A Relation Between Newton’s Method and Successive Approximations for Quadratic Irrationals

Abstract

A quadratic irrational, ¹ ζ, has the form (a + b ^1/2)/c, where a, b, c are integers, b > 0, b is not the square of an integer, and c ≠ 0. Let ζ⁺ and ζ⁻ be quadratic irrationals corresponding to ζ⁺ = (a + b ^1/2)/c and ζ⁻ (a − b ^1/2)/c. A quadratic polynomial that has these roots is p(z) = (z − ζ⁺)(z − ζ⁻), or

$$p(z) = z^2 - \frac{{2a}} {c}z + \frac{{a^2 - b}} {{c^2 }}$$