Achilles and the tortoise

The most celebrated of Zeno’s paradoxes pits Achilles (very fast) against the tortoise (very slow). If the tortoise starts with a head start, Zeno argues, Achilles will never be able to catch it: every time Achilles reaches where the tortoise was, it has already advanced a little.

History — Achilles and the tortoise

In the 5th century BC the Greek philosopher Zeno of Elea formulated four paradoxes that undermined the idea of motion (Boyer). In the most famous one, Achilles would never catch the tortoise because at each “reconnection” there always remains a small gap to close.

Let us put in some numbers: Achilles runs at $10$ m/s, the tortoise at $1$ m/s, with an initial head start of $100$ m. The times of the successive “reconnections” are

$t_1 = 10\,\text{s}\ (\text{Achilles covers }100\,\text{m},\ \text{the tortoise }10\,\text{m}), \quad t_2 = 1\,\text{s}\ (10\to 11), \quad t_3 = 0{,}1\,\text{s},\ \ldots$

The sum of all these times is a geometric series with ratio $1/10$:

$10 + 1 + 0{,}1 + 0{,}01 + \cdots = \sum_{k=0}^{\infty} 10^{1-k} = \frac{10}{1-1/10} = \frac{100}{9}\,\text{s} \approx 11{,}11\,\text{s}.$

A sum of infinitely many terms has a finite value: this is the answer given by the limit. Achilles catches the tortoise at time $100/9$ s, at position $1000/9$ m.

Links

Topics: Limits
Concepts: Limit · Geometric series
Skills: Modelling
People: Zeno