Behind Zeno’s Paradox of Achilles and the Tortoise


At least one moment in your life you are envying someone. But one thing about envying is that you are wasting your time. Let’s start discussion with one of famous Zeno’s paradoxes.

In 450 BC, Zeno of Elea is credited for publishing famous paradoxes and perhaps the best known is Achilles and the Tortoise paradox. Achilles was a Greek hero of Trojan War. He is invulnerable in all of his body except for his heel. Achilles’s heel has come to mean a person’s point of weakness.

One day Achilles met the Tortoise and challenged him to a race. Achilles was strong and fast, whereas the Tortoise was heavy and slow. “I am too fast. It would be a very quick race so I will give you ten meters start ahead”, asked Achilles. “Are you sure? You’ll never win this race if you give me a head start”, Tortoise laughed. “But I am ten times faster than you, Slow.” Achilles didn’t believe so. Then, The Tortoise convinced Achilles with his logical theory.

The start position of Achilles is A and the Tortoise is B. Point B is ten meters after point A. At the first trial Achilles reaching the Tortoise position, where is he? He is in point B. Where is the Tortoise? As the Tortoise is ten times slower than Achilles, he is 1 meter after point B. Then it’s 1 meter distance between the Tortoise and Achilles. At second trial Achilles will catch up that new distances. While Achilles doing so, The Tortoise shall have gone a little way farther, so that Achilles must then catch up the next distance again and again.


At each moment Achilles must be catching up the distance between them, yet at the same time, The Tortoise will be adding a new distance for catch up again. The Tortoise proved that Achilles can never catch him. That is the paradox.

Every time you compare your status with others, you will be Achilles trying to catch up The Tortoise. Even the people you’re catching up are slower than you, based on The Tortoise conclusion, you’ll never catch them.

But I will show you another view. At the first trial, the distance of Achilles and the Tortoise is ten meters. As Achilles move ten times faster than The Tortoise, the next distance will be a meter, then 0.1 m, 0.01 m and so on. The total distances will be S.

(1) S = 10 + 1 + 0.1 + 0.01 + 0.001 + …

multiple S with 10

(2) 10S = 100 + 10 + 1 + 0.1 + 0.01 + 0.001 + …

the (2) – (1) will be (3)

(3) 9S = 100, so S = 100/9 meters

The conclusion are:

  1. The number of your trials is infinite. 1st, 2nd, 3rd, 4th, 5th … Nth
  2. But the sum of total distance is exact. 1st + 2nd + 3rd + 4th + 5th + … + Nth => 10 + 1 + 0.1 + 0.01 + 0.001 + … = 100/9 meters

When you see Mark and Bill are more success than you, you envy them. Every single year you compare your current status with Mark. Bill’s goal is always part of your goal planning. You will catch up the distances with them by infinite number of trials. You’ll never know how much trials you have. But as long as you move faster than the people you’re catching up, certainly you can be as successful as them in one condition: you move faster. Your movement distance is always certain.

Don’t ever try to compare your current success with other people success because your energy to compare would be multiplied to infinite number of trials. You will be exhausted with your infinite number of comparisons. The comparison will be your Achilles’s heel. Focus with your movement and stay faster.