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π, e, φ, √2 as continued fractions · Convergents · Best rational approximations

Number line — convergents stepping toward target (drag to zoom)

Convergent denominators qₙ and errors |α − pₙ/qₙ|

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Current time as continued fraction:

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Number

Depth (terms): 12 Zoom level: 1.0x

Convergents p_n/q_n:

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Lagrange's theorem: Best rational approximations are exactly the convergents. |α − p/q| < 1/(2q²) for any convergent.
Golden ratio φ = [1;1,1,1,…] — hardest to approximate (Fibonacci denominators).