Reduced Surds
The quadratic surd is said to be reduced if ζ > 1 and its conjugate satisfies the inequalities −1 < η < 0. For instance, the golden ratio φ is a reduced surd because its conjugate ½(1 −√5) is greater than −1 and less than zero. On the other hand, the square root of two is not a reduced surd because its conjugate is less than −1.
Galois proved that the regular continued fraction which represents a quadratic surd ζ is purely periodic if and only if ζ is a reduced surd. In fact, Galois showed more than this. He also proved that if ζ is a reduced quadratic surd and η is its conjugate, then the continued fractions for ζ and for (−1/η) are both purely periodic, and the repeating block in one of those continued fractions is the mirror image of the repeating block in the other. In symbols we have
where ζ is any reduced quadratic surd, and η is its conjugate.
From these two theorems of Galois a result already known to Lagrange can be deduced. If r > 1 is a rational number that is not a perfect square, then
In particular, if n is any non-square positive integer, the regular continued fraction expansion of √n contains a repeating block of length m, in which the first m − 1 partial denominators form a palindromic string.
Read more about this topic: Periodic Continued Fraction
Famous quotes containing the word reduced:
““Write that down,” the King said to the jury, and the jury eagerly wrote down all three dates on their slates, and then added them up, and reduced the answer to shillings and pence.”
—Lewis Carroll [Charles Lutwidge Dodgson] (1832–1898)