Definition:Value of Continued Fraction

Definition
The value of a continued fraction is the number which results from calculating out the fractions.

Note that formally the continued fraction and its value are considered to be distinct; a continued fraction is an arithmetic representation of its value.

However, this is a nicety of interpretation and may usually be ignored - $x = \left[{a_1, a_2, a_3, \ldots, a_n}\right]$ is often used to mean that $x$ is the value of the given continued fraction.