Continued Fraction Algorithm/Examples
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Examples of Use of Continued Fraction Algorithm
Example: $\sqrt 2$
Applying the Continued Fraction Algorithm to $\sqrt 2$:
| \(\text {(1)}: \quad\) | \(\ds x_0\) | \(=\) | \(\ds \sqrt 2\) | |||||||||||
| \(\text {(2)}: \quad\) | \(\ds a_0\) | \(=\) | \(\ds \floor {x_0}\) | \(\ds = \floor {\sqrt 2}\) | step $(2)$ | |||||||||
| \(\text {(3)}: \quad\) | \(\ds \) | \(=\) | \(\ds 1\) | integer part of $\sqrt 2$ is $1$ | ||||||||||
| \(\text {(4)}: \quad\) | \(\ds x_{0 + 1}\) | \(=\) | \(\ds \frac 1 {x_0 - a_0}\) | \(\ds = \frac 1 {\sqrt 2 - 1}\) | step $(3)$ | |||||||||
| \(\text {(5)}: \quad\) | \(\ds x_1\) | \(=\) | \(\ds \frac 1 {\sqrt 2 - 1} \times \paren {\frac {\sqrt 2 + 1} {\sqrt 2 + 1} }\) | multiply by $1$ | ||||||||||
| \(\text {(6)}: \quad\) | \(\ds \) | \(=\) | \(\ds \sqrt 2 + 1\) | |||||||||||
| \(\text {(7)}: \quad\) | \(\ds a_1\) | \(=\) | \(\ds \floor {x_1}\) | \(\ds = \floor {\sqrt 2 + 1}\) | step $(2)$ | |||||||||
| \(\text {(8)}: \quad\) | \(\ds \) | \(=\) | \(\ds 2\) | integer part of $\paren {\sqrt 2 + 1 }$ is $2$ | ||||||||||
| \(\text {(9)}: \quad\) | \(\ds x_{1 + 1}\) | \(=\) | \(\ds \frac 1 {x_1 - a_1}\) | \(\ds = \frac 1 {\paren {\sqrt 2 + 1 } - 2}\) | step $(3)$ | |||||||||
| \(\text {(10)}: \quad\) | \(\ds x_2\) | \(=\) | \(\ds \frac 1 {\sqrt 2 - 1} \times \paren {\frac {\sqrt 2 + 1} {\sqrt 2 + 1} }\) | multiply by $1$ | ||||||||||
| \(\text {(11)}: \quad\) | \(\ds \) | \(=\) | \(\ds \sqrt 2 + 1\) | lines $7$ through $11$ repeat ad infinitum: $\sqrt 2 = \sqbrk {1, \sequence 2}$ |
$\blacksquare$