Lochs' theorem
Encyclopedia
In number theory
, Lochs' theorem is a theorem concerning the rate of convergence of the continued fraction
expansion of a typical real number. The theorem was proved by Gustav Lochs in 1964.
The theorem states that for almost all
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 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 for which each digit carries less information than one continued fraction quotient; going to base-11
(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
.
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
, Lochs' theorem is a theorem concerning the rate of convergence of the continued fraction
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 integer part and another reciprocal, and so on...
expansion of a typical real number. The theorem was proved by Gustav Lochs in 1964.
The theorem states that for almost all
Almost all
In mathematics, the phrase "almost all" has a number of specialised uses."Almost all" is sometimes used synonymously with "all but finitely many" or "all but a countable set" ; see almost....
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 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
Decimal
The decimal numeral system has ten as its base. It is the numerical base most widely used by modern civilizations....
system is the last positional system for which each digit carries less information than one continued fraction quotient; going to base-11
Undecimal
The undecimal positional notation system is based on the number eleven, rather than ten as in decimal or eight in octal and so on. It is not a commonly used system. Undecimal requires eleven symbols representing the decimal numbers 0 through 10...
(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
Lévy's constant
In mathematics Lévy's constant occurs in an expression for the asymptotic behaviour of the denominators of the convergents of continued fractions....
.