Neighbor says: >If we leave M and take A(M) and (M)B without it, then the two parts are not really halves of AB, for the sum of the two halves in which one thing is divided is the thing itself, but A(M)+(M)B will be less than AB, because AB = A(M)+M+(MB). Now if we take M on either AM or MB, let it be taken on AM, then when we separate AB we will have straight-lines AM and (M)B, but they will be unequal, inasmuch as A(M)=(M)B ==> AM>(M)B.
You are at least correct in discovering that you can't divide a mathematical real number line segment into two segments, such that they are both non-overlapping and "symmetrical" (in the sense that they are both either open or closed at the dividing point.)
That seems to bother you. I also get the feeling the very notion of an open ended interval bothers you. You could read up on it and think about it for awhile.