For example, the FCF representation of $\frac{17}{13}$ can be calculated as follows:

$a$ | $q$ | $b$ | $r$ |

$17$ | $1$ | $13$ | $4$ |

$13$ | $3$ | $4$ | $1$ |

$4$ | $4$ | $1$ | $0$ |

The value of the FCF is contained in the second column from top to bottom: $\frac{17}{13}$ is $\left[ 1,3,4 \right]$. This is clearly an application of Euclid's algorithm for calculating the GCD of two integers. The algorithm for calculating the GCD stops at row $3$ but by adding one more row containing $4 = 4 \times 1 + 0$ the column containing the FCF is complete.

See also:

- Continued fractions (1)

- Continued fractions (2)

- Continued fractions (2a)

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