3

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.

A

T

1

A

A

A

T

T

T

1/2

feet

1/41/8

1/2

1/4

1/8

seconds

Questions and comments to think about:

1.

With reference to the above sketch, do you agree that the following is an
expression for the sum of the distances traveled by Achilles? 1 + 1/2 + 1/4 + 1/8
+ 1/16 + ….

What do you suppose the three dots at the end stand for? [Don't worry if you don't
know how to do the arithmetic that the statement suggests. We will remedy that
below. In this question and the next three, you should just be taking information
from the drawing and putting it into rather suggestive mathematical statements.]

2.

Write an expression for the sum of the distances traveled by the tortoise:

3.

Write an expression for the sum of the intervals of time traveled by Achilles: