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?
Best Answer
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.