# Math Help - Finite Continued Fraction Question

1. ## Finite Continued Fraction Question

Show that every rational number has exactly two finite simple continued fraction expansions.

(Does this have something to do with how you handle the end of the continued fraction?)

2. $\displaystyle \frac{225}{157}$

This can be represented as $ \ \mbox{and} \ $

Therefore, our fraction can be represented like

$\displaystyle <1; 2,3,4,5>$

$\displaystyle 1+\frac{1}{2+\frac{1}{3+\frac{1}{4+\frac{1}{5}}}}$

or

$\displaystyle <1;2,3,4,4,1>$

$\displaystyle 1+\frac{1}{2+\frac{1}{3+\frac{1}{4+\frac{1}{4+\fra c{1}{1}}}}}}$