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A property of the golden ratio φ

Because the continued fraction expansion for φ doesn't use any integers greater than 1, φ is one of the most "difficult" real numbers to approximate with rational numbers. Hurwitz's theorem states that any irrational number k can be approximated by infinitely many rational m / n with

While virtually all real numbers k will eventually have infinitely many convergents m / n whose distance from k is significantly smaller than this limit, the convergents for φ (i.e., the numbers 5 / 3, 8 / 5, 13 / 8, 21 / 13, etc.) consistently "toe the boundary", keeping a distance of almost exactly 1 n 2 5 {\displaystyle {\scriptstyle {1 \over n^{2}{\sqrt {5}}}}} away from φ, thus never producing an approximation nearly as impressive as, for example, 355 / 113 for π. It can also be shown that every real number of the form a + b φ / c + d φ, where a, b, c, and d are integers such that a d − b c = ±1, shares this property with the golden ratio φ; and that all other real numbers can be more closely approximated.