Is there a number that's right in the middle of this interval $(0, 1)$? This might seem like a silly question, but is there a number that's right in the middle of this interval $(0, 1)$?

And the half-open intervals: $(0, 1]$, $[0, 1)$? I know for a fully closed interval $[0, 1]$ it's $1/2$ because that's half the length of the interval but for $(0 ,1)$, I can't say. What's the interval's length? Certainly not $1$.

Edit: In response to the comment below: I said, in my mind, it can't be $1$ because that that length of the interval $[0, 1]$, and that's not the same as $(0, 1)$. I suppose it approaches 1, but that's not the same, or is it?

And I think by 'length' I mean the the difference in the end points: $|x_2 - x_1|$. $(0, 1)$ does't include $0$ or $1$. There are no end-points to subtract!
To answer this question, you need to first answer the following question. $$\text{What is the `length' of a single point?}$$
If you take it to be zero, then note that $$[0,1] = \{0\} \cup (0,1) \cup \{1\}$$
And assuming that length of these disjoint sets can be added, we get that the length of the interval $(0,1)$ is the same as the length of the interval $[0,1]$.
No one has actually defined the length of an interval precisely. Any interval can be written in exactly on of the forms: (a,b) or (a,b] or [a,b) or [a,b], where b >= a. In any of the 4 cases, call the interval I, a subset of R. Then define the length of the interval to be sup{x: x is in I} - inf{x: x is in I}.