THE DEFINITION
√M := lim(p→∞) A_p / 2^p
where:
A_p = max{ a ∈ Z≥0 : a² ≤ M·2^(2p) }
= floor(√M · 2^p)
For √2: M = 2
where:
A_p = max{ a ∈ Z≥0 : a² ≤ M·2^(2p) }
= floor(√M · 2^p)
For √2: M = 2
CERTIFIED BOUNDS
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≤
√2
<
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Bracket width: 2^(-p)
CURRENT COMPUTATION
A_p (INTEGER)
92681
A_p²
---
2·2^(2p)
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GAP
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CONVERGENCE (BIT-BY-BIT)
| p | A_p | √2 ≈ A_p/2^p |
|---|
GEOMETRIC MEANING
Metric Completion:
• Squared distance d² is always integer
• d² = Δx² + Δy² + Δz² + ...
• Length = √(d²) needs sqrt!
√2 specifically:
• Diagonal of unit square
• √2 = √(1² + 1²)
• Bridges discrete → continuous
• Squared distance d² is always integer
• d² = Δx² + Δy² + Δz² + ...
• Length = √(d²) needs sqrt!
√2 specifically:
• Diagonal of unit square
• √2 = √(1² + 1²)
• Bridges discrete → continuous
NEWTON'S METHOD (INTEGER)
// All operations are integer!
X_{n+1} = (X_n + M·2^(2p)/X_n) >> 1
Converges in O(log p) iterations
Each iteration doubles correct bits
X_{n+1} = (X_n + M·2^(2p)/X_n) >> 1
Converges in O(log p) iterations
Each iteration doubles correct bits