# ACHILLES AND THE TORTOISE PARADOX

The Paradox of Achilles and the Tortoise is one of a number of theoretical discussions of movement put forward by the Greek philosopher Zeno of Elea in the 5th century BC. It begins with the great hero Achilles challenging a tortoise to a footrace. To keep things fair, he agrees to give the tortoise a head start of, say, 500m. When the race begins, Achilles unsurprisingly starts running at a speed much faster than the tortoise, so that by the time he has reached the 500m mark, the tortoise has only walked 50m further than him. But by the time Achilles has reached the 550m mark, the tortoise has walked another 5m. And by the time he has reached the 555m mark, the tortoise has walked another 0.5m, then 0.25m, then 0.125m, and so on. This process continues again and again over an infinite series of smaller and smaller distances, with the tortoise *always* moving forwards while Achilles *always* plays catch up.

Logically, this seems to prove that Achilles can never overtake the tortoise—whenever he reaches somewhere the tortoise has been, he will always have some distance still left to go no matter how small it might be. Except, of course, we know intuitively that he *can* overtake the tortoise. The trick here is not to think of Zeno’s Achilles Paradox in terms of distances and races, but rather as an example of how any finite value can always be divided an infinite number of times, no matter how small its divisions might become.

Achilles will actually meet up with the Tortoise at an exact distance of 556 meters. This you may think doesn’t sound logical. If we apply mathematics to the specified distances using the given set of parameters, we can actually find the mathematical distance at which the Achilles and the Tortoise meetup.

First off we must enter in the equation. 500 + 50 + 5 + 1/2 + 1/4 + 1/8 + 1/16 + ….. + 1/(2^n) = ? Now one can easily sum up the first few numbers 500 + 50 + 5 = 555. The next step is to sum up the infinite series being described here which is typically described in mathematics as Sn = [Σ] 1/(2^n) . One then proceeds to multiply both sides by 2 making it because of a special 2Sn = 2/2 + 2/4 + 4/8+…2/((2)^(n-1))+ 2/(2^n). This reveals an interesting relationship so the equation now becomes 2Sn = 1 + [1/2 + 1/4 + 1/8 + 1/16 + ….. + 1/((2)^(n-1)] = 1 + [Sn – 1/(2^n)]. One then subtracts both sides by Sn making the equation now Sn = 1 – 1/(2^n). Now by applying calculus and taking the limit of 1 – 1/(2^n) this then gives us the value of Sn. Sn = 1

So now we have our final term which is simply 1 So adding all the numbers together gives us 500 + 50 + 5 + 1 = 556 meters is the point at which Achilles will finally meet up with the Tortoise. Now this case only solves this particular paradox because of the poorly worded statement about the last few changes to the distance being a geometric sum 1/2 + 1/4 + 1/8 + 1/16 + ….. + 1/(2^n). Also this was not a solvable equation back in ancient Greece since the invention of calculus did not take place until the 1700s by Isaac Newton. But there you have it, the solution to an age old paradox.

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