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## Main Question or Discussion Point

Why is it that the distance between two real numbers ##a## and ##b## in an ordered interval of numbers, for example ##a<x_{1}<\ldots <x_{n-1}<b##, is given by $$\lvert a-b\rvert$$ when there are in actual fact $$\lvert a-b\rvert +1$$ numbers within this range?!

Is it simply that, when measuring the distance between two real numbers we are counting the number of unit intervals that separate the two of them, and there will always be one less unit interval between the two numbers than the range of numbers between them?!

For example, suppose I have the ordered set of integers ##(0,1,2,3,4,5)##, then the distance between 4 and 1 is of course ##\lvert 4-1\rvert = \lvert 1-4\rvert = 3##, which is to say, there are 3 unit intervals between 1 and 4. Equivalently, one could arrive at this result by counting the number elements between 1 and 4, including the endpoint (4) but not the start point (1). However, if one includes both the start point and the endpoint then the number of elements between 1 and 4 is 4. Is the qualitative difference here that in the former case I am determining a relative quantity - the separation between 1 and 4, whereas in the latter case I am determining an absolute quantity- the number of elements ranging from 1 to 4?!

Apologies if this is a really stupid question, but it's something that I've been thinking about recently, and how I would reason the answer.

Is it simply that, when measuring the distance between two real numbers we are counting the number of unit intervals that separate the two of them, and there will always be one less unit interval between the two numbers than the range of numbers between them?!

For example, suppose I have the ordered set of integers ##(0,1,2,3,4,5)##, then the distance between 4 and 1 is of course ##\lvert 4-1\rvert = \lvert 1-4\rvert = 3##, which is to say, there are 3 unit intervals between 1 and 4. Equivalently, one could arrive at this result by counting the number elements between 1 and 4, including the endpoint (4) but not the start point (1). However, if one includes both the start point and the endpoint then the number of elements between 1 and 4 is 4. Is the qualitative difference here that in the former case I am determining a relative quantity - the separation between 1 and 4, whereas in the latter case I am determining an absolute quantity- the number of elements ranging from 1 to 4?!

Apologies if this is a really stupid question, but it's something that I've been thinking about recently, and how I would reason the answer.