For a real number ,
let be the number of terms in the convergent
to a regular continued fraction that
are required to represent
decimal places of .
Then for almost all ,
Therefore, the regular continued fraction is only slightly more efficient at representing real numbers than is the decimal
expansion. The set of
for which this statement does not hold is of measure 0.
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276-285, 1936.Kraaikamp, C. "A New Class of Continued Fraction
Expansions." Acta Arith.57, 1-39, 1991.Lévy,
P. "Sur le developpement en fraction continue d'un nombre choisi au hasard."
Compos. Math.3, 286-303, 1936.Lochs, G. "Vergleich
der Genauigkeit von Dezimalbruch und Kettenbruch." Abh. Hamburg Univ. Math.
Sem.27, 142-144, 1964.Perron, O. Die
Lehre von den Kettenbrüchen, 3. verb. und erweiterte Aufl. Stuttgart,
Germany: Teubner, 1954-57.Sloane, N. J. A. Sequence A086819
in "The On-Line Encyclopedia of Integer Sequences."
,
let 
be the number of terms in the convergent
to a regular continued fraction that
are required to represent 
decimal places of 
.
Then for almost all 
,
for which this statement does not hold is of measure 0.
