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Difference between revisions of "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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076100/q0761001.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076100/q0761002.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076100/q0761003.png" /> are rational numbers, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076100/q0761004.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076100/q0761005.png" /> is an integer which is not a perfect square. A real number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076100/q0761006.png" /> is a quadratic irrationality if and only if it has an infinite periodic [[Continued fraction|continued fraction]] expansion.
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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|continued fraction]] expansion.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.Ya. Khinchin,  "Continued fractions" , Phoenix Sci. Press  (1964)  pp. Chapt. II, §10  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.Ya. Khinchin,  "Continued fractions" , Phoenix Sci. Press  (1964)  pp. Chapt. II, §10  (Translated from Russian)</TD></TR></table>

Revision as of 09:04, 22 January 2013


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.

References

[a1] A.Ya. Khinchin, "Continued fractions" , Phoenix Sci. Press (1964) pp. Chapt. II, §10 (Translated from Russian)
How to Cite This Entry:
Quadratic irrationality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quadratic_irrationality&oldid=29336
This article was adapted from an original article by A.I. Galochkin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article