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If distance could be divided into an infinite no of units or points, then it seems to me that motion would be impossible since a meter for instance, having an infinite no of points within it (and the points having an infinite point within it), would be un-breachable. It seems (intuitively) to me that for a measure of distance to be finite it would have to be made up of a finite unit of points however small those units were.

Otherwise, we are all teleporting at some point in the process of motion...

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The ancient Greek geometers had the same problem, and it all came down to this: they didn't fully appreciate that infinite sums can converge to finite values. – Chris White Jan 20 at 2:47
@Chris White, good remark (+1). Bill Bryson says in a short column, "There is always a little more toothpaste in the tube, think about it!" and the underlying trick is also an infinite convergent series. – Eduardo Guerras Valera Jan 20 at 3:35

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You have re-discovered the Arrow Paradox from Zeno of Elea (5th century BC).

Physics doesn't deal with that. There is obviously movement, and all we do is trying to resemble and predict it with mathematical models. That is how we face the question of understanding nature.

You may find a better answer among philosophers than physicists for that question. But if you insist in getting a physical answer, it might be that there exists a length that is considered in the order of the shortest meaningful distance, called Plank Length, around $10^{-35}$ m. There are (physical) theoretical speculations about the possibility that, at that scales, space together with time could be sort of discrete.

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Thanks Chris and Eduardo. Your answers make my question seem so smart :-). – gerald Jan 21 at 21:23
Thanks Chris and Eduardo.Your answers make my question seem so smart :-). Heard somewhere that at the plank, physics as we know it is suspended and that maybe at that level movement becomes 'digital' where the object is simply destroyed in one location and created in the next. Also that an object in motion is simply several 'copies' of that object in the instant, hence the 'extra' mass observed in momentum.Don't know how much of this is credible and whats just star trek. This is all intuition here - no understanding of the calculations tho I'm reading upon your references. cheers and thanks. – gerald Jan 21 at 21:36

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