Mathematics 5
Winter Term 2002
The World According to Mathematics
Friday Discussion: Week #4
Part 1: Geometric Sums
Consider for a positive number the geometric sum:
Exercise 1: What is the
value of the above sum for and ?
Exercise 2: By multiplying both sides above by , verify that:
[Here are the steps in the multiplication of the right-hand-side by :
Is this what you got when you did it?]
Now, dividing through by we get that:
You should find this formula to be useful when you analyze the formulations of Zeno's paradox that we will study next week.
Also, think about what it might mean if, starting with the above formula, we let (that is, we let get large without bound). In mathematical terms, we describe this process by writing the right-hand side as:
and by writing the rest as:
Exercise 3: If is a number such that , what do you suppose is the value of
Hint: Try first calculating
Part 2: Zeno's Paradox
This supposedly is what Zeno said:
"Achilles cannot overtake a fleeing tortoise because in the interval of time that he takes to get where the tortoise was, it can move away. But even if it should wait for him, Achilles must first reach the halfway mark between them and he cannot do this unless he first reaches the halfway mark to that mark, and so on indefinitely. Against such an infinite conceptual regression, he cannot even make a start, and so motion is impossible."
There actually are two different paradoxes contained in Zeno's statement. We will separate them, calling the first the Achilles Paradox and the second the Dichotomy Paradox.
1. Achilles Paradox: Achilles cannot overtake a fleeing tortoise because in the interval of time that he takes to get where the tortoise was, it can move away.
2. Dichotomy Paradox: There is no motion because that which is moved must arrive at the middle before it arrives at the end.
We are going to discuss the Achilles Paradox.
Suppose for sake of simplicity that although they start the race at the same time, the tortoise (T) starts one foot ahead of Achilles (A). Also, assume that the tortoise moves one-half foot, then one-quarter foot, then one-eighth foot, then one-sixteenth foot, and so on. Furthermore, assume that Achilles always moves to the position just vacated by the tortoise. That is, he is behind the tortoise by the amount the tortoise just moved. So, Achilles moves one foot, then one-half foot, then one-quarter foot, then one-eighth foot, then one-sixteenth foot, and so on. This seemingly leads to the paradoxical conclusion that Achilles never overtakes the tortoise.
Let us make one further assumption, relating the movements of Achilles and the tortoise to the passage of time. Assume that the first move of the two occurs in one-half second, the second move in one-quarter second, the third move in one-eighth second, and so on. This seems to lead to the paradoxical conclusion that the race is never ended.
Questions and comments to think about:
[With , we have for , and for . Verify these and then go ahead and try writing out the formula for and .]