. Apr 18 at 22:09
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I found this excellent paper (coincidentally where I go to school too!) and it provides an excellent introduction to the topic of hyperreals. Basically you have these new numbers that you can represent as infinite tuples, and you can compare two different hyperreal numbers by comparing their tuples. This allows for numbers less than any positive normal real number, but still greater than zero.

In the paper it is the example of $\epsilon = (\frac 12, \frac 14, \frac 18, \frac 1{16} \ldots )$ which eventually has infinite number of elements less than any real number $r$ represented in this tuple notation as $(r,r,r \ldots)$. We can also see that this special tuple is greater than $0$ for all elements. The tuple $(1,2,3 \ldots)$ in this system serves as the "infinity" or unlimited because it eventually gets bigger than any tuple $(r,r,r \ldots)$.

Furthermore, the paper proves that the properties of the real numbers still hold in the hyperreals, and thus don't contradict what we've already proved using the normal real numbers. I'm not sure whether or not sharing this paper is allowed, and of course there are more intricacies to non-standard analysis and other such numbering systems than just these.

https://sites.math.washington.edu/~morrow/336_15/papers/gianni.pdf

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  • $\begingroup$ This answer seems to imply that the only structures worth considering for this question are the hyperreals and the reals. I don't agree with that, personally. $\endgroup$ – Mark S. Apr 18 at 22:11
  • $\begingroup$ @MarkS. Unfortunately they are the only ones I know well, and I offered what I could $\endgroup$ – D.R. Apr 18 at 22:19
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    $\begingroup$ @MarkS. the question is what properties you are willing to give up. For example, there is smallest positive integer number. But integer numbers are, for example, not closed with respect to division. $\endgroup$ – mihaild Apr 18 at 22:33

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