Lochs' theorem, proved by Gustav Lochs in 1964, describes how quickly the continued fraction expansion of a typical real number converges compared to its decimal expansion. For almost every real number between 0 and 1, each additional term of the continued fraction yields on average about 0.97 of a decimal digit of accuracy, a limit equal to six times the natural logarithm of 2 times the natural logarithm of 10, divided by pi squared. The golden ratio is a notable exception, since its continued fraction consists entirely of ones and needs about 2.39 terms per decimal digit, the slowest possible rate, matching its reputation as the most irrational number. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/
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In number theory, Lochs's theorem concerns the rate of convergence of the continued fraction expansion of a typical real number.
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In number theory, Lochs's theorem concerns the rate of convergence of the continued fraction expansion of a typical real number.
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