A simple continued fraction is a special case of a generalized continued fraction for which the partial numerators
are equal to unity, i.e., for all , 2, .... A simple continued fraction is therefore an expression
of the form

(1)

When used without qualification, the term "continued fraction" is often used to mean "simple continued fraction" or,
more specifically, regular (i.e., a
simple continued fraction whose partial denominators ,
,
... are positive integer; Rockett and Szüsz 1992, p. 3). Care must therefore
be taken to identify the intended meaning based on the context in which such terminology
is encountered.

A simple continued fraction can be written in a compact abbreviated notation
as

(2)

or

(3)

where
may be finite (for a finite continued fraction) or (for an infinite continued fraction). In contexts where
only simple continued fractions are considered, the partial
denominators are often denoted instead of (e.g., Rockett and Szüsz 1992, p. 3),
a practice which unfortunately conflicts with the common notation for generalized
continued fractions in which denotes a partial numerator.

Further care is needed when encountering bracket notation for simple continued fractions since some authors replace the semicolon with a normal comma and begin indexing the
terms at
instead of ,
writing
instead of
or ,
causing ambiguity in the meaning of the initial term and resulting in the parity
of certain fundamental results in continued fraction theory to be reversed. To complicate
matters a bit further, Gaussian brackets use
the notation
to denote a different (but closely related) combination of partial denominators.

The terms
through
of the simple continued fraction of a number can be computed in the Wolfram
Language using the command ContinuedFraction[x,
n]. Similarly, the convergent of simple continued
fraction with partial denominators can be continued using ContinuedFractionK[a[k],
k,
n],
where
may be Infinity.