Let’s say that Zeno, a man from Elea (Turkey), wants to take a walk to the park to relax.

For Zeno to get to the park, he must first walk half way to the park. This portion of his journey will take some finite amount of time to complete.

Once Zeno gets to the half way point, he then has to walk half of the remaining distance. Again, this portion of the journey will take some finite amount of time.

Once he is there, Zeno still needs to walk half of the distance that is left, and this portion of the journey takes another finite amount of time.

Notice that this happens again and again and again, dividing the remaining distance to the park into smaller and smaller portions of the journey.

This means that Zeno must walk an infinite number of portions of the journey that each take a finite amount of time to complete.

Does this mean that Zeno will never get there?! Or another way to look at it; if Zeno has to walk an infinite number of portions of the journey to get to the park, does this mean that Zeno can finish an infinite process?! Is the walk to the park infinite, or can it be finished? A paradox.

## Here's the math behind it.

So first, Zeno travels ½ of the journey to the park.

½

Then, he travels the next ¼ of the journey to the park.

½ + ¼

Then, another ⅛ of the journey.

½ + ¼ + ⅛

And so on…

½ + ¼ + ⅛ + 1⁄16 + …

Let’s say that this infinite sum has a value, and let us label it 𝑆.

So

S = ½ + ¼ + ⅛ + 1⁄16 + …

Let’s label this equation (1).

Let us divide equation (1) throughout by 2 to get

½S = ¼ + ⅛ + 1⁄16 + …

Let’s label this equation (2).

Let us now subtract equation (2) from equation (1), i.e. (1) – (2)

½S = ½

And so

S = 1.

The math tells us that since

S = ½ + ¼ + ⅛ + 1⁄16 + …

And

S = 1,

Then

½ + ¼ + ⅛ + 1⁄16 + … = 1

So eventually, Zeno completes the whole (1) of the journey by walking infinitely many smaller and smaller portions of the journey, and does get to relax at the park! Mathematically, this tells us that an infinite sum of finite terms can be finite.

P.S. This is actually a famous mathematical paradox called ‘Zeno’s Dichotomy Paradox’, which means “the paradox of cutting in 2”. Zeno really existed – he was born in 495BC in the ancient Greek city of Elea and is known for coming up with such paradoxes that puzzled mathematicians for millennia.