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.
Read more about this topic: Periodic Continued Fraction
Famous quotes containing the words relation to and/or relation:
“Any relation to the land, the habit of tilling it, or mining it, or even hunting on it, generates the feeling of patriotism. He who keeps shop on it, or he who merely uses it as a support to his desk and ledger, or to his manufactory, values it less.”
—Ralph Waldo Emerson (1803–1882)
“Among the most valuable but least appreciated experiences parenthood can provide are the opportunities it offers for exploring, reliving, and resolving one’s own childhood problems in the context of one’s relation to one’s child.”
—Bruno Bettelheim (20th century)