What's the midpoint of $[a,b)$?

Question

While the midpoint of either $[a,b]$ or $(a,b)$ is $\dfrac{a+b}{2}$, what's the midpoint of $[a,b)$ if it has been defined at all?

score 6 · Accepted Answer · answered Oct 25 '15 at 05:06

6

It depends on how you want to define the so called "mid point" of an interval. One can simply define the mid point of an interval to be $\frac{a+b}2$ for any interval of the form $(a,b), (a,b], [a,b)$ or $[a,b]$. This definition would serve most general purposes quite well since a single point has measure zero.

However, suppose you want the define the mid point to have the property such that

for an interval $I$, $m$ is called the midpoint of $I$ if for any $x\in I$ there exist $x'\in I$ such that $x-m=m-x'$

then it's provable that $[a,b)$ has no mid point. (Hint: if such $m$ exists then $I$ must be of the form $(m-d,m+d)$ or $[m-d,m+d]$)

answered Oct 25 '15 at 05:06

BigbearZzz

15,084

Well I had the question because I knew your mentioned two definitions in my head but the question is which of them is the common accepted definition among the current mathematicians? Thank you. – Oct 25 '15 at 05:15
As other members has pointed out, we usually define it to be $\frac {a+b}2$. – BigbearZzz Oct 25 '15 at 05:16
Is there a citation for this alternate definition being in use someplace? – Daniel R. Collins Oct 26 '15 at 00:14
@DanielR.Collins I suppose there isn't since I made it up. It's a plausible property that a "mid point" ought to have in a naive sense (ever point has an opposite point) and I though it might be the reason that the OP was confused. – BigbearZzz Oct 26 '15 at 00:31
Downvoting because I don't think it's helpful to create a hypothetical nonstandard definition in response to this question that uses a standard piece of vocabulary ("midpoint"). – Daniel R. Collins Oct 26 '15 at 00:48
@DanielR.Collins It's alright if you want to do that. I though I made myself pretty clear when I said "However, suppose you want to define the mid point to have the property such that..." that the property was made up. By the way, I didn't define it out of the blue but in response to OP's confusion to what a "mid point" ought to behave. – BigbearZzz Oct 26 '15 at 00:52
Nor is the given $\frac {a+b}{2}$ taken as the standard definition, either. So basically this answer never touches the actual definition of midpoint. – Daniel R. Collins Oct 26 '15 at 01:59
@DanielR.Collins I claimed that $\frac {a+b}2$ is the standard definition as far as real analysis is concerned. You can chose not to believe me, however. My made up definition is not arbitrary either, it's also known as the point of reflection or hyperplane of symmetry, which for 1 dimensional Euclidean space, coincide with the term "mid point" pretty well. Since you seem to like reference from wiki-type page, I'm going to give some https://proofwiki.org/wiki/Definition:Real_Interval https://en.wikipedia.org/wiki/Interval_(mathematics)#Terminology. – BigbearZzz Oct 26 '15 at 10:14

score 2 · Answer 2 · answered Oct 25 '15 at 05:03

2

It's still $\displaystyle\frac{a+b}{2}$.

answered Oct 25 '15 at 05:03

Unit

7,601

score 1 · Answer 3 · answered Oct 25 '15 at 05:03

1

It's the same, $\dfrac{a+b}{2}$. It's still the point equidistance from the end points. (The "end points" don't actually have to be in the interval itself.)

answered Oct 25 '15 at 05:03

fleablood

124,253

goblin GONE · Answer 4 · 2015-10-26T00:30:38.343

Here's one possible definition. Let $A$ denote a measurable subset of $\mathbb{R}$. Then a midpoint of $A$ is an $x \in \mathbb{R}$ such that $\mu(A \cap (\infty,x]) = \mu(A \cap [x,\infty)),$ where $\mu$ denote the Lebesgue measure. Observe that a measurable subset of $\mathbb{R}$ may have many midpoints (for example, $[0,1] \cup [2,3]$) or none (for example, $[0,\infty).$) But all three of $[a,b]$, $[a,b)$ and $(a,b)$ have precisely one midpoint, namely $(a+b)/2$.

score 0 · Answer 5 · answered Oct 25 '15 at 05:07

0

I would say:

Midpoint=$(a+b)/2$. Think this way, if b) is infinitesimally close to b] the midpoint is the same as (a,b) or [a,b]. For example, if b=2.0 and you get to 1.99999999999....... the midpoint is the same as [a,b].

answered Oct 25 '15 at 05:07

Jonas

23

score -1 · Answer 6 · answered Oct 26 '15 at 00:17

The standard definition of the midpoint is, "The point on a line segment dividing it into two segments of equal length" or the equivalent. As a result, the midpoint is found by the formula $\frac{a + b}{2}$ for all types of intervals.

What's the midpoint of $[a,b)$?

6 Answers6