The continued fraction of a rational number terminates

Remarks

If $x = a/b$ with $b > 0$, the expansion follows the Euclidean algorithm on $a$ and $b$. So it stops after finitely many steps. Conversely, a finite simple continued fraction is a rational number. For example, $\frac{43}{19}$: $43 = 2 \cdot 19 + 5$, $19 = 3 \cdot 5 + 4$, $5 = 1 \cdot 4 + 1$, $4 = 4 \cdot 1$. So $\frac{43}{19} = [2; 3, 1, 4]$. A rational number has exactly two finite expansions, since $[a_0; \dots, a_n] = [a_0; \dots, a_n - 1, 1]$ when $a_n > 1$.