Complete quotient

In the metrical theory of regular continued fractions, the kth complete quotient ζ k is obtained by ignoring the first k partial denominators ai.

This result can be better understood by recalling that the successive convergents of an infinite regular continued fraction approach the value x in a sort of zig-zag pattern: so that when k is even we have Ak/Bk < x < Ak+1/Bk+1, and when k is odd we have Ak+1/Bk+1 < x < Ak/Bk.

We can define an equivalence relation on the set of real numbers by means of this group of linear fractional transformations.

Two irrational numbers x and y are equivalent under this scheme if and only if the infinitely long "tails" in their expansions as regular continued fractions are exactly the same.

Let x and y be two irrational (real) numbers, and let the kth complete quotient in the regular continued fraction expansions of x and y be denoted by ζ k and ψ k, respectively, Then x ~ y (under the equivalence defined in the preceding section) if and only if there are positive integers m and n such that ζ m = ψ n. The golden ratio φ is the irrational number with the very simplest possible expansion as a regular continued fraction: φ = [1; 1, 1, 1, …].