Why Does Summing Segment Lengths (x2-x1)+(x3-x2)+... Not Double-Count Points?

Hi guys;
I'm new here at this forum and I believe that I would boost my thinking by you guys at helping me a serious problem as a student I'm facing it.
I'm facing a problem which I don't know why I'm think like this but I will explain my problem and I hope you help me.
What I'm facing is like this: lets assume given four points and I need to add the length from 1 to 4 i.e:
1-----2----3----4, so I have to do for calculating the length from 1-4 like this:
 (1-0)+(2-1)+(3-2)+(4-3) = total length from 1----4
here is my problem, what's confusing me is: from 1-----2 the difference is 2-1=1, so we calculated the length between (2,1), for calculating the length between (3,2) we do (3-2)=1  ; but here at the "conjunction point" for instance in my case the number 2 is calculated twice, and we took it twice into consideration for calculating the length between 1----4, so we are adding it twice to the length and we mustn't do that because we will get wrong length, so how we are getting correct length from 1----4 although we are taking into consideration the "conjunction point" twice between two pieces ?!! thanks for helpers.
exactly my problem, at the "conjunction point" we are taking them twice into consideration once at the first piece and second one at the second piece i.e for instance:
1---2----3 = here one piece from (1,2) and the second piece (2,3) we are taking into consideration the point(conjunction point) "2" twice in calculating the length of between 1----3 and we get right answer, why? isn't taking into consideration "2" will add a lil amount twice to the length?!

Ryan,
Looking at your second example. To calculate the length of 1---2---3As you said first you calculate 2-1 then 3-2 then add them.So the total is (2-1)+(3-2)Since the order of addition doesn't matter we can also write (3-2) +(2-1)Subtraction can be considered adding the negative number so we have(3+(-2))+(2+(-1))
Because this is addition, we can rearrange the parenthesis like so3+((-2)+2)+(-1)Now you should be able to see that we're adding and subtracting the same number so it cancels out.

but lets assume I have any continuous function like x^2 and I take three points upon it , lets assume they are (respectively in this order): a , b , c .. so in order to calculate the difference between a and c I make c-a but what I don't understand is how c-a=  (b-a) + (c-b) , here we are adding the point b twice ! because what actually I take in first element is: from a I reached b then "included the point b" in the first difference, and in the second difference I say from b to c which's also including the point b .. so how the total(c-a) equal to b-a + c-b if I included the point b twice?! thanks!!!!

I think I see your confusion. The thing is that when you say c-a you are NOT including  point a but including everything from just beyond a to c.So b-a includes from just beyond a to b (including b) and c-b includes from just beyond b to c (NOT including b). So b is only included once.