Quadratic irrationality

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A root of a quadratic trinomial with rational coefficients which is irreducible over the field of rational numbers. A quadratic irrationality is representable in the form $a+b\sqrt{d}$, where $a$ and $b$ are rational numbers, $b\ne 0$, and $d$ is an integer which is not a perfect square. A real number $\alpha$ is a quadratic irrationality if and only if it has an infinite periodic continued fraction expansion.


[a1] A.Ya. Khinchin, "Continued fractions" , Phoenix Sci. Press (1964) pp. Chapt. II, §10 (Translated from Russian) Zbl 0117.28601
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Quadratic irrationality. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by A.I. Galochkin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article