Finding rational approximations to irrational numbers
Current Values:
Grid spacing:1/4 = 0.2500
√2 ≈1.414214
x = 1.2500with x² = 1.562500 < 2 ✓
x + ε = 1.5000with (x+ε)² = 2.250000 > 2 ✓
Key insight
As you increase q, the grid becomes finer, and x gets closer to √2 from below. For any ε = p/q, we can always find such an x because we're choosing the largest grid point whose square is still less than 2.