Quadratic Irrationals

The main result of this section is that the continued fraction expansion of a number is eventually repeating if and only if the number is a quadratic irrational. This can be viewed as an analogue for continued fractions of the familiar fact that the decimal expansion of $x$ is eventually repeating if and only if $x$ is rational. The proof that continued fractions of quadratic irrationals eventually repeats is surprisingly difficult and involves an interesting finiteness argument. Section 5.4.2 emphasizes our striking ignorance about continued fractions of real roots of irreducible polynomials over  $\mathbb {Q}$ of degree bigger than $2$ .

Definition 5.4 (Quadratic Irrational)   A real number $\alpha\in\mathbb{R}$ is a quadratic irrational if it is irrational and satisfies a quadratic polynomial with coefficients in  $\mathbb {Q}$ .

Thus, e.g., $(1+\sqrt{5})/2$ is a quadratic irrational. Recall that

$$\frac{1+\sqrt{5}}{2} = [1,1,1,\ldots].$$

The continued fraction of $\sqrt{2}$ is $[1,2,2,2,2,2,\ldots]$ , and the continued fraction of $\sqrt{389}$ is

$$[19,1,2,1, 1, 1, 1, 2, 1, 38, 1, 2, 1, 1, 1, 1, 2, 1, 38,\ldots].$$

Does the $[1,2,1, 1, 1, 1, 2, 1, 38]$ pattern repeat over and over again?

SAGE Example 5.4   We compute more terms of the continued fraction expansion of $\sqrt{389}$ using SAGE:

sage: def cf_sqrt_d(d, bits):
...   x = sqrt(RealField(bits)(d))
...   return continued_fraction(x)
sage: cf_sqrt_d(389,50)  
[19, 1, 2, 1, 1, 1, 1, 2, 1, 38, 1, 2, 1, 1, 1, 1, 2, 1, 38]
sage: cf_sqrt_d(389,100)   
[19, 1, 2, 1, 1, 1, 1, 2, 1, 38, 1, 2, 1, 1, 1, 1, 2, 1, 38, \
 1, 2, 1, 1, 1, 1, 2, 1, 38, 1, 2, 1, 1, 1, 1, 2, 1, 38, 1, 2, 1, 1]