N (intervals shown)
10
ε (total budget)
0.30
Total length covered
0.0000 / ε = 0.30
Dense everywhere, measure zero.
Formula
Total = ε × (1 − 1/2ᴺ)
As N → ∞, sum → ε
Each interval n:
length = ε / 2ⁿ
Each interval n:
length = ε / 2ⁿ
Running sum
0.2994
√2 − 1 ≈ 0.414 (irrational)
Odd-indexed intervals
Even-indexed intervals
Rationals are dense: between any two reals, there are infinitely many. Yet their total measure is exactly zero.