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The paradox of Achilles and the tortoise

Zeno of Elea lived in the fifth century BC. It is especially known for his mathematical paradoxes. Little is known about his life. The only sources are from the Parmenides of Plato and the writings of the Greek pre-Socratic to the Roman historian Diogenes Laertius ,.

Main article: brief history of mathematics

One of the paradoxes, the most famous perhaps concerned that arrow that never reaches its target: it must indeed still browse the moist path has left. If the distance to the target is, for example 100 meters, the arrow must travel 100/2 (half) + 100/4 (a quarter) + 100/8 (the eighth) + 100/16 (the sixteenth) + …. of the distance, and so to infinity. So il put a time infinite and therefore never reach its target.

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Zenon : Arrow

The paradox of Achilles and the tortoise

Another famous paradox portrays Achilles, the hero of the Trojan War, died after receiving a boom (driven by Paris) in the heel. Achilles and a tortoise will make a run. And the game is balanced, the turtle leaves with 100 meters ahead. Achilles Assume Achiles moves at 10 m / s. So Achilles takes 10 seconds to reach the starting point of the animal. During these 10 seconds, the turtle, for its part, advanced 1 meter (it moves 10 cm / s or 0.1 m / s). Achilles is going to put a tenth of a second to join the new starting point of the turtle located 1 meter in front of him. Meanwhile, the tortoise will have advanced further 10cm and etc …

Achilles
Achilles and his intrepid tortoise

Time puts Achilles to catch the runaway, also reduced as it is, is utilized by Tortoise to advance. Achilles will never catch the tortoise! Of course, this is wrong. But why ?

First stage

To browse the first 100 meters, 100 meters will Achilles / 10 meters per second, be 10 seconds . The turtle has meanwhile traveled 10 seconds × 0.1 meters per second, or 1 small meter.

Second step

To cross this meter, Achilles will then need to 1/10 = 0.1 seconds . The tortoise will therefore traveled 0.1 × 0.1 = 0.01 meter.

The following steps

Achilles will successively need 10 second s, then 0.1 second , then 0,001  second, then 0.00001 seconds and then 0.00000001 second … There is no effect due to the following stops. If we add all these fractions of time, we get a total of 10.10101010101010101 …… Seconds etc.

A little theory

In mathematics, the Greek symbol sigma (Σ) means « sum ». The Achilles necessary time can be written as follows:

  • 100/10 + 1/10 + 0.01 / 10 + …. or, what amounts to the same:
  • 10 + 10/100 + 10/10000 + 10/1000000 or with our Sigma:
  • Σ 10/100 n with n varying from 0 to infinity.

We recognize (for specialists) a sum of terms of a geometric sequence. A geometric sequence (for non-specialists) is a series of numbers, such as 1 to 2 – 4 – 8 – 16-32 …; or each number is derived from the previous multiplying by a certain figure (here 2). In our example, 4 = 2 × 2 and 4 × 2 = 8 and 16 = 8 × 2 …. We say that this sequence has first term 1 and as a result 2. The theory is that the sum of terms of a geometric sequence converges if:

  •  his first term is greater than 0;
  • and its purpose is less than 1.

The general term of our suite

  • Un 10/100 = n , we therefore
  • Un+1 = 10/100 n + 1 and
  • Un+1 = 10/100 × 10/100 n , or
  • Un+1 = 1/100 × a

The first term of the sequence (10) is greater than 0 and the result is less than 1 (1/100). The sum of terms converges to a finite value which is reassuring for Achilles! The time it takes to reach the tortoise is not infinite. The same theory gives a way to compute the convergence value!

Σ10/100 n = 10- (1/100) n + 1 / [1-1 / 100] = 10 / 0.99, because (1/100) n + 1 goes to 0 as n tends towards « infinity.

The sum of the terms of a geometric sequence

Indeed if there is a geometric sequence Uk reason q and first term is> 0 then

  • Uk = aq k
  • and Sn = a Σq k = ax [1-q n + 1 / (1-q)]

Sample the sum

  • S n = a + aq + aq aq² + … + n ;
  • we can multiply both sides by q and we obtain:
  • qS n = aq + aq aq² + … + n + aq n + 1
  • S n – q × S n = a – aq n + 1
  • S n = a (1-q n + 1 ) / (1-q)

Sample convergence

S n = a (1-q n + 1 ) / (1-q)

  • if q <1, then Sn tends to a / 1-q (q n + 1 approaches 0)
  • if q> 1, then (q n + 1 tends towards ∞) and thus Sn too.

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