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Zeno was a pre-Socratic great philosopher born in c. 495 BC. He is most famous for his Paradoxes of Motion generally known as “Zeno’s Paradoxes”. The paradoxes suggest that motion is not real – just an illusion.

Does it sound interesting?

These paradoxes are discussed by people from many backgrounds – Philosophy, Physics, Maths etc. Here – we will discuss the paradox from Physicist/Mathematician point of view.

The word Dichotomy means division into two parts/halves. As its name suggests – Its a Paradox about Motion that says the motion is an illusion and doesn’t really exist – using the idea that path can be divided into halves.

Most examples of Zeno’s Paradoxes include Greek characters like Atalanta, Achilles etc but let us step aside and take an example of an interesting cosmic journey.

Suppose we want to go to Moon. The average distance between Moon and Earth is 384,400 km. So our journey will be of 384,400 km.

Let’s divide this interesting and long journey into two parts – each of 192,200 km. Also, let the total distance be algebraically represented by “d”. It will help us to generalize the paradox.

We first travel 192,200 km – half of the total distance. That takes some time.

So we travelled half distance – now another half distance is left.
Let’s divide the remaining half parts into two parts. We travel the first half i:e 96,100 km. This journey will also take some time.

Now the remaining distance is 96,100 km. Let’s again divide that remaining journey of 96,100 km into two parts each of distance 48,050 km. This journey will also take some time.

Even though we have travelled this much – we still have to travel 48,050 km.

Let’s divide that journey into two parts – each of 24,025 km.

If we travel 24,025  km – we still have 24,025 km to travel.

Let’s divide that journey to half too – each of distance 12,012 km.  If we travel first 12,012 km – we still have more 12,012 km to travel.

If we divide that distance to two and travel the first half – we still have the second half to travel – and this process never ends. You can check yourself…

And as time is taken during all process – total time taken to reach the moon is the sum of all time taken.

So total time = t1 + t2 + t3 +t4 +…..

Since there are an infinite number of times – so total time should be infinite. Thus, we won’t ever reach the moon – that’s Zeno’s Dichotomy Paradox.

Not only moon, but Zeno’s paradox also implies we can’t go anywhere by moving.

Suppose you want to go to your college from home.

For that, you have to travel half of the distance for some time. And then travel half of the remaining half distance. Then you have to travel half of the half of the remaining half distance and so on…the process never ends – so you can’t go to your college.

The first asserts the non-existence of motion on the ground that that which is in locomotion must arrive at the half-way stage before it arrives at the goal.

– Aristotle Physics, 239b11

What Zeno suggests is – one can never move so motion must be an illusion.

Later on – Mathematicians found ways to add an infinite series of numbers – and this paradox got solved.

This type of series – where the term is obtained by multiplying the previous term with some number is called the Geometric Series.

An example of Geometric Series could be 1,2,4,8,16,32,64,… Here – term next to a number is obtained by multiplying the number by 2 – which is called the common ratio.

Later, mathematicians found out that if the common ratio is less than 1 – we can add infinite geometric series to get a finite answer.

In the Zeno’s Dichotomy Paradox – the distance to travel is obtained by halving the remaining distance. This means the common ratio is 1/2 i:e 0.5 which is less than 1.
So, we can add them to get a finite answer – and thus motion is not an illusion – we can go to the moon – we can go to college – we can go anywhere if we have enough resources.

We can mathematically proceed as follows:-

The way or formula to calculate the sum of infinite geometric series with a common ratio less than 1 is – you take the first term of series and divide it by (1 minus common ratio).

In our case, the first term is 1/2 and the common ratio is also 1/2 so the above series can be written as,

Doing some basic calculations –

And this totally makes sense – as we need to travel distance “d”. Same goes with time – so we get finite time.

You can read WikiPedia for more detailed information.

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