The Arrow's Dichotomy

The story of 'Achilles and the Tortoise' as mentioned in my blog on 2 August, 2010 is likened to the story of 'The Arrow's Dichotomy' in which it goes as follows:
For an arrow to be shot from one side of the room to the other, the arrow has first to travel one half of the room. When the arrow reaches one half of the room, it still has to travel one half of the remaining length (i.e. one quarter) of the room. When the arrow reaches that position, it still has to travel yet one half of the remaining length (i.e. one eighth) of the room, and so on and so forth. Thus, the total distance the arrow needs to travel in order to cross the room is an infinite series:

1/2 + 1/4 + 1/8 + ..............

Being an infinite series, there is no finite value for the sum. Rather, the limit of this sum is 1 meaning the more terms we add, the sum will get closer and closer to 1 but will never be 1 and cannot exceed 1. Hence, the total distance the arrow travels will never be equal to the length of the room. In other words, the arrow cannot get to the other side of the room.
If we replace the length of the room by some shorter length, say a metre, with similar arguments, we can say the arrow cannot get to the end of the metre. If we replace the metre by yet a shorter length, say an inch, with similar arguments, we can say the arrow cannot get to the end of an inch. If we go on with shorter and shorter length, we will come to the conclusion that the arrow cannot move at all. Finally we conclude motion is not possible, which obviously is a paradox.

Both 'Achilles and the Tortoise' and 'The Arrow's Dichotomy' are different representations of the Zeno's Paradox, and both stories concern about motion, position and time - i.e. physics. We shall discuss Zeno's Paradox from the physics perspective in some later blogs as time permits.