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!

Solution 1:

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 $[0,1]$.

Solution 2:

See Wikipedia for a discussion of an interval. The open interval $(1, 0)$ is a bounded set,

and therefore has diameter (length) = $\sup((0, 1)) - \inf((0, 1)) = |1 - 0| = 1$.

• For any set $A$, $\sup(A)$ is the least upper bound of $A$; $\inf(A)$ is the greatest lower bound of $A$.

Also note that the intervals $\left(0, 1\right]$ and $\left[0, 1\right)$ are also bounded and each is of length $1$.

Half the length of any interval of length $1$ is $\frac12$.

So $\frac12$ could be said to be in the "middle" of each of the above intervals,

since $\left(0, \frac12\right)$, $\left(0, \frac12\right]$, $\left[0, \frac12\right)$, $\left[0, \frac12\right]$, $\left(\frac12, 1\right)$, $\left(\frac12, 1\right]$, $\left[\frac12, 1\right)$, and $\left[\frac12, 1\right]$

each have diameter = length = $\frac12$.

Solution 3:

One sensible definition for the middle point of a bounded set $A \subseteq \mathbb{R}$ could be

$$\operatorname{middle}(A) = \frac{\sup(A) + \inf(A)}{2}$$

This is a good definition for intervals, and the middle point of $(0,1)$ would be $1/2$ with this definition. However, when $A$ is not an interval, $\operatorname{middle}(A)$ might not be a point of $A$.

Solution 4:

Any interval on the real line can be written as $(a,b)$,$[a,b)$,$(a,b]$ or $[a,b]$ with $a\leq b$ being extended real numbers (the set of all real numbers with $+\infty$ and $-\infty)$ called its endpoints. The length of the interval (possibly $0$ or $+\infty$) can then be defined as the difference between its endpoints. If we use Lebesgue measure on the real line, it can be shown that the measure of an interval is simply its length.

If number in the middle of an interval means the length on either side is equal, then indeed $\frac{1}{2}$ would be in the middle of the intervals $(0,1)$,$[0,1)$,$(0,1]$ and $[0,1]$, because for example the length of $(0,\frac{1}{2})$ and $(\frac{1}{2},1)$ are each $\frac{1}{2}$.