Graphing Continued Fractions of Quadratic Irrationals

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Let , . The continued fraction of is either finite (when is a perfect square so that is rational) or eventually periodic (when is not a perfect square so that is irrational).

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If is rational, the elements of its continued fraction are plotted.

If is irrational, let its continued faction be , where the repeating part under the bar starts as soon as possible. In that case, the plot is of the repeating part , with the initial elements ignored.

Sometimes is a palindrome; that is, is the same read from right to left as from left to right, , and its graph is symmetric. Often is the concatenation of two palindromes, like . If is rational and not a perfect square, then ; that is, is a palindrome concatenated with twice the integer part of , which is a trivial palindrome. Finally, there are cases where is not a palindrome. Colors distinguish the various cases.

The continued fraction is shown under the plot in the Mathematica notation .

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Contributed by: George Beck (July 2012)
Open content licensed under CC BY-NC-SA


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Reference

[1] E. R. Burger, "A Tail of Two Palindromes," The American Mathematical Monthly, 112, 2005 pp. 311–321. mathdl.maa.org/mathDL/22/?pa=content&sa=viewDocument&nodeId=3154.



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