Lochs' Theorem on:
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A prominent example of a number not exhibiting this behavior is the
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Math ...
, Lochs's theorem concerns the rate of convergence of the continued fraction
In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integ ...
expansion of a typical real number. A proof of the theorem was published in 1964 by Gustav Lochs
Gustav, Gustaf or Gustave may refer to:
*Gustav (name), a male given name of Old Swedish origin
Art, entertainment, and media
* ''Primeval'' (film), a 2007 American horror film
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.
The theorem states that for almost all
In mathematics, the term "almost all" means "all but a negligible amount". More precisely, if X is a set, "almost all elements of X" means "all elements of X but those in a negligible subset of X". The meaning of "negligible" depends on the mathem ...
real numbers in the interval (0,1), the number of terms ''m'' of the number's continued fraction expansion that are required to determine the first ''n'' places of the number's decimal expansion behaves asymptotically
In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related contexts, ...
as follows:
: .
As this limit is only slightly smaller than 1, this can be interpreted as saying that each additional term in the continued fraction representation of a "typical" real number increases the accuracy of the representation by approximately one decimal place. The decimal system is the last positional system
Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any base of the Hindu–Arabic numeral system (or decimal system). More generally, a positional system is a numeral system in which the ...
for which each digit carries less information than one continued fraction quotient; going to base-11
The undecimal numeral system (also known as the base-11 numeral system) is a positional numeral system that uses eleven as its base. While no known society counts by elevens, two are purported to have done so: the Māori, one of the two Polyne ...
(changing to in the equation) makes the above value exceed 1.
The reciprocal of this limit,
: ,
is twice the base-10 logarithm of Lévy's constant.
A prominent example of a number not exhibiting this behavior is the golden ratio
In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0,
where the Greek letter phi ( ...
—sometimes known as the " most irrational" number—whose continued fraction terms are all ones, the smallest possible in canonical form. On average it requires approximately 2.39 continued fraction terms per decimal digit.
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