Encyclopaedia Britannica, 11th Edition, "Constantine Pavlovich" to "Convention" / Volume 7, Slice 2

by Various

are two series of positive integers increasing without limit if the fraction does not terminate.

The general continued fraction

a1 +	b2		b3		b4		...
	a2	+	a3	+	a4	+

is evidently equal, convergent by convergent, to the continued fraction

a1 +	λ2b2		λ2λ3b3		λ3λ4b4		...,
	λ2a2	+	λ3a3	+	λ4a4	+

where λ₂, λ₃, λ₄, ... are any quantities whatever, so that by choosing λ₂b₂ = 1, λ₂λ₃b₃ = 1, &c., it can be reduced to any equivalent continued fraction of the form

a1 +	1		1		1		...,
	d2	+	d3	+	d4	+

Simple Continued Fractions.

1. The simple continued fraction is both the most interesting and important kind of continued fraction.

Any quantity, commensurable or incommensurable, can be expressed uniquely as a simple continued fraction, terminating in the case of a commensurable quantity, non-terminating in the case of an incommensurable quantity. A non-terminating simple continued fraction must be incommensurable.