Order axioms and completeness
General · Mathematics
Study notes
Q: Show ℚ has a 'gap' at √2: no rational squares to 2, but rationals approach it. 1.4² = 1.96 < 2; 1.42² = 2.0164 > 2. 1.414² = 1.999396 < 2; 1.415² = 2.002225 > 2. Rationals squeeze √2 from both sides, but √2 ∉ ℚ! ℝ fills the gap (completeness!): sup of {q: q²<2} = √2!