Theorem 5.2. A periodic continued fraction is a quadratic surd, i.e. an irrational root of a quadratic equation with integral coefficients. 25 25 Proof. Using the notation set up in the definition, we have = aLaL, aL+1,, aL+k-1, aL, aL+1,...] = [aL, AL+1, ..., aL+k−1, a'] = p'a +p" q'a'+q" where p"/q" and p'/q' are the last two convergents to [aL, AL+1, ..., AL+k−1]. This gives us the equation q'a+(a" p')a-p" = 0 We know that which gives us PL-1α + PL-2 x= IL-19 +91-2 a'L = PL-2 IL-22 QL-1x-PL-1 Substituting this value back in the quadratic equation and clearing the denominators gives us a quadratic equation in x with integer coefficients. Q.E.D.

Theorem 5.2. A periodic continued fraction is a quadratic surd, i.e. an irrational root of a quadratic equation with integral coefficients. 25 25 Proof. Using the notation set up in the definition, we have = aLaL, aL+1,, aL+k-1, aL, aL+1,...] = [aL, AL+1, ..., aL+k−1, a'] = p'a +p" q'a'+q" where p"/q" and p'/q' are the last two convergents to [aL, AL+1, ..., AL+k−1]. This gives us the equation q'a+(a" p')a-p" = 0 We know that which gives us PL-1α + PL-2 x= IL-19 +91-2 a'L = PL-2 IL-22 QL-1x-PL-1 Substituting this value back in the quadratic equation and clearing the denominators gives us a quadratic equation in x with integer coefficients. Q.E.D.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter2: Equations And Inequalities

Section2.1: Equations

Problem 46E

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