Zeno’s Dichotomy Paradox

By: Michael Mo

Once upon a time, an Ancient Greek philosopher named Zeno created a paradox that “denied” the possibility for movement. Here is how it goes:

If a person wants to travel from Point A to Point B, they must first walk half that distance in a finite amount of time. Thereafter, they must walk half the remaining distance again in another finite amount of time. Then, they must walk half of the remaining distance again, and so on. By continually halving the remaining distance, the person will walk an infinite interval of distances and still remain slightly away from their final destination. This also means that traveling between any two points will take an infinite amount of time. So in theory, all sorts of motion will be impossible.

In reality, however, motion is everywhere. Cars are running on the streets; people are moving to and fro; alas, your eyeballs are moving across the computer screen right now as you read this sentence! So Zeno was clearly having some issues with his thinking. In short, he lacked the understanding of a convergent geometric series.

Fun, Fun Math

A convergent geometric series is the sum of a series of terms where each term is obtained by multiplying the previous term by a common ratio between -1 and 1.

where a is the first term and r is the common ratio.

Notice that when we multiply the entire sum by r, we get:

When we subtract these equations and simplify, we get:

When n goes towards infinity, r raised to the n becomes zero (r is a fraction between 1 and -1). Therefore, a convergent geometric series of infinite terms is:

Let’s get back to Zeno’s problem. Imagine that a person wants to walk 2 meters. According to Zeno, they would walk

With our algebraically-derived formula, we can see that walking an infinite interval of “halves” would not only get you close to the total distance but equal to the total distance. Therefore, the person would definitely reach their destination.

The same can be said about the duration of their travel. By assuming that the person walks at a constant speed for t seconds, we will see that the total duration of travel equals to

Again, we can apply the same formula and discover that the total walking time comes to a finite value even though the person has walked an “infinite” interval of distances. And of course, this value is equal to t.

Conclusion

Zeno’s Dichotomy Paradox is a great example showing how an infinite amount of numbers do not always sum up to infinity. The next time you get confused about sequences and series in math class, think about Zeno. Perhaps you can then realize that the unintuitive approach is intuitive after all.