Continued fractions have a number of remarkable properties related to the Euclidean algorithm for integers or real numbers. Every rational number has two closely related expressions as a finite continued fraction, whose coefficients can be determined by applying the Euclidean algorithm to formula_6. The numerical value of an infinite continued fraction is irrational; it is defined from its infinite sequence of integers as the limit of a sequence of values for finite continued fractions. Each finite continued fraction of the sequence is obtained by using a finite prefix of the infinite continued fraction's defining sequence of integers. Moreover, every irrational number formula_7 is the value of a "unique" infinite continued fraction, whose coefficients can be found using the non-terminating version of the Euclidean algorithm applied to the incommensurable values formula_7 and 1. This way of expressing real numbers (rational and irrational) is called their "continued fraction representation".