Mathwords logoMathwords

Zeno's Paradoxes — Definition, Formula & Examples

Zeno's Paradoxes are a set of ancient Greek thought experiments that seem to prove motion is impossible by arguing that crossing any distance requires completing infinitely many smaller steps. They are resolved by the modern concept of convergent infinite series, where infinitely many terms can sum to a finite value.

Zeno's Paradoxes, attributed to Zeno of Elea (c. 490–430 BCE), are arguments that exploit the infinite divisibility of space and time to generate contradictions with observed motion. The most famous — the Dichotomy and Achilles paradoxes — implicitly assume that an infinite sum of positive durations must be infinite, an assumption refuted by the theory of limits, which shows that a geometric series with common ratio ∣r∣<1|r| < 1 converges to a finite sum.

Key Formula

∑n=1∞12n=121−12=1\sum_{n=1}^{\infty} \frac{1}{2^n} = \frac{\frac{1}{2}}{1 - \frac{1}{2}} = 1
Where:
  • nn = Step number in the sequence of halving distances
  • 12n\frac{1}{2^n} = Fraction of total distance covered at step n

How It Works

Consider Zeno's Dichotomy: to walk across a room, you must first cover half the distance, then half the remaining distance, then half again, and so on forever. Zeno argued you can never finish because there are infinitely many steps. The resolution comes from limits: the total distance is 12+14+18+⋯\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots, which is a geometric series that converges to exactly 1. Each step also takes proportionally less time, so the total time is also finite. The paradox dissolves once you accept that an infinite number of terms can have a finite sum.

Worked Example

Problem: Achilles runs at 10 m/s and a tortoise runs at 1 m/s with a 9 m head start. Using Zeno's reasoning, Achilles must first reach where the tortoise was, but the tortoise has moved. Show the total distance Achilles runs before catching the tortoise.
Step 1: Achilles runs 9 m to the tortoise's starting position. In that time, the tortoise moves 0.9 m ahead.
d1=9 md_1 = 9 \text{ m}
Step 2: Achilles runs 0.9 m to the tortoise's new position. The tortoise moves another 0.09 m.
d2=0.9 md_2 = 0.9 \text{ m}
Step 3: This continues as a geometric series. Sum all the distances Achilles must cover.
∑n=0∞9⋅(110)n=91−110=10 m\sum_{n=0}^{\infty} 9 \cdot \left(\frac{1}{10}\right)^n = \frac{9}{1 - \frac{1}{10}} = 10 \text{ m}
Answer: Achilles catches the tortoise after running exactly 10 m, which takes 1 second. The infinite sequence of "catch-up" steps sums to a finite distance.

Why It Matters

Zeno's Paradoxes are one of the oldest motivations for defining limits rigorously, which is the foundation of calculus. When you study convergent series in AP Calculus or a university analysis course, you are directly engaging with the mathematics that resolves these 2,400-year-old puzzles.

Common Mistakes

Mistake: Believing that adding infinitely many positive numbers must always give infinity.
Correction: A geometric series with ratio ∣r∣<1|r| < 1 converges to a finite sum. This is the key insight that resolves Zeno's Paradoxes.