Expand rationals and quadratic irrationals into continued fractions. The page shows the term list, a nested fraction preview, convergents, approximation errors, and the geometry behind the approximation.
continued fraction: [a₀; a₁, a₂, …] means a₀ + 1/(a₁ + 1/(a₂ + …)).
convergent pₙ/qₙ: truncate after aₙ. These fractions are often the best small-denominator approximations.
error plot: shows correct decimal digits, −log₁₀(|x − pₙ/qₙ|). Higher means better.
Each row truncates the continued fraction one term later. Watch the denominator grow and the error usually drop sharply.
A rising point means the convergent gained decimal accuracy. Flat or low points are normal for early terms.
Even and odd convergents approach from opposite sides. Each line spans two consecutive convergents; the target is trapped inside the shrinking intervals.