Hello,
I was doing a little research on the Tortoise Paradox and found out the following:
When solving this kind of problems it is better not to get a headache over them, but try to solve them the easiest way through logic.
First of all we are trying to find out how many miles does it take to Achilles to catch up the Tortiose? But let us see it this way better: How many miles would it take Achilles to finish the race if the race is one mile?
If we already know that every time he reaches half of the distance he has to run, he has 1/4 of the distance of that half he has left, and then as he continues he has 1/8 of the distance, by then running half of the 1/8 he has 1/16, 1/32 and so on. This creates infinte numbers that derive from a finite distance. The finite distance is 1 mile and is practically divided into small and infite numbers. What seems most logic is that Achilles' limit is probably an infinite number, but in fact it is not. On the other hand it is a finite sum. The distance Achilles will spend travelling 1 mile is exactly 1 mile.
Ok, to further understand why this is. First we realize that if we divide the finite distance (1 mile), it divides into infinite small distances, (1/2, 1/4...), then by logic if we add all those infinte numbers it equals us the finite distance we started with, or to say 1 mile!
When an infinite sum like the one in the Tortiose' paradox, also known as an infinite series, adds up to a finite number, the series is summable.
In Zeno's Paradox, Achilles gives the Tortiose a 10-meter head start, but as Achilles catches up with that head start, the tortiose has already walked 1 meter more. From that meter on, Achilles is always trying to catch up with the distance and every time the Tortiose is creating a new meter of difference between them. Therefore it is a finite number creating an infinite number of distances and to find the distance where they meet is equal to the one they started with. 1 meter.
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well, using the distances you gave I'm going to solve this problem. If they start out with a 10m distance between them, and after achilles travels the 10 meters the tortoise has traveled one, then achilles speed is 10 times that of the tortoise. Lets assume that it took 1 second for achilles to travel those ten meters and for the tortoise to travel that extra meter. Then on the next .1 second achilles would have travled 1 meter and the tortoise would have traveled .1 meter. If you use the equation to find the sum of a series you can solve to see at what distance aquiless would catch the tortoise.
The first term is 10 since thats the distance they started out with, the second term is one, the third is .1 etc. So ou series would look like this
10, 1, .1, .01, .001 etc, Now we can apply the formula
Sn = t1 (1 - r^n) / (1 - r)
In this case t1 = 10 r= 1/10 and n equals infinity. When we raise r to infinity the term would go to zero. Now we can solve using simpl algebra.
Sn = 10 (1 - 0) / (1 - 1/10)
Sn= 10/(9/10)
Sn= 100/9
If you plug in 100/9 in a calculator you would get 11.111111111111111111....meters and thats the distance were achilles would catch up to the tortoise and it would take achilles 1 and 1/9 seconds to catch him.
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