Calculating the quadratic irrationality whose simple continued fraction is given

The quadratic irrationality ω is given as a simple continued fraction:

ω = [a₀,...,a_p-1,\overline{b₀,...,b_q-1}].

Here the pre-period, a₀,...,a_p-1 consists of integers, with a₁,...,a_p-1 positive if p > 1.
Also the period b₀,...,b_q-1 consists of positive integers.

If there is no pre-period, simply enter 0 0 in the pre-period box below.

The construction.

Let ζ be the purely periodic reduced (ie. ζ > 1 & -1 < ζ′ < 0) quadratic irrational defined by

ζ = [\overline{b₀,...,b_q-1}].

Then

ζ = (A_q-1ζ + A_q-2)/(B_q-1ζ + B_q-2),

where A_n/B_n is the n-th convergent to ζ. (See O. Perron, Kettenbrüche, Zweite verbesserte Auflage, Seite 69.)

This gives a quadratic equation:

B_q-1ζ² + (B_q-2 – A_q-1)ζ – A_q-2 = 0.

Solving for ζ gives

ζ = (A_q-1 - B_q-2 + √{(A_q-1 – B_q-2)² + 4A_q-2B_q-1}) / 2B_q-1.

Then

ω = (A_p-1ζ + A_p-2)/(B_p-1ζ + B_p-2),

where A_n/B_n is the n-th convergent to ω. (See O. Perron, Kettenbrüche, Zweite verbesserte Auflage, Seite 85.)

We print ω in standard form (u+√d)/v, where v divides d - u² and gcd(u,v,(d-u²)/v = 1.

Last modified 4th February 2013
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