Zeno’s Paradox of Achilles and the Tortoise

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Achilles in a War Chariot by bazylek100

Achilles in a War Chariot Image by bazylek100

Zeno, Achilles and the Tortoise

Around 450 BCE, the Greek philosopher Zeno of Elea spread the vicious rumour that Achilles was unable to catch a tortoise. This story is often called “Zeno’s Paradox”, but it is only one of several attributed to him.

The Math Problem for Achilles and the Tortoise

Let’s put the story into the form of a modern math problem. Achilles can run ten times faster than the tortoise, which has a head start of 100m. (For those in the USA, “100 yards” is just as good a number). They both begin running at the same time. How far must Achilles run to exactly catch up to the tortoise?

The Paradox for Achilles and the Tortoise

"Turkish Tortoise" by frefran

“Turkish Tortoise” Image by frefran

After 100m, when Achilles reaches the tortoise’s starting point, he sees that the chelonian is now 10m farther ahead. When Achilles reaches the 110m mark, the tortoise has gained another metre. Each time, in fact, that Achilles reaches the place where the tortoise had been, it still is 10% farther ahead. Zeno’s “paradox” is that the swift Achilles cannot catch the plodding tortoise.

The Simple Modern Solution for Achilles and the Tortoise

"Zeno, Achilles and Tortoise Plotted" by Mike DeHaan

“Zeno, Achilles and Tortoise Plotted” Image by Mike DeHaan

How far must Achilles run to catch the tortoise? The simple solution is that Achilles will catch up after running exactly 111.11111… metres; or 10/9 the starting distance. How do we know this?

From physics, we know that d=vt, where “d” means distance, “v” the velocity (or speed), and “t” the elapsed time. At a constant velocity, the distance that something travels is equal to its velocity times the time it spent.

Since Achilles runs ten times as fast as the tortoise, then the distance that the tortoise would run is one-tenth as far as Achilles would run, in the same amount of time.

Let’s say that “dA” refers to the distance Achilles runs in some amount of time; and “dT” is the tortoise’s distance in the same time. Since the tortoise covers one-tenth the distance Achilles covers, we can say dT = dA/10.

Achilles gave the tortoise a much-needed 100m head start. We want to find how far Achilles must run to overcome that 100m and also the distance the tortoise ran. That is dA = 100 + dT.

But we know dT = dA/10. Substituting, we find dA = 100 + dA/10.

Subtract (dA/10) from both sides of the equation, giving dA – dA/10 = 100.

But dA = dA*1 = dA*(10/10), so let’s simplify the left side: dA*(10/10 – 1/10) = dA*(9/10), which = 100.

Multiply both sides by (10/9) to get just one dA on the left: dA*(9/10)*(10/9) = dA*1 = 100*(10/9) = 1000/9 = 111.1111…

Therefore Achilles will run 111.1111… metres; the tortoise will run one-tenth that distance, or 11.1111… metres; and that’s exactly where Achilles catches the tortoise.

Why did Zeno Create this Paradox?

"Zeno of Elea by Carducci or Tibaldi" by Bartolomeo Carducci or Pellegrino Tibaldi

“Zeno of Elea by Carducci or Tibaldi” Image by Bartolomeo Carducci or Pellegrino Tibaldi

It seems that Zeno’s broad philosophical outlook was that “motion” is an illusion. This paradox seems to disprove the obvious fact that a faster runner can overtake a slower one.

Why was Zeno’s Paradox Successful?

This paradox “succeeded” in the sense that other Greek philosophers could not refute his logic or his mathematics. One reason was that the Greek philosophers did not have the algebraic techniques to frame or solve the equations as shown. They had not developed a graphing technique to demonstrate where Achilles would catch the tortoise. Neither did they have a repeating-decimal fraction notation.

References:
Jon McLoone. Wolfram Demonstrations Project. “Zeno’s Paradox: Achilles and the Tortoise.” 2011) Accessed July 3, 2011.
Field, Paul and Weisstein, Eric W. MathWorld-A Wolfram Web Resource. “Zeno’s Paradoxes.” Accessed July 10, 2011.

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