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Thursday, August 04, 2005

Pujjle yagain

There is a very famous puzzle known as Zeno's paradox. It has remained one of my favourites thru the last few years - specially because of its simplicity. This is how it goes.


Suppose there is a guy, X, who is 10 metres in front of another guy Y. Further suppose that X and Y are running in the same direction; Y running 10 times as fast as X. Then,

when Y covers 10 metres, X will have covered one metre (so, he is one metre in front of Y). When Y covers that 1 metre, X would have moved a further 0.1 metres. When Y covers the 0.1 metres, X would have moved 0.01 metres ahead... till infinity.

If this goes on till infinity, how will Y ever catch up with X? Yet, we know thru common sense, that given the above initial conditions, Y will definitely overtake X. That too, pretty quickly!

That is the paradox:). Now, obviously, Zeno's reasoning was faulty. Where and what is the error?


Blogger majanx said...

Say speed of X is v m/sec. Then speed of Y is 10v m/sec. By simple mathematics, it can be found that Y will cross X in time 10v/9 seconds. So Y will not cross X before 10v/9 sec.

Now for the paradox:
In 1/v secs, X crosses 1m and Y crosses 10m. Y takes 0.1/v secs to cross that 1m , in which X crosses 0.1m. Again, Y takes 0.01/v secs to cross that 0.1m , in which X crosses 0.01m. This goes on till infinity.
Now if one adds up the time till infinity we have the total time = (1/v + 0.1/v + 0.01/v + 0.001/v + ....), which, when limit tends to infinity, will equal to 10v/9 seconds. This is in accordance to the above result that Y will not cross X till 10v/9 sec, but after that.

So Y does not cross X till 'infinity', only after that. (Here, infinity refers to infinity in above series and not in actual time).

Can't explain any better.

4:32 AM  
Blogger karthik durvasula said...

perfect, manka.

What Zeno didn't know was that an infinite (geometric) series can have a finite sum. He assumed it would be unbounded.

as we know now, an infinite gemoetric series is unbounded only if the multiplication factor is more-than-or-equal-to one.

This paradox was solved only after there was a proper understanding of infinite geometric series (which was in the second half of this millennium - i still believe that we are in the 20th century;))

5:46 AM  

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