ZENO'S DICHOTOMY PARADOX | ANCIENT MATH | MYTHOLOGY OF MATH

INTRODUCTION:

I am damn sure that you never heard this name before.If you know this before then you are special.

     Don't get sad we will discuss about it briefly and from basic.

ZENO:

Zeno of Elea was a pre-Socratic Greek philosopher of Magna Graecia and a member of the Eleatic School founded by Parmenides. Aristotle called him the inventor of the dialectic. He is best known for his paradoxes.

What is Paradox?

A paradox is a statement that contradicts itself, or that must be both true and untrue at the same time. ... But a key part of paradoxes is that they at least sound reasonable.

Dichotomy Paradox:

Dichotomy Paradox means before before an object can travel a given distance , it must travel a distance . In order to travel , it must travel , etc. Since this sequence goes on forever, it therefore appears that the distance cannot be traveled.

Simply,

In Dichotomy Paradox, Zeno argued that a runner will never reach the stationary goal line on a straight racetrack. The reason is that the runner must first reach half the distance to the goal, but when there he must still cross half the remaining distance to the goal, but having done that the runner must cover half of the new remainder, and so on.

Example-1:  If the goal is one meter away, the runner must cover a distance of 1/2 meter, then 1/4 meter, then 1/8 meter, and so on ad infinitum.So the runner cannot reach the final goal.

EXAMPLE-2: you may know about the story of Rabbit and Tortoise Race.

According to zeno's paradox if the Rabbit start the race after tortoise start then rabbit will not overtake the tortoise.

Don't Confuse with this because Dichotomy Paradox means The Contradiction of itself which it true and false at same time.

   I know this is funny but it is real.

Every time the hare reaches the place where the tortoise was, the tortoise has moved on a bit further. (Image)

And if the Pictures continue to infinity then you will find that Tortoise  will be bit ahead of Rabbit with less than a nanometer or picometer of whatever..

So Now the question must arises that

1. what is the time takes to reach at the goal?

2. In real life we reach at a goal in finite time , So in Zeno,s Paradox is it finite or infinite?

3. Is Zeno's Paradox is just a illusion or works in real life?

Time Management of Zeno's Dichotomy Paradox:

Let

      Distance=2KM

      Velocity = 1km/hr

From Zeno's Point of view

• for Covering 1st half i.e 1km will take 1hour
• For covering 2nd portion i.e; 500m will take 1/2 hr.
• For Covering 3rd portion i.e; 250m will take 1/4 hr.
• And So On.....

So Total Time will be (in Hours)

   = 1+1/2+1/4+1/8+1/16+........

   = 1+2-1 +2-2 +2-3 +....

   It is a Geometric Progression

Here a=1 and r=2-1

It is a Infinite series so 1/(1-r) will be the Formula

=1/1-2-1

=2  ( Proof)

Now you can see that the answer is correct according to real life.