Paradox of Achilles and the tortoise

In the paradox of Achilles and tortoise , formulated by Zénon d' Élée, it is known as that one day, the Greek hero Achille disputed a race on foot with the slow reptile. As Achille was famous being a very fast runner, it had granted gracefully to the tortoise an advance of one hundred meters.

Zénon d' Élée affirms whereas to it fast Achille forever been able to catch up with the tortoise. “Indeed, let us suppose to simplify the reasoning that each competitor court at constant speed, one very quickly, and the other very slowly; at the end of some Time, Achille will have filled his hundred meters of delay and will have reached the starting point of the tortoise; but during this time, the tortoise will have traversed a certain distance, certainly much shorter, but nonnull, let us say one meter. That will require then of Achille an additional time to traverse this distance, during which the tortoise will advance even further; and then another duration before reaching this third point, whereas the tortoise still progresses. Thus, all the times that Achille reaches the place where the tortoise found, it is found even further. Consequently, it fast Achilles been able forever and will be able to never catch up with the tortoise”.

The reasoning of Zénon appears impeccable and irrefutable; however, we know all that it is Achille who gained the famous race!

Resolution

In modern analysis, the Paradoxe is solved by using the fact basically that a series Infini E of strictly positive numbers can converge towards a finished result.

In fact, this paradox functions by cutting out a one duration event finished (Achilles catches up with the tortoise) in an infinity of increasingly short events (for example, Achille makes 99% of the missing distance). Then, the mathematical error introduced into the paradox consists in affirming that the sum of this infinity of increasingly short events tends towards the Infini, i.e. Achille never manages (infinite time) to catch up with the tortoise.

Numerically, let us admit that each stage is 100 times shorter than the preceding one. If it is considered that the first stage took 10 seconds, then the following one took 0,1 second and one obtains the following series: T = 10 + 0,1 + 0,001 + 0,00001… = 10,10101… seconds. This paradox thus shows simply that Achille cannot join the tortoise in less than 10, seconds. Mathematically, one can write the sum in this form:

T = \ sum_ {n=0} ^ {\ infty} \ frac {10} {100^n}

It is about a geometrical Série of reason and initial term 10. This ensures us of his convergence:

T = \ frac {10} {1 \ frac {1} {100}} = \ frac {1000} {99} = 10 {,} \ overline {10}

It will be also noted that through this paradox, exists a will to show that the infinitely small does not exist. Thought also shared by Démocrite: the inventor of the concept of Atom. The Quantum physics also goes it in this direction by admitting the existence of a unit of time and a unit of size both indivisible (roughly 10^-44 seconds and 10^-33 meters).

See too

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