Pell Number - Approximation To The Square Root of Two

Approximation To The Square Root of Two

Pell numbers arise historically and most notably in the rational approximation to the square root of 2. If two large integers x and y form a solution to the Pell equation

then their ratio provides a close approximation to . The sequence of approximations of this form is

where the denominator of each fraction is a Pell number and the numerator is the sum of a Pell number and its predecessor in the sequence. That is, the solutions have the form . The approximation

of this type was known to Indian mathematicians in the third or fourth century B.C. The Greek mathematicians of the fifth century B.C. also knew of this sequence of approximations: Plato refers to the numerators as rational diameters. In the 2nd century CE Theon of Smyrna used the term the side and diameter numbers to describe the denominators and numerators of this sequence.

These approximations can be derived from the continued fraction expansion of :

Truncating this expansion to any number of terms produces one of the Pell-number-based approximations in this sequence; for instance,

As Knuth (1994) describes, the fact that Pell numbers approximate allows them to be used for accurate rational approximations to a regular octagon with vertex coordinates and . All vertices are equally distant from the origin, and form nearly uniform angles around the origin. Alternatively, the points, and form approximate octagons in which the vertices are nearly equally distant from the origin and form uniform angles.