Periodic Continued Fraction - Relation To Quadratic Irrationals

Relation To Quadratic Irrationals

A quadratic irrational number is an irrational real root of the quadratic equation

where the coefficients a, b, and c are integers, and the discriminant, b2 − 4ac, is greater than zero. By the quadratic formula every quadratic irrational can be written in the form

where P, D, and Q are integers, D > 0 is not a perfect square, and Q divides the quantity P2 − D.

By considering the complete quotients of periodic continued fractions, Euler was able to prove that if x is a regular periodic continued fraction, then x is a quadratic irrational number. The proof is straightforward. From the fraction itself, one can construct the quadratic equation with integral coefficients that x must satisfy.

Lagrange proved the converse of Euler's theorem: if x is a quadratic irrational, then the regular continued fraction expansion of x is periodic. Given a quadratic irrational x one can construct m different quadratic equations, each with the same discriminant, that relate the successive complete quotients of the regular continued fraction expansion of x to one another. Since there are only finitely many of these equations (the coefficients are bounded), the complete quotients (and also the partial denominators) in the regular continued fraction that represents x must eventually repeat.