Mathemagics 4: Solving the Achille's Paradox

 

If you want to go to the nearby park which is, say at a distance of 1 km from your house, you can ne
ver complete the journey. What!? Isn’t that absurd? Well this is what we are going to learn today – Zeno’s Paradox.

In the fifth century, when the then great mathematicians like Democritus, Plato were trying to find the concept of infinity, Zeno was offering some paradoxes dealing with infinity. One of the paradoxes he came up with was the Achilles paradox.

This paradox states that Achilles start to chase a tortoise – which is obviously a slower runner that Achilles. As it is slower, it has a head start of 1 km. Now if Achilles wants to catch the tortoise it has to first move to the place where the tortoise is and for this he will take some time. But by the time he reaches there, the tortoise will have move to another place. So Achilles must run to that new place, but the tortoise meanwhile will move to a new place again. And this would repeat infinitely. And the conclusion is that Achilles would never catch the tortoise.

 

Achille's Paradox

 

To say this in simple language – if you want to move from A to B, or say a 1 km distance, first you have to cover half the distance which is ½ km. Then you have to cover half of the rest distance, which is ¼ km, then 1/8 km, and then 1/16 km and this would go on infinitely. Mathematically this is

1 + 1/2 + 1/4 + 1/8 +……

Now taking it in terms of time – to cover half the distance you take some finite time, for the one-fourth you take another finite time. Now as the distance consists of infinite number of fractions you would also need infinite number of time to cover the distance, which simple means that you will never get there – the one we talked about at the first. But that’s unrealistic in a way right?

So now let’s jump into the solution of the paradox. Well, it’s nothing complex – involves a little mathematics and nothing much.

Let’s take the expression again

x = 1 + 1/2 + 1/4 + 1/8 +…        ------ (1)

x – 1 = 1/2 + 1/4 + 1/8 +…        ------ (2)

By doing (1) – (2), we get

1 = (1 – ½) + (1/2 -1/4) + (1/4 – 1/8) + … = 1/2 + 1/4 + 1/8 +…       

And this term is again (x – 1)  [from (2)]

 

Hence, x – 1 = 1            or,              x = 2.

Thus it can be proved by simple mathematics. Hope you got to learn something new.